phd thesis mathemtical analysis and control …

PhD THESIS

MATHEMTICAL ANALYSIS AND CONTROL STRETEGIES OF HIV-1

INFECTION MODELS

By

NIGAR ALI

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF MALAKAND

SESSION 2013-2016

PhD THESIS

MATHEMATICAL ANALYSIS AND OPTIMAL CONTRO OF

HIV-1 INFECTION MODELS

BY

NIGAR ALI

SUPERVISED BY

PROF. DR. GUL ZAMAN

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF MALAKAND

SESSION 2013-2016

To everyone who supports me

iv

Table of Contents

Table of Contents v

List of Tables ix

List of Figures x

Abstract xiii

Certificate of publications from thesis xiv

1 Introduction 1

1.1 History of HIV-1 Infection . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Immunological Study of HIV-1 Infection . . . . . . . . . . . . . . . . 3

1.3 HIV Transmission and Risks . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Treating HIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Role of Mathematical Models in Understanding HIV-1 Infection . . 5

1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Preliminaries and Basic Materials 14

2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Epidemiological Models . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Transmission and Out Break . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Pandemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Endemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Basic Reproduction Number . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Autonomous System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Equilibrium Points and Stability . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . . . . 18

v

2.5.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Positive Definite . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.4 Lyapunov Stability Theorems . . . . . . . . . . . . . . . . . . . 20

2.5.5 Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.6 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Delay Differential Equations in Mathematical Biology . . . . . . . . . 22

2.6.1 Basic Properties of Delay Differential Equations(DDE) . . . . 23

2.6.2 Linear Delay Differential Equations with Constant Delays

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6.3 Local Stability of Delay Differential Equation . . . . . . . . . . 25

2.7 Some Basics on Optimal Control . . . . . . . . . . . . . . . . . . . . . 25

2.7.1 Optimal control problem: . . . . . . . . . . . . . . . . . . . . . 26

2.7.2 Lipschitz Function . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7.3 Pontryagin’s Maximum Principle: . . . . . . . . . . . . . . . . 26

2.7.4 Necessary condition . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.5 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.6 Characterization of the Optimal Control . . . . . . . . . . . . . 28

2.7.7 Optimality System . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.8 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . 29

2.8.1 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . 29

2.8.2 The Beta Function . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.3 Repeated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.4 Cauchy Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8.5 Caputo’s Approach to Fractional Calculus . . . . . . . . . . . 31

2.8.6 Fractional Integral . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.8.7 Caputo’s Definition of Fractional Derivative . . . . . . . . . . 31

3 Asymptotic behavior of discrete delayed HIV-1 models 33

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Analysis of HIV-1 Model with CTLs Immune Response . . . . . . . . 34

3.2.1 Positivity and Well- posedness of the Solution . . . . . . . . . 35

3.2.2 Dynamical Behavior of Disease-free Equilibrium . . . . . . . 37

3.2.3 Dynamical behavior of CTLs- absent Equilibrium . . . . . . . 40

3.2.4 Dynamical behavior of CTLs- present Equilibrium . . . . . . 45

3.2.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Analysis of Delayed HIV-1 Model for Recombinant Virus . . . . . . . 49

3.3.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

vi

3.3.2 Determination of Equilibria and Basic Reproductive Number 52

3.3.3 Local Dynamics of the Proposed Model . . . . . . . . . . . . . 54

3.3.4 Global Dynamics of the Proposed Model . . . . . . . . . . . . 58

3.3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Analysis of double delayed HIV-1 model . . . . . . . . . . . . . . . . 65

3.4.1 Positivity and Well-Posedness of the Solution . . . . . . . . . 65

3.4.2 Stability of the Disease-free Equilibrium . . . . . . . . . . . . . 68

3.4.3 Stability of Single Infected Equilibrium . . . . . . . . . . . . . 70

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Dynamics of Continuous Delayed HIV-1 Infection Models 78

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Asymptotic Properties of Single (Continuous) Delayed HIV-1 Mod-

els with Inhibiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.1 Existence, Uniqueness and Non-negativity of the Solution . . 80

4.2.2 A Systematic Approach to Local Stability . . . . . . . . . . . . 84

4.2.3 Global Asymptotic Stability of the Proposed Model . . . . . . 87

4.3 Stability Properties of Double (Continuous) delayed HIV-1 Model . . 92

4.3.1 Existence Uniqueness and Non- negativity of the Solutions . 94

4.3.2 Local Behavior of the Proposed Model . . . . . . . . . . . . . . 97

4.3.3 Global Behavior of the Proposed Model . . . . . . . . . . . . . 101


4.4 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 108

5 Global dynamics of HIV-1 models with cure rate and saturation response109

5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Study of HIV-1 Model with Effect of Cure Rate . . . . . . . . . . . . . 109

5.2.1 Preliminaries and Assumptions . . . . . . . . . . . . . . . . . 110

5.2.2 Existence of the Equilibria . . . . . . . . . . . . . . . . . . . . 111

5.2.3 Stability Results of Infection Free Equilibrium . . . . . . . . . 112

5.2.4 Stability Results of Single Infection Equilibrium . . . . . . . . 115

5.2.5 Numerical Discussion . . . . . . . . . . . . . . . . . . . . . . . 119

5.3 Dynamics of Dose Dependent Infection Rate in HIV-1 Model . . . . 124

5.3.1 Preliminarily Results . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3.2 Existence of the Equilibria and Basic Reproductive Numbers . 127

5.3.3 Global Dynamics of Proposed Model for the Case R0 < 1 . . 127

5.3.4 Global behavior of the model for the case R0 > 1 . . . . . . . . 129

5.3.5 Numerical Discussion . . . . . . . . . . . . . . . . . . . . . . . 131

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

vii

6 Optimal Control Strategies of HIV-1 Models 136

6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2 Control Strategy of HIV-1 Model . . . . . . . . . . . . . . . . . . . . . 137

6.2.1 Existence of the optimal Control Problem . . . . . . . . . . . . 138

6.2.2 Numerical Solution of the Optimal Control Problem . . . . . . 143

6.3 Control Strategy of the Single delayed HIV-1 model . . . . . . . . . . 147

6.3.1 Existence of the Control Pair . . . . . . . . . . . . . . . . . . . 148

6.3.2 Numerical Simulation of the Optimality System . . . . . . . . 154

6.4 Optimal Control Problem of Double Delayed HIV-1 Model . . . . . . 159

6.4.1 Existence of the Optimal Control Problem . . . . . . . . . . . 160

6.4.2 Numerical Results of the Optimal Strategy . . . . . . . . . . . 166

6.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 171

7 Analytical Study of Fractional Order HIV-1 Model 173

7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.1.1 Formulation of the Model . . . . . . . . . . . . . . . . . . . . . 173

7.1.2 Uniqueness of Solution . . . . . . . . . . . . . . . . . . . . . . 175

7.1.3 The Laplace Adomian Decomposition Method (L-ADM) . . . 175


7.1.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 184

7.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8 Summary and Suggestions for further Directions 187

8.1 Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Bibliography 193

viii

List of Tables

5.1 Parameters Values used for Numerical Simulation . . . . . . . . . . . 119

6.1 Numerical Values of the Parameters of the Model with Sources . . . 144

6.2 Values of the Parameters with Different Sources . . . . . . . . . . . . 155

6.3 The Values of Parameters Used for Numerical Simulation . . . . . . . 167

ix

List of Figures

3.1 Numerical solution of system (3.2.1) for different concentrations of

cells, CTLs and viruses with effect of time delay . . . . . . . . . . . . 48

3.2 The dynamics of the system (3.3.1) for τ = 1.5. . . . . . . . . . . . . . 62



4.1 The dynamics of the system (4.3.2) showing concentration of cells

and viruses with continuous time delay . . . . . . . . . . . . . . . . . 107

5.1 The dynamics of the system (5.2.1) for τ = 1.5, showing convergence

to the stable equilibrium E. . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 The dynamics of the system (5.2.1) for τ = 0.7 showing convergence

to the stable equilibrium ¯E. . . . . . . . . . . . . . . . . . . . . . . . . 122

5.3 The dynamics of the system (5.2.1) for τ = 0.4 showing oscillating

behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.4 The plot supports example (5.3.1) and shows the stability of E0 . . . . 132

5.5 The plot supports example (5.3.2) and shows the stability of E . . . . 133

5.6 The plot supports example (5.3.2) . . . . . . . . . . . . . . . . . . . . 134

6.1 The graph represents the uninfected target cells x(t) with and with-

out control. The concentration of uninfected target cells increases

after the optimal control strategy. . . . . . . . . . . . . . . . . . . . . . 144

x

6.2 The graph represents the density of infected cells y(t) with and with-

out control. The concentration of infected cells approaches to a small

number due to optimal control. . . . . . . . . . . . . . . . . . . . . . . 145

6.3 The graph represents the density of double infected cells z(t) with

and without control. The concentration of double infected cells ap-

proaches to a small number due to optimal control. . . . . . . . . . . 145

6.4 The figure shows the density of free virus v(t), with and without

control. The concentration of pathogen viruses reaches to a small

number due to optimal control strategy. . . . . . . . . . . . . . . . . . 146

6.5 The graph shows the density of recombinant virus, with and with-

out control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.6 The graph represents control in the density of uninfected cells verses

time t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.7 The graph represents control in the density of infected cells verses

time t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.8 The graph represents control in the density of double infected cells

verses time t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.9 The graph represents control in the density of pathogen virus verses

time t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.10 The graph represents control in the density of recombinant virus

verses time t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.11 The graph shows the control pair in the optimal control problem. . . 158

6.12 The graph represents the density of uninfected cells verses time t in

weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.13 The graph represents the density of infected cells verses time t in

weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.14 The graph represents the density of double infected cells verses time

t in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

xi

6.15 The graph represents the density of pathogen virus verses time t in

weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.16 The graph represents the density of recombinant virus verses time t

in weeks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.17 The graph shows the behavior of control pair in the optimal control

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.1 Solution behavior of the uninfected CD4+ T cells x for σ = 1 (solid

line) σ = 0.95 (dashed line) σ = 0.85 (dot-dashed line) σ = 0.75

(dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.2 Solution behavior of the productively infected CD4+ T cells y for

σ = 1 (solid line) σ = 0.95 (dashed line) σ = 0.85 (dot-dashed line)

σ = 0.75 (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.3 Solution behavior of double infected cells z for σ = 1(solid line)

σ = 0.95 (dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.4 Solution behavior of pathogen virus v for σ = 1(solid line) σ = 0.95

(dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line). . . . . 183

7.5 Solution behavior of the recombinant virus w for σ = 1 (solid line)

σ = 0.95(dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted

line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

xii

Abstract

In this research, we will present mathematical analysis and optmal control strate-

gies to control the spread of HIV-1 infection. For this, first we will develop a single

delayed HIV-1 and discuss the effect of time delay in controlling HIV-1 infection.

Once the complete dynamics of single delayed model is studied, we will extend

this study to double delayed HIV-1 infection models. Incorporated time delays

represent delay in contact process between pathogen virus and CD+4 cells (bind-

ing), latent period, virus production period and CTLs response. Then, to study the

effect of dose-dependent infection rate in reducing HIV-1 infection, a new model is

formulated and its stability will be discussed. The model will be extended by incor-

porating the effect of recovery rate of unproductively infected cells to uninfected

cells. Moreover, in order to reduce HIV-1 infection in the body, optimal control

strategies will be developed. The aim of control strategies is to minimize the con-

centration of infected cells and maximize the concentration of uninfected cells. For

this purpose optimal control variables will be incorporated in the proposed HIV-1

infection models with time delays and without time delays. To analyze the dy-

namical behavior of fractional order HIV-1 infection model, the proposed integer

order HIV-1 models will be converted into fractional order model and its solution

behavior will be discussed. Finally, to verify the derived theoretical results nu-

merical simulations will be carried out for both integer and fraction order HIV-1

models.

xiii

Certificate of publications from thesis

It is declared that the following research articles have been published or

submitted from this PhD thesis of the PhD Scholar Mr. Nigar Ali.

• N. Ali, G. Zaman, ”Asymptotic behavior of HIV-1 epidemic model with infi-

nite distributed intracellular delays”, SpringerPlus, 5(1), p.324,2016.

• N. Ali, G. Zaman, and M.I. Chohan, ”Dynamical behavior of HIV-1 epidemic

model with time dependent delay”, Journal of Mathematical and Computational

Science, 6(3), pp.377-389, 2016.

• N. Ali, G. Zaman, and O. Algahtani, ”Stability analysis of HIV-1 model with

multiple delays”, Advances in Difference Equations, (1), pp.1-12, 2016.

• N. Ali, G. Zaman, and M.I. Chohan, ”Mathematical analysis of delayed HIV-

1 infection model for the competition of two viruses”, Cogent Mathematics,

4(1), p.1332821,2017.

• N. Ali, G. Zaman, and M.I. Chohan, ”Global Stability of a Delayed HIV-1

Model with Saturations Response”, Appl. Math, 11(1), pp.189-194, 2017.

• N. Ali, G. Zaman, and A.S. Alshomrani, ”Optimal control strategy of HIV-1

epidemic model for recombinant virus. Cogent Mathematics, 4(1), p.1293468,

2017.

• N. Ali, G. Zaman, A.M. Abdullah, and A.S. Alshormani, ”The effects of time

lag and cure rate on the global dynamics of HIV-1 model”.

xiv

xv

• N. Ali, G. Zaman, ”Stability analysis of delay integro differential equation”,

(Accepted in Georgian journal of mathematics)

• M.I. Chohan, N. Ali, G. Zaman, ”Modeling of optimal control problem of

delayed HIV-1 model”, (International journal of computer mathematics)

• N. Ali, G. Zaman, Optimal control strategy of double delayed HIV-1 model,

• , N. Ali, G. Zaman A. ZEB AND V. S. ERTURKd, ”Analytical study of HIV-1

fractional order model”, (International journal of computer mathematics)

Supervisor

Dr.Gul Zaman

Chapter 1

Introduction

1.1 History of HIV-1 Infection

The history of HIV-1 infection and AIDS began in death, illness and fear

as the world faced an unknown and new virus. Although, technological advance-

ments, for example the improvement of antiretroviral (ART) drugs, have enabled

the HIV-1 infected people with access to therapy to live long and having healthy

lives. The origin of HIV was Kinshasa (Congo). In 1920, HIV crossed species from

chimpanzees to humans. Nobody knows that how many people developed HIV

or AIDS up until the 1980s. During that time HIV was unknown and no signs or

symptoms were seen. At that time in 1980, this disease were spread to five conti-

nents including North America, Europe, South America, Africa and Australia and

the people between 100, 000 and 300, 000 were infected [1]. In 1981, Pneumocys-

tis carinii pneumonia (PCP) were found in some young, which were previously

healthy gay men in Los Angeles[2]. At the same time, New York and California

much suffered from unusually aggressive cancer which is called Kaposi’s Sarcoma

[3]. The very first cases of PCP were reported in the number of people who in-

ject drugs in the period December 1981 [4]. At the end of this year, there were 270

1

2

cases of severe immune deficiency, specially among the gay men and at least 121 of

them had died [5]. In June 1982, in Southern California, it was investigated that the

cause of the immune deficiency was sexual and the related syndrome was called

gay-related immune deficiency ( GRID) [6]. Then, the this disease was reported

in haemophiliacs and Haitians which leading many to believe it had originated

in Haiti [7, 8]. Later, the CDC (Center for Disease Control) introduced the word

AIDS (Acquired Immune Deficiency Syndrome) [8]. AIDS related diseases were

also being reported in some of the European countries [9]. In January 1983, it was

concluded that this disease could be passed on via heterosexual sex [8]. At that

time, the number of AIDS infected people in the USA had risen up to 3, 064 and at

least 1, 292 people had died [6]. WHO, the World Heath Organization announced

the 1st December as the first World AIDS Day in 1988. It was declared by WHO

in 1999, that AIDS was the fourth biggest cause of death worldwide in Africa. At

least 33 million people were HIV infected and 14 million people had died from

AIDS since the start of the epidemic [10]. President George W. Bush announced

(PEPFAR) in January 2003. He donated at least 15 USD billion, 5-year plan to

combat AIDS, in countries with a high number of HIV infections[11]. WHO an-

nounced the 3 by 5 initiative to bring HIV treatment to 3 million people by 2005

[12]. In 2013, UNAIDS noted that death rate due AIDS had fallen 30 percent since

2005 [13]. The number of HIV infected people was estimated 35 million [13]. In

October 2015, UNAIDS announced the control strategy named as Sustainable De-

velopment Goals (SDGs) [14]. The newly released figures also showed 64 percent

of all new HIV diagnoses in Europe occurred in Russia [15]. Moreover, (UNAIDS)

announced that 18.2 million HIV infected people were on antiretroviral therapy

3

(ART). Among them 910000 were children which is double of the number of the

five years earlier. But, achieving increased ART access means a greater risk of drug

resistance. The WHO released a report on dealing with this growing issue [16].

1.2 Immunological Study of HIV-1 Infection

HIV (Human Immunodeficiency Virus) is a pathogen virus. It attacks

the immune system of the body which is the body natural defense system. One

who has weak immune system, has the trouble to fight off diseases. The infection

causes by HIV is called HIV infection. The important part of the immune system

is the white blood cells which are also called CD4+cells. HIV infects and destroys

these cells. If too many of these cells are destroyed, then the body can no longer

defend itself against this infection. AIDS, which is Acquired Immunodeficiency

Syndrome, is the last stage of HIV infection. The people with AIDS have a low

concentration of CD4+ cells and get infections or cancers. But it should be clear

that having HIV doesn’t mean that some one has AIDS. The patient can also live

even without treatment for a long time as the duration from HIV to AIDS takes

much time which is usually 10 to 12 years. The medicine can only slow or stop

the damage to the immune system. If AIDS does develop, medicines can help the

immune system return to a healthier state but with treatment people with HIV

are able to live long and active lives. It should be mentioned that there are two

types of HIV. One is HIV-1, which causes almost all the cases of AIDS all over the

world. The other one is HIV-2, which becomes the cause of AIDS-like illness but

HIV-2 infection is uncommon in North America. One of the main target of HIV

is the class of lymphocytes, called CD4+ cells. The person having (AIDS) if the

4

concentration of CD4+ T cells falls to 200mm−3. The the process of the duration

of HIV infection towards AIDS probably depends on the power of replacement of

cells which are destroyed by viruses. [17].

