Research ArticleThe Dynamics of Epidemic Model with Two Types of InfectiousDiseases and Vertical Transmission

Raid Kamel Naji and Reem Mudar Hussien

Department of Mathematics College of Science University of Baghdad Baghdad Iraq

Correspondence should be addressed to ReemMudar Hussien reemmhussiengmailcom

Received 27 August 2015 Accepted 10 December 2015

Academic Editor Zhen Jin

Copyright copy 2016 R K Naji and R M HussienThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

An epidemic model that describes the dynamics of the spread of infectious diseases is proposed Two different types of infectiousdiseases that spread throughboth horizontal and vertical transmission in the host population are consideredThebasic reproductionnumber 119877

0is determined The local and the global stability of all possible equilibrium points are achieved The local bifurcation

analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied Numerical simulations are used toconfirm our obtained analytical results

1 Introduction

Mathematical models can be defined as a method of emu-lating real life situations with mathematical equations toexpect their future behavior In epidemiology mathematicalmodels play role as a tool in analyzing the spread and controlof infectious diseases Although one of the most famousprinciples of ecology is the competitive exclusion principlethat stipulates ldquotwo species competing for the same resourcescannot coexist indefinitely with the same ecological nicherdquo [12] Volterra was the first scientist who used the mathematicalmodeling and showed that the indefinite coexistence of twoor more species limited by the same resource is impossible[3] Moreover Ackleh and Allen [4] were the first who usedthe competitive exclusion principle of the infectious diseasewith different levels in single host population

It is well known that one of the most useful parametersconcerning infectious diseases is called basic reproductionnumber It can be specific to each strain of an epidemicmodelIn fact the basic reproduction number of themodel is definedas the maximum reproduction numbers of other strains [5ndash7] Diekmann et al [8] had studied epidemicmodels with onestrain while Martcheva in [9] studied the 119878119868119878-type of diseasewith multistrain However Ackleh and Allen [10] studied119878119868119877-type of disease with n strain and vertical transmission

Keeping the above in view in our proposed model twostrains with two different types of infectious diseases are

considered Accordingly two different reproduction numbersare obtained and then competitive exclusion principle is pre-sented It is assumed that two different types of diseases trans-mission say horizontal and vertical transmission are usedtoo The horizontal transmission occurs by direct contactbetween infected and susceptible individuals while verticaltransmission occurs when the parasite is transmitted fromparent to offspring [11ndash13] The incidence of an epidemiolog-ical model is defined as the rate at which susceptible becomesinfectious Different types of incidence rates are introducedinto literatures [14ndash17] Finally two types of incidence ratessay bilinearmass action and nonlinear type are used with thehorizontal and vertical transmission respectively The localand global stability for all possible equilibria are carried outwith the help of Lyapunov function and LaSallersquos invariantprinciple [18] An application of Sotomayor theorem [19 20]for local bifurcations is used to study the occurrence of localbifurcations near the equilibriaTheHopf bifurcation [21 22]conditions are derived Finally numerical simulations areused to confirm our obtained analytical results and specifythe control set of parameters

2 Model Formulation

Consider a real world system consisting of a host population119873(119905) that is divided into four compartments 119878(119905) which

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2016 Article ID 4907964 16 pageshttpdxdoiorg10115520164907964

2 Journal of Applied Mathematics

represents the number of susceptible individuals at time119905 1198681(119905) and 119868

2(119905) that represent the number of infected

individuals at time 119905 for 119878119868119877119878-type of disease and 119878119868119878-type ofdisease respectively finally 119877(119905) that represents the numberof recovered individuals at time 119905 thus 119873(119905) = 119878(119905) + 119868

1(119905) +

1198682(119905) + 119877(119905) Now in order to formulate the dynamics of

the above system mathematically the following assumptionshave been adopted

(1) There is a constant number of the host populationsentering to the system with recruitment rate Λ gt 0

(2) There is a vertical transmission of both of the diseasesthat is the infectious host gives birth to a new infectedhost of rates 0 le 119901

1le 1 and 0 le 119901

2le 1 for the

diseases 1198681and 1198682 respectively Consequently119901

11198681and

11990121198682individuals enter into infected compartments

1198681and 119868

2 respectively and the same quantities are

disappearing from recruitment in the susceptiblecompartment

(3) The diseases are transmitted by contact according tothe mass action law between the individuals in the 119878-compartment and those in 119868

119894(119894 = 1 2) compartments

with nonlinear incidence rate for 1198681that is given

by 12057311198781198681(1 + 119868

1) in which 120573

1gt 0 represents the

infection force rate while 1(1 + 1198681) represents the

inhibition effect of the crowding effect of the infectedindividuals and linear incidence rate for 119868

2that is

given by 12057321198781198682 where 120573

2gt 0 represents the infection

rate(4) The individuals in the 119868

1compartment are facing

death due to the disease with infection death rate 1205721ge

0 They recover from disease and get immunity witha recovery rate 120575 gt 0

(5) The individuals in the 1198682compartment are facing

death due to the disease with infection death rate 1205722ge

0 They also recover from the disease but return backto be susceptible with recovery rate 120574 gt 0

(6) The individuals in the 119877 compartment are losing theimmunity from the 119868

1disease and return back to be

susceptible again with losing immunity rate 0 le 120578 lt

1(7) There is a natural death rate 120583 gt 0 for the individuals

in the host population Finally it is assumed thatboth the diseases cannot be transmitted to the sameindividual simultaneously

According to these assumptions the dynamics of the abovereal world system can be represented mathematically by thefollowing set of differential equations

119889119878

119889119905= Λ minus (

12057311198681

1 + 1198681

+ 12057321198682) 119878 + (120574 minus 119901

2) 1198682minus 120583119878 minus 119901

11198681

+ 120578119877

1198891198681

119889119905=

12057311198781198681

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1) 1198681

1198891198682

119889119905= 12057321198781198682minus (120583 + 120572

2+ 120574 minus 119901

2) 1198682

119889119877

119889119905= 1205751198681minus (120578 + 120583) 119877

(1)

with the initial condition 119878(0) gt 0 1198681(0) gt 0 119868

2(0) gt 0

and 119877(0) gt 0 Moreover to insure that the recruitment Λ inthe susceptible compartment is always positive the followinghypotheses are assumed to be holding always

120575 ge 1199011

120574 ge 1199012

(2)

Theorem 1 The closed setΩ = (119878 1198681 1198682 119877) isin R4

+ 119873 le Λ120583

is positively invariant and attracting with respect to model (1)

Proof Let (119878(119905) 1198681(119905) 1198682(119905) 119877(119905)) be any solution of system

(1) with any given initial condition Then by adding all theequations in system (1) we obtain that

119889119873

119889119905= Λ minus 120583119878 minus (120583 + 120572

1) 1198681minus (120583 + 120572

2) 1198682minus 120583119877

le Λ minus 120583119873

(3)

Thus from standard comparison theorem [20] we obtain

119873(119905) le 119873 (0) 119890minus120583119905

+Λ

120583(1 minus 119890

minus120583119905) (4)

Consequently it is easy to verify that

119873(119905) leΛ

120583 when (0) le

Λ

120583 (5)

Thus Ω is positively invariant Further when 119873(0) gt Λ120583then either the solution enters Ω in finite time or 119873(119905)

approaches Λ120583 as 119905 rarr infin Hence Ω is attracting (ie allsolutions inR4

+eventually approach enter or stay inΩ)

Therefore the system of equations given in model (1)is mathematically well-posed and epidemiologically reason-able since all the variables remain nonnegative forall119905 ge 0Further since the equations of model (1) are continuousand have continuously partial derivatives then they areLipschitzian In addition to that from Theorem 1 model (1)is uniformly bounded Therefore the solution of it exists andis unique Hence from now onward it is sufficient to considerthe dynamics of model (1) in Ω

3 Equilibrium Points andBasic Reproduction Number

Model (1) has four equilibrium points that are obtained bysetting the right hand sides of this model equal to zero Thefirst equilibrium point is the disease-free equilibrium (DFE)point that is denoted by 119864

0= (1198780 0 0 0) with 119878

0= Λ120583

Moreover the basic reproduction number ofmodel (1) whichis denoted by 119877

0 is the maximum eigenvalue of the next

Journal of Applied Mathematics 3

generation matrix (ie the maximum of the reproductionnumbers those computed of each disease) That is

1198770= max 119877

1 1198772 (6)

Here 1198771= (12057311198780+1199011)(120583 +120572

1+120575) and 119877

2= (12057321198780+1199012)(120583 +

1205722+ 120574)The other three equilibrium points can be described as

followsThe first disease-free equilibrium point which is located

in the boundary 1198781198682-plane is denoted by 119864

1= (119878 0 119868

2 0)

where

119878 =1198780

1198792

1198682=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

(7)

and here1198792= 12057321198780(120583+120572

2+120574minus119901

2) Clearly119864

1exists uniquely

in the interior of 1198781198682-plane provided that

1198792gt 1 (8)

The second disease-free equilibrium point that is located inthe boundary 119878119868

1119877-space is given by 119864

2= (

1 0 ) where

=

1198780(1 +

1)

1198791

1=

Λ (1 minus 11198791)

(120583 + 1205721+ 120575) minus 120578120575 (120578 + 120583) + Λ119879

1

=1

120578 + 120583

(9)

and here 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) Obviously 119864

2

exists uniquely in the interior of positive octant of 1198781198681119877-space

provided that

1198791gt 1 (10)

Finally the endemic equilibrium point which is denoted by1198643= (119878lowast 119868lowast

1 119868lowast

2 119877lowast) where

119878lowast=

1198780

1198792

119868lowast

1= (

1198791

1198792

minus 1)

119877lowast=

120575 ((11987911198792) minus 1)

120578 + 120583

119868lowast

2=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

minus((11987911198792) minus 1)

(120583 + 1205722)

[(120583 + 1205721+ 120575) minus

120578120575

120578 + 120583]

(11)

exists uniquely in the interior ofΩprovided that the followingconditions hold

1198791gt 1198792gt 1

Λ gtΛ

1198792

+ 119868lowast

1((120583 + 120572

1+ 120575) minus

120578120575

120578 + 120583)

(12)

Keeping the above in view it is easy to verify with the help ofcondition (2) that

119879119894gt 1 (119879

119894lt 1) lArrrArr 119877

119894gt 1 (119877

119894lt 1) 119894 = 1 2 (13)

Then directly we obtain 119879119894gt 1 (119879

119894lt 1) hArr 119877

0gt 1 (119877

0lt

1) Consequently 119879119894represent the threshold parameters for

the existence of the last three equilibrium points of model(1) Moreover it is well known that the basic reproductionnumber (119877

0) is representing the average number of secondary

infections that occur from one infected individual in contactwith susceptible individuals Therefore if 119877

0lt 1 then

each infected individual in the entire period of infectivitywill produce less than one infected individual on averagewhich shows the disease will be wiped out of the populationHowever if119877

0gt 1 then each infected individual in the entire

infection period having contact with susceptible individualswill produce more than one infected individual this leads tothe disease invading the susceptible population

4 Local Stability Analysis

In this section the local stability analyses of all possibleequilibrium points of model (1) are discussed by determiningthe Jacobian matrix with their eigenvalues Now the generalJacobian matrix of model (1) can be written

119869 =

[[[[[[[[[[[[

[

minus12057311198681

1 + 1198681

minus 12057321198682minus 120583 minus

1205731119878

(1 + 1198681)2minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

12057311198681

1 + 1198681

minus12057311198781198681

(1 + 1198681)2+ 119891 (119878 119868

1) 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(14)

4 Journal of Applied Mathematics

where 119891(119878 1198681) = 1205731119878(1+119868

1)minus (120583+120572

1+120575minus119901

1) Therefore the

local stability results near the above equilibrium points canbe presented in the following theorems

Theorem 2 The disease-free equilibrium 1198640

= (1198780 0 0 0) is

locally asymptotically stable when1198770lt 1 and unstable for119877

0gt

1

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at the disease-free equilibrium can be written as

(120582 + 120583) (120582 minus 12057311198780+ (120583 + 120572

1+ 120575 minus 119901

1))

sdot (120582 minus 12057321198780+ (120583 + 120572

2+ 120574 minus 119901

2)) (120582 + (120578 + 120583)) = 0

(15)

So if 1198770lt 1 then according to (6) (15) has four negative real

roots (eigenvalues) Hence the DFE is locally asymptoticallystable Further for 119877

0gt 1 (15) has at least one positive

eigenvalue and then the DFE is a saddle point

Theorem 3 The first disease-free equilibrium point 1198641

=

(119878 0 1198682 0) of model (1) is locally asymptotically stable provided

that

1198792gt 1 gt 119879

1 (16)

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

1can be written as

[120582 minus 1205731119878 + (120583 + 120572

1+ 120575 minus 119901

1)] [120582 + (120578 + 120583)]

sdot [1205822

minus 1198791199031120582 + 119863

1] = 0

(17)

here 1198791199031

= minus(12057321198682+ 120583) lt 0 and 119863

1= 12057321198682(1205732119878 minus (120574 minus

1199011)) gt 0 due to condition (16) Hence both the eigenvalues 120582

119878

and 1205821198682

which describe the dynamics in the 119878-direction and1198682-direction respectively have negative real parts Moreover

from (17) the eigenvalue in the 1198681-direction can be written as

1205821198681

= 1205731119878 minus (120583 + 120572

1+ 120575 minus 119901

1) = (1 minus

1198792

1198791

)1205731119878 (18)

