the dynamics of growing islets and transmission of schistosomiasis japonica in the yangtze river

Bull Math Biol (2014) 76:1194–1217DOI 10.1007/s11538-014-9961-7

ORIGINAL ARTICLE

The Dynamics of Growing Islets and Transmissionof Schistosomiasis Japonica in the Yangtze River

Chunhua Shan · Xiaonong Zhou · Huaiping Zhu

Received: 1 October 2012 / Accepted: 9 April 2014 / Published online: 24 April 2014© Society for Mathematical Biology 2014

Abstract We formulate and analyze a system of ordinary differential equations forthe transmission of schistosomiasis japonica on the islets in the Yangtze River, China.The impact of growing islets on the spread of schistosomiasis is investigated by thebifurcation analysis. Using the projection technique developed by Hassard, Kazarinoffand Wan, the normal form of the cusp bifurcation of codimension 2 is derived toovercome the technical difficulties in studying the existence, stability, and bifurcationof the multiple endemic equilibria in high-dimensional phase space. We show that themodel can also undergo transcritical bifurcations, saddle-node bifurcations, a pitchforkbifurcation, and Hopf bifurcations. The bifurcation diagrams and epidemiologicalinterpretations are given. We conclude that when the islet reaches a critical size, thetransmission cycle of the schistosomiasis japonica between wild rats Rattus norvegicusand snails Oncomelania hupensis could be established, which serves as a possiblesource of schistosomiasis transmission along the Yangtze River.

Keywords Schistosomiasis · Growing islet · Transmission dynamics · Cuspbifurcation of codimension 2 · Transcritical bifurcation · Saddle-node bifurcation ·Pitchfork bifurcation · Hopf bifurcation

1 Introduction

Schistosomiasis is a parasitic disease affecting at least 240 million people world-wide, and more than 700 million people living in endemic areas are at risk

C. Shan · H. Zhu (B)Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canadae-mail: [email protected]

X. ZhouNational Institute of Parasitic Disease, Chinese Center for Disease Control and Prevention,Shanghai, China

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Dynamics of Growing Islets and Transmission of Schistosomiasis Japonica 1195

(http://www.who.int/mediacentre/factsheets/fs115/en/index.html). It remains formi-dable to humans because of the complexities of parasitic adaptation to two or moredifferent hosts. The persistence of this disease in a locality depends on a complexcycle of events involving humans or mammals (the definitive host), certain parasiticflatworms (the schistosome), and particular species of snails (the intermediate host).In this cycle, humans or mammals will become infected when they contact watercontaining the cercariae released from the infected snails, while snails will becomeinfected when they are invaded by miracidia hatched from eggs, which are released byinfected humans or other mammals (Anderson and May 1991; Gryseels et al. 2006).

The prevalence of Schistosomiasis japonica in China can trace back to thousands ofyears ago by the traditional records. The control program of schistosomiasis in China,launched in the 1950s and sustained over the past 60 years, is widely acknowledgedas one of the most successful disease control programs (Zhou et al. 2005). However,the surveillance data suggest that schistosomiasis transmission has re-emerged in 38counties from 7 provinces along the Yangtze River since the late 1990s, as both snailhabitats and local transmission had been observed (Zhao et al. 2005; Zhou et al. 2002,2005). The disease control is still a great challenge in endemic regions due to climateand environmental changes. Understanding the impact of these factors on the diseasetransmission is essential to the development of comprehensive control strategies.

In order to understand the mechanism between environmental changes and theschistosomiasis disease, we assess the patterns of the schistosomiasis transmissionre-emerging in Qian islet and Zimu islet, two typical ones among the hundreds ofislets in the Yangtze River, China (Fig. 1). Because of the sedimentation, two isletsstarted to emerge from the water surface in 1970 and 1976, respectively. The areas oftwo islets have been expanding since their formulation (Xu et al. 1999). As of 2011,the area of Qian islet is 3.33 million m2, and the area of Zimu islet is 5 million m2.

The snails were discovered on Qian islet in 1980, 10 years after its emergencefrom the river, and the record showed that snails appeared on Zimu islet in 1984. Theinvestigation carried out by the researchers from Nanjing Institute of Parasitic Diseasesin the period of 1996–1998 indicated that schistosomiasis disease had presented inthe two islets, and concluded that rats Rattus norvegicus were the primary definitive

? Year

Kc

2011

.

Area(million m^2)

3.33

19961980(Snails were discovered ) (Disease had presented)

1970(Islet emerged)

Fig. 1 Schematic diagram of the history of the schistosomiasis disease in Qian islet (Color figure online)

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http://www.who.int/mediacentre/factsheets/fs115/en/index.html

1196 C. Shan et al.

hosts and snails Oncomelania hupensis were the intermediate hosts of Schistosomajaponicum (Xu et al. 1999). From the land size evolution of the islets and outbreak ofschistosomiasis, it is reasonable to connect the transmission of schistosomiasis withthe growing sizes of the islets. In this paper, we will use mathematical models toexplore the impact of the expansion of islets on the disease transmission and try toexplain the re-emergence mechanism of S. japonica along the Yangtze River from thelate 1990s.

There have been lots of mathematical studies of schistosomiasis since 1960s(Castillo-Chavez et al. 2008; Feng et al. 2002, 2005; Liang et al. 2002, 2007; Mac-Donald 1965; Nåsell and Hirsch 1973; Wu et al. 1987; Wu and Feng 2002; Zhang etal. 2007; Zhao and Milner 2008), etc. The first mathematical model of schistosomiasiswas proposed and studied by MacDonald (1965), which is dated back to 1965. Sincethen much has been done in terms of modeling and analyses of transmission dynamicsof schistosomiasis. Nåsell and Hirsch (1973) proposed a stochastic version of Mac-Donald’s model. Wu et al. (1987) built and analyzed a schistosomiasis model, and thenit was generalized with the multiple definitive hosts (Wu and Feng 2002). Mathemat-ical models with mass chemotherapy in human hosts were formulated and studied byFeng et al. (2002). The schistosomiasis models with multigroups or age structure forhuman hosts were studied in Feng et al. (2005), Liang et al. (2002), Liang et al. (2007),Zhang et al. (2007). The delay during the schistosomiasis transmission was consideredin Castillo-Chavez et al. (2008), Wu and Feng (2002). However, most of these modelshave ignored the impact of the environment, especially the natural carrying capacityfor both the definitive hosts and the intermediate hosts, on the transmission of thisdisease.