1.3 HIV Transmission and Risks

Epidemiological study shows that HIV is transferred only through the

exchange of body fluids, such as semen, vaginal secretion, blood and mother’s

milk [18]. Moreover, HIV is vertically transmitted disease which can is transmitted

from an infected mother to child at the time of pregnancy or at the time birth.

The other ways of transmission is a infected breast milk which can be transferred

form infected mother to her child. The other causes of transmissions are High-risk

behaviors such as sexual intercourse which is unprotected and drug used needles

or syringes. In the past, many people got HIV infection through the transfusions

of blood which is infected. Further, the other cause is the blood-clotting factors

before blood screening which is begun in 1986. Hence, nowadays blood is tested

in the all over the word and it is no more risk for the world.

1.4 Treating HIV

There are many types of the treatments, available now a days, which can help

the HIV infected people. These treatments are much better than in earlier times

and most HIV infected people live long and having healthy lives. ART is one of the

treatment which can suppress the virus production. Recently, the available drugs

cannot completely eradicate the infection of HIV. Therefore, it is needed to treat

HIV infected for long time. The main use of ART is to decrease the mortality and

5

morbidity. To achieve this target, a dose of three active drugs or more than three

drugs is recommended [19]. The high prevalence of HIV shows an urgent need to

allocate resources for the prevention and treatment of of HIV [20]. But sill there are

gaps, as much people are still deprived having such services. For example, Kenya

was offering ART to 190000 people in the mid-2008. But only 12 percent of 1.4

million HIV-infected adults who required co-trimoxazole were receiving it in 2007

[21]. ART drugs and drugs for treatment and prevention benefit communities and

individuals in reducing death rate due to AIDS in many regions and countries.

It also can help in increasing HIV prevalence [22]. Furthermore, the treatment

of HIV may include quarantine of all seropositive persons which is successfully

practiced by Cuba in 1986 − 1994. Half of all HIV-positive Cubans were still lived

in the sanatoriums [23]. Assessing the impact of treatment on HIV, the researcher

designed and analyzed many epidemiological mathematical models in he last two

decades [24, 25, 26].

1.5 Role of Mathematical Models in Understanding

the Infection of HIV-1

Mathematical models have been formulated and analyzed to study the

dynamics of many infectious diseases for more than a century. Nowadays math-

ematical modeling has got much attention. There has been a tremendous efforts

made in the mathematical modeling of the HIV since the early 1980s. Many math-

ematical models have been developed to understand the spread of HIV and to

develop control strategies for HIV-1 infection.

A simple model for HIV infection was developed by Perelson [27]. Perelson et al.

6

[28] study this model in more detail and extended it further. It was concluded that

this HIV model exhibits many of the symptoms of AIDS which are seen clinically.

Then, this model was elaborated to four compartments such as the concentration

of uninfected cells, actively infected cells, latently infected cells and free virus.

The dynamics of these cells populations was also studied in detail. Nowak and

Bangham [29] modified this model by introducing the concentration of cytotoxic T

lymphocytes CTLs and proved that (CTLs) play a critical role in antiviral defense

by attacking virus-infected cells. They obtained conditions for the existence of one,

two or three steady states, and analyzed the stability of these steady states. In re-

cent research work, to control the infection of HIV-1, a new approach of genetic

engineering which is technology of the modification of a viral genome has played

an important role. This technology is the modification of rhabdoviruses, like the

the vesicular stomatitis and rabies, which kill and infect cells which are previously

infected by HIV-1(for details, see [30]).

Revilla and Garcia-Ramos [25] proposed a new model by extending the model in

[31] for the better understand of potential therapy to control the viral infections

using genetically modified viruses. These viruses are specifically designed to in-

fect only HIV-1 infected cells. The authors in this paper used a system of ODEs to

discuss the dynamics of the concentration of the cells populations. The researchers

in [25] studied in detail the temporal dynamics of the HIV model and their derived

results conclude that genetically modified virus is an alternate approach to extend

the survival of HIV infected people. Some numerical results in [32] have been

confirmed by Fluctuation lemma and constructing a suitable Lyapunov function

7

and functionals in [33]. These numerical and theoretical results proved the useful-

ness of mathematical technique in HIV-1 control. Through the results of [31, 25],

we know that this treatment could be an effective therapy in reducing individual

HIV-1 load. These results established the usefulness of mathematical modeling

in HIV-1 control research. The HIV-1 models discussed in [25, 31 − 33] ignored a

replication time delay between the entry of virus into the cell and the production

of new virons. But this process of time delay is very important in modeling HIV-1

infection as the model can be destabilized by time delay. To model the real sit-

uation of this infection, there may be a time period between the time when the

virus is attached to the target CD+4 cells and the time, the cells to which the virus

is attached, becomes affected actively [33]. Moreover, there is also time period be-

tween the time when the virus has completely penetrated into the cell and the time

when the new virion are produced within the cell and are released from the cell.

Therefore, some HIV-1 mathematical models described by differential equations

with constant time delays have been considered in [33, 34].

Two kinds of time delays can be found in time delayed HIV models in literature.

These two types are constant time delays and continuously distributed time delays

which have been incorporated into biological models such as [35, 36, 37, 38]. It is

known that the dynamics of DDEs, (delay differential equations) are complicated

than ODEs (ordinary differential equation). Because time delay may be the cause

of lose of stability of stable steady state and cause the oscillation of cell population.

Perelson et al. [37] proposed there exist two kinds of time delay, one of which is a

pharmacological time delay and the other one is intracellular time delay. The first

one time delay represents ingestion of drugs and its appearance within CD4+ cells.

8

An intracellular time delay represents the infection of a host cell and the emission

of viral particles. Herz et al. [36] showed probably for the first that discrete intra-

cellular time delay in a HIV-1 infection model would shorten half life estimation

of free virus. Mittler et al. [38] studied a related HIV-1 model and observed that

a Gamma-distribution time delay would be more accurate representation of the

process of HIV-1 compared to discrete time delays. Grossman et al. [39] developed

a new HIV infection model by incorporating time delay in the cell death process.

Later, Ruan and Culshaw [40] simplified the model proposed in the work of Perel-

son et al. [28] by taking into count three components: the uninfected CD4+ T-cells,

the infected CD4+T-cells, and the density of free virus. The reduced model is thus

more tractable mathematically and its theoretical analysis is also studied using the

delay as the bifurcation parameter. Herz et al. [36] proposed the time delayed

HIV model describing the time duration between infection of CD4+ T-cells and

the release of new viruses. It was shown by Nelson et al. [41] that estimated val-

ues of both the viral clearance rate and the rate of loss of productively infected

T cells are affected by the delay when the HIV model was studied by using ex-

perimental data. Moreover, in [42], a time delay was incorporated into the model

which was presented in [25]. It is proposed in [42] that the rate of production of

new virion can satisfy an exponentially decay function. Because the time period

is needed for virus to be produced form productively infected cells. In this paper,

the complete analysis using DDEs has been done. They also showed the impor-

tance of time delays in HIV-1 infection model. From the above discussion, we see

that different researchers have worked on the different strains of HIV-1, in diverse

periods. The researchers have focussed on particular dynamics of the disease, in

9

accordance with available data of the time for discrete and continuous delays. But

they did not focus on the effects of double delays in controlling of HIV-1 model

of recombinant virus. Also, they did not take into account the effect of saturation

response and cure rate on the stability of HIV-1 model and also did not consider

the fraction order model for recombinant virus. Similarly, they did not concentrate

on the optimal control strategies for controlling this infection.

Therefore, we will include these factors to develop new models. In order to do this,

we will incorporate time delays in the production period of new virions. Moreover,

the infection process is considered as linear mass action principle in [42]. But it is

reported in [43, 44] that the infection rate is an increasing function of the para-

site dose. Therefore, we will develop a new model by introducing saturated mass

action. In [42] the therapeutic of infected cells was ignored but actually through

therapy a part of infected cells may also revert to the uninfected state when these

cells losing all covalently closed circular DNA (cccDNA) from their nuclei at a cer-

tain rate [45]. Therefore, we will incorporate recovery rate in the proposed model.

Also, in [42] constant time delay is considered but it is proved in [46] that constant

time delays are not biologically realistic. Therefore, we will replace discrete time

delays by continues time delay in infection term as well as in virus production

term. To control the spread of HIV-1 infection, we will apply optimal control the-

ory to the HIV-1 models. Finally , we will convert the proposed model to fraction

order and will discuss its solution behavior and stability analysis. For this purpose

we will divide our thesis in the following chapters.

10

1.6 Organization of the Thesis

The remaining thesis is divided into the following chapters.

Chapter 2: This chapter is devoted to some basics of mathematical models, delay

differential equations and fractional differential equations. Further, some funda-

mentals about the stability analysis of mathematical models are given. Moreover,

basic mathematical tools like equilibria and its boundedness and nonnegativity are

discussed. Stability theory and optimal control theory will be used to find the local

and global stability. Fractional differential equations are also briefly introduced.

Chapter 3: This chapter is dedicated to the formulation and mathematical analysis

of three discrete delayed HIV-1 infection models. The first delayed HIV-1 infection

model consists of four compartments, infected cells, uninfected cells, pathogen

virus, CTLs response and time delays in contact term, infection term, and CTLs re-

sponse term. This model is analyzed for local and global stability and it is shown

that the incorporated time delays can reduce the infection in the body. The second

proposed model consists of uninfected cells, infected cells, double infected cells ,

pathogen virus and recombinant virus, and time delays in contact term and laten

period. The effect of discrete time delays on the stability of the infected steady

states is investigated. Criteria are given to ensure that the infected and uninfected

steady states are asymptotically stable for all time delays. The third proposed

model of this chapter is formulated by two constant time delays in model. The

incorporated time lags for infection of T cells and the replication of virus. The

proposed model consists of three equilibria. The basic properties of the model are

presented. Then the global stabilities of all types of equilibria are proved. Numer-

ical representation is also given.

11

Chapter 4: This chapter is dedicated to the formulation of HIV-1 infection mod-

els which consist of continues time delays. One proposed model studies the effect

of continues time delay in latent period and the second one discusses the effect

of continuous time delay in latent period and virus production period. The ba-

sic reproduction number R0 is found and its threshold property is presented. The

dynamical behavior of these proposed models is completely analyzed in term of

basic reproductive numbers. Numerical simulations are carried out.

Chapter 5: This chapter is devoted to the formulation of two HIV-1 infection mod-

els. The first one describes the effect of cure rate of unproductively infected cells

in reducing the infection of HIV-1. The analysis of this model shows that if R0 is

less than one, then infected cells are cleared and the disease dies out. For the for-

mulation of the second model, we assume that the infection rate between healthy

and infected cells is a saturating function of cell concentration. Then this model is

also analyzed for local and global stability using stability analysis theory. Numer-

ical simulations show that depending on the fraction of cells surviving the latent

period, the solutions approach either an infected steady state or a periodic orbit.

For the support of theoretical results numerical simulation are given.

Chapter 6: This chapter discusses the role of optimal control strategies in reduc-

ing the infection of HIV-1. Three models are proposed in this chapter. First, the

optimal control theory is applied to HIV-1 model without time delay. The optimal

control policy is formulated and solved as an optimal control problem. Objective

functionals are constructed which aims to (i) minimize infected cells quantity; (ii)

minimize free virus particles number; and (iii) maximize healthy cells density in

12

blood. Then the single delayed HIV-1 infection model is considered. The Pontrya-

gin maximum principle with delay is used to characterize these optimal controls.

Numerical results are presented to illustrate the optimal solutions. The model is

used to predict the possible beneficial effects of anti-retroviral drug administration

in HIV-1 disease. In the last, the double delayed HIV-1 infection model is taken

into account. Two control variables are introduced. One control variable denotes

the decreasing of the density of infected cells and the other describe the increase

of uninfected cells. Existence for the optimal control pair is accomplished and the

Pontryagin’s Maximum Principle is used to characterize these optimal controls.

Objective functional is constituted for the purpose of minimizing infected cells

measure and free virus particles amount and maximizing healthy cells density in

blood. The optimality system is derived and solved numerically.

Chapter 7: In this chapter, the HIV model of ordinary differential equations equa-

tion is extended to fractional order model. Then, the Laplace-Adomian decom-

position method is used to find an analytical solution of the system of nonlinear

fractional differential equations. The convergence of the proposed method is pre-

sented. The obtained results are compared with the available results of integer-

order derivatives. Numerical results shown that the Laplace-Adomian decompo-

sition method (L-ADM) is very simple and accurate for solving fractional-order

HIV-1 infection model.

Chapter 8: This chapter is devoted to the summary of our research work and gives

some future directions to extend this study.

Chapter 2

Preliminaries and Basic Materials

In this chapter, some basic definitions related to mathematical models and the

techniques of the stability analysis of the models are presented. Moreover, delay

differential equations and their mathematical analysis are also introduced. Then,

optimal control theory and fractional differential equations are briefly introduced.

2.1 Mathematical Model

Mathematical models are used to describe real world problem in mathe-

matical language [47]. The process of formulating a mathematical model is known

as mathematical modeling. A model helps in explaining a system and in study-

ing the effects of different components, and to make predictions about behavior.

Mathematical language can be expressed in equation or computer code. Mathe-

matical model has many applications in natural science, biology, ecology, toxicol-

ogy, physics, evolution, immunology, meteorology, engineering, computer science

and social sciences. Mathematical models are used to describe the behavior of a

system.

13

14

2.1.1 Epidemiological Models

Epidemiological models are used to identify appropriate management re-

sponses to infectious disease outbreaks, inform public policy on disease manage-

ment in the event of future outbreaks, design and evaluate control strategies [47].

The goal of the epidemiological models program is to determine the combination

of health policies and intervention strategies that can lead to disease eradication.

Epidemiological model calculates how diseases spread in particular areas, analyze

the effects of current and future health policies and intervention strategies. The

models support infectious disease campaign, data gathering, new product devel-

opment and policy decisions like generic transmission types: vector-borne, water-

borne, airborne and sexually transmitted. In such models the whole population is

classified into different groups or subgroups, each shows a certain state of popula-

tion dynamics. Deterministic model describes and explains what happens on the

average at the population scale. Deterministic model is also known as compart-

mental model. Some population groups are given below

2.2 Transmission and Out Break

Transmission is the passing of communicable disease from an infected indi-

vidual to uninfected individual [48]. Transmission takes place in different ways as

given below .

• Direct contact transmission: Direct contact transmission occurs when there

is physical contact between an infected person and a susceptible person.

Physical contact means, touching, biting and sexual practices etc.

15

• Indirect contact transmission: Indirect contact transmission occurs when

there is no direct human-to-human contact. Contact occurs from a reservoir

to contaminated surfaces or objects or to vectors such as mosquitos, flies,

mites, fleas, ticks, rodents or dogs.

• Vertical transmission: In the case of some diseases such as AIDS and Hep-

atitis B, it is possible for the offspring of infected parents to be born infected.

This type of transmission is called vertical transmission.

• Horizontal transmission: Horizontal transmission of diseases among hu-

mans and animals occurs through direct or indirect physical contact with

infectious hosts or through infected vector such as mosquitos, ticks, flies or

other biting insects.

• Vector transmission: Diseases transmitted from human to human indirectly,

i.e. Malaria spread by way of mosquitoes, which is transmitted through a

vector.

2.2.1 Epidemic

An epidemic is the outbreak of an infectious disease that infects major seg-

ment of population in a community. The epidemic may occurs when a particular

number of the infected individuals enter into a community. No one is immune to

a disease when it begins to show its teeth. The scientific study of these diseases is

known as epidemiology. Epidemiology is concerned with how diseases out-break

and how to overcome them in human population.

16

2.2.2 Pandemic

An infection that extends its tentacles to almost every nook and corner of

the country or entire world is known as pandemic.

2.2.3 Endemic

The constant presence of a disease or infectious agent within a given

geographic area, or unusual prevalence of a given disease within such areas.

2.3 Basic Reproduction Number

There is a threshold quantity which determines whether an epidemic occurs

or the disease simply dies out. This quantity is called the basic reproduction num-

ber. In epidemiological models, the basic reproduction number is a key concept

and is defined as the average number of secondary infection arising from a single

infected individual introduced into the susceptible class during its entire infectious

period in a totally susceptible population [49]. If the basic reproductive number is

less than unity then the disease simply dies out. When the basic reproductive is

greater than unity then the disease spread in the susceptible population.

2.4 Autonomous System

An autonomous system is a system of ordinary differential equations of the

form

dx(t)

dt= f (x(t)),

17

where x represents the values in n-dimensional Euclidean space and t is time.

2.5 Equilibrium Points and Stability

For our proposed study, the following theorems and definition are required to

study the stability of he equilibrium points. [50].

2.5.1 Equilibrium Point

For description of equilibrium point, consider a differential equation

dy(t)

dt= F(y(t)). (2.5.1)

If we putdy(t)

dt = 0, then (2.5.1) becomes

F(y(t)) = 0 (2.5.2)

The values of y which satisfies (2.5.2) is called equilibrium point. It is also called

critical point or stationary point. Equilibrium points are three types, Maximum

point, Minimum point and point of inflexion.

2.5.2 Stability

Stability means that a small change in initial conditions produce a small effect

on solution of differential equation. The Theory of stability of equilibrium points

has been discussed in detail in the literature( see for detail [49, 50, 51, 52, 53, 54])

and for its applications (see [47]). Let us consider the system dynamics

X = f (X); f : D → Rn,

18

D: an open and connected subset of Rn. Let f (Xe) = 0 ⇒ Xe be the equilibrium

point.

• Definition The equilibrium point Xe is said to be stable if for any ǫ > 0, there

exists δ(ǫ) > 0 such that ‖ X(0)− Xe ‖< δ(ǫ) ⇒‖ X(t)− Xe ‖< ǫ ∀t > t0.

• Definition: The equilibrium point Xe which is not is stable is called unstable.

• Definition: The equilibrium point Xe is said to be convergent equilibrium if

∃ δ such that ‖ X(0)− Xe ‖< δ ⇒ limt→∞ X(t) = Xe

• Definition: The convergent and stable point Xe is called asymptotically sta-

ble .

2.5.3 Positive Definite

A positive semi definite function in D is a function V : D → R if it satisfies

the following [52]:

(i) 0 ∈ D implies that V(0) = 0,

(ii) ∀X ∈ D, V(X) ≥ 0,

V : D → R is said to positive definite in D if condition (ii) is replaced by V(X) > 0

in D − {0} and V : D → R is negative definite (semi definite) in D if −V(X) is

positive definite.