Thus under the given condition (16) we have 1205821198681

lt 0 while120582119877

= minus(120578 + 120583) is always negative Hence 1198641is locally

asymptotically stable

Theorem 4 The second disease-free equilibrium point 1198642

=

( 1 0 ) ofmodel (1) is locally asymptotically stable provided

that

1198791gt 1 gt 119879

2(1 +

1) (19a)

2gt 120575 (19b)

where 2is given in the proof

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

2can be written as

[ minus 1205732 + (120583 + 120572

2+ 120574 minus 119901

2)] [3

+ 1198602

+ 119861 + 119862]

= 0

(20)

here

119860 = (1+ 2) + (120578 + 120583)

119861 = (1+ 2) (120578 + 120583) + (

12+ 3)

119862 = (120578 + 120583) (12+ 3) minus 4

(21)

with

1= (

12057311

1 + 1

+ 120583) gt 0

2=

1205731 1

(1 + 1)2

gt 0

3= (

12057311

1 + 1

(12057311

(1 + 1)2+ 1199011)) gt 0

4=

12057812057512057311

1 + 1

gt 0

(22)

Clearly the eigenvalue 1198682

in the 1198682-direction can be written

as

1198682

= 1205732 minus (120583 + 120572

2+ 120574 minus 119901

2)

= (1 minus1198791

1198792(1 +

1)

)1205732

(23)

and thus 1198682

lt 0 under the condition (19a) In addition from(20) we have 119860 gt 0 always while 119862 can be written as

119862 = 5

2

120575+ 1205835

2

120575+ 2(1205732119868lowast

2(120578 + 120583))

+ 2120583 (120578 + 120583) +

3(120578 + 120583) +

4(2+ (120578 + 120583))

minus 5

(24)

Hence 119862 gt 0 provided that the sufficient condition (19b)holds Further it is easy to verify that

119860119861 minus 119862 = (1+ 2) (119861 + (120578 + 120583)

2) + 4gt 0 (25)

Hence due to the Routh-Hurwitz criterion the third-degreepolynomial term in (20) has roots (eigenvalues) with negativereal parts Hence 119864

2is locally asymptotically stable

Theorem 5 The endemic equilibrium point 1198643

=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) of model (1) is locally asymptotically stable

provided that

1198791gt 1198792gt 1 (26a)

119876 gt 1199025gt 120575 (26b)

where 119876 and 1199025are given in the proof

Journal of Applied Mathematics 5

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

3can be written as

120582lowast4

+ 1198601120582lowast3

+ 1198611120582lowast2

+ 1198621120582lowast+ 1198631= 0 (27)

Here

1198601= (1199021+ 1199022) + (120578 + 120583) gt 0

1198611= (120578 + 120583) (119902

1+ 1199022) + 11990211199022+ 1199023+ 1199024gt 0

1198621= (11990211199022+ 1199023) (120578 + 120583) + 119902

4(1199022+ (120578 + 120583)) minus 119902

5

1198631= 11990241199022(120578 + 120583) gt 0

(28)

with

1199021=

1205731119868lowast

1

1 + 119868lowast

1

+ 1205732119868lowast

2+ 120583

1199022=

1205731119878lowast119868lowast

1

(1 + 119868lowast

1)2

1199023=

1205731119868lowast

1

1 + 119868lowast

1

(1205731119878lowast

(1 + 119868lowast

1)2+ 1199011)

1199024= 1205732119868lowast

2[1205732119878lowastminus (120574 minus 119901

2)]

1199025=

1205781205751205731119868lowast

1

1 + 119868lowast

1

(29)

Obviously 119902119894

gt 0 119894 = 1 2 3 5 while 1199024is positive under

condition (26a) Now by using the values of 119902119894and the

sufficient condition (26b) then straightforward computationgives

1198621= 119876 + 119902

4(1199022+ (120578 + 120583)) minus 119902

5gt 0 (30)

and here 119876 = (11990211199022+ 1199023)(120578 + 120583) Moreover we have

119860111986111198621= [119876 + (119902

1+ 1199022)2(120578 + 120583) + 119902

41199021

+ (11990211199022+ 1199023+ (120578 + 120583)

2) (1199021+ 1199022)] [119876 minus 119902

5]

1198622

1+ 1198602

11198631= (119876 minus 119902

5)2+ [(1199021+ 1199022) + (120578 + 120583)]

2

sdot (11990241199022(120578 + 120583))

(31)

where 119876 = 119876 + 1199024(1199022+ (120578 + 120583)) Therefore we obtain that

119860111986111198621minus 1198622

1minus 1198602

11198631

= 119865119876 + 1199024119876 (1199021+ 1199022) + 119902411990211198601(120578 + 120583)

2

+ 1199022

41199021(1199022+ (120578 + 120583))

+ 11990241199022(1199021+ 1199022) (11990211199022+ 1199023) minus 119865119902

5+ 11986211199025

(32)

Here119865 = (1199021+1199022)2(120578+120583)+(119902

11199022+1199023+(120578+120583)

2)(1199021+1199022)+11990241199021

Hence according to condition (26b) it is easy to verifythat 119860

111986111198621minus 1198622

1minus 1198602

11198631gt 0 Therefore all the coefficients

of (27) are positive and 119860111986111198621

minus 1198622

1minus 1198602

11198631

gt 0Hence due to the Routh-Hurwitz criterion all the eigenvalues(120582lowast

119878 120582lowast

1198681

120582lowast

1198682

and 120582lowast

119877) of the Jacobianmatrix near the endemic

equilibrium point 1198643have negative real partsThus the proof

is complete

5 Global Stability Analysis

This section deals with the global stability of the equilibriumpoints of model (1) using Lyapunov methods with LaSallersquosinvariant principle The obtained results are presented in thefollowing theorems

Theorem 6 Assume that DFE 1198640

= (1198780 0 0 0) of model (1)

is locally asymptotically stable then it is global asymptoticallystable in Ω

Proof Consider Ω rarr R that is defined by

(119878 1198681 1198682 119877) =

2

sum

119894=1

119868119894 (33)

Computing the derivative of this positive semidefinite func-tion with respect to time along the solution of model (1) andthen simplifying the resulting terms give

119889

119889119905= (

1205731119878

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1)) 1198681

+ (1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

(34)

Since the solution of model (1) is bounded by 1198780

= Λ120583 as119905 rarr infin

119889

119889119905le (12057311198780minus (120583 + 120572

1+ 120575 minus 119901

1)) 1198681

+ (12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

= (120583 + 1205721+ 120575 minus 119901

1) (1198791minus 1) 1198681

+ (120583 + 1205722+ 120574 minus 119901

2) (1198792minus 1) 1198682

(35)

Since 119879119894lt 1 119894 = 1 2 due to the local stability condition of

1198640then 119889119889119905 lt 0 Also we have that 119889119889119905 = 0 on the

set (119878 1198681 1198682 119877) isin Ω 119868

1= 1198682

= 0 so 119889119889119905 is negativesemidefinite and hence according to Lyapunov first theorem1198640is globally stable point Now since on this set we have

119889119878

119889119905= Λ minus 120583119878 + 120578119877 = 0 (36)

if and only if 119878 = 1198780 119877 = 0 thus the largest invariant set

contained in this set is reduced to the disease-free equilibriumpoint119864

0 Hence according to LaSallersquos invariant principle [18]

1198640is attractive point and hence it is globally asymptotically

stable in Ω

Theorem 7 Assume that the first disease-free equilibriumpoint 119864

1= (119878 0 119868

2 0) is locally asymptotically stable then it

is global asymptotically stable in Ω provided that

1198682gt 1198682 (37)

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

represents the number of susceptible individuals at time119905 1198681(119905) and 119868

2(119905) that represent the number of infected

individuals at time 119905 for 119878119868119877119878-type of disease and 119878119868119878-type ofdisease respectively finally 119877(119905) that represents the numberof recovered individuals at time 119905 thus 119873(119905) = 119878(119905) + 119868

1(119905) +

1198682(119905) + 119877(119905) Now in order to formulate the dynamics of

the above system mathematically the following assumptionshave been adopted

(1) There is a constant number of the host populationsentering to the system with recruitment rate Λ gt 0

(2) There is a vertical transmission of both of the diseasesthat is the infectious host gives birth to a new infectedhost of rates 0 le 119901

1le 1 and 0 le 119901

2le 1 for the

diseases 1198681and 1198682 respectively Consequently119901

11198681and

11990121198682individuals enter into infected compartments

1198681and 119868

2 respectively and the same quantities are

disappearing from recruitment in the susceptiblecompartment

(3) The diseases are transmitted by contact according tothe mass action law between the individuals in the 119878-compartment and those in 119868

119894(119894 = 1 2) compartments

with nonlinear incidence rate for 1198681that is given

by 12057311198781198681(1 + 119868

1) in which 120573

1gt 0 represents the

infection force rate while 1(1 + 1198681) represents the

inhibition effect of the crowding effect of the infectedindividuals and linear incidence rate for 119868

2that is

given by 12057321198781198682 where 120573

2gt 0 represents the infection

rate(4) The individuals in the 119868

1compartment are facing

death due to the disease with infection death rate 1205721ge

0 They recover from disease and get immunity witha recovery rate 120575 gt 0

(5) The individuals in the 1198682compartment are facing

death due to the disease with infection death rate 1205722ge

0 They also recover from the disease but return backto be susceptible with recovery rate 120574 gt 0

(6) The individuals in the 119877 compartment are losing theimmunity from the 119868

1disease and return back to be

susceptible again with losing immunity rate 0 le 120578 lt

1(7) There is a natural death rate 120583 gt 0 for the individuals

in the host population Finally it is assumed thatboth the diseases cannot be transmitted to the sameindividual simultaneously

According to these assumptions the dynamics of the abovereal world system can be represented mathematically by thefollowing set of differential equations

119889119878

119889119905= Λ minus (

12057311198681

1 + 1198681

+ 12057321198682) 119878 + (120574 minus 119901

2) 1198682minus 120583119878 minus 119901

11198681

+ 120578119877

1198891198681

119889119905=

12057311198781198681

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1) 1198681

1198891198682

119889119905= 12057321198781198682minus (120583 + 120572

2+ 120574 minus 119901

2) 1198682

119889119877

119889119905= 1205751198681minus (120578 + 120583) 119877

(1)

with the initial condition 119878(0) gt 0 1198681(0) gt 0 119868

2(0) gt 0

and 119877(0) gt 0 Moreover to insure that the recruitment Λ inthe susceptible compartment is always positive the followinghypotheses are assumed to be holding always

120575 ge 1199011

120574 ge 1199012

(2)

Theorem 1 The closed setΩ = (119878 1198681 1198682 119877) isin R4

+ 119873 le Λ120583

is positively invariant and attracting with respect to model (1)

Proof Let (119878(119905) 1198681(119905) 1198682(119905) 119877(119905)) be any solution of system

(1) with any given initial condition Then by adding all theequations in system (1) we obtain that

119889119873

119889119905= Λ minus 120583119878 minus (120583 + 120572

1) 1198681minus (120583 + 120572

2) 1198682minus 120583119877

le Λ minus 120583119873

(3)

Thus from standard comparison theorem [20] we obtain

119873(119905) le 119873 (0) 119890minus120583119905

+Λ

120583(1 minus 119890

minus120583119905) (4)

Consequently it is easy to verify that

119873(119905) leΛ

120583 when (0) le

Λ

120583 (5)

Thus Ω is positively invariant Further when 119873(0) gt Λ120583then either the solution enters Ω in finite time or 119873(119905)

approaches Λ120583 as 119905 rarr infin Hence Ω is attracting (ie allsolutions inR4

+eventually approach enter or stay inΩ)

Therefore the system of equations given in model (1)is mathematically well-posed and epidemiologically reason-able since all the variables remain nonnegative forall119905 ge 0Further since the equations of model (1) are continuousand have continuously partial derivatives then they areLipschitzian In addition to that from Theorem 1 model (1)is uniformly bounded Therefore the solution of it exists andis unique Hence from now onward it is sufficient to considerthe dynamics of model (1) in Ω

3 Equilibrium Points andBasic Reproduction Number

Model (1) has four equilibrium points that are obtained bysetting the right hand sides of this model equal to zero Thefirst equilibrium point is the disease-free equilibrium (DFE)point that is denoted by 119864

0= (1198780 0 0 0) with 119878

0= Λ120583

Moreover the basic reproduction number ofmodel (1) whichis denoted by 119877

0 is the maximum eigenvalue of the next

Journal of Applied Mathematics 3

generation matrix (ie the maximum of the reproductionnumbers those computed of each disease) That is

1198770= max 119877

1 1198772 (6)

Here 1198771= (12057311198780+1199011)(120583 +120572

1+120575) and 119877

2= (12057321198780+1199012)(120583 +

1205722+ 120574)The other three equilibrium points can be described as

followsThe first disease-free equilibrium point which is located

in the boundary 1198781198682-plane is denoted by 119864

1= (119878 0 119868

2 0)

where

119878 =1198780

1198792

1198682=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

(7)

and here1198792= 12057321198780(120583+120572

2+120574minus119901

2) Clearly119864

1exists uniquely

in the interior of 1198781198682-plane provided that

1198792gt 1 (8)

The second disease-free equilibrium point that is located inthe boundary 119878119868

1119877-space is given by 119864

2= (

1 0 ) where

=

1198780(1 +

1)

1198791

1=

Λ (1 minus 11198791)