For a finite or infinite dimensional dynamical system, the center manifold theo-rem plays an important role, which reduces the system to a lower dimension to makethe analysis easier (Carr 1981). There are no theoretical difficulties in center manifoldcalculation. However, the calculation is rather complicated for a high or infinite dimen-sional system, and that is the reason why the bifurcation analyses of high codimensionwere carried out in the planar system in most papers. With the adjoint operator theoryand the spectral decomposition, the projection technique developed in Hassard et al.(1981) to study the Hopf bifurcation in infinite dimensional system is very powerful forthe center manifold calculation. In this paper, we will apply the projection techniqueto the bifurcation analysis due to the high dimension of our model.

The paper is organized as follows. In Sect. 2, we formulate a system of ordinary dif-ferential equations incorporating the impact of growing islets to model the mouse–snailcycle of schistosomiasis. This high-dimensional model can exhibit multiple endemicequilibria which results in complexity of dynamics and technical difficulties in study-ing the existence and stability of endemic equilibria. Therefore, in Sect. 3, we calculatethe center manifold and normal form of the cusp bifurcation of codimension 2 to over-come these difficulties, and then study the Hopf bifurcation. The effect of the isletexpansion on the disease transmission is obtained from the bifurcation analysis. InSect. 4, numerical simulations are carried out to supplement and illustrate the theoret-ical results. The conclusion and control strategies are given in Sect. 5.

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2 Formulation of the Model

According to the investigations by the researchers from Nanjing Institute of ParasiticDiseases (Xu et al. 1999), there are no human residents, livestock, or wild animalson Qian and Zimu islets. The rats R. norvegicus are the only definitive hosts, and thesnails O. hupensis serve as the intermediate hosts. The ecosystem on these two isletsis so ideal for our modeling, and we study the schistosomiasis transmission on anyone of these two islets.

Suppose Ms and Mi are the numbers of susceptible and infected mice, respectively.Let Ss and Si represent the numbers of susceptible and infected snails, respectively.

The growth of host population will be limited by intraspecific competitions for finiteresources, so we assume that the populations of mice and snails on the islet follow thelogistic growth. For the snails, more than 97 % of the infected female snails lose thereproduction ability because of the lesions and atrophy of gonads (Zhou 2005). Thereare few literatures concerning the impact of the disease on the reproduction ability ofmammals. However, some researches have shown that schistosomiasis can damage thegenital system of human beings, and results in ectopic pregnancy, infertility, abortion,increased infant mortality rate, cervical lesions, and cervical cancer (Poggensee andFeldmeier 2001; Poggensee et al. 1999), (http://www.who.int/mediacentre/factsheets/fs115/en/index.html). Therefore, we suppose that the infected snails and mice cannotreproduce, and the growth of the populations of the snails and mice is contributed bythe susceptible classes.

According to the life cycle of schistosomiasis transmission, we know that the defin-itive hosts and intermediate hosts can infect each other via the cercariae and miracidia(Anderson and May 1991; Gryseels et al. 2006), and we assume that the cross-infectionbetween mice and snails is subject to the mass action principle. Therefore, we formu-late the schistosomiasis model involving the two hosts as follows:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Ms = rm Ms

(

1 − Ms + δ1 Mi

Km

)

− dm Ms − βm MsSi ,

Mi = βm MsSi − (dm + μm)Mi ,

Ss = rsSs

(

1 − Ss + δ2Si

Ks

)

− dsSs − βs Mi Ss,

Si = βs Mi Ss − (ds + μs)Si .

(1)

Here, rm and rs are the intrinsic growth rates of mice and snails, respectively. βm andβs are the transmission rates from the infected snails to susceptible mice and fromthe infected mice to susceptible snails, respectively. dm and ds denote the per capitanature death rates of mice and snails, and μm and μs are the per capita death rates ofmice and snails induced by infection, respectively.

In model (1), Km and Ks in the logistic growth terms are the spatial environmentalcapacities of the islet, which correspond to the maximum populations of mice andsnails, respectively. Since the spatial environmental capacity depends on the physicalsize of the islet, if we let K (m2) be the physical size of the islet, and denote themaximum densities of mice and snails per m2 as km and ks, then we can write Km =km K and Ks = ks K .

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1198 C. Shan et al.

It is apparent that the infected mice and snails do not share the same resources onthe islet, so we introduce two positive parameters δ1 and δ2 in model (1) to measurethe competition abilities between infected and susceptible classes of mice and snails,respectively.

For the infected snails, their digestive system especially the digestive gland isdestroyed, and some organs loss their functions due to the connective tissue hyperplasia(Zhou 2005). For the infected mammals, the disease can induce the impaired physicaland cognitive development. Severely infected animals deteriorate rapidly and usuallydie within a few months of infection, while those less heavily infected ones will developchronic disease with the growth retardation.

Therefore, it is reasonable to presume that for the infected mice and snails, theyhave no enough energy and mobility to compete with the susceptible species for thesustaining resources, and we have 0 ≤ δ1 � 1, 0 ≤ δ2 � 1. In the following analysis,we will consider a simplified version of model (1) by assuming that δ1 = 0 and δ2 = 0and study the corresponding simplified model, and we have the following proposition.

Proposition 2.1 For system (1), R4+ is invariant, and the solution exists globally.

Proof First, the smoothness of functions at the right-hand side of system (1) insuresthe local existence and uniqueness of the system with the initial condition.

Second, Ms|Ms=0 = 0, Mi |Mi =0 = βm MsSi > 0, Ss|Ss=0 = 0, Si |Si =0 =βs Mi Ss > 0, so the solution with the initial value in R4+ will remain nonnegative forall t ≥ 0.

From (1), we have

Ms ≤ rm Ms

(

1 − Ms

Km

)

− dm Ms and Ss ≤ rsSs

(

1 − Ss

Ks

)

− dsSs.

Let Nm = Ms + Mi and Ns = Ss + Si . From system (1), we have

Nm ≤ rm Ms

(

1 − Ms

Km

)

− dm Nm − μm Mi ≤ 1

4rm Km − dm Nm,

Ns ≤ rsSs

(

1 − Ss

Ks

)

− ds Ns − μsSi ≤ 1

4rs Ks − ds Ns.

Therefore, all the solutions in the nonnegative cone approach, enter, or stay inside theregion.

� ={

0 ≤ Ms ≤ U1, 0 ≤ Ss ≤ U2, 0 ≤ Nm ≤ rm Km

4dm, 0 ≤ Ns ≤ rs Ks

4ds

}

,

where U1 = max{0, Km(rm−dm)rm

} and U2 = max{0, Ks(rs−ds)rs

}. ��

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3 Dynamics of the Model

3.1 Existence of Equilibria

System (1) has at most four equilibria on the boundary of the nonnegative cone: E0 =(0, 0, 0, 0), representing the extinction of both species; Eb1 = (

Km(rm−dm)rm

, 0, 0, 0),representing the extinction of the snails and existence of the mice; Eb2 =(0, 0, Ks(rs−ds)

rs, 0), representing the extinction of the mice and existence of the snails;

and EDFE = (Km(rm−dm)

rm, 0, Ks(rs−ds)

rs, 0), representing the coexistence of both species

in the absence of disease.According to the next generation matrix method (van den Driessche and Watmough

2002), we calculate the basic reproduction number

R0 =√βm Km(rm − dm)

rm(dm + μm)

βs Ks(rs − ds)

rs(ds + μs). (2)

R0 is well defined if rm > dm and rs > ds. If rm ≤ dm or rs ≤ ds, then either mice orsnails will be extinct, and the disease cannot outbreak. From now on, we study system(1) when R0 is well defined. One can obtain the following proposition based on thestandard stability analysis.