2.5.4 Lyapunov Stability Theorems

Theorem 2.5.1. (Stability) [53] Let X = Xe be an equilibrium point of X = f (X), f :

D → Rn. Further, let V : D → R be a function such that it is continuously differentiable

function and:

19

(i) V(Xe) = 0,

(ii) V(X) > 0, in D − Xe,

(iii) V(X) ≤ 0, in D − Xe.

Then X = Xe is stable.

Theorem 2.5.2. (Asymptotically stable) Let V : D → R be a continuously differentiable

function and X = Xe be an equilibrium point of X = f (x), f : D → Rn such that

(i) V(Xe) = 0,

(ii) V(X) > 0, in D − Xe,

(iii) V(X) < 0, in D − Xe.

Then, X = Xe is asymptotically stable.

Theorem 2.5.3. (Globally asymptotically stable) Suppose that V : D → R is con-

tinuously differentiable function and X = Xe be an equilibrium point of X = f (x),

f : D → Rn, such that :

(i) V(Xe) = 0;

(ii) V(X) > 0, in D − Xe;

(iii) V(X) is “radially unbounded”;

(iv) V(X) < 0, in D − Xe.

Then X = Xe is globally asymptotically stable.

20

2.5.5 Jacobian Matrix

We considered n-dimensional system of nonlinear first order ordinary differ-

ential equations in Rn, which is given by

X(t) = F(X(t)), X(t0) = X0, (2.5.3)

where ”.” denotes the derivative with respect to time, and F is nonlinear n × 1

vector continuous function in X and X is 1 × n state vectors.

F = [ f1, f2, f3, ..., fn], X =

x1

x2

...

xn

.

The variational matrix or Jacobian matrix of F(x1, x2, . . . , xn) denoted by J is de-

fined as follows

J =

∂ f1∂x1

∂ f1∂x2

· · ·∂ f1∂xn

∂ f2∂x1

∂ f2∂x2

· · ·∂ f2∂xn

......

. . ....

∂ fn

∂x1

∂ fn

∂x2· · · ∂ fn

∂xn

,

where fi are the components of F and∂ fi∂xi

are partial derivative of fi with respect to

xi for i = 1, 2, . . . , n. The stability of an equilibrium point can be checked via using

Jacobian matrix. The importance of the Jacobian matrix lies in the fact that it repre-

sents the best linear approximation to a differentiable function near an equilibrium

point.

21

2.5.6 Invariance Principle

It is necessary to consider that associated population sizes can never be neg-

ative because general epidemiological models monitor human populations. Thus,

these models should be considered in regions which are feasible ‘and where the

property of non-negativity is preserved.

• Definition: An invariant set is a set M such that for all x ∈ M we have

φ(x; t) ∈ M for all t. A set is invariant positively or negatively if for all

x ∈ M, φ(x; t) ∈ M for all t > 0 or (t < 0).

Suppose that the equilibrium point of autonomous system of ordinary differential

equations x = f (x); x ∈ Rn : is x = 0. Further, V is a Lyponove function in the

neighborhood U of x = 0. If x0 ∈ U has its forward bounded trajectory along with

the limit points in U and the largest invariant set M is of E = { f x ∈ U : V(x) = 0};

then φ(x; t) ∈ M as t → ∞.

2.6 Delay Differential Equations in Mathematical Bi-

ology

It is a long history for the use of ordinary and partial differential equa-

tions to model biological systems, dating to Malthus, Verhulst, Lotka and Volterra.

Another approach is of delay differential equations. These equations are are ob-

tained by incorporating time delay terms in the differential equations. The time

delays or lags represent incubation periods, gestation times and transport delays.

22

Delay models are becoming more common, appearing in many branches of bio-

logical modeling. They have been used for describing several aspects of infectious

disease dynamics such as primary infection and dug therapy [55, 56].

2.6.1 Basic Properties of Delay Differential Equations(DDE)

DDEs have some characteristic different form ODEs due to which the analysis

of these equations are more complicated. For example consider the example

x(t) = f (x(t), x(t − τ)). (2.6.1)

DDEs requires some more information than an analogous problem for a system

without delays. For the solution of an ordinary differential system, an initial point

in Euclidean space is required at an initial time t0. One needs information on the

whole interval [t0−τ , t0]. Hence, to find the rate of change at t0, one should required

to know about x(t0 and x(t0 − τ). Further, for x(t0 + ω), one should has the in-

formation about x(t0 + ω) and x((t0 + ω)− ω). Each such initial function gives a

unique solution to DDE. For the initial continuous function, the space of solutions

has the same dimensionality as C([t0 − τ, t0], R). One can say that it is infinite

dimensional. This property of DDE is apparent in linear systems. For ordinary

differential equations, one seeks exponential solutions and finds a characteristic

equation. But for DDE, we get a transcendental equation of the form

Q0(ρ) + Q1(ρ)e−ρτ = 0, (2.6.2)

where Q0 and Q1 are polynomial in ρ. This equation has infinite solutions[57].

The nature of DDE is worse than that of ordinary differential equations, this is not

always the case. The solutions to x(t) = (x(t))2 diverge to infinity in finite time

23

but solutions to DDE x(t) = x(t − τ(t))2 , however, are continuable for all time if

ρ(t) is positive ∀ t.

2.6.2 Linear Delay Differential Equations with Constant Delays

Coefficients

Consider a first order delay differential equation system

x(t) =n

∑i=1

Aix(t − τ),

where Ai is a constant matrix of order n × n, ∀ i, and i ≤ τi ≤ τ, ∀ i and some fixed

constant τ. The auxiliary equation of this system is

det

(λI −

n

∑i=1

Aie−τ

)= 0. (2.6.3)

We state the theorems below without proof [58].

Theorem 2.6.1. The above characteristic equation has at most a finite number of solutions

λ for any real number ρ, such that Reλ ≥ ρ.

Most of the roots of the (2.6.3) have negative real part. Furthermore, the roots

cannot accumulate except about Re(λ) = −∞. In our coming study, we will use

C([−τ, 0], R), showing all initial functions, when endowed with the norm.

||φ|| = supt∈[−τ,0]

φ(t), (2.6.4)

this is a Banach space.

24

2.6.3 Local Stability of Delay Differential Equations

The local stability of a steady state of ordinary differential equations depends

on the location of roots of the characteristic function. If all the roots have negative

real parts then the steady state is said to be stable . Similarly, Routh-Hurwitz crite-

ria can also give the nature of the roots. But for DDEs, this function takes the form

transcendental equation. Thus, there are infinitely many roots and the the men-

tioned criteria is failed here. Many search work done for the stability of DDEs can

be found such as Pontriagin Criteria, Chebotarev’s Theorem, Domain Subdivision

Frequency Methods and the Tsypkin Criterion.

2.7 Some Basics on Optimal Control

Optimal control theory is a powerful mathematical tool that can be used

to make decisions involving complex biological situations. The behavior of the

underlying dynamical system is described by a state variable(s). We assume that

there is a way to steer the state by acting upon it with a suitable control function(s).

The control enters the system of ODEs and affects the dynamics of the state system.

The gaol of using control variables is just to maximize (or minimize) the given

objective functional. Before beginning, we establish some definitions and concepts

from [59].

25

2.7.1 Optimal control problem:

The general optimal control problem is given by

min[φ(T, y(T)) +∫ t

0g(t, y(t), u(t)dt)],

where, y = [y1(t), y2(t), ..., yn(t)] and u = [u1(t), u2(t), ..., um(t)] are state and con-

trol variables respectively. The state and control variables of the dynamics of first

order differential equations of the following system are given by

dy

dt= f (t, y(t, u(t))), y0 = y(0), 0 ≤ t ≤ T.

2.7.2 Lipschitz Function

A function k is called Lipschitz if there exists a constant m such that |k(t1)−

k(t2)| ≤ m|t1 − t2| for all points t1, t2 in the domain of k.

2.7.3 Pontryagin’s Maximum Principle:

The Pontryagin’s Maximum Principle converts the minimization(maximization)

of an objective functional together with the state variables into minimizing (maxi-

mizing) point-wise the Hamiltonian H along the control variable u.

Theorem 2.7.1. Let y∗(t) and u∗(t) are state and control variables respectively for control

problem, then there exists adjoint variable which is denoted by λ(t), satisfying

H(t, y(t), u(t), λ(t)) ≤ H(t, y∗(t), u∗(t), λ(t)),

where H can be defined as

H = f((t, y(t), u(t)) + λ(t)g(t, y(t), u(t))

),

26

and

dλ(t)

dt= −

∂H

∂y, λ(T) = 0,

where λ(t) is the adjoint or co-state variable and T is the final time.

2.7.4 Necessary condition

The necessary conditions are given as if y∗(t) and u∗(t) are optimal state and

control variables, respectively then they satisfy the following conditions

dλ(t)dt = − ∂H

∂y ,

λ(T) = 0,

∂H∂u = 0.

(2.7.1)

2.7.5 Sufficient Conditions

The sufficient conditions are given as if y∗(t), u∗(t) and λ(T) satisfy the

conditions given in (2.7.1), then the control and the state variables are optimal. We

apply optimal control theory to mathematical models in the following steps

(i) The Hamiltonian for the proposed model is constructed.

(ii) The set of adjoint differential equations is derived. Transversality boundary

conditions are adjusted under some constraints. Then, the optimality con-

ditions are placed. So that we have three unknowns variables, u∗, x∗ and

λ.

(iii) u∗ is eliminated by applying the optimality condition H(u) = 0.

27

(iv) Then the two differential equations x∗ and λ with the two boundary condi-

tions is solved. Further, u∗ is substituted in the differential equations with

the equation for the optimal control by using the previous step.

(v) Once the optimal state and adjoint is found, then , it can be solved for the

optimal control.

2.7.6 Characterizing the Optimal Control

To characterize the optimal control problem, the values of the control pair u∗

is found.

2.7.7 Optimality System

The optimality system is the set of state equations, adjoint equations and optimal

control pair.

2.7.8 Fractional Calculus

Fractional calculus is the study of derivatives and integrals of arbitrary order.

This subject has a long mathematical history. Over the centuries many Mathe-

maticians have got a lot of mathematical knowledge on fractional integrals and

derivatives.

28

2.8 Fractional Differential Equations

In recent years it has been pointed out that most of the physical problems

in Engineering, Physics, Chemistry, Electromagnetic Theory, Circuit Theory, Biol-

ogy, Atmospheric Physics and other sciences can be presented very successfully

by models using mathematical tools from fractional calculus. Fractional differen-

tial equations are generalizations of ordinary differential equations to an arbitrary

(non-integer) order. Fractional differential equations have attracted considerable

interest because of their ability to model complex phenomena. We will state with

some definitions and results which are represented in [60].

2.8.1 The Gamma Function

The gamma function denoted by Γ(α) can be defined by the improper integral

Γ(α) =∫ ∞

0e−ttα−1dt (α > 0), (2.8.1)

which is meaningful only if α > 0.

Integration by parts gives the important functional relation of the gamma function

Γ(α + 1) = αΓ(α). (2.8.2)

From equation (2.8.1) we have Γ(1) = 1. Hence, if α is any positive integer, say k

then by the repeated applications of equation (2.8.2), we obtain

Γ(k + 1) = k!. (2.8.3)

This can be observed that gamma function is the generalization of the elementary

factorial function.

29

2.8.2 The Beta Function

Beta function is defined is given by

B(x, y) =∫ 1

0tx−1(1 − t)y−1dt, (x > 0, y > 0),

and its relation with gamma function is given by

B(x, y) =Γ(x)Γ(y)

Γ(x + y).

2.8.3 Repeated Integrals

Suppose we have a function f (x) defined for x > 0, we can form the indefinite

integral of f (x) from 0 to x, we call this (I f )(x) and is given by

(I f )(x) =∫ x

0f (t)dt.

If we integrate (I f )(x) again, we get the second integral

(I2 f )(x) =∫ x

0(I f )(t)dt =

∫ x

0

( ∫ t

0f (s)ds

)dt.

And another integration gives the third integral, i.e.

(I3 f )(x) =∫ x

0[∫ t

0(∫ s

0f (u)du)ds]dt.

Similarly, nth order integral is given by Such integrals is called repeated integrals.

2.8.4 Cauchy Result

As to find nth integral of a function f (x) is much complicated if n is large. It

was Cauchy who showed that how we can define the nth integral of an equation.

(In f )(x) =1

(n − 1)!

∫ x

0(x − t)n−1 f (t)dt. (2.8.4)

Thus Cauchy’s result reduce the n repeated integrals into just one integral.

30

2.8.5 Caputo’s Approach to Fractional Calculus

In past, fractional calculus was primarily a study reserved for the best minds

in Mathematics. Some of the Mathematicians like Fourier, Euler, Laplace presented

the theory of fractional calculus. They have used their own notation and method-

ology, definitions those fit the concept of a non-integer order integral or derivative.

The most famous of these definitions that have been popularized in the world of

fractional calculus are the Riemann-Liouville and Grunwald-Letnikov definitions.

But there was certain disadvantages of these definitions when trying to model real-

world phenomena with fractional differential equations. Caputo reformulated the

more classic definition of the fractional derivative in order to use integer order

initial conditions to solve the fractional order differential equations.

2.8.6 Fractional Integral

The fractional integral of order α > 0 for a function f : R+ → R is defined by

Iα f (t) =1

Γ(α)

∫ t

0(t − τ)α−1 f (τ)dτ.

2.8.7 Caputo’s Definition of Fractional Derivative

As discussed earlier that there are many definitions of fractional derivatives.

Perhaps the best known and the most common definition is the Caputo definition,

since it is widely used in real applications. The Caputo’s fractional order derivative

of order α ∈ (n − 1, n) for f (t) is defined as

Dα f (t) = In−αDn f (t),

31

where In−α denotes the fractional integral of order n− α and D represents the ordi-

nary derivative, further n − 1 is the integer part of α. This definition was proposed

by M. Caputo in his work, while solving an initial value problem, whose solution

is mathematically possible by Riemann-Lioville definition but practically useless.

Under condition α → n on the function f (t), the Caputo derivative becomes a

conventional nth derivative of f (t). There are more properties of Caputo deriva-

tive such as the derivative of a constant is zero etc. A comprehensive detail can be

found in [60].

Chapter 3

Asymptotic behavior of discretedelayed HIV-1 models

3.1 Overview

In this chapter, the effect of discrete time delays in reducing HIV-1 infection

will be studied. These time delays represent time duration in the contact between

virus and uninfected CD+4 cells, time duration between contact and infection of

CD+4 cells and time duration between infection and the production of new virus

from infected cells. Three delayed HIV-1 infection models will be proposed to in-

corporate all the above mentioned time delays. The first proposed HIV-1 infection

model is related to CTLs response and discusses time delay in contact process,

latent period and virus production period. The second HIV-1 infection model is

related to recombinant virus and discusses the effect of time delays in binding and

infection process. Then, this HIV-1 delayed model further proposed for time de-

lays in latent period and virus production period. These formulated model mainly

focus in reducing the density of infected cells and in the density of pathogen virus.

The positivity and well posedness of the three proposed models will be discussed.

32

33

The basic reproduction numbers will be found. Further the local behavior of this

model is found using Routh Hurwits criterion and global behavior is determined

by using Lyapunov functional theory. Numerical simulations will be presented to

support the derived theoretical results.

3.2 Analysis of HIV-1 Model with CTLs Immune Re-

sponse

The new model is formulated by proposing the contact process between

the uninfected and unproductively infected cells is not instantaneous. Most of the

previous work has considered this process instantaneously and considered time

delay only in disease transmission term. Therefore, time delay is incorporated into

the contact term in model presented in [61], similar to disease transmission term.

By introducing these terms the proposed model becomes

dx

dt= λ − dx(t)− βe−aτ1 x(t − τ1)v(t − τ1),

dy

dt= βe−mτ1 x(t − τ1)v(t − τ1)− ay(t) − py(t)i(t),

dv

dt= ky(t) − uv(t), (3.2.1)

di

dt= cy(t − τ2)i(t − τ2)− bi(t).

Here x(t) is the concentration of healthy CD4+ T cells, y(t) is the concentration of

uninfected CD4+ T cells and v(t) is the density free virus at time t. Here a mass

action infection mechanism is adopted, with constant infection rate β. The natural

production rate of T cells is given by λ. a is the removal rate of of infected cells.

k is virus production rate. i represents Cytotoxic T lymphocytes (CTLs) response

34

cells. which has the major in reducing virus concentration through the period of

infection of HIV. We assume that the viruses are produced after the constant time

delay τ1, when the virus entered into the cell. The production of virus-producing

cells at time t is given by the number of the uninfected CD4+ T cells that were

newly infected at time t− τ1 and are still alive at time t [62]. e−aτ1 is the probability

of surviving of infected cells through the time period from t − τ1 to t, where m is

assumed to be constant death rates for infected CD4+ T cells. τ2 is the time pe-

riod in which antigenic stimulation generates CTLs but the concentration of CTLs

depends on the density of antigen at time t which is denoted by (t − τ2)[63].

3.2.1 Positivity and Well- posedness of the Solution

In this section, the positivity and well-posedness of the solutions of the

proposed model will be discussed.

Theorem 3.2.1. Under the given initial conditions which are non-negative and bounded,

the solution of the system (3.2.1) is non-negative.

Proof. For the system (3.2.1), let the initial conditions be given as:

x(φ) ≥ 0, y(φ) ≥ 0, i(φ) ≥ 0, v(φ) ≥ 0, φ ∈ [−τ, 0]. (3.2.2)

Here(

x(φ), y(φ), v(φ), i(φ))∈ X and τ = max{τ1, τ2} and X = C([−τ, 0]; R4) be

the space of continuous mapping from [−τ, 0] to R4 which is Banach Space and

equal to suprimum-norm. Using the theory of functional differential equations

(see, e.g. [64]). One can show the existence of unique solution of the system (3.2.1)

35

under the given initial conditions in (6.3.2). Solving for x(t), y(t), i(t),and v(t) di-

rectly, we have

x(t) = x(0)e−∫ t

0 (d)dζ +∫ t

0(λ − x(t − τ1)v(t − τ1)βe−aτ1)e−

∫ t0 (d)dζdη,

y(t) = y(0)e−∫ t

0 (a+pi(ζ))dζ +∫ t

0

(βe−aτ1 x(t − τ1)v(t − τ1)

)e−∫ t

η (a+pv(ζ))dζdη,

v(t) = v(0)e−ut +∫ t

0ky(η)e−u(t−η)dη,

i(t) = i(0)e−bt +∫ t

0cy(t − τ2)i(t − τ2)e

−b(t−η)dη.