(120583 + 1205721+ 120575) minus 120578120575 (120578 + 120583) + Λ119879

1

=1

120578 + 120583

(9)

and here 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) Obviously 119864

2

exists uniquely in the interior of positive octant of 1198781198681119877-space

provided that

1198791gt 1 (10)

Finally the endemic equilibrium point which is denoted by1198643= (119878lowast 119868lowast

1 119868lowast

2 119877lowast) where

119878lowast=

1198780

1198792

119868lowast

1= (

1198791

1198792

minus 1)

119877lowast=

120575 ((11987911198792) minus 1)

120578 + 120583

119868lowast

2=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

minus((11987911198792) minus 1)

(120583 + 1205722)

[(120583 + 1205721+ 120575) minus

120578120575

120578 + 120583]

(11)

exists uniquely in the interior ofΩprovided that the followingconditions hold

1198791gt 1198792gt 1

Λ gtΛ

1198792

+ 119868lowast

1((120583 + 120572

1+ 120575) minus

120578120575

120578 + 120583)

(12)

Keeping the above in view it is easy to verify with the help ofcondition (2) that

119879119894gt 1 (119879

119894lt 1) lArrrArr 119877

119894gt 1 (119877

119894lt 1) 119894 = 1 2 (13)

Then directly we obtain 119879119894gt 1 (119879

119894lt 1) hArr 119877

0gt 1 (119877

0lt

1) Consequently 119879119894represent the threshold parameters for

the existence of the last three equilibrium points of model(1) Moreover it is well known that the basic reproductionnumber (119877

0) is representing the average number of secondary

infections that occur from one infected individual in contactwith susceptible individuals Therefore if 119877

0lt 1 then

each infected individual in the entire period of infectivitywill produce less than one infected individual on averagewhich shows the disease will be wiped out of the populationHowever if119877

0gt 1 then each infected individual in the entire

infection period having contact with susceptible individualswill produce more than one infected individual this leads tothe disease invading the susceptible population

4 Local Stability Analysis

In this section the local stability analyses of all possibleequilibrium points of model (1) are discussed by determiningthe Jacobian matrix with their eigenvalues Now the generalJacobian matrix of model (1) can be written

119869 =

[[[[[[[[[[[[

[

minus12057311198681

1 + 1198681

minus 12057321198682minus 120583 minus

1205731119878

(1 + 1198681)2minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

12057311198681

1 + 1198681

minus12057311198781198681

(1 + 1198681)2+ 119891 (119878 119868

1) 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(14)

4 Journal of Applied Mathematics

where 119891(119878 1198681) = 1205731119878(1+119868

1)minus (120583+120572

1+120575minus119901

1) Therefore the

local stability results near the above equilibrium points canbe presented in the following theorems

Theorem 2 The disease-free equilibrium 1198640

= (1198780 0 0 0) is

locally asymptotically stable when1198770lt 1 and unstable for119877

0gt

1

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at the disease-free equilibrium can be written as

(120582 + 120583) (120582 minus 12057311198780+ (120583 + 120572

1+ 120575 minus 119901

1))

sdot (120582 minus 12057321198780+ (120583 + 120572

2+ 120574 minus 119901

2)) (120582 + (120578 + 120583)) = 0

(15)

So if 1198770lt 1 then according to (6) (15) has four negative real

roots (eigenvalues) Hence the DFE is locally asymptoticallystable Further for 119877

0gt 1 (15) has at least one positive

eigenvalue and then the DFE is a saddle point

Theorem 3 The first disease-free equilibrium point 1198641

=

(119878 0 1198682 0) of model (1) is locally asymptotically stable provided

that

1198792gt 1 gt 119879

1 (16)

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

1can be written as

[120582 minus 1205731119878 + (120583 + 120572

1+ 120575 minus 119901

1)] [120582 + (120578 + 120583)]

sdot [1205822

minus 1198791199031120582 + 119863

1] = 0

(17)

here 1198791199031

= minus(12057321198682+ 120583) lt 0 and 119863

1= 12057321198682(1205732119878 minus (120574 minus

1199011)) gt 0 due to condition (16) Hence both the eigenvalues 120582

119878

and 1205821198682

which describe the dynamics in the 119878-direction and1198682-direction respectively have negative real parts Moreover

from (17) the eigenvalue in the 1198681-direction can be written as

1205821198681

= 1205731119878 minus (120583 + 120572

1+ 120575 minus 119901

1) = (1 minus

1198792

1198791

)1205731119878 (18)

Thus under the given condition (16) we have 1205821198681

lt 0 while120582119877

= minus(120578 + 120583) is always negative Hence 1198641is locally

asymptotically stable

Theorem 4 The second disease-free equilibrium point 1198642

=

( 1 0 ) ofmodel (1) is locally asymptotically stable provided

that

1198791gt 1 gt 119879

2(1 +

1) (19a)

2gt 120575 (19b)

where 2is given in the proof

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

2can be written as

[ minus 1205732 + (120583 + 120572

2+ 120574 minus 119901

2)] [3

+ 1198602

+ 119861 + 119862]

= 0

(20)

here

119860 = (1+ 2) + (120578 + 120583)

119861 = (1+ 2) (120578 + 120583) + (

12+ 3)

119862 = (120578 + 120583) (12+ 3) minus 4

(21)

with

1= (

12057311

1 + 1

+ 120583) gt 0

2=

1205731 1

(1 + 1)2

gt 0

3= (

12057311

1 + 1

(12057311

(1 + 1)2+ 1199011)) gt 0

4=

12057812057512057311

1 + 1

gt 0

(22)

Clearly the eigenvalue 1198682

in the 1198682-direction can be written

as

1198682

= 1205732 minus (120583 + 120572

2+ 120574 minus 119901

2)

= (1 minus1198791

1198792(1 +

1)

)1205732

(23)

and thus 1198682

lt 0 under the condition (19a) In addition from(20) we have 119860 gt 0 always while 119862 can be written as

119862 = 5

2

120575+ 1205835

2

120575+ 2(1205732119868lowast

2(120578 + 120583))

+ 2120583 (120578 + 120583) +

3(120578 + 120583) +

4(2+ (120578 + 120583))

minus 5

(24)

Hence 119862 gt 0 provided that the sufficient condition (19b)holds Further it is easy to verify that

119860119861 minus 119862 = (1+ 2) (119861 + (120578 + 120583)

2) + 4gt 0 (25)

Hence due to the Routh-Hurwitz criterion the third-degreepolynomial term in (20) has roots (eigenvalues) with negativereal parts Hence 119864

2is locally asymptotically stable

Theorem 5 The endemic equilibrium point 1198643

=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) of model (1) is locally asymptotically stable

provided that

1198791gt 1198792gt 1 (26a)

119876 gt 1199025gt 120575 (26b)

where 119876 and 1199025are given in the proof

Journal of Applied Mathematics 5

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

3can be written as

120582lowast4

+ 1198601120582lowast3

+ 1198611120582lowast2

+ 1198621120582lowast+ 1198631= 0 (27)

Here

1198601= (1199021+ 1199022) + (120578 + 120583) gt 0

1198611= (120578 + 120583) (119902

1+ 1199022) + 11990211199022+ 1199023+ 1199024gt 0

1198621= (11990211199022+ 1199023) (120578 + 120583) + 119902

4(1199022+ (120578 + 120583)) minus 119902

5

1198631= 11990241199022(120578 + 120583) gt 0

(28)

with

1199021=

1205731119868lowast

1

1 + 119868lowast

1

+ 1205732119868lowast

2+ 120583

1199022=

1205731119878lowast119868lowast

1

(1 + 119868lowast

1)2

1199023=

1205731119868lowast

1

1 + 119868lowast

1

(1205731119878lowast

(1 + 119868lowast

1)2+ 1199011)

1199024= 1205732119868lowast

2[1205732119878lowastminus (120574 minus 119901

2)]

1199025=

1205781205751205731119868lowast

1

1 + 119868lowast

1

(29)

Obviously 119902119894

gt 0 119894 = 1 2 3 5 while 1199024is positive under

condition (26a) Now by using the values of 119902119894and the

sufficient condition (26b) then straightforward computationgives

1198621= 119876 + 119902

4(1199022+ (120578 + 120583)) minus 119902

5gt 0 (30)

and here 119876 = (11990211199022+ 1199023)(120578 + 120583) Moreover we have

119860111986111198621= [119876 + (119902

1+ 1199022)2(120578 + 120583) + 119902

41199021

+ (11990211199022+ 1199023+ (120578 + 120583)

2) (1199021+ 1199022)] [119876 minus 119902

5]

1198622

1+ 1198602

11198631= (119876 minus 119902

5)2+ [(1199021+ 1199022) + (120578 + 120583)]

2

sdot (11990241199022(120578 + 120583))

(31)

where 119876 = 119876 + 1199024(1199022+ (120578 + 120583)) Therefore we obtain that

119860111986111198621minus 1198622

1minus 1198602

11198631

= 119865119876 + 1199024119876 (1199021+ 1199022) + 119902411990211198601(120578 + 120583)

2

+ 1199022

41199021(1199022+ (120578 + 120583))

+ 11990241199022(1199021+ 1199022) (11990211199022+ 1199023) minus 119865119902

5+ 11986211199025

(32)

Here119865 = (1199021+1199022)2(120578+120583)+(119902

11199022+1199023+(120578+120583)

2)(1199021+1199022)+11990241199021

Hence according to condition (26b) it is easy to verifythat 119860

111986111198621minus 1198622

1minus 1198602

11198631gt 0 Therefore all the coefficients

of (27) are positive and 119860111986111198621

minus 1198622

1minus 1198602

11198631

gt 0Hence due to the Routh-Hurwitz criterion all the eigenvalues(120582lowast

119878 120582lowast

1198681

120582lowast

1198682

and 120582lowast

119877) of the Jacobianmatrix near the endemic

equilibrium point 1198643have negative real partsThus the proof

is complete

5 Global Stability Analysis

This section deals with the global stability of the equilibriumpoints of model (1) using Lyapunov methods with LaSallersquosinvariant principle The obtained results are presented in thefollowing theorems

Theorem 6 Assume that DFE 1198640

= (1198780 0 0 0) of model (1)

is locally asymptotically stable then it is global asymptoticallystable in Ω

Proof Consider Ω rarr R that is defined by

(119878 1198681 1198682 119877) =

2

sum

119894=1

119868119894 (33)

Computing the derivative of this positive semidefinite func-tion with respect to time along the solution of model (1) andthen simplifying the resulting terms give

119889

119889119905= (

1205731119878

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1)) 1198681

+ (1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

(34)

Since the solution of model (1) is bounded by 1198780

= Λ120583 as119905 rarr infin

119889

119889119905le (12057311198780minus (120583 + 120572

1+ 120575 minus 119901

1)) 1198681

+ (12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

= (120583 + 1205721+ 120575 minus 119901

1) (1198791minus 1) 1198681

+ (120583 + 1205722+ 120574 minus 119901

2) (1198792minus 1) 1198682

(35)

Since 119879119894lt 1 119894 = 1 2 due to the local stability condition of

1198640then 119889119889119905 lt 0 Also we have that 119889119889119905 = 0 on the

set (119878 1198681 1198682 119877) isin Ω 119868

1= 1198682

= 0 so 119889119889119905 is negativesemidefinite and hence according to Lyapunov first theorem1198640is globally stable point Now since on this set we have

119889119878

119889119905= Λ minus 120583119878 + 120578119877 = 0 (36)

if and only if 119878 = 1198780 119877 = 0 thus the largest invariant set

contained in this set is reduced to the disease-free equilibriumpoint119864

0 Hence according to LaSallersquos invariant principle [18]

1198640is attractive point and hence it is globally asymptotically

stable in Ω

Theorem 7 Assume that the first disease-free equilibriumpoint 119864

1= (119878 0 119868

2 0) is locally asymptotically stable then it

is global asymptotically stable in Ω provided that

1198682gt 1198682 (37)

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

generation matrix (ie the maximum of the reproductionnumbers those computed of each disease) That is

1198770= max 119877

1 1198772 (6)

Here 1198771= (12057311198780+1199011)(120583 +120572

1+120575) and 119877

2= (12057321198780+1199012)(120583 +

1205722+ 120574)The other three equilibrium points can be described as

followsThe first disease-free equilibrium point which is located

in the boundary 1198781198682-plane is denoted by 119864

1= (119878 0 119868

2 0)

where

119878 =1198780

1198792

1198682=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

(7)

and here1198792= 12057321198780(120583+120572

2+120574minus119901

2) Clearly119864

1exists uniquely

in the interior of 1198781198682-plane provided that

1198792gt 1 (8)

The second disease-free equilibrium point that is located inthe boundary 119878119868

1119877-space is given by 119864

2= (

1 0 ) where

=

1198780(1 +

1)

1198791

1=

Λ (1 minus 11198791)

(120583 + 1205721+ 120575) minus 120578120575 (120578 + 120583) + Λ119879

1

=1

120578 + 120583

(9)

and here 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) Obviously 119864

2

exists uniquely in the interior of positive octant of 1198781198681119877-space

provided that

1198791gt 1 (10)

Finally the endemic equilibrium point which is denoted by1198643= (119878lowast 119868lowast

1 119868lowast

2 119877lowast) where

119878lowast=

1198780

1198792

119868lowast

1= (

1198791

1198792

minus 1)

119877lowast=

120575 ((11987911198792) minus 1)

120578 + 120583

119868lowast

2=

Λ

(120583 + 1205722)(1 minus

1

1198792

)

minus((11987911198792) minus 1)