Proposition 3.1 For system (1),

Equilibrium rm < dm, rs < ds rm > dm, rs > ds

R0 < 1 R0 = 1 R0 > 1

E0 stablenode saddle saddle saddle

Eb1 notexist saddle saddle saddle

Eb2 notexist saddle saddle saddle

EDF E notexist stablenode saddle-node saddle

For any endemic equilibrium, one can find that its coordinates should satisfy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Mi = g1(Si ) �

C︷︸︸︷βm

dm + μm

Km

rmβm

⎛

⎜⎜⎜⎝

D︷︸︸︷rm − dm

βm−Si

⎞

⎟⎟⎟⎠

Si ,

Si = g2(Mi ) � βs

ds + μs

Ks

rsβs

︸︷︷︸A

⎛

⎜⎜⎜⎝

rs − ds

βs︸︷︷︸B

−Mi

⎞

⎟⎟⎟⎠

Mi .

(3)

Eliminating the variable Si in (3), we can obtain the following cubic equation withrespect to Mi :

123

1200 C. Shan et al.

G(Mi ) := A2C M3i − 2A2 BC M2

i + (A2 B2C + AC D)Mi + 1 − R20 = 0. (4)

For the existence of the endemic equilibrium point, we have the following proposition.

Proposition 3.2 If R0 ≤ 1, there is no endemic equilibrium; If R0 > 1, then therewill be at least one endemic equilibrium and three endemic equilibria at most.

Proof Two parabolas Mi = g1(Si ) and Si = g2(Mi ) have no intersection in thefirst quadrant of (Mi , Si ) plane if and only if ∂g2(Mi )

∂Mi(0) ≤ (

∂g1(Si )∂Si

(0))−1, which isequivalent to R0 ≤ 1. Therefore, there is no endemic equilibrium if R0 ≤ 1.

If R0 > 1, then G(0) = 1 − R20 < 0. By intermediate value theorem, equation (4)

has at least one positive root. Equation (4) has at most three positive roots according tothe fundamental theorem of algebra, so system (1) has at most three endemic equilibria.

��

Proposition 3.3 If R0 < 1, then limt→+∞

(Mi (t)Si (t)

)

= 0 for ∀ (Ms(0),Mi (0), Ss(0),

Si (0)) ∈ R4+.

Proof From proposition 2.1, we can restrict our analysis in region �.

Let Q =(

−(dm + μm)βm Km(rm−dm)

rmβs Ks(rs−ds)

rs−(ds + μs)

)

. Since R0 < 1, one can verify that all

the eigenvalues of Q have negative real parts. The variation of constant formula yieldsto

0 ≤(

Mi (t)Si (t)

)

= eQt(

Mi (0)Si (0)

)

−t∫

0

eQ(t−τ)⎛

⎝βm Si

(Km(rm−dm)

rm− Ms(τ )

)

βs Mi

(Ks(rs−ds)

rs− Ss(τ )

)

⎞

⎠ dτ

≤ eQt(

Mi (0)Si (0)

)

−→ 0 (t → +∞).

Therefore, the disease is eradicated when R0 < 1. ��Define �1 = � ∩ {Mi = Si = 0}. From Proposition 3.3, if R0 < 1, then we only

need to analyze system (1) in the region �1 by the limit system theory.Let�b1 = � ∩ {Mi = Si = Ss = 0} and�b2 = � ∩ {Mi = Si = Ms = 0}. In�1,

two equilibrium points Eb1 and Eb2 are saddles with the stable manifold in �b1 and�b2, respectively, and E0 is a repelling node. There is no limit cycle in�1, which canbe proved by contradiction. Hence, we can obtain the following proposition.

Proposition 3.4 If R0 < 1, then EDFE is globally asymptotically stable in �1 \(�b1

⋃�b2).

Recall that Km = km K and Ks = ks K , where K is the area of islet, and from (2),

we have R0 = K√βmkm(rm−dm)rm(dm+μm)

βsks(rs−ds)rs(ds+μs)

.Hence, given the demographic parameters

and contact transmission rates, R0 is proportional to the size K of the islet.

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Let R0 = 1, we have

Kc =√

rm(dm + μm)

βmkm(rm − dm)

rs(ds + μs)

βsks(rs − ds). (5)

It is the threshold size of the islet for the prevalence of the disease. From Proposition3.2 and Proposition 3.3, if K < Kc, then the disease will die out; if K > Kc, then thedisease will outbreak.

According to the existence of the equilibrium and the stability analysis in Proposi-tion 3.1 and Proposition 3.2, the steady state bifurcations outlined below occur.

Theorem 3.5 (i) System (1) undergoes a transcritical bifurcation involving E0 andEb1 (or Eb2) when rm = dm (or rs = ds), and E0 changes its stability from anattracting node to a saddle while increasing the parameter rm (or rs).

(ii) System (1) undergoes a transcritical bifurcation involving EDFE and an endemicequilibrium point when R0 = 1, and EDFE changes its stability from an attractingnode to a saddle point while increasing R0.

3.2 Stability and Bifurcations

In this section, we study the stability of endemic equilibria and possible bifur-cations under the condition R0 > 1. Denote any endemic equilibrium point asE = (Ms, Mi , Ss, Si ). The Jacobian at E is given by

T =

⎛

⎜⎜⎜⎜⎜⎝

− rmKm

Ms 0 0 −βm Ms

βm Si −(dm + μm) 0 βm Ms

0 −βs Ss − rsKs

Ss 0

0 βs Ss βs Mi −(ds + μs)

⎞

⎟⎟⎟⎟⎟⎠

,

so the eigenvalues of T are the roots of

λ4 + p3λ3 + p2λ

2 + p1λ+ p0 = 0, (6)

where

p0 = (dm + μm)(ds + μs)

(rmβs

KmMs Mi + rsβm

KsSs Si − βmβs Mi Si

)

,

p1 = rm

Km

rs

KsMs Ss(dm+μm + ds + μs)+ (βm Si + βs Mi )(dm + μm)(ds + μs) > 0,

p2 = rm

KmMs

rs

KsSs +

(rm

KmMs + rs

KsSs

)

(dm + μm + ds + μs) > 0,

p3 = rm

KmMs + rs

KsSs + dm + μm + ds + μs > 0.