It can be seen that all the solutions are positive. In order to check the solution to be

bounded, we define

M∗(t) = x(t) + y(t) +a

2kv(t) +

p

ci(t + τ2).

Calculating the derivative of the above equation, we get

d

dt(M∗(t)) =

d

dt(x(t)) +

d

dt(y(t)) +

a

2k

d

dt(v(t)) +

p

c

d

dt(i(t + τ2)). (3.2.3)

Using equation (3.2.1) in equation (3.2.3), get

d

dt(M∗(t)) = λ − dx(t)− βe−aτ1 x(t − τ1)v(t − τ1) + βe−mτ1 x(t − τ1)v(t − τ1)

−ay(t) − py(t)i(t) +a

2k

(ky(t) − uv(t)

)+

bk

2

(ci(t)− qw(t))

= λ −(dx(t) +

a

2y(t) + u

a

2kv(t) + b

p

ci(t + τ2)

)

≤ λ − ǫM⋆(t).

where ǫ = min(d, a2 , u, b). This shows that M∗(t) is bounded for large time t. So

that all the individual solution is bounded.

The model (3.2.1) has disease-free equilibrium E0(x0, y0, v0, i0), the CTLs-absent

equilibrium E1(x1, y1, v1, i1) and CTLs present equilibrium E2(x2, y2, v2, i2), can be

36

determined as follows:

E0 =

(λ

d, 0, 0, 0, 0

),

E1 =

(au

βke−aτ1,

ud(R0 − 1)

βke−mτ1,

d(R0 − 1)

βe−mτ1, 0

),

E2 =

(Nuc

βkbe−mτ1 + ucd,

b

c,

bk

uc,

βkλcae−aτ1 − a(cdu + bkβe−aτ1)

p(cdu + kbβe−mτ1)

),

where R0 [61, 65] is the reproduction number and is given by

R0 =kβλe−aτ1

adu.

Here λd shows the average density of healthy CD4+ T cells which are available for

infection. Further,βe−aτ1

a represents the average density of cells that is infected by

HIV. ku is the production rate of viruses. If R0 < 1, then E0 biologically meaningful

equilibrium and for R0 > 1, the equilibria E1 and E2 which can also be interpret

biologically.

3.2.2 Dynamical Behavior of Disease-free Equilibrium

In this section, we show the dynamical behavior of the system (3.2.1) at

disease free equilibrium.

Theorem 3.2.2. For R0 < 1, the disease-free equilibrium E0 is locally asymptotically

stable while for R0 > 1, E0 loses it stability.

37

Proof. The characteristic equation of the Jacobian matrix corresponding to the lin-

earized system of (3.2.1) can be determined as follows

det[ρI − J(E0)] = det

ρ + d 0 βx0e−τ1(ρ+a) 0

0 ρ + a −βe−τ1(ρ+a)x0 p

0 −k ρ + u 0

0 0 0 ρ + b

= 0. (3.2.4)

After some simplification, equation (3.2.4) takes the form

det[ρI − J(E0)] = (b + ρ)(d + ρ)

[(a + ρ)(u + ρ)−

λ

dβke−τ1(ρ+m)

]= 0. (3.2.5)

Here ρ1 = −b and ρ2 = −d are the two roots of the equation(3.2.5) and the remain-

ing roots can be determined as follows:

(a + ρ)(u + ρ) =λ

dβke−τ1(ρ+m). (3.2.6)

If ρ has non- negative real part then modulus of the left hand side of the equation

(3.2.6) satisfies

|(a + ρ)(u + ρ)| ≥ au.

While modulus of the right hand side of (3.2.6) gives

λ

dβk|e−τ(ρ+a)| = |auR0 | < au,

which is contradiction. Therefore, for R0 < 1, E0 is locally asymptotically stable

because the real part of ρ is negative, while for R0 > 1, we have

h(ρ) = (a + ρ)(u + ρ)−λ

dβke−τ1(ρ+a),

= (a + ρ)(u + ρ)− aue−τ1aR0.

38

Here h(0) = au(1 − R0) < 0 and limρ→∞ h(ρ) = +∞. Then, by the continuity of

h(η) there exists at least one positive root of h(ρ) = 0 . Therefore, the E0 is unstable

for R0 > 1 [66].

Theorem 3.2.3. When R0 < 1, the disease-free equilibrium E0 is globally asymptotically

stable.

Proof. Consider the following Lyapunov functional

V0(t) = x0(x

x0− ln(

x

x0)− 1) + y(t) +

a

kv(t) +

p

ci(t)

+x0βe−aτ1

∫ t

t−τ1

x(ζ)v(ζ)d(ζ)

x(ζ + τ1)+ p

∫ t

t−τ2

y(ζ)i(ζ)d(ζ). (3.2.7)

The derivative of equation (3.2.7) and the use of equation (3.2.1), gives the follow-

ing equation

d

dtV0(t) = (1 −

x0

x)

(λ − dx(t)− βe−mτ1 x(t − τ1v(t − τ1)

)

+

(βx0e−mτ1y(t − τ1)i(t − τ1)− ay(t) − py(t)i(t)

)+

a

k

(ky(t) − uv(t)

)

+p

c

(cy(t − τ2)i(t − τ2)− bi(t)) + βx0e−aτ1

(xt)v(t)

x(t + τ1)−

x(t − τ1))v(t − τ1)

x(t)

)

+p

(y(t)i(t) − y(t − τ2)i(t − τ2)

). (3.2.8)

Re-arranging and simplifying (3.2.8), we get

d

dtV0(t) = −

d

x(x(t)− x0)

2 +(

βλ

de−aτ1

x(t)

x(t + τ1)−

au

k

)v(t)−

bp

ci(t).

If τ1 is very large, then the rate of infection as well as the chance of contact between

virus and host cells are very small but if τ1 is very small, then this infection well

spread more rapidly, so we take x(t) = x(t + τ1). Using this assumption ddt V0(t)

39

can be written as:

d

dtV0(t) = −

d

x(x(t)− x0)

2 +au

k(R0 − 1)v(t) −

bp

ci(t).

This implies that ddt V0(t) ≤ 0 holds only if R0 < 1 and equality holds when x0 = λ

d ,

y(t) = 0, i(t) = 0, v(t) = 0. Thus, by LaSalli invariance principle (see [64]), Thus,

E∗ is globally stable when R0 < 1.

3.2.3 Dynamical behavior of CTLs- absent Equilibrium

This section, is devoted to the stability of CTLs- absent equilibrium E1.

Theorem 3.2.4. This theorem can be proved in the following two parts:

(i) When τ1 = 0, then CTLs- absent equilibrium E1 is locally stable for R0 > 1. But

if τ1 > 0 , then E1 is locally asymptotically stable if 1 < R0 < 1 + (a+u)2aud provided

that aud2 < 1, while for R0 > 1 + (a+u)

2aud , E1 becomes unstable.

(ii) When τ2 ≥ 0,then E1 is locally asymptotically stable if 1 < R0 < 1 +bβke−aτ1

cdu .

Proof. The characteristic equation of the Jacobian matrix corresponding to the lin-

earized system of (3.2.1) can be obtained as

det[ρI − J(E∗)]

= det

ρ + d + βv1e−τ1(a+ρ) 0 βx1e−τ1(ρ+a) 0

−βv1e−τ1(a+ρ) ρ + a −βe−τ1(ρ+a)x1 p

0 −k ρ + u 0

0 0 0 ρ + b − cy1e−ρτ2

= 0.

(3.2.9)

40

After some fundamental calculation of equation (3.2.9), we get

(ρ+ b− cy1e−ρτ2)

[ρ3 + a0ρ2 + a1ρ+ a2 + (b0ρ2 + b1ρ+ b2)e

−τ1(m+ρ)

]= 0, (3.2.10)

where

a0 = a + u + d, a1 = (a + u)d + au, a2 = aud,

b0 = βv1, b1 = (a + u)βv1 − kβx1, b2 = auβv1 .

The second factor of equation (3.2.10) can be written as

ρ3 + a0ρ2 + a1ρ + a2 + (b0ρ2 + b1ρ + b2)e−τ1(m+ρ) = 0.

Now, we discuss the roots of the above equation for two cases of time delay τ1.

When τ1 = 0, then equation (3.2.10) becomes

ρ3 + c0ρ2 + c1ρ + c2 = 0, (3.2.11)

where

c0 = a0 + b0 = a + u + d + d(R0 − 1) > 0,

c1 = a1 + b1 = (a + u)d + (a + u)d(R0 − 1) > 0,

c2 = a2 + b2 = aud + aud(R0 − 1) > 0,

c0c1 = dR0(a2 + (a + u)(u + dR0) > c2 = aud + aud(R0 − 1).

Thus, by using Routh- Hurtwitz criterion [67], we see that (3.2.11) has no positive

roots when τ1 = 0 and R0 > 1. Now, we consider the distribution of roots when

τ1 6= 0. If ικ(κ > 0) is a solution of (3.2.11) then separating real and imaginary

parts, we get the following equations

a1κ − κ3 = (b2 − b0κ2) sin κτ1 − b1κ cos κτ1,

a0κ2 − a2 = (b2 − b0κ2) cos κτ1 + b1κ sin κτ.

41

By squaring and adding the above two equations, we get

κ6 + m1κ4 + m2κ2 + m3 = 0, (3.2.12)

m1 = a20 − 2a1 − b2

0, m2 = a21 − 2a0a2 + 2b0b2 − b2

1, m3 = a22 − b2

2.

Let us suppose that σ = κ2> 0, then equation (3.2.12) becomes

σ3 + m1σ2 + m2σ + m3 = 0,

where

m1 = a20 − 2a1 − b2

0 = a2 + u2 + d2 − (R0 − 1)2)d2> 0,

m2 = a21 − 2a0a2 + 2b0b2 − b2

1 > 0,

m2 = (ad)2 + (ud)2 + 2au(d(R0 − 1))2 + (a + u)d(R0 − 1)

×(2au − (a + u)d(R0 − 1)

)> 0,

m3 = (aud)2R0(2 − R0) > 0.

Similarly, we get

m1m2 − m3 = b2(1 − a21) + a2(2a1a0 − a2) + (a2

1 + 2b0b2)(a20 − 2a1) + b2

0(b1 − 2b0b2)

+b21(2a1 − a2

2) + 2a0a2(b0 − a21).

= ((1 − ((a + u)d + au)2) + aud

[aud + 2a2d + au

42

+2(u + d)((a + u)d + au)

]+ ((a + u)d + au)2

+2β2v2au)(a2 + u2 + d2) + (R0 − 1)(βv)2

[(a + u)d

−au(1 + 2d2(R0 − 1))

]+ ((a + u)βv − kβx)2[2au + 2ud

+2ad − (aud)2 ] + 2aud(a + u + d)

[d(R0 − 1)− ((a + u)d

+au)

]> 0.

Hence, all the roots of equation (3.2.12) have negative real part for τ1 6= 0, when

1 < R0 < 1 + (a+u)2aud provided that aud

2 < 1 (see [67]). Now consider the second

factor of equation (3.2.11)

ρ + b − cy1e−ρτ2 = 0. (3.2.13)

If τ2 = 0, then for 1 < R0 < 1 +bβke−aτ1

cdu , we obtain

ρ = cy1 − b =c

kβe−aτ1

(R0 − (1 +

bβke−mτ1

cdu))< 0.

This shows that the roots of equation (3.2.13) are negative for τ2 = 0. Next, we

discuss the roots in case τ2 > 0. Let ρ = κι(κ > 0) be pure imaginary root of

(3.2.13). Then, we get

κ = c(βke−aτ1 − aud

aβke−aτ1) sin κτ2,

b = c(βke−aτ1 − aud

aβke−aτ1) cos κτ2.

43

which implies that

κ2 = (βke−aτ1 − aud

aβke−aτ1)2 − b2.

But 1 < R0 < 1 + bβke−aτ1

cdu which implies that κ2< 0, which is a contradiction.

Therefore, we conclude that the all the roots have negative real parts and from [64]

it can be concluded that equilibrium point E1 is locally asymptotically stable.

Theorem 3.2.5. For 1 < R0 < 1 +bβke−aτ1

cdu , the CTLs present equilibrium E1 is globally

asymptotically stable while E1 is unstable for R0 > 1 +bβke−aτ1

cdu .

Proof. Denote f (ψ) = ψ − 1 − ln ψ, ψ ∈ R+. Let us construct the Lyapunov func-

tional

Vs(t) = x1 f (x

x1) + y1 f (

x

y1) +

a

kv1 f (

v

v1) +

p

ci1 f (

i

i1)

+βx1v1e−aτ1

∫ t

t−τ1

f (x(µ)v(µ)

x(τ1 + µ)v1)dµ + p

∫ t

t−τ2

y(µ)i(µ)dµ

(3.2.14)

By taking derivative of equation (3.2.14)

d

dtVs(t) = (1 −

x1

x)(λ − dx(t)− βe−aτ1 x(t − τ1)v(t − τ1)

)

+(1 −y1

y)(

βe−aτ1 x(t − τ1)v(t − τ1)− ay(t) − py(t)i(t))

+p

c(1 −

i

i1)(cy(t − τ2)i(t − τ2)− bi(t)

)+

a

k(1 −

v1

v)(ky(t) − uv(t)

)

+βx1e−aτ1x(t)v(t)

x(τ1 + t)− βx1e−aτ1v1 ln(

x(t)v(t)

x(τ1 + t)v1)

−βx1e−aτ1x(t − τ1)v(t − τ1)

x(τ1 + t)) + βx1e−aτ1v1 ln(

x(t − τ1)v(t − τ1)

x(τ1 + t)v1).

44

After some simplification, equation (3.2.15) becomes

d

dtVs(t) = e−aτ1(2 −

x1

x−

x

x1) + βx1v1e−aτ1

(3 −

x1

x−

yv1

y1v−

y1x(t − τ1)v(t − τ1)

x1v1y

+ ln(x(t − τ1)v(t − τ1x(t + τ1)

x2(t)v1)

)+

bp

c

(R0 − (1 +

bβke−mτ1

cdu))i

+(au

k

x(t)

x(t + τ1)−

au

k)v,

where we have used the following identities for simplification: λ = dx1 + βx1v1,

ay1 = βe−aτ1 x1v1, ky1 = uv1. For further simplification, we suppose that τ1 is very

large then x(t) = x(t + τ1). Therefore, we get

d

dtVs(t) = e−aτ1(2 −

x1

x−

x


(3 −

x1

x−

yv1

y1v−

y1x(t − τ1)v(t − τ1)

x1v1y

+ ln(x(t − τ1)v(t − τ1

x(t)v1)

)+

bp

c

(R0 − (1 +

bβke−aτ1

cdu))i

≤ 0.

The above inequality holds by using the following results (see [42])

e−aτ1(2 −x1

x−

x

x1) ≤ 0,

(3 −

x1

x−

yv1

y1v−

y1x(t − τ1)v(t − τ1)

x1v1y− ln(

x2(t)v(t)

x(t − τ1)v(t − τ1)x(t + τ1)

)≤ 0.

Therefore, ddt Vs(t) ≤ 0, when R0 < 1+

bβke−mτ1

auc . Moreover the equality holds when

x = x1 and y = y1, v = v1 and i = 0. Thus, we get E1 to be globally stable[64].

3.2.4 Dynamical behavior of CTLs- present Equilibrium

Theorem 3.2.6. If τ1 6= 0 and τ2 = 0 and R0 > 1 + bβke−mτ1

cdu , then E2 is globally

asymptotically stable.

45

Proof. : Let us construct the Lyapunov functional.

L2(t) = x2 f (x

x2) + y1 f (

x

y2) +

a

kv2 f (

v

v2) +

p

cz2 f (

z

z2)

+βx2v2e−mτ1

∫ t

t−τ1

f (x(µ)v(µ)

x(τ1 + µ)v2)dµ + p

∫ t

t−τ2

y(µ)z(µ)dµ.

By taking derivative of the above equation, we get

dL2

dt= e−mτ1(2 −

x2

x−

x

x2) + βx2v1e−mτ1

(3 −

x2

x−

yv2

y2v−

y2x(t − τ1)v(t − τ1)

x2v2y

+ ln(x(t − τ1)v(t − τ1

x(t)v(t))

)+ (βx2e−mτ1

x(t)

x(t + τ1)−

au

k)v

−py2z2 − pyz2 + pz2v2y

v+ py2z2

v

v2,

where we have used the following identities: λ − dx2 = βx2v2e−mτ1 ,

βx2v2e−mτ1 = ay2 + py2z2, ky2 = uv2, by2 = c.

Let us suppose that τ1 is very large, then x(t) = x(t + τ1) and dL2dt (t) equation can

be written as

dL2

dt= e−mτ1(2 −

x2

x−

x

x2) + βx2v2e−mτ1

(3 −

x2

x−

yv2

y2v−

y2x(t − τ1)v(t − τ1)

x2v2y

− ln( x(t)v(t)

x(t − τ1)v(t − τ1)

))

−p

acdu

(R0 − (1 +

bβke−mτ1

cdu

))((

v2

v− 1)y − (

v

v2− 1)y2

),

where we have used the following inequalities

(2 −

x1

x−

x

x1

)≤ 0,

(3 −

x1

x−

yv1

y1v−

y1x(t − τ1)v(t − τ1)

x1v1y− ln(

xt(t)v(t)

x(t − τ1)v(t − τ1)

)≤ 0.

Therefore, dL2dt ≤ 0, when R0 < 1 +

bβke−mτ1

auc . Moreover, the equality holds when

x = x2 and y = y2, v = v2 and w = w2. Hence, we get the conclusion that that E2

is globally stable.

46

3.2.5 Numerical Simulation

In the previous sections, we studied dynamical behaviors of the system (3.2.1) and

obtained some important results. For the purpose of simulating the model, we

have focused on the following standard parameters.

Table 1: The values of the parameters used in numerical results.