(120583 + 1205722)

[(120583 + 1205721+ 120575) minus

120578120575

120578 + 120583]

(11)

exists uniquely in the interior ofΩprovided that the followingconditions hold

1198791gt 1198792gt 1

Λ gtΛ

1198792

+ 119868lowast

1((120583 + 120572

1+ 120575) minus

120578120575

120578 + 120583)

(12)

Keeping the above in view it is easy to verify with the help ofcondition (2) that

119879119894gt 1 (119879

119894lt 1) lArrrArr 119877

119894gt 1 (119877

119894lt 1) 119894 = 1 2 (13)

Then directly we obtain 119879119894gt 1 (119879

119894lt 1) hArr 119877

0gt 1 (119877

0lt

1) Consequently 119879119894represent the threshold parameters for

the existence of the last three equilibrium points of model(1) Moreover it is well known that the basic reproductionnumber (119877

0) is representing the average number of secondary

infections that occur from one infected individual in contactwith susceptible individuals Therefore if 119877

0lt 1 then

each infected individual in the entire period of infectivitywill produce less than one infected individual on averagewhich shows the disease will be wiped out of the populationHowever if119877

0gt 1 then each infected individual in the entire

infection period having contact with susceptible individualswill produce more than one infected individual this leads tothe disease invading the susceptible population

4 Local Stability Analysis

In this section the local stability analyses of all possibleequilibrium points of model (1) are discussed by determiningthe Jacobian matrix with their eigenvalues Now the generalJacobian matrix of model (1) can be written

119869 =

[[[[[[[[[[[[

[

minus12057311198681

1 + 1198681

minus 12057321198682minus 120583 minus

1205731119878

(1 + 1198681)2minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

12057311198681

1 + 1198681

minus12057311198781198681

(1 + 1198681)2+ 119891 (119878 119868

1) 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(14)

4 Journal of Applied Mathematics

where 119891(119878 1198681) = 1205731119878(1+119868

1)minus (120583+120572

1+120575minus119901

1) Therefore the

local stability results near the above equilibrium points canbe presented in the following theorems

Theorem 2 The disease-free equilibrium 1198640

= (1198780 0 0 0) is

locally asymptotically stable when1198770lt 1 and unstable for119877

0gt

1

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at the disease-free equilibrium can be written as

(120582 + 120583) (120582 minus 12057311198780+ (120583 + 120572

1+ 120575 minus 119901

1))

sdot (120582 minus 12057321198780+ (120583 + 120572

2+ 120574 minus 119901

2)) (120582 + (120578 + 120583)) = 0

(15)

So if 1198770lt 1 then according to (6) (15) has four negative real

roots (eigenvalues) Hence the DFE is locally asymptoticallystable Further for 119877

0gt 1 (15) has at least one positive

eigenvalue and then the DFE is a saddle point

Theorem 3 The first disease-free equilibrium point 1198641

=

(119878 0 1198682 0) of model (1) is locally asymptotically stable provided

that

1198792gt 1 gt 119879

1 (16)

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

1can be written as

[120582 minus 1205731119878 + (120583 + 120572

1+ 120575 minus 119901

1)] [120582 + (120578 + 120583)]

sdot [1205822

minus 1198791199031120582 + 119863

1] = 0

(17)

here 1198791199031

= minus(12057321198682+ 120583) lt 0 and 119863

1= 12057321198682(1205732119878 minus (120574 minus

1199011)) gt 0 due to condition (16) Hence both the eigenvalues 120582

119878

and 1205821198682

which describe the dynamics in the 119878-direction and1198682-direction respectively have negative real parts Moreover

from (17) the eigenvalue in the 1198681-direction can be written as

1205821198681

= 1205731119878 minus (120583 + 120572

1+ 120575 minus 119901

1) = (1 minus

1198792

1198791

)1205731119878 (18)

Thus under the given condition (16) we have 1205821198681

lt 0 while120582119877

= minus(120578 + 120583) is always negative Hence 1198641is locally

asymptotically stable

Theorem 4 The second disease-free equilibrium point 1198642

=

( 1 0 ) ofmodel (1) is locally asymptotically stable provided

that

1198791gt 1 gt 119879

2(1 +

1) (19a)

2gt 120575 (19b)

where 2is given in the proof

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

2can be written as

[ minus 1205732 + (120583 + 120572

2+ 120574 minus 119901

2)] [3

+ 1198602

+ 119861 + 119862]

= 0

(20)

here

119860 = (1+ 2) + (120578 + 120583)

119861 = (1+ 2) (120578 + 120583) + (

12+ 3)

119862 = (120578 + 120583) (12+ 3) minus 4

(21)

with

1= (

12057311

1 + 1

+ 120583) gt 0

2=

1205731 1

(1 + 1)2

gt 0

3= (

12057311

1 + 1

(12057311

(1 + 1)2+ 1199011)) gt 0

4=

12057812057512057311

1 + 1

gt 0

(22)

Clearly the eigenvalue 1198682

in the 1198682-direction can be written

as

1198682

= 1205732 minus (120583 + 120572

2+ 120574 minus 119901

2)

= (1 minus1198791

1198792(1 +

1)

)1205732

(23)

and thus 1198682

lt 0 under the condition (19a) In addition from(20) we have 119860 gt 0 always while 119862 can be written as

119862 = 5

2

120575+ 1205835

2

120575+ 2(1205732119868lowast

2(120578 + 120583))

+ 2120583 (120578 + 120583) +

3(120578 + 120583) +

4(2+ (120578 + 120583))

minus 5

(24)

Hence 119862 gt 0 provided that the sufficient condition (19b)holds Further it is easy to verify that

119860119861 minus 119862 = (1+ 2) (119861 + (120578 + 120583)

2) + 4gt 0 (25)

Hence due to the Routh-Hurwitz criterion the third-degreepolynomial term in (20) has roots (eigenvalues) with negativereal parts Hence 119864

2is locally asymptotically stable

Theorem 5 The endemic equilibrium point 1198643

=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) of model (1) is locally asymptotically stable

provided that

1198791gt 1198792gt 1 (26a)

119876 gt 1199025gt 120575 (26b)

where 119876 and 1199025are given in the proof

Journal of Applied Mathematics 5

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

3can be written as

120582lowast4

+ 1198601120582lowast3

+ 1198611120582lowast2

+ 1198621120582lowast+ 1198631= 0 (27)

Here

1198601= (1199021+ 1199022) + (120578 + 120583) gt 0

1198611= (120578 + 120583) (119902

1+ 1199022) + 11990211199022+ 1199023+ 1199024gt 0

1198621= (11990211199022+ 1199023) (120578 + 120583) + 119902

4(1199022+ (120578 + 120583)) minus 119902

5

1198631= 11990241199022(120578 + 120583) gt 0

(28)

with

1199021=

1205731119868lowast

1

1 + 119868lowast

1

+ 1205732119868lowast

2+ 120583

1199022=

1205731119878lowast119868lowast

1

(1 + 119868lowast

1)2

1199023=

1205731119868lowast

1

1 + 119868lowast

1

(1205731119878lowast

(1 + 119868lowast

1)2+ 1199011)

1199024= 1205732119868lowast

2[1205732119878lowastminus (120574 minus 119901

2)]

1199025=

1205781205751205731119868lowast

1

1 + 119868lowast

1

(29)

Obviously 119902119894

gt 0 119894 = 1 2 3 5 while 1199024is positive under

condition (26a) Now by using the values of 119902119894and the

sufficient condition (26b) then straightforward computationgives

1198621= 119876 + 119902

4(1199022+ (120578 + 120583)) minus 119902

5gt 0 (30)

and here 119876 = (11990211199022+ 1199023)(120578 + 120583) Moreover we have

119860111986111198621= [119876 + (119902

1+ 1199022)2(120578 + 120583) + 119902

41199021

+ (11990211199022+ 1199023+ (120578 + 120583)

2) (1199021+ 1199022)] [119876 minus 119902

5]

1198622

1+ 1198602

11198631= (119876 minus 119902

5)2+ [(1199021+ 1199022) + (120578 + 120583)]

2

sdot (11990241199022(120578 + 120583))

(31)

where 119876 = 119876 + 1199024(1199022+ (120578 + 120583)) Therefore we obtain that

119860111986111198621minus 1198622

1minus 1198602

11198631

= 119865119876 + 1199024119876 (1199021+ 1199022) + 119902411990211198601(120578 + 120583)

2

+ 1199022

41199021(1199022+ (120578 + 120583))

+ 11990241199022(1199021+ 1199022) (11990211199022+ 1199023) minus 119865119902

5+ 11986211199025

(32)

Here119865 = (1199021+1199022)2(120578+120583)+(119902

11199022+1199023+(120578+120583)

2)(1199021+1199022)+11990241199021

Hence according to condition (26b) it is easy to verifythat 119860

111986111198621minus 1198622

1minus 1198602

11198631gt 0 Therefore all the coefficients

of (27) are positive and 119860111986111198621

minus 1198622

1minus 1198602

11198631

gt 0Hence due to the Routh-Hurwitz criterion all the eigenvalues(120582lowast

119878 120582lowast

1198681

120582lowast

1198682

and 120582lowast

119877) of the Jacobianmatrix near the endemic

equilibrium point 1198643have negative real partsThus the proof

is complete

5 Global Stability Analysis

This section deals with the global stability of the equilibriumpoints of model (1) using Lyapunov methods with LaSallersquosinvariant principle The obtained results are presented in thefollowing theorems

Theorem 6 Assume that DFE 1198640

= (1198780 0 0 0) of model (1)

is locally asymptotically stable then it is global asymptoticallystable in Ω

Proof Consider Ω rarr R that is defined by

(119878 1198681 1198682 119877) =

2

sum

119894=1

119868119894 (33)

Computing the derivative of this positive semidefinite func-tion with respect to time along the solution of model (1) andthen simplifying the resulting terms give

119889

119889119905= (

1205731119878

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1)) 1198681

+ (1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

(34)

Since the solution of model (1) is bounded by 1198780

= Λ120583 as119905 rarr infin

119889

119889119905le (12057311198780minus (120583 + 120572

1+ 120575 minus 119901

1)) 1198681

+ (12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

= (120583 + 1205721+ 120575 minus 119901

1) (1198791minus 1) 1198681

+ (120583 + 1205722+ 120574 minus 119901

2) (1198792minus 1) 1198682

(35)

Since 119879119894lt 1 119894 = 1 2 due to the local stability condition of

1198640then 119889119889119905 lt 0 Also we have that 119889119889119905 = 0 on the

set (119878 1198681 1198682 119877) isin Ω 119868

1= 1198682

= 0 so 119889119889119905 is negativesemidefinite and hence according to Lyapunov first theorem1198640is globally stable point Now since on this set we have

119889119878

119889119905= Λ minus 120583119878 + 120578119877 = 0 (36)

if and only if 119878 = 1198780 119877 = 0 thus the largest invariant set

contained in this set is reduced to the disease-free equilibriumpoint119864

0 Hence according to LaSallersquos invariant principle [18]

1198640is attractive point and hence it is globally asymptotically

stable in Ω

Theorem 7 Assume that the first disease-free equilibriumpoint 119864

1= (119878 0 119868

2 0) is locally asymptotically stable then it

is global asymptotically stable in Ω provided that

1198682gt 1198682 (37)

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

where 119891(119878 1198681) = 1205731119878(1+119868

1)minus (120583+120572

1+120575minus119901

1) Therefore the

local stability results near the above equilibrium points canbe presented in the following theorems

Theorem 2 The disease-free equilibrium 1198640

= (1198780 0 0 0) is

locally asymptotically stable when1198770lt 1 and unstable for119877

0gt

1

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at the disease-free equilibrium can be written as

(120582 + 120583) (120582 minus 12057311198780+ (120583 + 120572

1+ 120575 minus 119901

1))

sdot (120582 minus 12057321198780+ (120583 + 120572

2+ 120574 minus 119901

2)) (120582 + (120578 + 120583)) = 0

(15)

So if 1198770lt 1 then according to (6) (15) has four negative real

roots (eigenvalues) Hence the DFE is locally asymptoticallystable Further for 119877

0gt 1 (15) has at least one positive

eigenvalue and then the DFE is a saddle point

Theorem 3 The first disease-free equilibrium point 1198641

=

(119878 0 1198682 0) of model (1) is locally asymptotically stable provided

that

1198792gt 1 gt 119879

1 (16)

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

1can be written as

[120582 minus 1205731119878 + (120583 + 120572

1+ 120575 minus 119901

1)] [120582 + (120578 + 120583)]

sdot [1205822

minus 1198791199031120582 + 119863

1] = 0

(17)

here 1198791199031

= minus(12057321198682+ 120583) lt 0 and 119863

1= 12057321198682(1205732119878 minus (120574 minus

1199011)) gt 0 due to condition (16) Hence both the eigenvalues 120582

119878

and 1205821198682

which describe the dynamics in the 119878-direction and1198682-direction respectively have negative real parts Moreover

from (17) the eigenvalue in the 1198681-direction can be written as

1205821198681

= 1205731119878 minus (120583 + 120572

1+ 120575 minus 119901

1) = (1 minus

1198792

1198791

)1205731119878 (18)

Thus under the given condition (16) we have 1205821198681

lt 0 while120582119877

= minus(120578 + 120583) is always negative Hence 1198641is locally

asymptotically stable

Theorem 4 The second disease-free equilibrium point 1198642

=

( 1 0 ) ofmodel (1) is locally asymptotically stable provided

that

1198791gt 1 gt 119879

2(1 +

1) (19a)