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1202 C. Shan et al.

Since p1 > 0, the characteristic equation (6) cannot have double-zero eigenvalues;then, the Bogdanov-Takens bifurcation and 1 : q (q ∈ N ) resonance bifurcation cannotoccur. System (1) cannot present the Hopf-Hopf bifurcation, because the characteristicequation (6) does not have two pairs of pure imaginary roots due to p3 > 0. TheLemma 3.6 in section 3.2.1 asserts that the cusp bifurcation and the Hopf bifurcationcannot happen simultaneously. Therefore, as the breakthrough point, we study the cuspbifurcation which is related to the bifurcation of the equilibria, and then study the Hopfbifurcation which can change the stability of the equilibria and induce oscillations.

3.2.1 Cusp Bifurcation of Codimension 2

It is impossible to solve (4) to get the equilibrium or study its stability from (6).Alternatively, from the viewpoint of bifurcation, we first study the singularity withmultiplicity 3, and then obtain the existence and stability of all the possible equilibriafrom the bifurcation analysis.

A dynamical system restricted on the center manifold is generally of lower dimen-sion than the original system, and the center manifold theorem (Carr 1981) shows thatqualitative behaviors in a neighborhood of a non-hyperbolic critical point are deter-mined by its behaviors on the center manifold. There are no theoretical difficultiesin the center manifold calculation, but calculations are very complicated in higher orinfinite dimensional systems, and that is the reason why the bifurcation analyses ofhigh codimension are directly carried out for the planar system in most papers. Thehigh dimension of model (1) brings more difficulties that we cannot easily transformthe linear part of the system into a canonical form.

The projection technique, derived from the adjoint operator and spectral decompo-sition theory, avoiding changing the linear part into a canonical form, was originallydeveloped to study bifurcations in partial differential equations using the Lyapunov–Schmidt reduction. With this technique, Hassard et al. studied the center manifold andnormal form of Hopf bifurcation in finite and infinite dimensional systems Hassard etal. (1981). Here, we apply this technique to the center manifold computation, becausewe have no idea about the other three eigenvalues except the sign of their real parts asshown in Lemma 3.6.

The endemic equilibrium of multiplicity 3 implies that

g(Mi ) = g′(Mi ) = g′′(Mi ) = 0. (7)

From the above equations, we obtain this critical equilibrium point.

E∗ = (M∗s ,M∗

i , S∗s , S∗

i ) =(

dm + μm

βm

3

AB,

2B

3,

ds + μs

βs

AB

3,

2AB2

9

)

.

It follows from (4) and (7) that the parameters of model (1) in this case should satisfy

AB2 − 3D = 0, A2 B3C − 27 = 0, (8)

and we have the following lemma.

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Lemma 3.6 For the endemic equilibrium E∗, λ = 0 is a simple eigenvalue, and allother eigenvalues have negative real parts.

Proof Let g(λ) = λ4 + p3λ3 + p2λ

2 + p1λ+ p0. For the endemic equilibrium E∗,we have g(0) = p0 = 0, and g′(0) = p1 > 0, so λ = 0 is a simple root of g(λ) = 0.Furthermore, p3 > 0 and p2 p3 − p1 > 0. By Routh-Hurwitz criteria, we prove thelemma. ��

From conditions (8), by means of Implicit Function Theorem, there existtwo smooth functions K ∗ = K ∗(rm, rs, βm, βs, μm, μs, km, ks, ds) and d∗

m =d∗

m(rm, rs, βm, βs, μm, μs, km, ks, ds). When K = K ∗ and dm = d∗m, three endemic

equilibria coalesce at E∗. In the following, we prove that E∗ is a cusp bifurcationpoint of codimension 2, and use a series of transformations to reduce system (1) to thenormal form, by which the bifurcation and dynamics of system (1) will be presented.First, we need the following preliminaries.

Lemma 3.7 Let

φ =(

−βm Km

rm,−βm Km(rm − dm)

rm(dm + μm),

3(ds + μs)

rs − ds, 1

)′,

=(

−2βs Ks(rs − ds)

3rs(dm + μm),− βs Ks(rs − ds)

3rs(dm + μm), 2, 1

)′,

α = 〈,φ〉−1, and � = αφ; then � and are the eigenvectors corresponding toλ = 0 of T and T ∗ which satisfy 〈,�〉 = 1. Here, 〈·, ·〉 is the inner product in R4

and T ∗ is the adjoint operator of T .

Define the space T su = {Y ∈ R4|〈,Y 〉 = 0}, and then we have the followingtheorem.

Theorem 3.8 For system (1),

(i) The endemic equilibrium E∗ is a cusp bifurcation point of codimension 2.(ii) The endemic equilibrium E∗ is a stable equilibrium.

Proof (i) It has been shown in Chow et al. (1994), Kuznetsov (1998) that any systemwhich has a cusp bifurcation point of codimension 2 is C∞ equivalent to

u = −u3 + o(u3). (9)

Thus, it is sufficient to show that there exist smooth coordinate changes whichtransform system (1) into (9) on the center manifold with conditions (8). Thetranslation

x1 = Ms − M∗s , x2 = Mi − M∗

i , x3 = Ss − S∗s , x4 = Si − S∗

i , (10)

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1204 C. Shan et al.

brings E∗ to the origin, and we obtain X = T X + f (X), where X =(x1, x2, x3, x4)

′ and

f (X) =(

− rm

Kmx2

1 − βmx1x4, βmx1x4,− rs

Ksx2

3 − βs x2x3, βs x2x3

)′.

Using the Fredholm Alternative Theorem, we can decompose any vector X ∈ R4

as X = u�+ Y with {u = 〈, X〉,Y = X − 〈, X〉�.