Notations Parameter definition Value Source

λ Recruitment rate 160 [61]

d death rate of uninfected target cells 0.16 [Assumed]

β infection rate of uninfected cells by virus 0.002 [61]

a death rate of productively infected cells 1.85 [61]

p killing rate of infected cells by CTL response cells 0.2 [assumed]

k rate of the virus particles produced by infected cells 1200 [61]

u viral clearance rate constant, 8 [Assumed]

c rate at which the CTL response is produced 0.2 [61]

b death rate of the CTL response 0.4 [Assumed]

τ1 intracellular delay 0.2 [61]

τ2 delay in antigenic stimulation 2.4 [61]

47

0 10 20 30 40 50 60 700

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

xyvz

Figure 3.1: Numerical solution of system (3.2.1) for different concentrations of cells,CTLs and viruses with effect of time delay

48

3.3 Analysis of Delayed HIV-1 Model for Recombi-

nant Virus

In this section, a delayed HIV-1 infection model is studied. Double de-

layed HIV-1 model is taken into account. The firs time delay is used for latent

period and the other one is used for contact term. Tian et al.[42] considered de-

lay only in infection term. However, the case where the contact process between

the uninfected cells and pathogen virus is not instantaneous [68], also should be

examined. Therefore, we assume the same values of delays in both terms. This

assumption is for simplicity and it is valuable to analyze the case where the two

types of time delays do not have the same values. This process was considered

directly in [42]. Also, in [61], the authors considered delays in the model dealing

with the investigations of global stability of the system of DDEs but ignored delay

in rate of contact between virus and target cells. But our proposed model inves-

tigate both the local and global dynamic for the model of DDEs which discusses

the recombinant virus interactions. Here, we introduce time delay, similar to the

disease transmission term, in the rate of contact term. By introducing delay in the

mentioned term, our proposed model becomes

dx

dt= λ − dx(t)− βe−aτx(t − τ)v(t − τ),

dy

dt= βe−aτx(t − τ)v(t − τ)− ay(t) − αw(t)y(t),

dz

dt= αw(t)y(t) − bz(t),

dv

dt= ky(t) − pv(t),

dw

dt= cz(t)− qw(t), (3.3.1)

49

where x(t), y(t) and v(t) have been defined in the previous section and some pa-

rameters are also considered in detail. The new incorporated variables such as w(t)

represents the concentration of double-infected cells and z(t) represent the densi-

ties of genetically modified virus. The infection rate of double-infected cells is de-

noted by αw(t)y(t). The removal rate of recombinant virus is qw(t). bz represents

the death rate of recombinant infected cells. which can also release recombinant

at rate cz(t). τ is the duration od infection of cell. e−aτ is t surviving probability

of of cells in the time period from t − τ to t. The dynamical behavior of the pro-

posed model, will be studied and the influence of the time delays on the stability

of the model will be discussed. The well-posdeness of the solutions of model and

the stability of equilibria points will be found. Moreover, the basic reproduction

numbers will be investigated. It will be shown that infection free equilibrium E∗

is locally as well as globally asymptotically stable. We will prove that Es (single

infection equilibrium) is locally as well as globally asymptotically stable.

3.3.1 Basic Results

In this section, we will discuss the well-posedness, basic reproduction

numbers and the existence of equilibria of the proposed model (3.3.1)).

Theorem 3.3.1. All the solutions of the system (3.3.1) are non-negative under the given

initial conditions

Proof. Consider the following initial conditions:

x(φ) ≥ 0, y(φ) ≥ 0, z(φ) ≥ 0, v(φ) ≥ 0, w(φ) ≥ 0, φ ∈ [−τ, 0]. (3.3.2)

50

for the system (3.3.1), where(

x(φ), x(φ), x(φ), x(φ), x(φ))∈ X and the description

of X = C([−τ, 0]; R5) is given earlier.

The fundamental theory of functional differential equations (see, e.g. [69]) give

the existence of solution which is unique. Using constant of variation formula, we

get the following solutions of the system (3.3.1).

x(t) = e−∫ t

0 (d+βv(ζ))dζ x(0) + λ∫ t

0βe−aτx(t − τ)v(t − τ)e

−∫ t

η (d+βv(ζ))dζdη,

y(t) = y(0)e−∫ t

0 (a+αz(ζ))dζ +∫ t

0

(βe−aτx(t − τ)v(t − τ)

)e−∫ t

η (a+αv(ζ))dζdη,

z(t) = z(0)e−bt +∫ t

0αw(t)y(t)e

−∫ t

η −b(t−ζ)dζdη,

v(t) = v(0)e−pt +∫ t

0ke−p(t−η)dη,

w(t) = w(0)e−qt +∫ t

0cz(η)e−q(t−η)dη.

Which clearly indicates that all the solutions are positive. Let us define the follow-

ing function

B(t) = ckx(t) + cky(t) + ckz(t) +ac

2v(t) +

bk

2w(t). (3.3.3)

Calculating the derivative of equation (3.3.3) and using the system (3.3.1)

d

dtB(t) = ck

(λ − dx(t)− βx(t − τ)v(t − τ)e−aτ

)

+ck(

βe−aτx(t − τ)v(t − τ)− ay(t) − αw(t)y(t))

+ck(

aw(t)y(t) − bz(t))+

ac

2

(ky(t) − pv(t)

)+

bk

2

(cz(t)− qw(t))

= ckλ −(

dckx(t) +a

2cky(t) +

b

2ckz(t) + q

bk

2w(t) + p

ac

2v(t)

)

≤ ckλe−aτ − εB(t).

Here ε = min{d, a2 , b

2 , q, p}. This means that B(t) is bounded. Therefore, all the

solutions are bounded.

51

3.3.2 Determination of Equilibria and Basic Reproductive Num-

ber

The system (3.3.1) has the following three possible biologically meaningful equilib-

ria: disease-free equilibrium [68] E∗(x0, y0, z0, v0, w0), single infection equilibrium

Es(x1, y1, z1, v1, w1) and double infection equilibrium Ed(x2, y2, z2, v2, w2). The value

of each equilibrium point is given below

E0 =

(λ

d, 0, 0, 0, 0

),

Es =

(ap

βke−aτ,

λkβe−aτ − adp

kaβe−aτ, 0,

λkβe−aτ − adp

paβe−aτ, 0

),

Ed =

((αλc + γbq)p

αcdp + βkqbe−aτ,

bq

αc,

q

αc(

ckλαβe−a(τ) − aαcdp − abqkβe−aτ

αcdp + bkqβe−aτ),

kqb

αcp

,αckβλe−a(τ) − aαcdp − abqkβe−aτ

α(αcdp + bkqβe−aτ)

).

The biological interpretation of each equilibrium point can be described as: E0 is

an infection-free equilibrium corresponding to maximal levels of healthy CD+4 T

cells. The second equilibrium Es correspond to positive levels of infected CD+4

T cells. Ed represent the concentration recombinant virus. The basic reproduction

number can be defined as

R0 =kβλe−aτ1

adp,

where the ration of each parameters involved in the reproduction formula has been

described before. If R0 < 1, then E0 exists and if R0 > 1, then Es occurs. But Ed

enters if Rd > 1 , where

Rd =αβλcke−aτ − αcdpa

βbkqae−aτ=

aαcdp

βbkqe−aτ(R0 − 1).

Let Rs = 1 + βbkqe−aτ

αcdp , then Rd > 1 if and only if R0 > Rs.

52

3.3.3 Local Dynamics of the Proposed Model

In this section, we will show the local dynamical behavior of the proposed

model (3.3.1).

Theorem 3.3.2. For R0 < 1, E0 is locally asymptotically stable. When R0 > 1, then E0

becomes unstable and inters into Es .

Proof. : The model (3.3.1) around E0 can be linearized as follows

dx

dt= −dx(t)− βe−aτ λ

dv(t − τ),

dy

dt= βe−aτ λ

dv(t − τ)− ay(t),

dz

dt= −bz(t), (3.3.4)

dv


dw

dt= cz(t)− qw(t).

The characteristic equation corresponding to the Jacobian matrix of linearized sys-

tem (3.3.4) is given by

(b + ρ)(d + ρ)(q + ρ)

[(a + ρ)(p + ρ)−

λ

dβke−τ(ρ+a)

]. (3.3.5)

where ρ stands for an eigne value. The three negative roots of the characteristic

equation (3.3.5) are −b,−d and −q and the remaining two roots can be determined

from the following equation

(a + ρ)(p + ρ) =λ

dβke−τ(ρ+a). (3.3.6)

If ρ has non- negative real part, then modulus of the left hand side of equation

(3.3.6) satisfies

|(a + ρ)(p + ρ)| ≥ ap.

53

While modulus of the right hand side of (3.3.6) gives

λ

dβk|e−τ(ρ+a)| = |apR0 | < ap.

This leads to contradiction. Thus, when R0 < 1, then all the eigne values have

negative real part. Therefore, the infection free state E0 is locally asymptotically

stable. For R0 > 1, we have

g(ρ) = (a + ρ)(p + ρ)−λ

dβke−τ(ρ+a).

Now g(0) = ap(1 − R0) < 0 and limρ→∞ g(ρ) = +∞. By the continuity of g(ρ)

there exists at least one positive root of g(ρ) = 0. Thus, the infection-free equilib-

rium E0 is unstable if R0 > 1.

Theorem 3.3.3. If 1 < R0 < Rs, then the recombinant present equilibrium Es is locally

asymptotically stable while Es become unstable for R0 > Rs.

Proof. The linearized system of (3.3.1) at Es(x1, y1, z1, v1, w1) is given by

dx

dt= −dx(t)− βe−aτ(x1v(t − τ) + v1x(t − τ)),

dy

dt= βe−aτ(x1v(t − τ) + v1x(t − τ)− ay(t) − αy1w(t),

dz

dt= αy1w(t)− bz(t), (3.3.7)

dv


dw

dt= cz(t) − qw(t).

The characteristic equation corresponding to the Jacobian matrix of the system

(3.3.7) can be written in simplified form as f1(ρ) f2(ρ) = 0,

54

where

f1(ρ) = ρ2 + (b + q)ρ + bq −cα(λkβe−aτ − adp)

akβe−aτ,

f2(ρ) = ρ3 +

(a + p +

kβλ

ape−aτ

)ρ2 +

[kβλ

ape−aτ(a + p) + ap

]ρ

+kβλe−aτ − a(ρ + d)pe−ρτ .

Now f1(ρ) can be simplified as

f1(ρ) = ρ2 + (b + q)ρ + bq(1 − Rd),

which represents f1(ρ) = 0 has the two roots the real part of each is negative which

is possible only if Rd< 1 (i.e. R0 < Rs), but if for Rd > 1 one root will be positive

and the other one will be nagative. Therefore, if R0 > Rs, then the single infection

equilibrium Es is unstable. After some simplification f2(ρ) = 0, can be written as

ρ3 + a2(τ)ρ2 + a1(τ)ρ + a0(τ)− (c1ρ + c2)e

−ρτ = 0, (3.3.8)

where

a2(τ) = a + p +kβλ

ape−aτ, a1(τ) =

kβλ

ape−aτ(a + p) + ap,

a0(τ) = kβλe−aτ , c1 = ap, c2 = apd.

ρ = 0 is not a root of (3.3.8) if R0 > 1, since

a0(τ)− c2 = kβλe−aτ − apd = apd(R0 − 1) > 0.

When τ = 0, then (3.3.8) becomes

ρ3 + a2(0)ρ2 + (a1(0)− c1)ρ + a0(0)− c2 = 0. (3.3.9)

55

We know that all the roots of the equation (3.3.9) have negative real parts (see [67]),

as

a2(0) = a + p +kβλ

ap> 0,

a1(0)− c1 =kβλ

ap(a + p) > 0,

a0(0)− c2 = apd(R0 |τ=0 − 1) > 0.

Similarly,

a2(0)(a1(0)− c1)− (a0(0)− c2) =k2β2λ2

(a2 p2(a + p) +

kβλ

ap(a + p)2 + apd > 0.

Hence, for τ = 0, any root of (3.3.8) will have negative real part. Now, we consider

the distribution of the roots when τ > 0. Let ρ = iν(ν > 0) be the purely imaginary

root of (3.3.8), then

−iν3 − a2(τ)ν2 + ia1(τ)ν + a0(τ)− (ic1ν + c2)e

−iντ = 0.

The modula of the above equation results in

G(ν2) = ν6 + [a22(τ)− 2a1(τ)]ν

4 + [a21(τ)− 2a0(τ)a2(τ)− c2

1]ν2 + a2

0(τ)− c22 = 0.

(3.3.10)

The value of each co-efficient can be determined as follows:

a22(τ)− 2a1(τ)] = a2 + p2 + d2R2

0 > 0,

a21(τ)− 2a0(τ)a2(τ)− c2

1 = d2[(a2 + p2]R20 > 0,

a20(τ)− c2

2 = a2 p2d2(R20 − 1) > 0.

Thus it can be concluded that for τ > 0, all the roots of (3.3.10) have negative real

parts if R0 > 1.

56

3.3.4 Global Dynamics of the Proposed Model

In this section, we will study the global stability of equilibria of the system

(3.3.1), by using suitable Lyapunov functionals and LaSalle’s invariant principle.

Theorem 3.3.4. When R0 < 1, then the disease-free equilibrium E0 is globally asymptot-

ically stable.

Proof. Construct the Lyapunov functional

V0(t) =1

2(x(t)−

λ

d)2 +

λ

dy(t) +

λ

dz(t) +

aλ

kdv(t) +

bλ

cdw(t)

+βλ

de−aτ

∫ t

t−τx(ζ)v(ζ)d(ζ), (3.3.11)

where V0(t) stands for Lyapunov functional at E0. The derivative of (3.3.11) and

the use of (3.3.1), yield the following equation

dV0

dt= (x(t)−

λ

d)

(λ − dx(t)− βe−aτx(t − τ)v(t − τ)

)

+λ

d

(βe−aτx(t − τ)v(t − τ)− ay(t) − αw(t)y(t)

)

+λ

d

(αw(t)y(t) − bz(t)

)+

aλ

kd

(ky(t) − pv(t)

)

+bλ

cd

(cz(t)− qw(t)

)+

βλ

de−aτ

∫ t

t−τx(ζ)v(ζ)d(ζ).

After further simplification, the above equation becomes

dV0

dt= −(x(t)−

λ

d)

((x(t) −

λ

d) + βe−aτx(t − τ)v(t − τ)

)

−apλ

dk(

kβλe−aτ

adp− 1)v(t) −

qbλ

cdw(t)

= −(x(t)−λ

d)

((x(t) −

λ


)

−apλ

dk(1 − R0)v(t)−

qbλ

cdw(t). (3.3.12)

57

Thus, when R0 < 1, then equation (3.3.12) implies that dV0dt < 0. Therefore, by (see

[70]), we have E0 is globally asymptotically stable when R0 < 1.

Theorem 3.3.5. For 1 < R0 < Rs, the single infection equilibrium Es is globally asymp-

totically stable.

Proof Let us construct the Lyapunove functional

Vs(t) = (x − x1 ln x) + (y − y1 ln y) + z +a

k(v − v1 ln v) +

b

cw

+x1v1βe−aτ

t∫

t−τ

(x(θ)v(θ)

v1x(θ + τ)− ln x(θ)v(θ)

)dθ, (3.3.13)

where Vs(t) stands for Lyapunov functional at single infection equilibrium Es. The

derivative of equation (3.3.13) yields

dVs

dt= (1 −

x1

x)x + (1 −

y1

y)y + z +

a + γ

k(1 −

v1

v)v +

b

cw

+ x1v1βe−aτ

(x(t)v(t)

x(t + τ)v1

)−

(x(t − τ)v(t − τ)

x(t)v1− ln(x(t)v(t))

+ ln(x(t − τ)v(t − τ))

),

= (1 −x1

x)

(λ − dx(t)− βe−aτx(t − τ)v(t − τ) + γy(t)

)

+ (1 −y1

y)

(βe−aτx(t − τ)v(t − τ)− (a + γ)y(t) − αw(t)y(t)

)

+ αw(t)y(t) − bz(t) +a

k(1 −

v1

v)(ky(t) − pv(t)

)+

b

c

(cz(t)− qw(t)

)

+ x1v1βe−aτ

(x(t)v(t)

x(τ + t)v1−

x(t − τ)v(t − τ)

x(t)v1+ ln

x(t − τ)v(t − τ))

(x(t)v(t)

).

(3.3.14)

The proposed model at single infection equilibrium Es(x1, y1, z1, v1, w1), gives the

following identities: λ = dx1 + βe−aτx1v1, βe−aτx1v1 = ay1, and ky1 = pv1. Fur-

ther, if τ is very large, i.e, when delay in contact of targeted cells with virus and

58

the latent period is very large, then the rate of infection will be very small and con-

trarily if τ is very small, then the infection will spread more rapidly. Therefore, we

suppose that time delay is very large, then

limτ→∞

(x(t + τ)) = x(t).

By considering the above identities and assumption, equation (3.3.14) becomes

dVs

dt= dx1

(2 −

x

x1−

x1

x

)+ βe−aτx1v1

(3 −

x1

x−

yv1

y1v−

y1x(t − τ)v(t − τ)

yx1v1

+ lnx(t − τ)v(t − τ)

xv

)+

αdp

βk(R0 − Rs)w(t). (3.3.15)

The following inequalities hold (see [42])

2 −x

x1−

x1

x≤ 0,

(3 −

x1

x−

yv1

y1v−

y1x(t − τ)v(t − τ)

yx1v1+ ln

x(t − τ)v(t − τ)

xv

)≤ 0.

By using the above inequalities, equation (3.3.15) implies that dVsdt < 0 when R0 <

Rs. Then,[70], one can get Es to be globally asymptotically stable.