2gt 120575 (19b)

where 2is given in the proof

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

2can be written as

[ minus 1205732 + (120583 + 120572

2+ 120574 minus 119901

2)] [3

+ 1198602

+ 119861 + 119862]

= 0

(20)

here

119860 = (1+ 2) + (120578 + 120583)

119861 = (1+ 2) (120578 + 120583) + (

12+ 3)

119862 = (120578 + 120583) (12+ 3) minus 4

(21)

with

1= (

12057311

1 + 1

+ 120583) gt 0

2=

1205731 1

(1 + 1)2

gt 0

3= (

12057311

1 + 1

(12057311

(1 + 1)2+ 1199011)) gt 0

4=

12057812057512057311

1 + 1

gt 0

(22)

Clearly the eigenvalue 1198682

in the 1198682-direction can be written

as

1198682

= 1205732 minus (120583 + 120572

2+ 120574 minus 119901

2)

= (1 minus1198791

1198792(1 +

1)

)1205732

(23)

and thus 1198682

lt 0 under the condition (19a) In addition from(20) we have 119860 gt 0 always while 119862 can be written as

119862 = 5

2

120575+ 1205835

2

120575+ 2(1205732119868lowast

2(120578 + 120583))

+ 2120583 (120578 + 120583) +

3(120578 + 120583) +

4(2+ (120578 + 120583))

minus 5

(24)

Hence 119862 gt 0 provided that the sufficient condition (19b)holds Further it is easy to verify that

119860119861 minus 119862 = (1+ 2) (119861 + (120578 + 120583)

2) + 4gt 0 (25)

Hence due to the Routh-Hurwitz criterion the third-degreepolynomial term in (20) has roots (eigenvalues) with negativereal parts Hence 119864

2is locally asymptotically stable

Theorem 5 The endemic equilibrium point 1198643

=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) of model (1) is locally asymptotically stable

provided that

1198791gt 1198792gt 1 (26a)

119876 gt 1199025gt 120575 (26b)

where 119876 and 1199025are given in the proof

Journal of Applied Mathematics 5

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

3can be written as

120582lowast4

+ 1198601120582lowast3

+ 1198611120582lowast2

+ 1198621120582lowast+ 1198631= 0 (27)

Here

1198601= (1199021+ 1199022) + (120578 + 120583) gt 0

1198611= (120578 + 120583) (119902

1+ 1199022) + 11990211199022+ 1199023+ 1199024gt 0

1198621= (11990211199022+ 1199023) (120578 + 120583) + 119902

4(1199022+ (120578 + 120583)) minus 119902

5

1198631= 11990241199022(120578 + 120583) gt 0

(28)

with

1199021=

1205731119868lowast

1

1 + 119868lowast

1

+ 1205732119868lowast

2+ 120583

1199022=

1205731119878lowast119868lowast

1

(1 + 119868lowast

1)2

1199023=

1205731119868lowast

1

1 + 119868lowast

1

(1205731119878lowast

(1 + 119868lowast

1)2+ 1199011)

1199024= 1205732119868lowast

2[1205732119878lowastminus (120574 minus 119901

2)]

1199025=

1205781205751205731119868lowast

1

1 + 119868lowast

1

(29)

Obviously 119902119894

gt 0 119894 = 1 2 3 5 while 1199024is positive under

condition (26a) Now by using the values of 119902119894and the

sufficient condition (26b) then straightforward computationgives

1198621= 119876 + 119902

4(1199022+ (120578 + 120583)) minus 119902

5gt 0 (30)

and here 119876 = (11990211199022+ 1199023)(120578 + 120583) Moreover we have

119860111986111198621= [119876 + (119902

1+ 1199022)2(120578 + 120583) + 119902

41199021

+ (11990211199022+ 1199023+ (120578 + 120583)

2) (1199021+ 1199022)] [119876 minus 119902

5]

1198622

1+ 1198602

11198631= (119876 minus 119902

5)2+ [(1199021+ 1199022) + (120578 + 120583)]

2

sdot (11990241199022(120578 + 120583))

(31)

where 119876 = 119876 + 1199024(1199022+ (120578 + 120583)) Therefore we obtain that

119860111986111198621minus 1198622

1minus 1198602

11198631

= 119865119876 + 1199024119876 (1199021+ 1199022) + 119902411990211198601(120578 + 120583)

2

+ 1199022

41199021(1199022+ (120578 + 120583))

+ 11990241199022(1199021+ 1199022) (11990211199022+ 1199023) minus 119865119902

5+ 11986211199025

(32)

Here119865 = (1199021+1199022)2(120578+120583)+(119902

11199022+1199023+(120578+120583)

2)(1199021+1199022)+11990241199021

Hence according to condition (26b) it is easy to verifythat 119860

111986111198621minus 1198622

1minus 1198602

11198631gt 0 Therefore all the coefficients

of (27) are positive and 119860111986111198621

minus 1198622

1minus 1198602

11198631

gt 0Hence due to the Routh-Hurwitz criterion all the eigenvalues(120582lowast

119878 120582lowast

1198681

120582lowast

1198682

and 120582lowast

119877) of the Jacobianmatrix near the endemic

equilibrium point 1198643have negative real partsThus the proof

is complete

5 Global Stability Analysis

This section deals with the global stability of the equilibriumpoints of model (1) using Lyapunov methods with LaSallersquosinvariant principle The obtained results are presented in thefollowing theorems

Theorem 6 Assume that DFE 1198640

= (1198780 0 0 0) of model (1)

is locally asymptotically stable then it is global asymptoticallystable in Ω

Proof Consider Ω rarr R that is defined by

(119878 1198681 1198682 119877) =

2

sum

119894=1

119868119894 (33)

Computing the derivative of this positive semidefinite func-tion with respect to time along the solution of model (1) andthen simplifying the resulting terms give

119889

119889119905= (

1205731119878

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1)) 1198681

+ (1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

(34)

Since the solution of model (1) is bounded by 1198780

= Λ120583 as119905 rarr infin

119889

119889119905le (12057311198780minus (120583 + 120572

1+ 120575 minus 119901

1)) 1198681

+ (12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

= (120583 + 1205721+ 120575 minus 119901

1) (1198791minus 1) 1198681

+ (120583 + 1205722+ 120574 minus 119901

2) (1198792minus 1) 1198682

(35)

Since 119879119894lt 1 119894 = 1 2 due to the local stability condition of

1198640then 119889119889119905 lt 0 Also we have that 119889119889119905 = 0 on the

set (119878 1198681 1198682 119877) isin Ω 119868

1= 1198682

= 0 so 119889119889119905 is negativesemidefinite and hence according to Lyapunov first theorem1198640is globally stable point Now since on this set we have

119889119878

119889119905= Λ minus 120583119878 + 120578119877 = 0 (36)

if and only if 119878 = 1198780 119877 = 0 thus the largest invariant set

contained in this set is reduced to the disease-free equilibriumpoint119864

0 Hence according to LaSallersquos invariant principle [18]

1198640is attractive point and hence it is globally asymptotically

stable in Ω

Theorem 7 Assume that the first disease-free equilibriumpoint 119864

1= (119878 0 119868

2 0) is locally asymptotically stable then it

is global asymptotically stable in Ω provided that

1198682gt 1198682 (37)

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Proof The characteristic equation of the Jacobian matrix ofmodel (1) at 119864

3can be written as

120582lowast4

+ 1198601120582lowast3

+ 1198611120582lowast2

+ 1198621120582lowast+ 1198631= 0 (27)

Here

1198601= (1199021+ 1199022) + (120578 + 120583) gt 0

1198611= (120578 + 120583) (119902

1+ 1199022) + 11990211199022+ 1199023+ 1199024gt 0

1198621= (11990211199022+ 1199023) (120578 + 120583) + 119902

4(1199022+ (120578 + 120583)) minus 119902

5

1198631= 11990241199022(120578 + 120583) gt 0

(28)

with

1199021=

1205731119868lowast

1

1 + 119868lowast

1

+ 1205732119868lowast

2+ 120583

1199022=

1205731119878lowast119868lowast

1

(1 + 119868lowast

1)2

1199023=

1205731119868lowast

1

1 + 119868lowast

1

(1205731119878lowast

(1 + 119868lowast

1)2+ 1199011)

1199024= 1205732119868lowast

2[1205732119878lowastminus (120574 minus 119901

2)]

1199025=

1205781205751205731119868lowast

1

1 + 119868lowast

1

(29)

Obviously 119902119894

gt 0 119894 = 1 2 3 5 while 1199024is positive under

condition (26a) Now by using the values of 119902119894and the

sufficient condition (26b) then straightforward computationgives

1198621= 119876 + 119902

4(1199022+ (120578 + 120583)) minus 119902

5gt 0 (30)

and here 119876 = (11990211199022+ 1199023)(120578 + 120583) Moreover we have

119860111986111198621= [119876 + (119902

1+ 1199022)2(120578 + 120583) + 119902

41199021

+ (11990211199022+ 1199023+ (120578 + 120583)

2) (1199021+ 1199022)] [119876 minus 119902

5]

1198622

1+ 1198602

11198631= (119876 minus 119902

5)2+ [(1199021+ 1199022) + (120578 + 120583)]

2

sdot (11990241199022(120578 + 120583))

(31)

where 119876 = 119876 + 1199024(1199022+ (120578 + 120583)) Therefore we obtain that

119860111986111198621minus 1198622

1minus 1198602

11198631

= 119865119876 + 1199024119876 (1199021+ 1199022) + 119902411990211198601(120578 + 120583)

2

+ 1199022

41199021(1199022+ (120578 + 120583))

+ 11990241199022(1199021+ 1199022) (11990211199022+ 1199023) minus 119865119902

5+ 11986211199025

(32)

Here119865 = (1199021+1199022)2(120578+120583)+(119902

11199022+1199023+(120578+120583)

2)(1199021+1199022)+11990241199021

Hence according to condition (26b) it is easy to verifythat 119860

111986111198621minus 1198622

1minus 1198602

11198631gt 0 Therefore all the coefficients

of (27) are positive and 119860111986111198621

minus 1198622

1minus 1198602

11198631

gt 0Hence due to the Routh-Hurwitz criterion all the eigenvalues(120582lowast

119878 120582lowast

1198681

120582lowast

1198682

and 120582lowast

119877) of the Jacobianmatrix near the endemic

equilibrium point 1198643have negative real partsThus the proof

is complete

5 Global Stability Analysis

This section deals with the global stability of the equilibriumpoints of model (1) using Lyapunov methods with LaSallersquosinvariant principle The obtained results are presented in thefollowing theorems

Theorem 6 Assume that DFE 1198640

= (1198780 0 0 0) of model (1)

is locally asymptotically stable then it is global asymptoticallystable in Ω

Proof Consider Ω rarr R that is defined by

(119878 1198681 1198682 119877) =

2

sum

119894=1

119868119894 (33)

Computing the derivative of this positive semidefinite func-tion with respect to time along the solution of model (1) andthen simplifying the resulting terms give

119889

119889119905= (

1205731119878

1 + 1198681

minus (120583 + 1205721+ 120575 minus 119901

1)) 1198681

+ (1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

(34)

Since the solution of model (1) is bounded by 1198780

= Λ120583 as119905 rarr infin

119889

119889119905le (12057311198780minus (120583 + 120572

1+ 120575 minus 119901

1)) 1198681

+ (12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2)) 1198682

= (120583 + 1205721+ 120575 minus 119901

1) (1198791minus 1) 1198681

+ (120583 + 1205722+ 120574 minus 119901

2) (1198792minus 1) 1198682

(35)

Since 119879119894lt 1 119894 = 1 2 due to the local stability condition of

1198640then 119889119889119905 lt 0 Also we have that 119889119889119905 = 0 on the

set (119878 1198681 1198682 119877) isin Ω 119868

1= 1198682

= 0 so 119889119889119905 is negativesemidefinite and hence according to Lyapunov first theorem1198640is globally stable point Now since on this set we have

119889119878

119889119905= Λ minus 120583119878 + 120578119877 = 0 (36)

if and only if 119878 = 1198780 119877 = 0 thus the largest invariant set

contained in this set is reduced to the disease-free equilibriumpoint119864

0 Hence according to LaSallersquos invariant principle [18]

1198640is attractive point and hence it is globally asymptotically

stable in Ω

Theorem 7 Assume that the first disease-free equilibriumpoint 119864

1= (119878 0 119868

2 0) is locally asymptotically stable then it

is global asymptotically stable in Ω provided that

1198682gt 1198682 (37)

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Proof Consider that 119871 Ω rarr R that is defined by

119871 (119878 1198681 1198682 119877) =

1

2[(119878 minus 119878) + 119868

1+ (1198682minus 1198682) + 119877]

2

+(2120583 + 120572

1)

1205731

1198681

+(2120583 + 120572

2)

1205732

(1198682minus 1198682minus 1198682ln 1198682

1198682

)

+(2120583 + 120572

1)

21205751198772

(38)

Clearly 119871 is continuous and positive definite function Nowby taking the derivative of 119871 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119871

119889119905= minus120583 (119878 minus 119878)

2

minus 1205831198772minus 2120583119877 (119878 minus 119878)

minus (2120583 + 1205721) (119878 minus 119878) 119868

1

minus (2120583 + 1205722) (119878 minus 119878) (119868

2minus 1198682)

minus (2120583 + 1205721+ 1205722) (1198682minus 1198682) 1198681

minus (2120583 + 1205721) 1198681119877 minus (120583 + 120572

2) (1198682minus 1198682)2

minus (2120583 + 1205722) (1198682minus 1198682) 119877 minus (120583 + 120572

1) 1198682

1

119889119871

119889119905le minus120583 [(119878 minus 119878) + 119877]