The scalar u and vector Y ∈ T su can be considered as new coordinates, and system(1) can be written as

{u = 〈, f (u�+ Y )〉,Y = T Y + f (u�+ Y )− 〈, f (u�+ Y )〉�. (11)

From Lemma 3.6 and the center manifold theorem, there exists a one-dimensionalcenter manifold of system (11) which can be represented as follows:

W c(0) = {(u,Y ) ∈ (R, T su)|Y = h(u), |u| < δ, h(0) = 0, Dh(0) = 0}

for δ sufficiently small. Hence, h(u) = au2 + O(u3), which satisfies

Dh(u)(〈, f (u�+ Y )〉) = T Y + f (u�+ h(u))− 〈, f (u�+ h(u))〉�, (12)

where a = (a1, a2, a3, a4)′ ∈ T su ⊂ R4, which will be determined later. Therefore,

system (1) restricted on the center manifold is given by

u = −(

rm

Kmψ1φ

21 + βmψ2φ1φ4 + rs

Ksψ3φ

23 + βsψ4φ2φ3

)

u2

−[

2

(rm

Kmψ1φ1a1 + rs

Ksψ3φ3a3

)

+ βmψ2(φ1a4 + φ4a1)

+βsψ4(φ2a4 + φ3a2)] u3 + o(u3). (13)

Here, ψi and φi (i = 1, 2, 3, 4) are the i th components of the vectors and �,respectively. Comparing the coefficients of each power of u from both sides of equation(12), we obtain

T a =

⎛

⎜⎜⎝

0−βmφ1φ4

0−βsφ2φ3

⎞

⎟⎟⎠ ⇒ a = (I −� ′)

⎛

⎜⎜⎝

0−α2C

27α2 ds+μsβs AB3

0

⎞

⎟⎟⎠ ,

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where I ∈ R4×4 is a unit matrix. Substituting a into (13), we obtain the reducedsystem on the center manifold

u = −6α3(ds + μs)C

Bu3 + o(u3). (14)

The coefficient of u3 is negative since α = 〈,φ〉−1 > 0. Hence, E∗ is a cuspbifurcation point of codimension 2.

(i i) From (14), one can see that E∗ is stable on the center manifold, and accordingto Lemma 3.6, E∗ is a stable equilibrium. ��

By Theorem 3.8, if K = K ∗ and dm = d∗m, then the endemic equilibrium E∗ is

a cusp bifurcation point of codimension 2. In order to investigate the impact of theexpansion of the islet and design control strategies on schistosomiasis transmission,we take (K , dm) as bifurcation parameters and develop a generic unfolding of thecusp bifurcation for system (1), when (K , dm) is perturbed near the point (K ∗, d∗

m).Substituting {

K = K ∗ + ε1,

dm = d∗m + ε2,

(15)

into system (1), we obtain the perturbed system

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

Ms = rm Ms

(1 − Ms

km(K ∗+ε1)

)− (d∗

m + ε2)Ms − βm MsSi ,

Mi = βm MsSi − (dm + μm)Mi ,

Ss = rsSs

(1 − Ss

ks(K ∗+ε1)

)− dsSs − βs Mi Ss,

Si = βs Mi Ss − (ds + μs)Si ,

(16)

where ε = (ε1, ε2) is sufficiently small.

Theorem 3.9 For the parameter ε = (ε1, ε2) sufficiently small, system (16) is ageneric unfolding of the cusp bifurcation of codimension 2.

The unfolding of codimension 2 cusp bifurcation is given in the bifurcation diagramof Fig. 2 with parameters K and dm.

Proof It has been shown in Chow et al. (1994), Kuznetsov (1998) that a genericunfolding with parameters (ν0, ν1) of codimension 2 cusp bifurcation is C∞ equivalentto

v = ν0 + ν1v − v3, (17)

so we will prove that system (16) with the parameter (ε1, ε2) is also a generic unfold-ing of codimension 2 cusp bifurcation by showing that there exist smooth coordinatechanges which take system (16) into (17) on the center manifold. Applying the trans-lation (10) and expanding system (16) in the power series at origin, we have

{X = T X + F(X, ε),ε = 0,

(18)

123

1206 C. Shan et al.

)

K

I II III

O

SN1

SN2

III

Ro=1ΓSN1

SN2

mr

dm

* *(K, dm

* *(K, dm

)

Fig. 2 Bifurcation diagram in the (K , dm) plane and the phase portrait on the one-dimensional centermanifold in a small neighborhood of (K ∗, d∗

m) (Color figure online)

and F(X, ε)

=

⎛

⎜⎜⎜⎜⎝

−M∗s ε2 − rm x2

1K ∗

m− βmx1x4 − ε2x1 + rmε1(M∗2

s +2M∗s x1+x2

1 )

K ∗m K ∗

−M∗i ε2 + βmx1x4 − ε2x2

− rsx23

K ∗s

− βs x2x3 + rsε1(S∗2s +2S∗

s x3+x23 )

K ∗s K ∗

βsx2x3

⎞

⎟⎟⎟⎟⎠

+O(|ε|2, |ε|2 X, |ε|2 X2),

where K∗m = km K ∗ and K

∗s = ks K ∗. Following the same procedure in Theorem 3.8,

system (18) can be written as

⎧⎨

⎩

u = 〈, F(u�+ Y )〉,Y = T Y + F(u�+ Y )− 〈, F(u�+ Y )〉�,ε = 0,

(19)

and the center manifold has the representation

Y = h(u, ε1, ε2) = c1ε1 + c2ε2 + au2 + b1uε1 + b2uε2 + O(|ε1, ε2|2)+O(|u, ε1, ε2|3).

Since

Duh(u, ε1, ε2)u + Dε1 h(u, ε1, ε2)ε1 + Dε2 h(u, ε1, ε2)ε2

= Dh(u)(〈, F(u�+ h(u, ε1, ε2))〉),

h(u, ε1, ε2) should satisfy

123


Dh(u)(〈, F(u�+ h(u, ε1, ε2))〉)= T h(u, ε1, ε2)+ F(u�+ h(u, ε1, ε2))− 〈, F(u�+ h(u, ε1, ε2))〉�,

(20)

where a, b1, b2, c1, c2 ∈ T su ⊂ R4, and · = d/dt .According to the equation (20), we have

T a = ζ0, T c1 = ζ1, and T c2 = ζ2, (21)

where

ζ0 =

⎛

⎜⎜⎝

0−βmφ1φ4

0−βsφ2φ3

⎞

⎟⎟⎠ , ζ1 =

⎛

⎜⎜⎝

− (ds+μs)AB2

9K ∗0

− (dm+μm)B3K ∗0

⎞

⎟⎟⎠ ,

ζ2 =

⎛

⎜⎜⎝

−φ1(ψ1 M∗s + ψ2 M∗

i )+ M∗s

−φ2(ψ1 M∗s + ψ2 M∗

i )+ M∗i−φ3(ψ1 M∗

s + ψ2 M∗i )−φ4(ψ1 M∗

s + ψ2 M∗i )

⎞

⎟⎟⎠ .

By solving linear systems (21), we obtain that

a = (I −� ′)

⎛

⎜⎜⎜⎝

0−α2C

27α2(ds+μs)

βs AB3

0

⎞

⎟⎟⎟⎠, c1 = (I −� ′) 1

K ∗

⎛

⎜⎜⎝

M∗s

M∗i−S∗s

0

⎞

⎟⎟⎠ ,

c2 = (I −� ′)

×

⎛

⎜⎜⎜⎜⎝

K ∗m

rm

(ψ1φ1 − 1 + ψ2φ1

M∗i

M∗s

)

1dm+μm

[K ∗

mβm S∗i

rm

(ψ1φ1 − 1 + ψ2φ1

M∗i

M∗s

)+ φ2(ψ1 M∗

s + ψ2 M∗i )− M∗

i

]

− 3((ψ1φ3+ψ1φ4)M∗s +(ψ2φ3+ψ2φ4)M∗

i )

βs B0

⎞

⎟⎟⎟⎟⎠.