3.3.5 Numerical Results

In this section, we illustrate the theoretical results obtained in previous sec-

tions numerically. We discuss some numerical results and simulations. It is clear

from the derived results that time delays play a major role in determining the dy-

namic behavior of the HIV-1 modeling. The time delay can change the dynamical

behavior quantitatively. For numerical simulation, we have taken some of the val-

ues estimated and some from available literature in experimental basis. Let λ = 2 (

59

Density of CD4+ T cells in the healthy human blood is X = 1000cell/mm3 [71]. As-

sumed equilibrium, their production λ equal loss λ = Xd. Assumed that a fraction

µ = 0.2 of new generated cells are activated λ = λµ = 2 [72]). d = 0.01(Average

life span of CD4+ T cell is two years, so d = 0 : 0014 [71]. From modeling, d = 0.01

[73]). β = 0.004mm3/vir Assumed indirectly as a small value that preserves both

infections. For single infection β = 0 : 00027 [71], β = 0 : 00065 [73]). a = 0.5

((Based on life span of HIV-1 infected cells of three days [? ]. Also, Other esti-

mates: a = 0.49 [37], a = 0.39 [71])). α = 0.004 (Estimated indirectly as a small

value that preserves the double infection. Taken identical to β). b = 2 (Based

on observations of virus release within 8 h of infection before lysis [74]). p = 3

(Based on life span of 1/2 day [71]. Another value, p = 3 [37]). k = 50vir/cell

(k = n1a. n1 is total number of infectious HIV-1 produced by a cell: n1 ∼ 140 [75]).

c = 2000vir/cell (c = n2b. n2 is total number of infectious recombinant produced

by a double-infected cell. In vitro total number of recombinant per cell is ∼ 3333

[74]. Assumed n2 = 1000). q = p ( estimated identical to p). τ = 1.0 ∼ 1.5 days

Estimated).

Fig 3.1 shows the simulation of the system (3.3.1) at τ = 1.5 and represents

convergence to the stable equilibrium Es. If we decrease the value further, that

is, τ = 0.7, then Es will lose its stability and the double-infection equilibrium E2

will occur, which is shown in Fig 3.2. Simulation of the system (3.3.1) for τ = 0.4

are shown in Fig 3.3. Comparing the results in Fig 3.3 with that in Fig 3.1 shows

that the solution trajectory takes longer to converge to its steady-state value, and

they take long time to enters into E2. Also, it can be noted that the amplitudes

of the oscillations increases. Therefore, the incorporation of small delay in model

60

0 100 200 300 400 50010

20

30

40

50

60

70Virus−free host cells

time t

x(t)

(a) y(t) (density of infected cells) verses t in days

0 100 200 300 400 5000

1

2

3

4

5

6

7

8

9Infected cells

time t

y(t)

(b) x(t) (density of uninfected cells) verses t) indays

0 100 200 300 400 5000

1

2

3

4

5

6

7Double−infected cells

time t

z(t)

(c) z(t) (density of double infected cells) verses tin days

0 100 200 300 400 5000

20

40

60

80

100

120

140Pathogen virus

time t

v(t)

(d) v(t) (density of pathogen virus) verses t indays

0 100 200 300 400 5000

500

1000

1500

2000

2500

3000

3500Recombinant (genetically modified) virus

time t

w(t

)

(e) w(t) (density of recombinant virus) verses tin days

Figure 3.2: The dynamics of the system (3.3.1) for τ = 1.5.

61

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

7

8Infected cells

time t

y(t)

(a) y(t) (density of infected cells) verses t in days

0 100 200 300 400 500 600 700 80010

15

20

25

30

35

40


time t

x(t)


0 100 200 300 400 500 600 700 8000

1

2

3

4

5


time t

z(t)


0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120Pathogen virus

time t

v(t)


0 100 200 300 400 500 600 700 8000

500

1000

1500

2000

2500


time t

w(t

)



62

0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

7Infected cells

time t

y(t)

(a) y(t) (density of infected cells) verses t in indays

0 200 400 600 800 1000 1200 140010

15

20

25

30


time t

x(t)


0 200 400 600 800 1000 1200 14000

1

2

3

4

5


time t

z(t)


0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120Pathogen virus

time t

v(t)


0 200 400 600 800 1000 1200 14000

500

1000

1500

2000

2500


time t

w(t

)



63

(3.3.1) can produce changes in its solutions. This significance of delay can not be

seen from the non-delayed model. Hence, time delays are very important for the

modeling of HIV-1 infection and cannot be ignored.

3.4 Analysis of double delayed HIV-1 model

In the previous model, the delayed HIV-1 infection model was consid-

ered for time delay incorporated in contact term. In this section, a double delayed

HIV-1 infection model will be considered. The new incorporated time delay τ2

shows time lag in the production of virus from the infected cells as time duration

is needed for the generation of new virus from the infected cells. By incorporating

the mentioned time delay, the new model is formulated as follows:

dx

dt= λ − dx(t)− βx(t)v(t),

dy

dt= βe−aτ1 x(t − τ1)v(t − τ1)− ay(t) − αw(t)v(t),

dz

dt= aw(t)y(t) − bz(t), (3.4.1)

dv

dt= ke−aτ2 y(t − τ2)− pv(t),

dw

dt= cz(t)− qw(t),

where all the parameters and variables have been defined in the section (3.4.1)

and time delay τ2 is the duration for the production of new virus from the infected

cells.

3.4.1 Positivity and Well-Posedness of the Solution

In this section, we discuss the positivity and well-posedness of the solution.

64

Theorem 3.4.1. The system (3.4.1) has non-negative solution if the given conditions are

non-negative and bounded.

Proof. Consider the initial conditions (x(φ), y(φ), z(φ), v(φ), w(φ)) ∈ X

x(φ) ≥ 0, y(φ) ≥ 0, z(φ) ≥ 0, v(φ) ≥ 0, w(φ) ≥ 0, φ ∈ [−max(τ1, τ2), 0], (3.4.2)

The fundamental theory of functional differential equations [64] can be used to

prove existence of the uniqueness of the solution of the model. The positivity of

the solutions y(t), z(t), w(t) have been proved in section (3.4.2) and the positivity

of the solutions x(t) and v(t) of can be determined as follows:

x(t) = x(0)e−∫ t

0 (d+βv(ζ))dζ + λ∫ t

0e−∫ t


v(t) = v(0)e−pt +∫ t

0ke−aτ2 y(t − τ2)e

−p(t−η)dη,

w(t) = w(0)e−kt +∫ t

0cz(η)e−k(t−η)dη.

It can be seen that each solution is positive. Let us define the following functional

G⋆(t) = cke−aτ1 x(t − τ1) + cky(t) + ckz(t) +ac

2eaτ2v(t + τ2)

+bk

2w(t). (3.4.3)

65

Calculating the derivative of the equation (3.4.3)

dG⋆

dt= cke−aτ1

(λ − dx(t − τ1)− βx(t − τ1)v(t − τ1)

)

+ ck

(βe−aτ1 x(t − τ1)v(t − τ1)ay(t) − αw(t)v(t)

)

+ ck

(aw(t)y(t) − bz(t)

)+

ac

2eaτ2

(ke−aτ2 y(t)− pv(t + τ2)

)

+bk

2

(cz(t)− qw(t)

),

= ckλe−aτ1 −

(dcke−aτ1 x(t − τ1) +

a

2cky(t) +

b

2kcz(t) + q

bk

2w(t)

+ pac

2eaτ2 v(t + τ2)

)

≤ ckλe−aτ1 − µG⋆(t).

Here µ = min{d, a2 , b

2 , q, p}. Thus G⋆(t) is bounded. Hence the solutions are

bounded.

The system (3.4.1) has three equilibria: disease-free equilibrium E1(x0, y0, z0, v0, w0),

which has been found in section (3.2.2). The single-infection equilibrium Es(x1, y1, z1, v1, w1)

and double-infection equilibrium E2(x2, y2, z2, v2, w2), are given below:

Es =( ape−a(τ1+τ2)

βk,

kβλe−a(τ1+τ2) − adp

aβke−aτ2, 0,

kβλe−a(τ1+τ2) − adp

apβ, 0).

E2 = (αλcp

αcdp + βkqbe−aτ2,

qb

αc,

q(kβλae−a(τ1+τ2) − adp)

apβ,

kqbe−aτ2

αcp

,αcβkλe−a(τ1+τ2) − αcadp − abqkβe−aτ2

αcdp + bkqβe−aτ2).

We define (see [42]),

R0 =λβke−a(τ1+τ2)

apd

66

If R0 < 1, then E1 is the only equilibrium which is biologically meaningful. But if

R0 > 1, then there is another equilibrium Es (single-infection equilibrium), which

is biologically interpretable. By simple calculation, it can be shown that E2 exists if

and only if Rd > Rs , where

Rs =αcdp

βbkq(R0 − 1).

Hence, Rd > 1 if and only if R0 > Rs, where Rs = 1 +βbkqαcdp .

To find the local stability of the equilibria, we find the characteristic equation

det[ρI − J(E2)] corresponding to the system (3.4.1) at any equilibrium point as

det

−ρ − (d + βv) 0 0 −βx 0

−βe−τ1(η+a)v −ρ − a 0 −βe−τ1(η+a)x + αw −αw

0 αw −ρ − b 0 αy

0 ke−τ2(η+a) 0 −ρ − p 0

0 0 c 0 −ρ − q

= 0.

3.4.2 Stability of the Disease-Free Equilibrium

In this section, we show the dynamical behavior of the system (3.4.1) at E1.

Theorem 3.4.2. When R0 < 1, E1 is locally asymptotically stable; while for R0 < 1, E1

is unstable and the single-infection equilibrium Es occurs.

Proof. The characteristic equation for the Jacobian matrix corresponding to system

(3.4.1) at E1(x0, y0, z0, v0, w0) is

det[ρI − J(E1)] = (b+ η)(d+ η)(q+ η)

[(a + η)(p + η)−

λ

dβke−τ1(η+a)e−τ2(η+a)

]= 0.

The first three factors of the above characteristic equation has the following roots

η1 = −b and η2 = −d and η3 = −q and the remaining two roots can be determined

67

by using

(a + η)(p + η) =λ

dβke−(τ1+τ2)(η+a). (3.4.4)

If η has non negative real part then modulus of the left hand side of (3.4.4) is sim-

plified to the following equation

|(a + η)(p + η)| ≥ ap.

While modulus of the right hand side of (3.4.4) satisfies

λ

dβk|e−(τ1+τ2)(η+a)| = |apR0| < ap.

This leads to contradiction. Thus, when R0 < 1, then all the eigne values have

negative real part, and hence the infection free state E1 is locally asymptotically

stable provided that τ1 ≥ 0 and τ2 ≥ 0.

When R0 > 1, we have

g(η) = (a + η)(p + η)−λ

dβke−(τ1+τ2)(η+a).

Now g(0) = ap(1 − R0) < 0 and limη→∞ g(η) = +∞. By the continuity of g(η)

there exists at least one positive root of g(η) = 0. Thus, the infection-free equilib-

rium is unstable if R0 > 1.

Theorem 3.4.3. When R0 < 1, then E1 will be globally asymptotically stable provided

provided that e−aτ2 < 1.

Proof. Let us consider the following Lyapunov functional

V0(t) = 2e−aτ1(x(t)−λ

d)2 +

λ

d)2y(t) +

λ

dz(t) +

λ

dv(t) +

aλ

kdw(t)

+N6

∫ t

t−τ1

x(ζ)v(ζ)d(ζ) + N7

∫ t

t−τ2

y(ζ)dζ.

68

Taking derivative of equation (3.4.5) using (3.4.1), we have

dV0

dt= 2e−aτ1

(x(t)−

λ

d)2(λ − dx(t)− βx(t)v(t)

)+

λ

d


−ay(t) − αw(t)v(t))+

λ

d

(w(t)y(t) − bz(t)

)

+aλ

kd

(ke−aτ2 y(t − τ2)− pv(t)

)+

bλ

cd

(cz(t)− qw(t)

)

+βλ

d

∫ t

t−τ1

x(ζ)v(ζ)d(ζ) +aλ

de−aτ2

∫ t

t−τ2

y(ζ)dζ. (3.4.5)

After some simplification, the equation (3.4.5) becomes

dV0

dt= −e−aτ1(x(t)−

λ

d)2(d + βv(t)) −

qbλ

cdw(t)

−apeaτ2 λ

dk

(1 −

kβλe−a(τ1+τ2)

adp

)v(t) +

aλ

d(e−aτ2 − 1)y(t),

= −e−aτ1(x(t)−λ

d)2(d + βv(t)) −

qbλ

cdw(t)−

apeaτ2 λ

dk(1 − R0)v(t)

−aλ

d(1 − e−aτ2)y(t).

Thus dV0dt ≤ 0, when R0 < 1 and e−aτ2 < 1, which is possible only if τ2 is very

large. But the equality will be satisfied if x0 = λd , y(t) = 0, z(t) = 0, v(t) = 0,

w(t) = 0. Then, by LaSalli invariance principle (see [70]), we conclude that E1 is

globally asymptotically stable when R0 < 1 and τ2 is very large.

3.4.3 Stability of Single Infected Equilibrium

Here, we discuss the single infected-free equilibrium Es.

Theorem 3.4.4. When 1 < R0 < Rs the single infection-free equilibrium Es is locally

asymptotically stable; while for R0 > Rs, Es loses stability and genetically modified virus

may persist.

69

Proof. The characteristic equation corresponding to the Jacobian matrix of the sys-

tem (3.4.1) at Es is given by

det[ρI − J(E)] = ((b + η)(q + η)− αy1c)

[(d + βv1 + η)kβx1e−(τ1+τ2)(a+η)

−β2x1kv1e−(τ1+τ2)(a+η) − (d + βv1 + η)(a + η)(b + η)

]= 0.

We can write equation (3.4.6) in form f1(η) f2(η) = 0, where

f1(η) = (b + η)(q + η)− αy1c.

f2(η) = (d + βv1 + η)kβx1e−(τ1+τ2)(η+a) − β2x1kv1e−(τ1+τ2)(η+a)

−(d + βv1 + η)(a + η)(b + η).

Now f1(η) can be written as

f1(η) = η2 + (b + q)ρ + bq(1 − Rd),

which shows that f1(η) = 0 has the roots the real part of which are negative if and

only if Rd < 1. The single infection free equilibrium E1 is unstable. Also f2(η) = 0

can be written as

η3 + c2(τ1 + τ2)η2 + c1(τ1 + τ2)η + c0(τ1 + τ2)− (m1η + m2)e

−η(τ1+τ2) = 0. (3.4.6)

where

c2(τ1 + τ2) = a + p + d +dp

ke−aτ2(R0 − 1),

c1(τ1 + τ2) = (a + p)(d +dp

ke−aτ2(R0 − 1),

c0(τ1 + τ2) = ap(d + (dp

ke−aτ2)(R0 − 1)),

m1 = ap,

m2 = apd.

70

Since the stability analysis of the HIV-1 infection model depends on the time delays

τ1 and τ2. Therefore, we will consider different values of time delays. When τ1 =

τ2 = 0, then (3.4.6) after some simplification can be written as

η3 + c2(0)η2 + (c1(0)− m1(0))η + c0(0)− m2(0) = 0. (3.4.7)

The (3.4.7) has negative real part if R0 > 1 (see [67]). This can be proved by using

the values of c0, c1, c2, m1 and m2.

c2(0) = a + p + d +pd

k((R0)τ1=τ2=0 − 1) > 0,

c1(0)− m1(0) = (a + p)(d +dp

k((R0)τ1=τ2=0 − 1) > 0,

c0(0)− m2(0) = ap(d +dp

k((R0)τ1=τ2=0 − 1) > 0.

Similarly.

c2(c1 − m1)− (c0 − m2) = (d +dp

k((R0)τ1=τ2=0 − 1)2(a + p)) + (a + p)2

×(d +dp

k

((R0)τ1=τ2=0 − 1

)+ apd + ap(a + p) > 0.

Therefore, (3.4.7) has negative real part roots when τ1 = τ2 = 0. Now we consider

the case of the roots when τ1 = τ2 6= 0. Let us suppose that iν(ν > 0) be the roots

of (3.4.6), then separating real and imaginary parts, it follows that

c0 − c2nu2 = m2 cos(τ1 + τ2)− m1 sin(τ1 + τ2),

c1ν − nu3 = wm1 cos(τ1 + τ2) + m2 sin(τ1 + τ2).

By squaring and adding the above equations, we get

ν6 + (c22 − 2c1)ν

4 + (c21 − 2c0c2 − m2

1)ν2 + (c2

0 − m22) = 0. (3.4.8)

71

Let us suppose r = ν2> 0, then (3.4.8) becomes

r3 + (c22 − 2c1)r

2 + (c21 − 2c0c2 − m2

1)r + (c20 − m2

2) = 0,

where

c22 − 2c1 = (d + βv1)

2 + a2 + p2> 0,

c21 − 2c0c2 − m2

1 = (d + βv1)2(a2 + p2) > 0,

c20 − m2

2 = a2 p2β2v21 + 2βd

dp

ke−aτ2(R0 − 1)a2 p2

> 0,

(c22 − 2c1)(c

20 − m2

2)− (c20 − m2

2) = (a2 + p2)(d + βv1)4 + (a4 + p4)(d + βv1)

2

+(apβv1)2> 0.

Therefore, [67]) implies the roots of (3.4.8) have negative real parts for τ1 = τ2 6= 0

if R0 > 1.

Theorem 3.4.5. If 1 < R0 < Rs, provided that e−aτ2 = 1, then the single infection free

equilibrium Es is globally asymptotically stable.

Proof. The lyapunove functional theory is used to prove the required result. There-

fore, the following lyapunove functional is constructed

Vs(t) = W1(t) + βx1v1e−aτ1W2(t) + ae−aτ2W3(t). (3.4.9)

where

W1(t) = e−aτ1(x(t)− x1 ln x(t)) + (y(t) − y1 ln y(t) + z +a

k(v(t) − v1 ln v(t))

+b

cw(t).

W2(t) =∫ t

t−τ1

(x(ξ)v(ξ)

x1v1− ln(

x(ξ)v(ξ)

x1v1))dξ.

W3(t) =∫ t

t−τ2y(ξ)dξ.

72

By taking derivative of (3.4.9), we get

dVs

dt= W1(t) + βx1v1e−aτ1W2(t) + ae−aτ2W3(t). (3.4.10)

Now taking derivative of W1, W2 and W3, we have

dW1

dt= e−aτ1(1 −

x1

x)x. + (1 −

y1

y)y. + z. +

a

k(1 −

v1

v)v. +

b

cw.

dW2

dt=

(x(t)v(t)

x1v1− ln(

x(t)v(t)

x1v1)−

x(t − τ1)v(t − τ1)

x1v1+ ln(

x(t − τ1)v(t − τ1)

x1v1)

).

dW3

dt= y(t)− y(t − τ2).