2

minus 120583 [1198681+ (1198682minus 1198682)]2

minus 2120583 (1198682minus 1198682) 119877

+2120583 (120583 + 120572

1+ 120575 minus 119901

1)

1205731

[1198791

1198792

minus 1] 1198681

minus120583 (120578 + 120583)

1205751198772

(39)

Hence according to local stability condition (16) along withthe sufficient condition (37) it obtains that 119889119871119889119905 is negativedefinite function Thus due to Lyapunov second theorem 119864

1

is global asymptotically stable in Ω

Theorem 8 Assume that the second disease-free equilibriumpoint 119864

2= (

1 0 ) of model (1) is locally asymptotically

stable then it is global asymptotically stable in Ω if

1199022

12lt 1199021111990222 (40a)

1199022

14lt 1199021111990244 (40b)

1199022

23lt 1199022211990233 (40c)

1199022

34lt 1199023311990244 (40d)

where

11990211

= 120583

11990222

= (120583 + 1205721) +

(1 + 1) (1 + 119868

1)

11990233

= (120583 + 1205722)

11990244

=(2120583 + 120572

1) (120578 + 120583)

2120575+ 120583

11990212

=1

1 + 1198681

minus (2120583 + 1205721)

11990214

= 2120583

11990223

= (2120583 + 1205721+ 1205722)

11990234

= (2120583 + 1205722)

(41)

Proof Consider the function 119882 Ω rarr R that is defined by

119882(119878 1198681 1198682 119877)

=1

2[(119878 minus ) + (119868

1minus 1) + 1198682+ (119877 minus )]

2

+1

1205731

(1198681minus 1minus 1ln 1198681

1

) +(2120583 + 120572

2)

1205732

1198682

+(2120583 + 120572

1)

2120575(119877 minus )

2

(42)

Clearly 119882 is continuous and positive definite function Nowby taking the derivative of 119882 with respect to time along thesolution of model (1) we get after simplifying the resultingterms that

119889119882

119889119905= minus11990211

(119878 minus )2

+ 11990212

(119878 minus ) (1198681minus 1)

minus 11990214

(119878 minus ) (119877 minus ) minus 11990222

(1198681minus 1)2

minus 11990223

(1198681minus 1) 1198682minus 119902331198682

2minus 11990234

(119877 minus ) 1198682

minus 11990244

(119877 minus )2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(43)

Now by using the given conditions (40a)ndash(40d) we get that

119889119882

119889119905le minus [radic

11990211

2(119878 minus ) minus radic

11990222

2(1198681minus 1)]

2

minus [radic11990222

2(1198681minus 1) + radic

11990233

21198682]

2

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

minus [radic11990211

2(119878 minus ) + radic

11990244

2(119877 minus )]

2

minus [radic11990244

2(119877 minus ) + radic

11990233

21198682]

2

+2120583 (120583 + 120572

2+ 120574 minus 119901

2)

1205732

[

1198792(1 +

1)

1198791

minus 1] 1198682

(44)

Hence according to local stability condition (19a) it obtainsthat 119889119882119889119905 is negative definite function Thus due to Lya-punov second theorem 119864

1is global asymptotically stable in

Ω

Theorem 9 Assume that the endemic equilibrium point 1198643=

(119878lowast 119868lowast

1 119868lowast

2 119877lowast) ofmodel (1) is locally asymptotically stable then

it is global asymptotically stable in Ω if

1205722

14lt 1205721112057244 (45a)

1205722

12lt 1205721112057222 (45b)

1205722

24lt 1205722212057244 (45c)

where

12057211

= (12057311198681

1 + 1198681

+ 12057321198682+ 120583)

12057222

= 1205731119878lowast

12057244

= (120578 + 120583)

12057212

=1205731

1 + 1198681

minus (1205731119878lowast

(1 + 1198681) (1 + 119868

lowast

1)+ 1199011)

12057224

= 120575

12057214

= 120578

(46)

Proof Consider the function 119881 Ω rarr R that is defined by

119881 (119878 1198681 1198682 119877) =

(119878 minus 119878lowast)2

2+ (1198681minus 119868lowast

1minus 119868lowast

1ln 1198681

119868lowast

1

)

+(2120583 + 120572

2)

1205732

(1198682minus 119868lowast

2minus 119868lowast

2ln 1198682

119868lowast

2

)

+(119877 minus 119877

lowast)2

2

(47)

Clearly the function 119881 is continuous and positive definitefunction By taking the derivative of 119881 with respect to time

along the solution of model (1) we get after simplifying theresulting terms that

119889119881

119889119905= minus12057211

(119878 minus 119878lowast)2minus 12057222

(1198681minus 119868lowast

1)2

minus 12057244

(119877 minus 119877lowast)2+ 12057212

(119878 minus 119878lowast) (1198681minus 119868lowast

1)

+ 12057214

(119877 minus 119877lowast) (119878 minus 119878

lowast)

+ 12057224

(1198681minus 119868lowast

1) (119877 minus 119877

lowast)

(48)

Thenby using the given conditions (45a)ndash(45c)we obtain that

119889119881

119889119905lt minus [radic120572

11(119878 minus 119878

lowast) + radic120572

22(1198681minus 119868lowast

1)]2

minus [radic12057211

(119878 minus 119878lowast) + radic120572

44(119877 minus 119877

lowast)]2

minus [radic12057244

(119877 minus 119877lowast) + radic120572

22(1198681minus 119868lowast

1)]2

(49)

Hence 119889119881119889119905 is negative semidefinite and 119889119881119889119905 = 0 on theset (119878 119868

1 1198682 119877) isin Ω 119878 = 119878

lowast 1198681

= 119868lowast

1 1198682

gt 0 119877 = 119877lowast

so according to Lyapunov first theorem 1198643is globally stable

point Further since on this set we have

119889119878

119889119905= Λ minus (

1205731119868lowast

1

1 + 119868lowast

1

+ 12057321198682) 119878lowast+ (120574 minus 119901

2) 1198682minus 120583119878lowast

minus 1199011119868lowast

1+ 120578119877lowast= 0

(50)

if and only if 1198682

= 119868lowast

2 then the largest compact invariant set

contained in this set is reduced to the endemic equilibriumpoint119864

3 Hence according to LaSallersquos invariant principle [18]

1198643is attractive point and hence it is globally asymptotically

stable in Ω

6 Bifurcation Analysis

In this section the local bifurcations near the equilibriumpoints of model (1) are investigated as shown in the followingtheorems with the help of Sotomayor theorem [20] Note thatmodel (1) can be rewritten in a vector form 119889119883119889119905 = 119891(119883)where 119883 = (119878 119868

1 1198682 119877)119879 and 119891 = (119891

1 1198912 1198913 1198914)119879 with

119891119894 119894 = 1 2 3 4 are given in the right hand side of model (1)

Moreover straightforward computation gives that the generalsecond derivative of the Jacobian matrix (14) can be written

1198632119891 (119883 120573) (119880119880)

=

[[[[[[[[[[[

[

minus2120573112057711205772

(1 + 1198681)2

minus 2120573212057711205773+

21205731119878 (1205772)2

(1 + 1198681)3

2120573112057711205772

(1 + 1198681)2minus

21205731119878 (1205772)2

(1 + 1198681)3

2120573212057711205773

0

]]]]]]]]]]]

]

(51)

where 120573 is any bifurcation parameter and 119880 = (1205771 1205772 1205773 1205774)119879

is any eigenvector

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

Theorem 10 Assume that 1198792

lt 1 then as 1198791passes through

the value 1198791

= 1 model (1) near the disease-free equilibrium1198640has

(1) no saddle-node bifurcation

(2) a transcritical bifurcation

(3) no pitchfork bifurcation

Proof Since 1198791

= 12057311198780(120583 + 120572

1+ 120575 minus 119901

1) = 1 then 120573

1=

(120583 + 1205721+ 120575 minus 119901

1)1198780= 120573lowast

1 Now straightforward computation

shows that the Jacobianmatrix ofmodel (1) at1198640with120573

1= 120573lowast

1

has zero eigenvalue (1205821198681

= 0) and can be written as follows

1198690

=

[[[[

[

minus120583 minus120573lowast

11198780minus 1199011

minus12057321198780+ (120574 minus 119901

2) 120578

0 0 0 0

0 0 12057321198780minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]

]

(52)

Let 119881 = (V1 V2 V3 V4)119879 be the eigenvector corresponding to

1205821198681

= 0 Thus 1198690119881 = 0 gives

119881 =

[[[[[

[

119886V2

V2

0

119887V2

]]]]]

]

(53)

where V2is any nonzero real number 119886 = (120578120575(120578 + 120583) minus (120583 +

1205721+ 120575))120583 and 119887 = 120575(120578 + 120583)Similarly 119882 = (119908

1 1199082 1199083 1199084)119879 represents the eigen-

vector corresponding to eigenvalue 1205821198681

= 0 of 119869119879

0 Hence

119869119879

0119882 = 0 gives that

119882 =

[[[[[

[

0

1199082

0

0

]]]]]

]

(54)

and here 1199082is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(55)

thus 1198911205731

(1198640 120573lowast

1) = (0 0 0 0)

119879 which gives 1198821198791198911205731

(1198640 120573lowast

1) =

0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near DFE at1205731= 120573lowast

1

Now since

1198631198911205731

(1198640 120573lowast

1) =

[[[[

[

0 minus1198780

0 0

0 1198780

0 0

0 0 0 0

0 0 0 0

]]]]

]

(56)

then119882119879(1198631198911205731

(1198640 120573lowast

1)119881) = 119878

0V21199082

= 0 Now by substituting1198640and 120573

lowast

1in (51) we get

1198632119891 (1198640 120573lowast

1) sdot (119881 119881)

=

[[[[[

[

minus2120573lowast

1V1V2minus 21205732V1V3+ 2120573lowast

11198780(V2)2

2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2

21205732V1V3

0

]]]]]

]

(57)

Therefore

119882119879(1198632119891 (1198640 120573lowast

1) sdot (119881 119881))

= (2120573lowast

1V1V2minus 2120573lowast

11198780(V2)2)1199082

= [2120573lowast

1(V2)2(119886 minus 119878

0)] 1199082

= 0

(58)

So according to Sotomayorrsquos theorem model (1) has atranscritical bifurcation at 119864

0with parameter 120573

1= 120573lowast

1asymp

(1198791= 1) while the pitchfork bifurcation cannot occur

Note that similar results as those of Theorem 10 areobtained at 119879

2= 1 or 120573

2= 120573lowast

2= (120583 + 120572

2+ 120574 minus 119901

2)1198780

Theorem 11 Assume that 1198791

= 1198792 then model (1) near the

first disease-free equilibrium point 1198641= (119878 0 119868

2 0) has

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof There are two cases in the first case it is assumed that1198791

= 1198792

= 1 then straightforward computation shows that119869(1198641) = 119869(119864

0) that is 119868

2= 0 So (by Theorem 10) model (1)

has no bifurcation and then the proof is completeNow in the second case it is assumed that 119879

1= 1198792

= 1 orequivalently120573

1= 120573lowastlowast

1= 1205732(120583+1205721+120575minus119901

1)(120583+120572

2+120574minus119901

2) So

straightforward computation shows that the Jacobian matrixof model (1) at 119864

1with 120573

1= 120573lowastlowast

1has zero eigenvalue (120582

1198681

= 0)

and can be written as follows

1198691=

[[[[[

[

minus12057321198682minus 120583 minus120573

lowastlowast

1119878 minus 1199011

minus1205732119878 + (120574 minus 119901

2) 120578

0 0 0 0

12057321198682

0 1205732119878 minus (120583 + 120572

2+ 120574 minus 119901

2) 0

0 120575 0 minus (120578 + 120583)

]]]]]

]

(59)

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

Let 119883 = (1199091 1199092 1199093 1199094)119879 be the eigenvector corresponding to

1205821198681

= 0 which satisfies 1198691119883 = 0 so we get

119883 =

[[[[

[

0

1199092

11988611199092

1198871199092

]]]]

]

(60)

where 1199092is any nonzero real number 119886

1= minus[(120573

1119878 + 1199011) minus

120578120575(120578 + 120583)]1205731119878 minus (120574 minus 119901

2) and 119887 = 120575(120578 + 120583)

Similarly the eigenvector 119884 = (1199101 1199102 1199103 1199104)119879 that is

corresponding to the eigenvalue 1205821198681

= 0 of 1198691198791satisfies 119869119879

1119884 =

0 so we get

119884 =

[[[[

[

0

1199102

0

0

]]]]

]

(61)

and here 1199102is any nonzero real number Now since

119889119891

1198891205731

= 1198911205731

(119883 1205731) = (

minus1198681119878

1 + 1198681

1198681119878

1 + 1198681

0 0)

119879

(62)

then by substituting the values of 1198641and 120573

lowastlowast

1we obtain

that 1198911205731

(1198641 120573lowastlowast

1) = (0 0 0 0)

119879 and hence we get that1198841198791198911205731

(1198641 120573lowastlowast

1) = 0

Thus according to Sotomayorrsquos theorem for local bifur-cation model (1) has no saddle-node bifurcation near 119864