Substituting a, c1, and c2 into the first equation of (19), we obtain

u = μ0(ε1, ε2)+ μ1(ε1, ε2)u + μ2(ε1, ε2)u2 + μ3(ε1, ε2)u

3 + O(u4), (22)

where

μ0(ε1, ε2) =(

2(ds + μs)

βm+ 2K ∗

s (rs − ds)2

(d∗m + μm)rs

)

ε2 + O(|ε1, ε2|2),

μ1(ε1, ε2) = 6αds + μs

K ∗ ε1 + C0ε2 + O(|ε1, ε2|2),

123

1208 C. Shan et al.

μ2(ε1, ε2) = O(|ε1, ε2|),μ3(ε1, ε2) = −6α3(ds + μs)

C

B+ O(|ε1, ε2|),

and

C0 = −α2(ds + μs)

[63(ds + μs)

AB2(d∗m + μm)βm

+ 12(ds + μs)

d∗m + μmβs B

+ 324(ds + μs)

β2m A2 B4

+ ds + μs

(d∗m + μm)2

+ 5

d∗m + μm

+ 27

βm AB2

]

< 0.

Let v = −μ133 u − 1

3μ2μ

− 23

3 . This translation brings system (22) to

v = ν0(ε1, ε2)+ ν1(ε1, ε2)v − v3 + O(v4), (23)

where

ν0(ε1, ε2) = μ0 + 2μ32

27μ23

− μ1μ2

3μ3= ξ0ε2 + O(|ε1, ε2|2),

ν1(ε1, ε2) = − μ1

μ1/33

+ μ22

3μ4/33

= ξ1ε1 + ξ2ε2 + O(|ε1, ε2|2),

and

ξ0 = 2(ds + μs)

βm+ 2K ∗

s (rs − ds)2

(d∗m + μm)rs

> 0,

ξ1 = (6(ds + μs))23

K ∗ (CB

) 13

> 0, ξ2 = C0

α(6(ds + μs)

CB

) 13

< 0.

Therefore,

∂(ν0, ν1)

∂(ε1, ε2) |ε=(0,0)= −ξ0ξ1 �= 0.

Using the Malgrange Preparation Theorem, system (23) is topologically equivalent to

v = ν0(ε1, ε2)+ ν1(ε1, ε2)v − v3.

Hence, system (16) with parameter ε = (ε1, ε2) is a generic unfolding of codimension2 cusp bifurcation. ��

From Theorem 3.9, one can immediately obtain the following corollary.

123


Corollary 3.10 For system (1), the saddle-node bifurcation and pitchfork bifurcationcan occur. Furthermore, two branches of saddle-node bifurcation curves intersectingat (K ∗, d∗

m) can be represented in terms of K and dm as follows:

SN1,2

={

(K , dm)|ξ0(dm − d∗m) = ±

√4

27(ξ1(Km − K ∗

m)+ ξ2(dm − d∗m))

3 + O(|ε1, ε2|2)}

.

3.2.2 Hopf Bifurcation

In this section, we study the Hopf bifurcation. Since p3 > 0, the characteristic equation(6) cannot have two pairs of pure imaginary roots simultaneously, so the Hopf-Hopfbifurcation cannot occur. We choose K as the bifurcation parameter, then coefficientsp1, p2, p3, and p4 are functions of K . Let

ϕ(K ) = (p2(K )p3(K )− p1(K ))p1(K )− p23(K )p0(K ).

Theorem 3.11 Suppose there exists a K > 0 such that p0(K ) > 0, ϕ(K ) = 0 anddϕ(K )

dK∣∣K=K

�= 0, then the Hopf bifurcation occurs and a family of periodic solutions

appear from the endemic equilibrium point E when K passes through K .

Proof If there exists a K > 0 such that p0(K ) > 0 and ϕ(K ) = 0, then

p2(K )p3(K )− p1(K ) = p23(K )p0(K )/p1(K ) > 0.

Hence, equation (6) has four roots, ±iω0, λ1, and λ2, where ω0 =√

p1(K )/p3(K ),Re(λ1) < 0, and Re(λ2) < 0. To prove the occurrence of Hopf bifurcation at theendemic equilibrium E , we need to verify the transversality condition.

Consider the roots of (6) as a function of K . When 0 < |K − K | � 1, equation (6)has four roots, α(K )± iω(K ), λ1(K ), and λ2(K ), which satisfy α(K ) = 0, ω(K ) =ω0, λ1(K ) = λ1, and λ2(K ) = λ2. Substitute λ = α(K ) + iω(K ) into equation (6)and differentiate equation (6) with respect to K . Straightforward calculation leads to

dα(K )

dK∣∣K=K

= − p3

2[(p2 p3 − 2p1)2 + p0 p23]

dϕ(K )

dK∣∣K=K

.

Therefore,

Sign

{dα(K )

dK∣∣K=K

}

= −Sign

{dϕ(K )

dK∣∣K=K

}

�= 0.

Thus, a pair of complex eigenvalues cross the imaginary axis transversally whenK = K , and Hopf bifurcation occurs. ��

123

1210 C. Shan et al.

If dm is chosen as the bifurcation parameter, then we will have the similar resultas Theorem 3.11. Figure 3a, b in Sect. 4 show that the Hopf bifurcation occurs whenconsidering K and dm as bifurcation parameters, respectively.

3.3 Bifurcation Diagram

In this section, we present the bifurcation diagram in the region {(K , dm)|K > 0, dm >

0}.First, dm = rm defines a dash curve as shown in Fig. 2 across which the transcritical

bifurcation occurs according to Theorem 3.5 (i).Denote γ = βmkmβsks(rs−ds)

rmrs(ds+μs). Solve dm in terms of K from the equation R0 = 1,

and we get

�R0=1 ={

(K , dm)|dm = rm − rm + μm

γ K 2 + 1

}

.

It is a strictly increasing curve of K with a unique inflection ( 1√3γ, 1

4rm − 34μm) and

a horizontal asymptote dm = rm as K → +∞. The transcritical bifurcation occurswhen crossing the curve �R0=1.

In the following, we determine the position of (K ∗, d∗m). Conditions (8) define the

following two curves:

�R0=3 ={

(K , dm)|dm = rm − 9(rm + μm)

γ K 2 + 9

}

,

L ={

(K , dm)|dm = rm − βmks(rs − ds)2

(ds + μs)rsK

}

.