Then equation (3.4.10) becomes

dVs

dt= e−aτ1(1 −

x1

x)

(λ − dx(t)− βx(t)v(t)

)+ (1 −

y1

y)

(βe−aτ1 x(t − τ1)v(t − τ1)

−ay(t) − αw(t)v(t)

)+


)+

a

k(1 −

v1

v)

(ke−aτ2 y(t − τ2)

−pv(t)

)+

b

c

(cz(t)− qw(t)

)+ βx1v1e−aτ1

(x(t)v(t)

x1v1)− ln(

x(t)v(t)

x1v1)

−x(t − τ1)v(t − τ1)

x1v1+ ln(

x(t − τ1)v(t − τ1)

x1v1

)

+ae−aτ2(y(t)− y(t − τ2)

). (3.4.11)

Next, substituting the single infection equilibrium E(x1, y1, z1, v1, w1) in the system

(3.4.1), we obtain the following identities

λ = dx1 − βx1v1,

βe−aτ1 x1v1 = ay1,

ke−aτ2 y1 = pv1.

73

Now using the above equations in equation (3.4.11), we get after some simplifica-

tion

dVs

dt= e−aτ1(2 −

x1

x−

x


(3 −

x1

x−

y1x(t − τ1)v(t − τ1)

x1v1y

+ ln(x(t − τ1)v(t − τ1)

x1v1)−

v1y(t − τ2)

vy1

)−

αdp

aβkeατ2(Rs − R0)

−a(1 − e−aτ2)y(t). (3.4.12)

Let us suppose that e−aτ2 = 1, then equation (3.4.12) becomes

dVs

dt= e−aτ1(2 −

x1

x−

x


(3 −

x1

x−

y1x(t − τ1)v(t − τ1)

x1v1y

+ ln(x(t − τ1)v(t − τ1)

x1v1−

v1y(t − τ2)

vy1

)−

αdp

aβk(Rs − R0). (3.4.13)

By using, the following inequalities, (see [42])

e−aτ1(2 −x1

x−

x

x1) ≤ 0,

(3 −

x1

x−

y1x(t − τ1)v(t − τ1)

x1v1y + ln(

x(t − τ1)v(t − τ1)

x1v1)−

v1y(t − τ2)

vy1

)≤ 0,

Therefore, (3.4.13) implies that dVsdt ≤ 0, when R0 ≤ Rs. Also, by LaSalles invari-

ance principle [70], we get E2 is globally asymptotically stable.

3.5 Conclusion

Different delayed HIV-1 infection models were studied. It has been shown

that our first proposed delayed HIV-1 infection model with time delays has three

equilibrium solutions: E0, E1, and E2. Also, it has been shown that a series of bifur-

cations occur as R0 is increased. It is noted that to reduce the density of pathogen

virus, one must have the value of R0 to below one. From the derived formula of

74

R0, one can see that the value of R0 can be reduced by increasing either of the two

time delays. It is observed that if all the parameters values are kept fixed, the larger

value of τ can reduce the value of R0 to below one. Form which one can conclude

the infection free equilibrium to be globally stable. It has been shown that E1 is

locally as well as globally asymptotically stable for R0 ∈ (0, 1), and loses stability

at R0 = 1, and bifurcates into E1, which is stable for R0 ∈ (1, Rs). First a delayed

HIV-1 infection model with CTLs response is discussed. Dynamical behavior of the

proposed model shows that time delays play an important role in the stability of

the HIV-1 model. The detailed analytic study has shown that the proposed model

in this study also has three equilibrium solutions. E1, E1, and E2. It has shown

that E1 is globally asymptotically stable for R0 ∈ (0, 1) and unstable at R0 = 1

and bifurcates into globally stable equilibrium E1,for R0 > 1. However, it losses it

stability at an other bifurcation point R0 > 1 + bβke−aτ1

cdu and E2 occurs. Also, it has

been shown that E2 is also globally stable. Finally, through numerical simulations,

it can be concluded that time delays in the infection process and virus produc-

tion period play an important role in the disease control. Time delays may change

dynamic behavior quantitatively and qualitatively even in the normal range of

values. Therefore, time delay is a very important fact and cannot be ignored for

reducing the infection of HIV-1. The second proposed model is proposed for intro-

ducing time delays in contact process and infection process. Then, this model has

been studied for local and global stability in detail. It is proved that large time de-

lays in the mentioned processes can reduce the density of infected cells in the body.

The third proposed HIV-1 infection model was considered for the introduction of

fixed time delays accounting for the time between viral entry into a target cell and

75

infection of cells and the time between infection of a cell and the emission of viral

particles. It was shown that if the basic reproduction ratio R0 is less than unity, the

infection-free equilibrium is globally asymptotically stable using the correspond-

ing characteristic equations. If the basic reproduction ratio R0 is greater than unity,

then the chronic-infection equilibrium exists and is locally asymptotically stable.

The global stability of the infection-free equilibrium followed if the virus produc-

tion delay is very large and R0 < 1. Similarly chronic-infection equilibrium of

proposed system has been completely established by using the Lyapunov-LaSalle

type theorem and proved that for 1 < R0 < Rs, the chronic-infection equilibrium

is globally asymptotically stable provided that e−a2τ2 = 1. Our results show that

globally stability of the proposed system depends on the virus production delay.

When there is large time in virus production, then no pathogen virus will be cre-

ated into the cells and no further infection will be spread.

Chapter 4

Dynamics of Continuous DelayedHIV-1 Infection Models

4.1 Overview

In this chapter, the analysis of continues delayed HIV-1 infection models

will be discussed. Two controls which measure the efficacy of reverse transcriptase

and protease inhibitors, respectively are also used. The time delay in first proposed

HIV-1 infection model accounts for latent period for cell infection. The time delay

in the second proposed HIV-1 infection model explains the average time required

for the the production of new viruses after the penetration of pathogen viruses into

the cell and the production of new viruses. The positivity and well posedness of

the two proposed models will be discussed. The basic reproduction numbers will

be found. Further, the local behavior of this model is found using Routh Hurwits

Criterion and global behavior is determined by using Lyapunov functional the-

ory. Numerical simulations will be carried out to support the derived theoretical

results.

76

77

4.2 Asymptotic Properties of Single (Continuous) De-

layed HIV-1 Models with Inhibiters

The HIV-1 infection model with distributed continuous intracellular

delays and the measure of the efficacious of the protease inhibiter and the reverse

transcriptase inhibiter will be presented. In our proposed model the delay term

represents the latent period. The dynamics is represented by the following system

of delay differential equations

dx

dt= λ − dx(t)− (1 − nrt)βx(t)v(t),

dy

dt= (1 − nrt)β

∫ ∞

0e−aτ f1(τ)x(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t),

dz

dt= aw(t)y(t) − bz(t), (4.2.1)

dv

dt= (1 − np)ky(t) − pv(t),

dw

dt= cz(t)− qw(t).

The measure of the efficacious of the protease inhibiter and the reverse transcrip-

tase inhibiter are denoted by np and nrt, respectively. Also, it is assumed that the

infected cells becomes productively infected, τ units latter, where τ is distributed

according to the probability distribution f (τ). The recruitment of virus producing

cells at time t is given by the number of the cells that was infected at time t − τ

and are still alive at time t. Here a is constant death rate for infected cells but

yet not virus producing cells. Therefore, e−aτ is the probability of surviving the

time period from t − τ to t. We may propose our probability distribution functions

like f (τ) = δ(t − τ). The remaining variables and parameters are discussed in

the section (3.4.1). After some manipulation our proposed model in general form

78

becomes

dx


dy

dt= β

∫ ∞

0e−aτ f1(τ)x(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t),

dz

dt= aw(t)y(t) − bz(t), (4.2.2)

dv

dt= ky(t)− pv(t),

dw

dt= cz(t) − qw(t),

where β = (1 − nrt)β and k = (1 − np)k. In the system (4.2.2), the delay kernel is

assumed to be piecewise continuous to satisfy∫ ∞

0 f (τ)dτ = 1 and∫ ∞

0 τ f (τ)dτ < 1. Let the initial conditions for the system (4.2.2) to be supposed in

the form

x(ζ) = ψ1(ζ), y(ζ) = ψ2(ζ), z(ζ) = ψ3(ζ), v(ζ) = ψ4(ζ),

w(ζ) = ψ5(ζ), ζ ∈ [−∞, 0], (4.2.3)

where (ψ1, ψ2, ψ3, ψ4, ψ5) ∈ C[(−∞, 0), R5+0] is the space of continuous functions

mapping the interval (−∞, 0] into R5+0, where ψi(ζ) ≥ 0, i = 1, 2, , ..., 5 and R5

+0 =

{(x1, x2, x3, x4, x5); xi ≥ 0, i = 1, 2, ..., 5}. [76].

4.2.1 Existence, Uniqueness and Non-negativity of the Solution

In this section, we will discuss uniqueness and non-negativity and bounded-

ness of the solution. It is well known by the fundamental theory of functional dif-

ferential equations [64], that the system (4.2.2) admits a solution (x(t), y(t), v(t), z(t)

and w(t) which is unique and satisfying the conditions (4.2.3). It is easy to show

79

that all solutions of the system (4.2.2) are defined on [0,+∞) and remain posi-

tive for all t ≥ 0. The following theorem gives boundedness and positivity of the

solution.

Theorem 4.2.1. The solutions of the system (4.2.2) are non-negative, when the initial

conditions are bounded and non-negative .

Proof. We obtain the following solution of the system (4.2.2), by using constant of

variation formulae

x(t) = x(0)e−∫ t

0 (d+βv(η))dη + λ∫ t

0e−∫ t


y(t) = y(0)e−∫ t

0 (a+αw(ζ))dζ + β∫ t

0e−∫ t

η (a+αw(ζ))dζ∫ ∞

0e−aτ f1(τ)x(t − τ)v(t − τ)dτdη,

z(t) = z(0)e−bt +∫ t

0αw(t)y(t)e

−∫ t

η b(t−ζ)dζdη,

v(t) = v(0)e−pt + k∫ t

0e−p(t−η)y(t)dη,

w(t) = w(0)e−kt + c∫ t

0z(η)e−k(t−η)dη.

Which shows the positivity of the solutions x(t), y(t), v(t), z(t) and w(t). Next, we

show the boundedness of the solution. To do this, we define

H⋆(t) = ck∫ ∞

0e−aτ f (τ)x(t − τ)dτ + cky(t) + ckz(t) +

ac

2v(t) +

bk

2w(t).

Calculating the derivative of the above equation and using the system (4.2.2), we

80

have

H⋆(t) = ck∫ ∞

0e−aτ f (τ)

(λ − dx(t − τ)− βx(t − τ)v(t − τ)

)dτ

+ck

( ∫ ∞

0e−aτ f (τ)βx(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t)

)

+ck


)+

ac

2

(ky(t)− pv(t)

)+

bk

2

(cz(t)− qw(t)

)

= λck∫ ∞

0e−aτ f (τ)x(t − τ)dτ −

[dck

∫ ∞

0e−aτ f (τ)x(t − τ)dτ +

a

2cky(t)

+b

2ckz(t) + q

bk

2w(t)dτ + p

ac

2v(t)

]

≤ λck∫ ∞

0e−aτ f (τ)dτ − χH⋆(t)

< 0, for H⋆(t) >λck

∫ ∞

0 e−aτ f (τ)dτχ

> 0, for H⋆(t) <λck

∫ ∞

0 e−aτ f (τ)dτχ ,

where χ = min{d, a2 , b

2 , q, p}. This implies that H⋆(t) is bounded. Thus, all the cells

x(t), y(t), v(t), z(t) and w(t) are bounded. In order analyze the proposed model,

we use stability analysis theory. The model (4.2.2) has three possible equilib-

ria, disease-free equilibrium E0(x0, y0, z0, v0, w0). The single-infection equilibrium

Es(x1, y1, z1, v1, w1) and double-infection equilibrium Ed(x2, y2, z2, v2, w2), which

are given below

E0 =

(λ

d, 0, 0, 0

),

Es =

(ap

kβM(τ),

λβkM(τ)− apd

aβk, 0,

λβkM(τ)− apd

aβp, 0

),

Ed =

(λαcp

dαcp + βbkq,

qb

αc,

q(αβλckM(τ)− βabkq − aαcdp)

ac(βbkq + αcdp),

kqb

αcp

,αβλckM(τ)− βabkq − aαcdp

α(βbkq + αcdp)

),

where M(τ) =∫ ∞

0 e−aτ f (τ)dτ. The steady state with the pathogen presence is

possible when the equilibrium density of the pathogen is greater than zero (v1 >

81

0). This leads to a condition for invasion of the pathogen, hence we can write

R0 =λβk

∫ ∞

0 eaτ f (τ)dτ

apd> 0.

Here, R0 is called the basic reproduction ratio of the model which represents the

average number of secondary virus produced from a single virus for the system.

We can easily conform the above formula [26]. It turns out that the value of R0

determines the existence of the single-infection equilibrium Es which is exists if

and only if R0 > 1. The double-infection equilibrium exists if and only if S > 0

where

S = αβλckM(τ)− βabkq − aαcdp.

Therefore, the second basic reproductive number becomes

Rd =αcdp

βbkq(R0 − 1).

Hence, Rd > 1 if and only if R0 > Rs, where Rs = 1+βbkqαcdp . Noting that

∫ ∞

0 τ f (τ)dτ =

1 and if a > 0, then∫ ∞

0 eaτ f (τ)dτ < 1. More precisely, the delay term represent-

ing the time between the entry of a virion into a cell and the creation and release

of new virion from this cell. To analyze the stability of the equilibria, we need to

calculate the characteristic equation of the Jacobian matrix of the system (4.2.2) at

any equilibrium point E(x, y, z, v, w) as below

det[ξ I − J] = det

ξ + d + βv 0 0 βx 0

−vβN(ξ) ξ + a 0 −xβN(ξ) + wα wα

0 −wα ξ + b 0 −yα

0 −k 0 ξ + p 0

0 0 c 0 ξ + q

= 0.

where N(ξ) =∫ ∞

0 e−aτe−ξτ f (τ)dτ.

82

4.2.2 A Systematic Approach to Local Stability

In this section, we find the local stability of the system (4.2.2).

Theorem 4.2.2. When R0 < 1, the disease-free equilibrium E0 is locally asymptotically

stable while for R0 > 1, E0 becomes unstable and Es occurs.

Proof. The characteristic equation of the Jacobian matrix corresponding to the sys-

tem (4.2.2) at E0(λd , 0, 0, 0, 0) is

det[ξ I − J(E0)] = (b + ξ)(d + ξ)(q + ξ)

[(a + ξ)(p + ξ)−

λ

dβkN(ξ)

]= 0. (4.2.4)

The roots of the first three factors of the equation (4.2.4) are ξ1 = −b, ξ2 = −d

and ξ3 = −q which are negative and the remaining two equations are given by the

following equation

(a + ξ)(p + ξ)−λ

dβkN(ξ) = 0. (4.2.5)

Let us rewrite the equation (4.2.5) as

g(ξ) = (a + ξ)(p + ξ)−λ

dβkN(ξ). (4.2.6)

Noting that |N(ξ)| ≤ 1,. Let us assume g(0) = ap(1 − R0) < 0 and limξ→∞ g(ξ) =

+∞. By the continuity of g(ξ) there exist at least one positive root of g(ξ) = 0.

Thus, the infection-free equilibrium E0 is unstable if R0 > 1. If we choose the

direct delta function f (τ) = δ(t), (i = 1, 2), then we obtain N(ν) = 1. In this case

equation (4.2.5) becomes

ν2 + (a + p)ξ + ap(1 − R0) = 0. (4.2.7)

Thus, if R0 < 1, then equation (4.2.7) has two non-negative roots. Hence, the

equilibrium E0 is locally asymptotically stable for fi(τ) = δ(t). If ιν(ν > 0) is a

83

solution of equation (4.2.5), it follows that

−ν2 + (a + p)νι + ap − kβλ

dN(ιν) = 0. (4.2.8)

By taking modulai of equation (4.2.8), we have

ν4 + (a2 + p2)ν2 + (ap)2 − (kβλ

d)2|N(ιν)|2 = 0. (4.2.9)

We know that

|N(ιν)| =

∣∣∣∣∫ ∞

0e−aτ

(cos(ιξτ) − sin(ιξτ)

)f (τ)dτ

∣∣∣∣ ≤∫ ∞

0e−aτ f (τ)dτ. (4.2.10)

Therefore, we have

(ap)2 − (kβλ

d)2|N(ιν)|2 ≥ (ap)2(1 − R2

0).

Hence, if R0 < 1, then equation (4.2.9) has no positive roots. Therefore, by using

the general theory on characteristic equations of delay differential equations [64], it

is clear that E0 is always locally asymptotically stable for f (τ) = δ(t), if R0 < 1.

Theorem 4.2.3. For 1 < R0 < Rs the single infection-free equilibrium Es is locally

asymptotically stable, while Es becomes unstable for R0 > Rs and recombinant virus may

persist.

Proof. The characteristic equation corresponding to the Jacobian matrix of the sys-

tem is given by

det[ξ I − J(E1)] =((b + ξ)(q + ξ)− αy1c

)[(d + βv1 + ξ)kβx1N(ξ) − β2x1kv1N(ξ)

−(d + βv1 + ξ)(a + ξ)(b + ξ)

]= 0,

84

The above equation can be written in the form P1(ξ)P2(ξ) = 0, where

P1(ξ) = (b + ξ)(q + ξ)− αy1c,

P2(ξ) = (d + βv1 + ξ)kβx1N(ξ) − β2x1kv1N(ξ) − (d + βv1 + ξ)(a + ξ)(b + ξ).

Now P1(ξ) can be written as

P1(ξ) = ξ2 + (b + q)ξ + bq(1 − Rd),

which shows that p1(ξ) = 0 has the roots having negative real parts iff Rd < 1.

Thus, Es is unstable. Also P2(ξ) = 0, can be written as

ξ3 + a2(τ)ξ2 + a1(τ)ξ + a0(τ) + (b1ξ + b2)N(ξ) = 0, (4.2.11)

where

a2(τ) = a + p + d + βv1,

a1(τ) = (a + p)(d + βv1) + ap,

a0(τ) = ap(d + βv1),

b1(τ) = −kβx1,

b0(τ) = −dkβx1.

When f (τ) = δ(τ), we have N(ξ) = 1. In this case equation (4.2.11) becomes

ξ3 + a2(τ)ξ2 + (a1(τ) + b1(τ))ξ + a0(τ) + b2(τ) = 0. (4.2.12)

The equation (4.2.12) have negative real parts if R0 > 1 [67] as

a2(τ) = a + p + d + d(R0 − 1) > 0,

a1(τ) + b1(τ) = (a + p)(d + d(R0 − 1) > 0,

a0(τ) + b2(τ) = apd(R0 − 1) > 0.