1at

1205731= 120573lowastlowast

1 Now since

1198631198911205731

(1198641 1205731) =

[[[[[[

[

0 minus119878 0 0

0 119878 0 0

0 0 0 0

0 0 0 0

]]]]]]

]

(63)

then 119884119879(1198631198911205731

(1198641 120573lowastlowast

1)119883) = 119878119909

21199102

= 0 Thus by substituting1198641and 120573

lowastlowast

1in (51) we get

1198632119891 (1198641 120573lowastlowast

1) sdot (119883119883)

=

[[[[[[

[

minus2120573lowastlowast

111990911199092minus 2120573211990911199093+ 2120573lowastlowast

1119878 (1199092)2

2120573lowastlowast

111990911199092minus 2120573lowastlowast

1119878 (1199092)2

2120573211990911199093

0

]]]]]]

]

(64)

Therefore [119884119879(1198632119891(1198641 120573lowastlowast

1)sdot(119883119883))] = minus2120573

lowastlowast

1119878(1199092)21199102

= 0So model (1) has a transcritical bifurcation at 119864

1with

parameter 1205731= 120573lowastlowast

1asymp (1198791= 1198792= 1) while the pitchfork bi-

furcation cannot occur and hence the proof is complete

Theorem 12 Assume that condition (19b) holds and let 1198791

=

1198792(1 +

1) then model (1) near the second disease-free

equilibrium point 1198642= (

1 0 ) undergoes

(1) no saddle-node bifurcation(2) a transcritical bifurcation(3) no pitchfork bifurcation

Proof From 1198791= 1198792(1 +

1) it is obtained that

1205732= 120573lowastlowast

2=

1205731(120583 + 120572

2+ 120574 minus 119901

2)

(120583 + 1205721+ 120575 minus 119901

1) (1 +

1)

(65)

and then straightforward computation shows that the Jaco-bian matrix of model (1) at 119864

2with 120573

2= 120573lowastlowast

2has zero

eigenvalue (1205821198682

= 0) and can be written as follows

1198692

=

[[[[[[[[[[[[

[

minus12057311

1 + 1

minus 120583minus1205731

(1 + 1)2minus 1199011

minus120573lowastlowast

2 + (120574 minus 119901

2) 120578

12057311

1 + 1

minus1205731 1

(1 + 1)2

0 0

0 0 0 0

0 120575 0 minus (120578 + 120583)

]]]]]]]]]]]]

]

(66)

Further the eigenvector 119871 = (1198971 1198972 1198973 1198974)119879 that is correspond-

ing to 1205821198682

= 0 satisfies 1198692119871 = 0 so we get

119871 =

[[[[[[[[[

[

1 + 1

1198972

1198972

minus119864

120573lowastlowast

2 minus (120574 minus 119901

2)

1198972

1198871198972

]]]]]]]]]

]

(67)

where 1198972is any nonzero real number and 119864 = (120573

1(1 +

1)2)(1+ 1) + 1205831 +

1+ 1199011minus 120578119887

Similarly the eigenvector 119870 = (1198961 1198962 1198963 1198964)119879 that is

corresponding to eigenvalue 1205821198682

= 0 of 1198691198792satisfies 119869

119879

2119870 = 0

so we get

119870 =

[[[[[

[

0

0

1198963

0

]]]]]

]

(68)

and here 1198963is any nonzero real number Now since 119889119891119889120573

2=

1198911205732

(119883 1205732) = (minus119868

2119878 1198682119878 0 0)

119879 therefore 1198911205732

(1198642 120573lowastlowast

2) =

(0 0 0 0)119879 which yields 119870

1198791198911205732

(1198642 120573lowastlowast

2) = 0 Consequently

according to Sotomayorrsquos theorem for local bifurcationmodel (1) has no saddle-node bifurcation near 119864

2at 1205732=

120573lowastlowast

2Now since

1198631198911205732

(1198642 1205732) =

[[[[[[

[

0 0 minus 0

0 0 0 0

0 0 0

0 0 0 0

]]]]]]

]

(69)

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

then 119870119879(1198631198911205732

(1198642 120573lowastlowast

2)119871) = 119897

31198963

= 0 Now by substituting1198642and 120573

lowastlowast

2in (51) we get

1198632119891 (1198642 120573lowastlowast

2) sdot (119871 119871)

=

[[[[[[[[[[[

[

minus1205731

(1 + 1)211989711198972minus 2120573lowastlowast

211989711198973minus

120573111989711198972

(1 + 1)2+

21205731119878 (1198972)2

(1 + 1)3

12057311198972

(1 + 1)211989711198972+

1205731

(1 + 1)211989711198972minus

21205731119878 (1198972)2

(1 + 1)3

2120573lowastlowast

211989711198973

0

]]]]]]]]]]]

]

(70)

Therefore [119870119879(1198632119891(1198642 120573lowastlowast

2) sdot (119871 119871))] = 2120573

lowastlowast

2119897111989731198963

= 0Thus model (1) undergoes a transcritical bifurcation at 119864

2

with parameter 1205732

= 120573lowastlowast

2asymp (1198791

= 1198792(1 +

1)) while the

pitchfork bifurcation cannot occur

Moreover the following results are obtained too

(1) Although 1198791= 1198792

= 1 gives 1205732= lowastlowast

2 model (1) does

not undergo any of the above types of bifurcation nearthe equilibrium point 119864

1with parameter 120573

2= lowastlowast

2

(2) Although 1198791

= 1198792(1 +

1) gives 120573

1= lowastlowast

1 model (1)

does not undergo any of the above types of bifurcationnear the equilibrium point 119864

2with parameter 120573

1=

lowastlowast

1

(3) The determinant of the Jacobian matrix at 1198643 say

119869(1198643) cannot be zero and hence it has no real zero

eigenvalue So there is no bifurcation near 1198643

Keeping the above in view in the following theoremwe detectof the possibility of having Hopf bifurcation

Theorem 13 Assume that condition (26a) holds and let thefollowing conditions be satisfied Then model (1) undergoesHopf bifurcation around the endemic equilibrium point whenthe parameter 120573

2crosses a critical positive value

lowast

2

1198732gt 0

1198734lt 0

(71a)

1205901gt 21205902 (71b)

120575 lt 1199025lt 11990241199022 11990241199021 (71c)

1199022gt 1 (71d)

and here 1205901= 1199022

1+ 1199022

2+ (120578 + 120583)

2 and 1205902= (1199021+ 1199022)(120578 + 120583) +

211990211199022+ 21199023+ 21199024 while 119873

2and 119873

4are given in the proof

Proof It is well known that in order for Hopf bifurcation infour-dimensional systems to occur the following conditionsshould be satisfied [21 22]

(1) The characteristic equation given in (28) has two realand negative eigenvalues and two complex eigenval-ues say 120582lowast(120573

2) = 1205911(1205732) plusmn 1205912(1205732)

(2) 1205911(lowast

2) = 0

(3) (1198891198891205732)1205911(1205732)|1205732=lowast

2

= 0 (The transversality condi-tion)

Accordingly the first two points are satisfied if and only if

Δ2(lowast

2) = 119860

111986111198621minus 1198622

1minus 1198602

11198631= 0

1198603

1minus 4Δ1gt 0

(72)

while the third condition holds provided that

119889

1198891205732

1205911(1205732)

100381610038161003816100381610038161003816100381610038161205732=lowast

2

=minus (Ψ (120573

2)Θ (120573

2) + Φ (120573

2) Γ (1205732))

Ψ (1205732)2+ Φ (120573

2)2

= 0

(73)

That means Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

Here

Ψ (1205732) = (4120591

3

1minus 1212059111205912

2) + 3119860

1(1205912

1minus 1205912

2) + 2119861

11205911

+ 1198621

Φ (1205732) = (12120591

2

11205912minus 41205913

2) + 6119860

112059111205912+ 211986111205912

Θ (1205732) = 119860

1(1205913

1minus 312059111205912

2) + 1198611(1205912

1minus 1205912

2) + 11205911

+ 1198631

Γ (1205732) = 119860

1(31205912

11205912minus 1205913

2) + 2 119861

112059111205912+ 11205912

(74)

Now straightforward computation shows the condition

Δ2(lowast

2) = 0

gives that 11987311205733

2+ 11987321205732

2+ 11987331205732+ 1198734= 0

(75)

where

1198731= [1199022(120578 + 120583) ((120578 + 120583) + 119902

2) + (120578 + 120583) (120583 + 120572

2) ((120578

+ 120583) + 21199022+ (120583 + 120572

2))] 119868lowast3

2gt 0

1198732= [(120578 + 120583) [119902

2(120578 + 120583) (3119902

6+ 2 + (120578 + 120583))

+ 1199023((120578 + 120583) + 2 (120583 + 120572

2) + 2119902

2) + (120583 + 120572

2)

sdot [1199026(2 (120578 + 120583) + (120583 + 120572

2))

+ 1199022((120578 + 120583) + 4119902

6+ 1199022)] + 119902

2

2(31199026+ 1199022) + (120583

+ 1205722) (120578 + 120583)

2minus 1199025] + (120583 + 120572

2) [1199022

2(1 + 119902

2)

+ 1199022(1199023+ 1199026(120583 + 120572

2))] minus 119902

21199025] 119868lowast2

2

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

1198733= [(120578 + 120583)

2[1199023

2+ 1199023(21199026+ 21199022+ (120578 + 120583))

+ 311990261199022minus 1199025+ (1199026+ 1199022) 1199026(120583 + 120572

2)] + (120578 + 120583)

sdot [11990261199022

2(31199026+ 21199022) + (2119902

6+ 1199022) [211990221199023

+ 1199022(120578 + 120583)

2] + (120583 + 120572

2) [1 + 119902

6(120578 + 120583)

2

+ (21199026+ 1199022) (11990261199022+ 1199023minus 211990261199025) minus 11990221199025]] + 119902

2(120583

+ 1205722) [11990261199022(1 + 119902

2) + (119902

6+ 1199022) 1199023+ 1] minus (119902

2(21199026

+ 1199022) + 1199023) 1199025] 119868lowast

2

1198734= [(120578 + 120583)

2[11990261199022(1199022

6+ 11990261199022+ 1199022

2) + 11990231199026(1199026

+ 21199022) + (119902

6+ 1199022) (120578 + 120583) (119902

61199022+ 1199023) + 11990231199022

2

minus 1199022] + (119902

6+ 1199022) (120578 + 120583) [119902

2

61199022

2+ 2119902611990221199023+ 1199022

3]

+ ((120578 + 120583) (1199023minus 11990261199022) minus 1199026(120578 + 120583) (119902

6+ (120578 + 120583))

minus 1199022

2(120578 + 120583) minus (119902

6+ 1199022) (11990261199022+ 1199023)) 1199025minus 1199022

5]

(76)

with 1199021 1199022 1199023 1199024 and 119902

5given in (28) and 119902

6= 1199021minus 1205732119868lowast

2

Clearly fromcondition (71a) there is unique positive root say1205732= lowast

2 Consequently by using Δ

2= 0 in the characteristic

equation and then doing some algebraic computation we getfour roots

12058212

(lowast

2) = plusmn119894radic

1198621

1198601

= 1205912(lowast

2)

12058234

=1

2(minus1198601plusmn radic119860

2

1minus 4

Δ1

1198601

)

(77)

Now it is easy to verify that 1205823and 120582

4are real and negative

provided that (71b)Further for 120573

2isin (lowast

2minus 120598

lowast

2+ 120598) the general form of

complex eigenvalues can be written as

1205821= 1205911(1205732) + 1198941205912(1205732)

1205822= 1205911(1205732) minus 1198941205912(1205732)

(78)

Substituting 120582(1205732) = 120591

1(1205732) + 119894120591

2(1205732) into characteristic

equation and after that calculating the derivative with respectto 1205732and then comparing the real and imaginary parts give

that

Ψ (1205732) 1205911(1205732) minus Φ (120573

2) 1205912(1205732) = minusΘ (120573

2)

Φ (1205732) 1205911(1205732) + Ψ (120573

2) 1205912(1205732) = minusΓ (120573

2)

(79)

Moreover by solving the above linear system for 1205911(1205732) and

1205912(1205732) then we get that

Ψ (1205732)Θ (120573

2) + Φ (120573

2) Γ (1205732) = 11990211199022[1199023(120583 + 120572

2)

+ 2 (120578 + 120583)3

+ (120583 + 1205722) (120578 + 120583) ((120578 + 120583) + (3119902

2+ 31199021minus 1))]

+ 11990221199023[1199022(120583 + 120572

2)

+ (120578 + 120583) ((120583 + 1205722) + (2119902

2+ 21199021minus 1))] + 119902

1(120578

+ 120583) (120583 + 1205722) [(120578 + 120583) + 2119902

3+ (120578 + 120583)

2+ 1199024] + (120578

+ 120583)3[1199022(120583 + 120572

2) + 1199022

2+ 1199023+ 1199024] + 1199025(120583 + 120572

2)

sdot [21199022+ (120578 + 120583)] + 119902

11199023(120578 + 120583)

2+ 1199022(120578 + 120583) (120583

+ 1205722) [1199022(1199022minus 1) + (120578 + 120583) (2119902

2+ 21199021minus 1)]

+ 1199021(120583 + 120572

2) [11990211199022

2+ 1199023

2+ 11990221199024minus 1199025] + 1199021(120578 + 120583)

sdot [211990211199022(1 + 119902

2) + 2119902

3

2+ 11990221199024minus 1199025] + (119902

21199024minus 1199025)

sdot (11990211199022+ 1199022

2) + (120578 + 120583)