�R0=3 is a strictly increasing curve of K passing (0,−μm)with a horizontal asymptotedm = rm as K → +∞. L is a strictly decreasing straight line passing (0, rm).Therefore, they intersect at the half parameter plane K > 0, and (K ∗, d∗

m) is theunique intersection point.

�R0=3 and L intersect the K -axis at (K1, 0) and (K2, 0) with K1 =√

9μmγ rm

and

K2 = rmrs(ds+μs)

ksβm(rs−ds)2, respectively. If K1 < K2, then d∗

m > 0. If K1 ≥ K2, then d∗m ≤ 0,

which means that three endemic equilibria cannot coalesce at E∗ in reality. Hence, weonly consider the case K1 < K2. Therefore, the location of (K ∗, d∗

m) is determined,and one can verify that it is under the curve �R0=1.

From the Corollary 3.10, we can plot two branches of saddle-node bifurcationcurves, and they intersect at (K ∗, d∗

m). According the above analysis, we obtain thebifurcation diagram in the (K , dm) plane and have the following theorem.

Theorem 3.12 For system (1), using (K , dm) as parameters, the bifurcation diagramis given in Fig. 2.

(i) �R0=1 and {(K , dm)|dm = rm, K > 0} are transcritical bifurcation curves.

123


(ii) SN1 and SN2 are saddle-node bifurcation curves intersecting at the cusp point(K ∗, d∗

m). On SN1,2, there are two endemic equilibria, one is stable and the otheris a saddle-node.

(iii) In region I, there is no endemic equilibrium, and the disease free equilibrium isstable. In region II, there is only one stable endemic equilibrium. In region III,there are three endemic equilibria, two of them are stable and the third one isunstable.

Remark 3.13 If K1 > K2, (K ∗, d∗m) is located in the fourth quadrant of the parameter

space, yet we can still plot the bifurcation diagram as Fig. 2, and the only differenceis that the curve SN1 distinguishes the region II and III.

Remark 3.14 The bifurcation analysis is local, so the stability of the endemic equi-librium is valid in a neighborhood of (K ∗, d∗

m) in Theorem 3.12. The stability ofequilibrium changes with the occurrence of other bifurcations, for example the Hopfbifurcation.

From the characteristic equation (6), for the bifurcation of codimension 2 or evenhigher codimension bifurcation, the possibility of Bogdanov–Takens bifurcation andHopf–Hopf bifurcation is excluded. However, the degenerate Hopf bifurcation andZero-Hopf bifurcation could occur when (K , dm) is far from (K ∗, d∗

m). Nevertheless,it is still difficult to prove that theoretically, since it is hard to get the coordinates ofthe equilibrium and determine the eigenvalues from the cubic equation (4) and quarticequation (6). In Sect. 4, we will take advantage of numerical simulations to get moreinsights of system (1), and we do see that these types of bifurcations occur.

4 Simulations

In order to investigate the impact of growing islets and design control strategies onschistosomiasis transmission, we regard the physical size of islet K and the naturaldeath rate of mice dm as parameters. The range of other parameters are listed in Table1.

We choose parameter values as follows: km = 9, ks = 80, rm = 0.02, rs = 0.03,βm = 8.00 × 10−6, βs = 2.25 × 10−5, ds = 0.0075, μm = 0.000608, and μs =0.0195.

The disease will die out when dm ≥ 0.02, so we consider dm ∈ (0, 0.02). Usingthe parameter values listed above, we can easily find the cusp point (K ∗, d∗

m) =(120, 4 × 10−3).

(i) Pitchfork bifurcation. We fix dm = 4 × 10−3 and consider K as a parameter.When 0 < K < 40, there is no endemic equilibrium. System (1) undergoes trans-critical bifurcation when K = 40. When 40 < K < 120, there is only one stableendemic equilibrium denoted as E . The pitchfork bifurcation occurs as K passingthrough the bifurcation value K = 120 with another two stable endemic equilibriabifurcating from E , and the equilibrium E loses its stability and becomes a saddle. AsK increases, two stable endemic equilibria will lose their stability and supercriticalHopf bifurcations occur when K = 257.49 and K = 278.94, respectively. See Fig.3a.

123

1212 C. Shan et al.

Table 1 Parameters, interpretations and values of model (1)

Parameter Description Range Reference

K Area of islet (m2) Parameter

km Maximum mice population (per m2) 5–15 Xu et al. (1999)

ks Maximum snails population (per m2) 50–100 Xu et al. (1999)

rm Intrinsic growth rate of mice (per day) 0.019–0.2 Estimated

rs Intrinsic growth rate of snails (per day) 0.01–0.55 Guo (1991), Liang et al. (2002)

βm Contact transmission rate fromsnails to mice

0.0000005–0.001 Estimated

βs Contact transmission rate frommice to snails

0.000001–0.0005 Spear et al. (2002)

dm Nature death rate of mice (per day) Parameter

ds Nature death rate of snails (per day) 0.005–0.033 Anderson and May (1991),Liang et al. (2002)

μm Death rate of mice induced by infection(per day)

0.0005–0.005 Estimated

μs Death rate of snails induced byinfection (per day)

0.01–0.55 Guo (1991), Liang et al. (2002)

(ii) Saddle-node bifurcation. We fix K = 160 and consider dm as a parameter whichvaries from 0 to 0.02. One can find that the stable periodic solution disappears fromthe supercritical Hopf bifurcation. The number of endemic equilibria changes from1 → 2 → 3 → 2 → 1 → 0 via two saddle-node bifurcations and a transcriticalbifurcation as shown in Fig. 3b.(iii) Hopf and degenerate Hopf bifurcation. We have observed the Hopf bifurcationin cases (i) and (ii). If we fix the parameter K = 2600 and change dm, then fromFig. 3c, we can see a family of stable periodic solutions and a family of unstableperiodic solutions when dm ∈ (0.011, 0.012). Moreover, we can find the multiplelimit cycles region between one Hopf curve and the semi-stable limit cycle curvewhen K > 1651.77 as indicated in Fig. 4. The intersection of Hopf curve and semi-stable limit cycle curve is the degenerate Hopf bifurcation point.(iv) Bifurcation diagram. We plot the bifurcation curves in the (K , dm) plane in Fig.4, and it is consistent with the diagram in Fig. 2. The transcritical bifurcation curve�R0=1 is the threshold condition for the outbreak of the disease in terms K and dm.The blue curves in Fig. 4 are exactly the same saddle-node curves SN1,2 in Fig. 2.From simulations, we can see more sophisticated dynamical behaviors. In Fig. 4, twobranches of Hopf bifurcation curves are denoted with red lines. We can observe thatone branch of Hopf curves is tangent to the saddle-node curve SN1 indicating theoccurrence of the Zero-Hopf bifurcation. We can also find that the system presentsthe degenerate Hopf bifurcation, while K is large as we mentioned in case (iii).