2[311990211199022

2+ 1199023

2+ 11990211199024minus 1199025]

(80)

Thus it is easy to verify that Ψ(1205732)Θ(1205732) + Φ(120573

2)Γ(1205732) = 0

provided that (71c) and (71d)Thus the proof is complete

7 Numerical Simulations

In this section the global dynamics of model (1) is investi-gated numerically for different sets of initial values and dif-ferent sets of parameters values The objectives of such inves-tigation are to determine the effect of varying the parametersvalues and confirm our obtained results It is observed thatfor the following biologically feasible set of hypotheticalparameters values

Λ = 20

1205731= 075

1205732= 01

120574 = 075

1199011= 001

1199012= 001

120583 = 03

120578 = 05

1205721= 01

1205722= 06

120575 = 07

(81)

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Applied Mathematics

Started at 30Started at 20Started at 10

1000 20000Time

10

20

30

S

(a)

Started at 5Started at 10Started at 15

I1

1000 20000Time

5

125

20

(b)

Started at 10Started at 5Started at 15

1000 20000Time

5

10

15

20

I2

(c)

Started at 10Started at 5Started at 20

1000 20000Time

5

10

15

20

R

(d)

Figure 1 Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81) started from different sets of initialpoint

the solution of model (1) approaches asymptotically to theendemic equilibrium point 119864

3= (164 1028 918 899) as

shown in Figure 1 started from different sets of initial pointsClearly Figure 1 confirms our obtained analytical results

regarding existence of a globally asymptotically stable pos-itive equilibrium point when the parameters values aresatisfying 119877

1gt 1198772gt 1 On the other hand model (1) for the

following set of hypothetical data approaches asymptoticallyto the DFE as shown in Figure 2

Λ = 20

1205731= 005

1205732= 005

120574 = 06

1199011= 001

1199012= 003

120578 = 05

120583 = 09

1205721= 01

1205722= 01

120575 = 03

(82)

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 13

1000 2000 30000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

Figure 2 Time series of the solution of model (1) that approachesasymptotically to DFE for the data (82)

It is easy to verify that for the data (82) we have1198770= 086 lt 1

and the solution approaches to 1198640= (2222 0 0 0)

Now in order to investigate the effect of varying oneparameter value at a time on the dynamical behavior ofmodel(1) the following results are observed

(i) Varying of the parameters values (Λ 120574 1199012 120578 1205721 1205722 120575)

does not affect the dynamical behavior of model (1)that is the system still approaches to coexistenceequilibrium point

(ii) For the data (81) with 1205731

le 0015 the solutionof model (1) approaches asymptotically to 119864

1=

(119878 0 1198682 0) in the interior of positive quadrant of 119878119868

2-

plane with 1198771

lt 1 lt 1198772 However for 0015 lt

1205731lt 0066 the solution of the model still approaches

asymptotically to 1198641even when 1 lt 119877

1lt 1198772 Finally

when 1205731ge 0066 the solution of model (1) approaches

to coexistence equilibrium point with 1 lt 1198772lt 1198771as

shown in Figure 3(iii) Similar results are obtained in case of varying the

parameter 1205732keeping the rest of parameters values

in (81) fixed In fact for 1205732

le 0024 we have 1198772

lt

1 lt 1198771 and the solution of model (1) approaches

asymptotically to 1198642

= (119878 1198681 0 119877) However for

0024 lt 1205732

lt 006 we have 1 lt 1198772

lt 1198771 and it is

observed that the solution of model (1) approachesasymptotically to 119864

2= (119878 119868

1 0 119877) too while when

1205732

ge 006 the solution of model (1) approachesasymptotically to 119864

3= (119878 119868

1 1198682 119877) as shown in

Figure 4(iv) Now decreasing the parameter 120583 keeping the rest

of the parameters values in (81) fixed gives similardynamical behavior as that of varying 120573

2 Further it

is observed that when 120583 gt 09 for which 1198772lt 1 lt 119877

1

the solution of model (1) approaches asymptoticallyto 1198642

= (119878 1198681 0 119877) however when 057 lt 120583 lt 09

we have 1 lt 1198772

lt 1198771 and the solution of model

(1) still approaches to 1198642

= (119878 1198681 0 119877) Finally for

120583 le 057 that satisfies 1 lt 1198772

lt 1198771 the solu-

tion of model (1) approaches asymptotically to 1198643

=

(119878 1198681 1198682 119877) Clearly this confirmed our obtained exis-

tence conditions (10) and (12) as well as stabilityconditions of these points

(v) Finally varying the parameter 1199011 keeping the rest of

the parameters values in (81) fixed showed that for1199011

ge 06 that satisfies 1 lt 1198772

lt 1198771 the solution

of model (1) approaches asymptotically to 1198642

= (1198781198681 0 119877) however when 119901

1lt 06 which satisfies

1 lt 1198772lt 1198771too the solution of model (1) approaches

asymptotically to 1198643

= (119878 1198681 1198682 119877) as shown in

Figure 5

8 Conclusion

In this paper we proposed and analyzed an epidemic modelinvolving vertical and horizontal transmission of infectionwith nonlinear incidence rate It is assumed that the ratesof infections 119901

1 1199012are less than the recovery rates 120575

and 120574 respectively According to the diseases in model (1)the population is divided into four subclasses susceptibleindividuals that are represented by 119878(119905) infected individualsfor 119878119868119877119878-type of disease that are represented by 119868

1(119905) infected

individuals for 119878119868119878-type of disease that are represented by1198682(119905) and recovery individuals that are denoted by 119877(119905) The

boundedness and invariant of the model are discussed Thebasic reproduction number of the model and the associatedthreshold parameter values namely 119879

119894 119894 = 1 2 are deter-

mined It is observed that if the basic reproduction numberis less than unity then the diseases are eradicated fromthe model The competitive exclusion principle occurred inmodel (1) such that only the second disease-free equilibriumpoint appeared in case of 119877

2lt 1 lt 119877

1 However only the first

disease-free equilibrium point appeared in case of 1198771lt 1 lt

1198772 Finally the coexistence of both the diseases occurred in

case of 1 lt 1198772

lt 1198771and the sufficient condition (12) holds

The dynamical behavior of model (1) has been investigatedlocally as well as globally using Routh-Hurwitz criterionand Lyapunov function respectively The local bifurcationsof model (1) and the Hopf bifurcation around the endemicequilibriumpoint are studied Finally to understand the effectof varying each parameter on the global dynamics of system(1) and to confirm our obtained analytical results model (1)has been solved numerically and the following results areobtained for the set of hypothetical parameters values givenby (81)

(1) Model (1) approaches asymptotically to a globallyasymptotically stable point 119864

3= (164 1028 918

899)(2) Varying one of the parameters values (Λ 120574 119901

2 120578

1205721 1205722 120575) at a time keeping other parameters fixed has

no effect on the dynamical behavior of the model(3) As the infection rate of the first disease (120573

1) decreases

keeping other parameters fixed as in (81) the solution

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Journal of Applied Mathematics

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(b)

1000 20000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(c)

Figure 3 Time series of the solution ofmodel (1) for the data given by (81) (a) For 1205731= 0015 themodel approaches to119864

1= (164 0 1675 0)

(b) For 1205731= 006 the model approaches to the same point (c) For 120573

1= 03 the model approaches to 119864

3= (164 351 141 307)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(a)

S

I1 R

I2

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

(b)

S

I1 R

I2

2000 40000Time

0

5

10

15

20

25

30

Popu

latio

ns

(c)

Figure 4 Time series of the solution of model (1) for the data given by (81) (a) For 1205732

= 0024 the model approaches to 1198642

=

(2733 1780 0 155) (b) For 1205732

= 0059 the model approaches to the same point (c) For 1205732

= 0065 the model approaches to 1198643

=

(252 163 176 143)

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 15

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

ns

S

I1 R

I2

(a)

4000 80000Time

0

5

10

15

20

25

30

Popu

latio

nsS

I1 R

I2

(b)

Figure 5 Time series of the solution ofmodel (1) for the data given by (81) (a) For1199011= 06 themodel approaches to119864

2= (159 229 0 2008)

(b) For 1199011= 05 the model approaches to 119864

3= (164 195 240 1706)

of model (1) approaches asymptotically to the equilib-rium point 119864

1= (164 0 1675 0) However in case

of increasing this parameter the model is still globallyasymptotically stable in the interior of R4

+

(4) As the infection rate of the second disease (1205732)

decreases keeping other parameters fixed as in (81)the solution of model (1) approaches asymptoticallyto the equilibrium point 119864

2= (2733 1780 0 155)

However in case of increasing this parameter themodel is still globally asymptotically stable in theinterior of R4

+

(5) As the mortality rate (120583) increases keeping otherparameters fixed as in (81) the solution of model (1)approaches asymptotically to the equilibrium point1198642

= (138 516 0 25) and when 120583 decreases themodel is still globally asymptotically stable in theinterior of R4

+ Further it is observed that 119901

1has the

same effect as 120583 on the dynamical behavior of model(1)

(6) For the parameter set given in (82) the solution ofmodel (1) approaches asymptotically to DFE

(7) Finally although for our selected parameters valuesmodel (1) does not undergo periodic dynamics themodel still has possibility to have periodic dynamicsfor other sets of parameters especially Hopf bifurca-tion existing analytically

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Hardin ldquoThe competitive exclusion principlerdquo Science vol131 no 3409 pp 1292ndash1297 1960

[2] A Iggidr J-C Kamgang G Sallet and J-J Tewa ldquoGlobalanalysis of new malaria intrahost models with a competitiveexclusion principlerdquo SIAM Journal on AppliedMathematics vol67 no 1 pp 260ndash278 2006

[3] V Volterra ldquoVariations and fluctuations of the number ofindividuals in animal species living togetherrdquo Journal du ConseilInternational pour lrsquoExploration de la Mer vol 3 no 1 pp 3ndash511928

[4] A S Ackleh and L J Allen ldquoCompetitive exclusion in SIS andSIR epidemic models with total cross immunity and density-dependent host mortalityrdquo Discrete and Continuous DynamicalSystemsmdashSeries B vol 5 no 2 pp 175ndash188 2005

[5] F Brauer J Wu and P van den Driessche MathematicalEpidemiology Springer Berlin Germany 2008

[6] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[7] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 no 4 pp 281ndash293 2005

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

16 Journal of Applied Mathematics

[8] O Diekmann J A P Heesterbeek and J A J Metz ldquoOnthe definition and the computation of the basic reproductionratio 119877

119900in models for infectious diseases in heterogeneous

populations in models for infectious diseases in heterogeneouspopulationsrdquo Journal of Mathematical Biology vol 28 no 4 pp365ndash382 1990

[9] M Martcheva ldquoA non-autonomous multi-strain SIS epidemicmodelrdquo Journal of Biological Dynamics vol 3 no 2-3 pp 235ndash251 2009

[10] A S Ackleh and L J S Allen ldquoCompetitive exclusion andcoexistence for pathogens in an epidemic model with variablepopulation sizerdquo Journal of Mathematical Biology vol 47 no 2pp 153ndash168 2003

[11] M Lipsitch S Siller and M A Nowak ldquoThe evolution of vir-ulence in pathogens with vertical and horizontal transmissionrdquoEvolution vol 50 no 5 pp 1729ndash1741 1996

[12] D Bichara A Iggidr and G Sallet ldquoGlobal analysis of multi-strains SIS SIR and MSIR epidemic modelsrdquo Journal of AppliedMathematics and Computing vol 44 no 1-2 pp 273ndash292 2014

[13] M Y Li H L Smith and L Wang ldquoGlobal dynamics an SEIRepidemic model with vertical transmissionrdquo SIAM Journal onApplied Mathematics vol 62 no 1 pp 58ndash69 2001

[14] W M Liu H W Hethcote and S A Levin ldquoDynamicalbehavior of epidemiological models with nonlinear incidenceratesrdquo Journal of Mathematical Biology vol 25 no 4 pp 359ndash380 1987

[15] O Adebimpe A AWaheed and B Gbadamosi ldquoModeling andanalysis of an SEIRS epidemic model with saturated incidencerdquoJournal of Engineering Research and Application vol 3 no 5 pp1111ndash1116 2013

[16] R K Naji and A N Mustafa ldquoThe dynamics of an eco-epidemiological model with nonlinear incidence raterdquo Journalof Applied Mathematics vol 2012 Article ID 852631 24 pages2012

[17] C V De-Leon ldquoOn the global stability of SIS SIR and SIRSepidemic models with standard incidencerdquo Chaos Solitons ampFractals vol 44 no 12 pp 1106ndash1110 2011

[18] J P LaSalle ldquoStability theory for ordinary differential equa-tionsrdquo Journal of Differential Equations vol 4 no 1 pp 57ndash651968

[19] E Shim An epidemic model with immigration of infectives andvaccination [MS thesis] Department of Mathematics Instituteof Applied Mathematics University of British Columbia Van-couver Canada 2004

[20] L Perko Differential Equation and Dynamical SystemsSpringer New York NY USA 3rd edition 2001

[21] M M A El-Sheikh and S A A El-Marouf ldquoOn stabilityand bifurcation of solutions of an SEIR epidemic model withvertical transmissionrdquo International Journal ofMathematics andMathematical Sciences vol 2004 no 56 pp 2971ndash2987 2004

[22] X Zhou and J Cui ldquoAnalysis of stability and bifurcation for anSEIV epidemicmodelwith vaccination andnonlinear incidenceraterdquo Nonlinear Dynamics vol 63 no 4 pp 639ndash653 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of