123


0

500

1000

1500

0 50 100 150 200 250 300

500

1000

1500

2000

2500

3000

3500

4000

4500

K

Mi

Pitctfork bifurcationK=120

Supercritical Hopf bifurcationK=278.94

Transcritical bifurcationK=40

Si

unstable fix point

Supercritical Hopf bifurcationK=257.49

stable fix point

(a)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

200

400

600

800

1000

1200

1400

1600

1800

dm

SN

transcritical bifurcation dm=0.01632

SN

unstable fix point

Mi

stable fix point

supercritical Hopf bifurcationdm=0.00204

(b)

0.0090.01

0.0110.012

02000

40006000

800010000

0

2000

4000

6000

8000

10000

dm

Mi

stable limit cycleunstable limit cyclestable fix pointunstable fix point

Si

(c)Fig. 3 a Pitchfork bifurcation and Hopf bifurcation diagram considering K as a parameter with dm = 0.004.b Saddle-node bifurcation and Hopf bifurcation diagram considering dm as a parameter with K = 160. cTwo limit cycles, one stable and the other unstable, bifurcate from the degenerate Hopf bifurcation withK = 2600 and dm ∈ (0.11, 0.12) (Color figure online)

0 500 1000 1500 20000

0.005

0.01

0.015

0.02

0.025

K

multiple limit cycle region

CuspPoint

transcritical bifurcation curve

II

dm

Zero−Hopf bifurcation point

Degenerate Hopf bifurcation point

saddle−node bifurcation curve

Hopf bifurcation curve

IIII

Fig. 4 Bifurcation diagram considering K and dm as parameters (Color figure online)

123

1214 C. Shan et al.

5 Conclusion and Discussion

Successful interventions in the transmission of schistosomiasis require a better under-standing of the influences of the biological and ecological factors. These factors andtheir interactions can be represented by the parameters in model (1). In this paper, weformulate and analyze a system of ODE model incorporating the impact of the growingislets on the S. japonica in the Yangtze River. The center manifold and normal form ofthe cusp bifurcation of codimension 2 are derived to obtain the existence and stabilityof the multiple endemic equilibria in the high-dimensional phase space. The bifurca-tion analysis is carried out to study the influence of the environment, especially thespatial carrying capacity. We give the threshold condition for the outbreak of the dis-ease in terms of the land size. Several kinds of bifurcations are found theoretically andnumerically, from which we expand the horizon on the mechanism of schistosomiasistransmission.

Due to the significance ofR0 in epidemiology, it is meaningful to plot the bifurcationcurves in the (R0,Mi ) plane by changing K or dm and fixing the other parameters.We list some typical bifurcation diagrams (a)− (d) in Fig. 5, which agree with Figs.2, 4.

According to the analyses and simulations in Sects. 3 and 4, it is a challenge topredict, control, and eliminate this disease due to the expansion of islets. The largerthe islet becomes, the more complicated the mechanism of the disease transmissionwill be. Nevertheless, from the formula of R0, Proposition 3.3, and Fig. 5, we candevelop two control strategies as follows:

(1) Use pesticides to kill miracidia, larval, and introduce the predators of snails andmice to decrease the number of these two hosts.

(2) Separate the islet into several small isolated parts by trenches and fences to cutoff the movement of mice and snails among these small partitions, and the areaof each isolated part is smaller than Kc.

System (1) undergoes several types of bifurcations when K > Kc. Biologicallyspeaking, it means different parameters and initial conditions will result in differentepidemic levels. The cusp bifurcation of codimension 2 is found, which suggests thatthe disease can prevail in several different scales. We also observe that the Hopf bifur-cation occurs in the model. The existence of Hopf bifurcation indicates the periodicphenomena. Therefore, the recurrence of the schistosomiasis is possible when theinitial infection density is in some certain ranges.

The expanding islet provides an advantageous condition for the disease transmis-sion. When the islet reaches a critical size Kc, the transmission of the schistosomiasisbetween the mice R. norvegicus and the snails O. hupensis will persist, which agreeswith the history of the occurrence of schistosomiasis disease in Qian and Zimu isletsas shown in Fig. 1. The prevalence of the disease on these islets serves as a possi-ble source of schistosomiasis in the Yangtze valley. Although the disease has beencontrolled or eradicated, it can still outbreak due to the water flow carrying the schis-tosome and snails from these endemic islets to the lower reaches of the Yangtze River.A typical example is that the re-emergence of schistosomiasis along the Yangtze River

123


0 1 2 3 4 5 60

100200300400500600700800900

1000

Ro

Ro

Ro

forward transcritical bifurcation

Mi S−N bifurcation

(a)

0 1 2 3 4 5 60

100200300400500600700800900

1000

Ro


pitchfork bifurcation

Mi

(b)

0 1 2 3 4 5 60

200

400

600

800

1000

1200


H

S−N bifurcation

Hopf bifurcationM

i

(c)

0 1 2 3 4 5 60

200

400

600

800

1000

1200

1400

Hopf bifurcation

S−N bifurcation

S−N bifurcation


Mi

(d)Fig. 5 Some typical bifurcation diagrams in the (R0,Mi ) plane considering K as a parameter in (a − c)and dm as a parameter in (d). The blue curves represent the stable fix points or limit cycles, and red curvesrepresent the unstable fix points. In fact, all stable fix points in case (a − d) lose their stability via Hopfbifurcation when R0 is sufficiently large (Color figure online)

was closely associated with flooding events occurred in 1998, which resulted in schis-tosome and snails diffusion (Zhao et al. 2005; Zhou et al. 2002, 2005).

Based on the life cycle of schistosomiasis transmission and the cross-infectionbetween the definitive hosts R. norvegicus and the intermediate hosts O. hupensis,we formulate the model (1), which captures the special feature of schistosomiasisdisease. This model is formulated specifically for the schistosomiasis; however, it maybe applicable for the other parasitic diseases, which share the common features withschistosomiasis transmission. For example, intermediate hosts and definitive hosts areinvolved during the transmission of these parasitic diseases, and they can infect eachother.

In this paper, we only took into account the influence of the expansion of the islet onthe schistosomiasis disease. There are many other natural factors, such as vegetation,soil types, moisture of earth surface, etc., which may contribute to the transmission ofS. japonica as well. We have also ignored some integrated factors, such as the diffusionof mice and snails, the delay in the disease transmission as well as the possible climatechange on the transmission in our study. More insights may be obtained if these factorsare included which we leave for future studies.

123

1216 C. Shan et al.

Acknowledgments This research was supported by NSERC and ERA, an Early Researcher Award ofMinistry of Research and Innovation of Ontario, Canada and NSFC-11171267 of China.

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