85

Finally, we have

a2(a1 + b1)− (a0 + b2) =

(a3 + (a + p)(p + dR0)dR0

)> 0.

Therefore, the equilibrium Es is locally asymptotically stable when f (τ) = δ(τ). If

iν for ν > 0 is a solution of equation (4.2.12) it follows that

−iν3 + a2(τ)ν2 + a1(τ)iν + a0(τ) + (b1iν + b2)N(iν) = 0. (4.2.13)

After some simplification of equation (4.2.13), we get

ν6 + (a22 − 2a1)ν

4 + (a21 − 2a0a2)ν

2 + a20 − (b2

2 + b21ν2)|N(iν)|2 = 0, (4.2.14)

where

a22 − 2a1 = (d + βv1)

2 + a2 + p2> 0,

a21 − 2a0a2 − b2

1|N(iν)|2 = (d + βv1)2(a2 + p2) + a2 p2 − (kβx1)|N(iν)|2

≥ (a2 + p2)(d + βv1)2,

(a20 − b2

2)|N(iν)|2 = (a2 + p2)(ap(d + βv1))2 − (dkβx1)

2|N(iν)|2

≥ apβv1

[ap(d + βv1) + dkβx1|N(iν)|2

].

Hence, if R0 > 1, then equation (4.2.14) has no positive roots. Therefore, the

general theory of characteristic equations of delay differential equations [64] im-

plies that the chronic infection equilibrium Es is locally asymptotically stable when

f (τ) = δ(τ).

4.2.3 Global Asymptotic Stability of the Proposed Model

In this section, we will study the global behavior of the system (4.2.3). To

do this, we will use Lyapunove functionals theory and Lasali invariance principle.

86

Theorem 4.2.4. The disease-free equilibrium E0 is globally asymptotically stable when

R0 < 1.

Proof. Consider the Lyapunove functional

V0(t) = V1(t) + V2(t), (4.2.15)

where

V1(t) = x − x0 − lnx

x0+ 11y(t) + r1z(t) + r1v(t) + r1

b

cV(t),

V2(t) = r1β∫ ∞

0f (τ)e−aτ

( ∫ t−τ

tx(ξ)v(ξ)

)dξdτ,

with r1 = 1∫ ∞

0 eaτ f (τ)dτ. By taking derivative of V1(t) along the positive solution of

the system (4.2.2), we have

dV1

dt= (x(t)−

x

x0)


)+ u1

(β∫ ∞

0f (τ)e−aτ x(t − τ)v(t − τ)dτ

−ay(t) − αw(t)y(t)

)+ u1

(aw(t)y(t) − bz(t) + u1

a

k

(ky(t)− pv(t)

)

+u1b

c

(cz(t)− qw(t)

).

On substituting λ = dx0, and simplifying we get

dV1

dt= −d

(x(t) − x0)2

x− βx(t)v(t) + r1

(β∫ ∞

0f1(τ)e

−aτ x(t − τ)v(t − τ)dτ


)+ r1


)+ r1

a

k

(ky(t)− pv(t)

)

+r1b

c

(cz(t) − qw(t)

),

=−d(x(t) − x0)

2

x− βx(t)v(t) + r1β

∫ ∞

0f (τ)eaτ x(t − τ)v(t − τ)dτ

−u1bq

cw(t)− r1

ap

kv(t).

87

The derivative of V2(t) yields the following equation

dV2

dt= u1 β

∫ ∞

0f (τ)eaτ

(x(t)v(t) − x(t − τ)v(t − τ)

)dτ.

By using the values of dV1dt and dV2

dt , we can find the derivative of V0(t) after some

simplification as

dV0

dt= −d

(x(t) − x0)2

x+

ap(R0 − 1)v(t)

k∫ ∞

0 f (τ)eaτdτ− r1

bq

cw(t). (4.2.16)

Thus, equation (4.2.16) implies that dV0dt ≤ 0 only if R0 ≤ 1. Moreover the equality

will be satisfied if x0 = λd , y(t) = 0, z(t) = 0, v(t) = 0, w(t) = 0. Hence (see [70]),

we conclude that E0 is globally stable for R0 < 1.

Theorem 4.2.5. If 1 < R0 < Rs, then the single infection free equilibrium Es is globally

asymptotically stable.

Proof. Let (x(t), y(t), z(t), v(t), w(t)) be any positive solution of the system with

initial conditions . Let us consider the following Lyapunove functional

Vs(t) =(

x(t)− x1 ln x(t))+ r1

(y(t)− y1 ln y(t)

)+ r1

a

k

(v(t)− ln v(t)

)

+r1z(t) + r1b

cw(t), (4.2.17)

where r1 is defined before. Now taking derivative of Vs(t), we get

dVs

dt= (1 −

x1

x)


)

+r1(1 −y1

y)

(β∫ ∞

0eaτ f (τ)dτx(t − τ)v(t − τ)


)+ r1


)

+r1a

k(1 −

v1

v)

(ky(t) − pv(t)

)+ r1

b

c

(cz(t)− qw(t)

). (4.2.18)

88

The following identities can be obtained by using Es in the system (4.2.2)

dx1 − βx1v1 = λ,

βx1v1

∫ ∞

0eaτ f (τ)dτ = ay1,

ky1 = pv1,

Using these identities in equation (4.2.18), we obtain

dVs

dt= −

(x(t) − x1)2

x− βx(t)v(t) + βx1v1

(1 −

x1

x

)

+r1β∫ ∞

0eaτ f (τ)x(t − τ)v(t − τ)dτ

−r1βy1

y

∫ ∞

0e−aτ f1(τ)x(t − τ)v(t − τ)v(t − τ)dτ + βx1v1

−r1av1

v(t)+ βx1v1 + r1(αy1 −

bq

c)w.

Let us define

Vss(t) = Vs(t) + r1βx1v1

∫ ∞

0eaτ f1(τ)

∫ t

t−τ1

(x(ξ)v(ξ)

x1v1− ln(

x(ξ)v(ξ)

x1v1)

)dξdτ.

(4.2.19)

By taking derivative of equation (4.2.19), we get

dVss

dt= Vs(t) +

(x(t)v(t)

x1v1)− ln(

x(t)v(t)

x1v1)−

x(t − τ1)v(t − τ1)

x1v1)

+ ln(x(t − τ1)v(t − τ1)

x1v1

). (4.2.20)

89

Using the value of dV1dt in equation (4.2.20), we get

dVss

dt= −

(x(t)− x1)2

x+ βx1v1

(1 −

x1

x

)

−r1βy1

y

∫ ∞

0ea1τ f1(τ)x(t − τ)v(t − τ)v(t − τ)dτ + βx1v1

−βx1v1v1y

vy1+ βx1v1 + r1(αy1 −

bq

c)w(t)

+βx1v1

∫ ∞

0eaτ f1(τ) ln

(x(t − τ1)v(t − τ1)

x(t)v(t)

)dτ, (4.2.21)

After some simplification equation (4.2.21) becomes

dVss

dt= dx1(2 −

x1

x−

x

x1) + βx1v1

(3 −

x1

x−

v1y

vy1

−r1

∫ ∞

0eaτ f (τ)

y1x(t − τ1)v(t − τ1)

x1v1ydτ

+r1

∫ ∞

0eaτ f (τ) ln(

x(t − τ1)v(t − τ1)

x1v1)dτ

)

+r1αdp

aβk(R0 − Rs)w(t). (4.2.22)

Now using

(2 −x1

x−

x

x1) < 0,

and(

3 −x1

x−

v1y

vy1− r1

∫ ∞

0eaτ f (τ)

y1x(t − τ1)v(t − τ1)

x1v1ydτ

+r1

∫ ∞

0eaτ f (τ) ln(

x(t − τ1)v(t − τ1)

x1v1)dτ

)< 0.

Equation (4.2.22) implies that dVsdt ≤ 0 for x1, y1, v1 > 0 provided that R0 ≤ Rs.

Also, equality holds when x = x1 and y = y1 v = v1, z = z1 and w = 0. Thus,

the solution limit to the largest invariant subset of ˙Vss(t) = 0. Then, by LaSalle’s

invariance principle [70], we conclude that Es is globally asymptotically stable.

This completes the proof.

90

4.3 Stability Properties of Double (Continuous) delayed

HIV-1 Model

This section focuses on the dynamical behavior of the system with

delays and studies the analysis of their equilibrium solutions. In the proposed

model one delay term represents the latent period which is the time that the target

cells are contacted by the virus particles and the time the contacted cells become

actively infected. While the second delay term represents the virus production

period which means the time during which new virions are created within the cell

and are released from the cell. Then the new system obtained as follows

dx

dt= λ − dx(t)− (1 − nrt)βx(t)v(t),

dy

dt= (1 − nrt)β

∫ ∞

0e−a1τ f1(τ)x(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t),

dz

dt= aw(t)y(t) − bz(t), (4.3.1)

dv

dt= (1 − np)k

∫ ∞

0f2(τ)y(t − τ)dτ − pv(t),

dw

dt= cz(t) − qw(t).

The measure of the efficacious of the protease inhibiter and the reverse transcrip-

tase inhibiter are denoted by np and nrt, respectively. It is also assumed that the

infected cells becomes productively infected, τ units later, where τ is distributed

according to the probability distribution f1(τ). The recruitment of virus producing

cells at time t is given by the number of cells that were infected at time t− τ and are

still alive at time t. Here a1 is the constant death rate for infected cells but which

are not virus producing cells as yet. Therefore, eaτ is the probability of surviving

in the time period from t − τ to t. Also it is assumed that τ units later the virus

91

penetrated into the cell at time t, where τ is distributed according to the probabil-

ity distribution f2(τ). The probability distribution functions like f1(τ) = δ(t − τ1),

f2(τ) = δ(τ) or f1(τ) = δ(τ), f2(τ) = δ(t − τ2), are proposed. After some manip-

ulation our proposed model in general form can be written as follows:

dx


dy

dt= β

∫ ∞

0e−a1τ f1(τ)x(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t),

dz

dt= aw(t)y(t) − bz(t), (4.3.2)

dv

dt= k

∫ ∞

0e−a2τ f2(τ)y(t − τ)dτ − pv(t),

dw

dt= cz(t)− qw(t),

where β = (1 − nrt)β and k = (1 − np)k. The term e−a2τ is the probability of

surviving from time t − τ to time t, where a2 is the death rate of infected but not

yet virus-producing cells. In the system (4.3.2), the delay kernel is assumed to be

piecewise continuous to satisfy∫ ∞

0 fi(τ)dτ = 1 and∫ ∞

0 τ fi(τ)dτ < 1, i = 1, 2. The

initial conditions for the system (4.3.2) become

x(ζ) = ψ1(ζ), y(ζ) = ψ2(ζ), z(ζ) = ψ3(ζ), v(ζ) = ψ4(ζ), w(ζ) = ψ5(ζ),

ζ ∈ [−∞, 0], (4.3.3)

where (ψ1, ψ2, ψ3, ψ4, ψ5) ∈ C[(−∞, 0), R5 ] is the space of continuous functions

mapping the interval (−∞, 0] into R5, where ψi(ζ) ≥ 0, i = 1, 2, , ..., 5 and R5 =

{(x1, x2, x3, x4, x5); xi ≥ 0, i = 1, 2, ..., 5}. It is easy to show that all solutions of the

system (4.3.2) with initial conditions (4.3.3) are defined on [0,+∞) and remain pos-

itive for all t ≥ 0. We will address the well-posedness of the model by proving the

positivity and boundedness of solutions. We also identify the basic reproduction

92

number R0 which determines whether or not there is an infected equilibrium. It

will be proved that disease free equilibrium is locally stable if R0 < 1 and chronic-

infection equilibrium is locally stable if R0 > 1. The global stability of and numer-

ical simulations will be carried out.

4.3.1 Existence Uniqueness and Non- negativity of the Solutions

In this section, we will discuss positivity and boundedness of the solution.

According to the fundamental theory of functional differential equations [64], the

system (4.3.2) admits a unique solution and satisfies the initial conditions (4.3.3).

The following theorem gives boundedness and positivity of the solution.

Theorem 4.3.1. The solutions of the system (4.3.2) are non-negative, given that the inial

conditions are non-negative and bounded.

Proof. We get the following solution of the system (4.3.2)

x(t) = x(0)e−∫ t


0e−∫ t

ξ (d+βv(ζ))dζdη,

y(t) = y(0)e−∫ t

0 (a+αw(ζ))dζ + β∫ t

0e−∫ t

η (a+αw(ζ))dζ∫ ∞

0e−a1τ f1(τ)x(t − τ)v(t − τ)dτdη,

z(t) = z(0)e−bt +∫ t

0αw(t)y(t)e

−∫ t

η b(t−ζ)dζdη,

v(t) = v(0)e−pt + k∫ t

0e−p(t−η)

∫ ∞

0e−a2τ f2(τ)y(t − τ)dτdη,

w(t) = w(0)e−kt + c∫ t

0z(η)e−k(t−η)dη.

Which shows the positivity of the solution of each solution x(t), y(t), v(t), z(t) and

93

w(t). Next, we show the boundedness of the solution. We define

D(t) = ck∫ ∞

0e−a2τ f2(τ)

∫ ∞

0e−a1τ f1(τ)x(t − τ)dτdτ + ck

∫ ∞

0e−a2τ f2(τ)y(t)dτ

+ck∫ ∞

0e−a2τ f2(τ)z(t)dτ +

ac

2v(t + τ)

+bk

2

∫ ∞

0ea2τ f2(τ)w(t)dτ. (4.3.4)

Calculating the derivative of equation (4.3.4) and using the system (4.3.2), we have

d

dtD(t) = ck

∫ ∞

0ea2τ f2(τ)

∫ ∞

0e−a1τ f1(τ)

(λ − dx(t − τ)− βx(t − τ)v(t − τ)

)dτdτ

+ck∫ ∞

0e−a2τ f2(τ)

(β∫ ∞

0e−a1τ f1(τ)βx(t − τ)v(t − τ)dτ − ay(t)

−αw(t)y(t)

)dτ + ck

∫ ∞

0ea2τ f2(τ)


)dτ

+ac

2

(k∫ ∞

0ea2τ f2(τ)y(t)dτ − pv(t + τ)

)dτ

+bk

2

∫ ∞

0e−a2τ f2(τ)

(cz(t)− qw(t)

)dτ

= λck∫ ∞

0e−a2τ f2(τ)

∫ ∞

0ea1τ f1(τ)dτdτ −

[dck

∫ ∞

0e−a2τ f2(τ)

∫ ∞

0e−a1τ f1(τ)

x(t − τ)dτdτ +a

2ck

∫ ∞

0e−a2τ f2(τ)y(t)dτ +

b

2ck

∫ ∞

0ea2τ f2(τ)z(t)dτ

+qbk

2

∫ ∞

0e−a2τ f2(τ)w(t)dτ + p

ac

2v(t)

]

≤ λck∫ ∞

0e−a2τ f2(τ)

∫ ∞

0e−a1τ f1(τ)dτdτ

−θD(t)

< 0, forD(t) >λck

∫ ∞

0 e−a2τ f2(τ)∫ ∞

0 e−a1τ f1(τ)dτdτθ ,

> 0, forD(t) <λck

∫ ∞

0 e−a2τ f2(τ)∫ ∞

0 e−a1τ f1(τ)dτdτθ .

where θ = min{d, a2 , b

2 , q, p}. This implies that D(t) is bounded. Thus, all the

solutions x(t), y(t), v(t), z(t) and w(t) are bounded.

In order to study the asymptotic behavior of the proposed model, we find all

94

the possible equilibria of the system (4.3.2). The disease-free equilibrium E0(x0, y0, z0, v0, w0),

single-infection equilibrium Es(x1, y1, z1, v1, w1) and double-infection equilibrium

Ed(x2, y2, z2, v2, w2) which can be determined as follow

E0 =

(λ

d, 0, 0, 0, 0

),

Es =

(ap

kβM1M2

,λβkM1M2 − apd

aβk, , 0,

λβkM1M2 − apd

aβp, 0

),

Ed =

(λαcp

dαcp + βbkq,

qb

αc,

q(αβλckM1M2 − βabkq − aαcdp)

ac(βbkq + αcdp),

kqb

αcp,

kqb

αcp,

αβλckM1M2 − βabkq − aαcdp

α(βbkq + αcdp)

),

where Mi =∫ ∞

0 eaiτ fi(τ)dτ, (i = 1, 2). The steady state with the pathogen presence

is possible when the equilibrium density of the pathogen is greater than zero. This

leads to a condition for invasion of the pathogen. Therefore, we can define

R0 =λβk

∫ ∞

0 e−a1τ f1(τ)dτ∫ ∞

0 e−a2τ f2(τ)dτ

apd> 0.

Here, R0 is called the basic reproduction ratio of the model (4.4.2) which repre-

sents the average number of secondary virus produced from a single virus for

the system (4.3.2). Noting that∫ ∞

0 τ fi(τ)dτ = 1 and if ai > 0(i = 1, 2), then∫ ∞

0 e−aiτ fi(τ)dτ < 1. It is clear that increasing either of the delay may decrease the

basic reproduction ratio R0. It turns out that the value of R0 determines the exis-

tence of the single-infection equilibrium, that is E0 exists if and only if R0 > 1. For

the third equilibrium to exist, the density of the recombinant virus must be greater

than zero (w2 > 0) and this leads to the condition

Rd =αcdp

βbkq(R0 − 1).

95

Hence, Rd > 1 if and only if R0 > Rs, where Rs = 1+ βbkqαcdp . To analyze the stability

of the equilibria, we need to calculate the characteristic equation of the Jacobian

matrix of the system ((4.3.2) at equilibrium point E(

x, y, z, v, w)

as below

det[ξ I − J] = det

η + d + βv 0 0 βx 0

−vβN1(ξ) ξ + a + αw 0 −xβN1(ξ) yα

0 −wα ξ + b 0 −yα

0 −kN2(ξ) 0 ξ + p 0

0 0 −c 0 ξ + q

= 0.

where Ni(ξ) =∫ ∞

0 eaiτe−ξτ fi(τ)dτ, (i = 1, 2).

4.3.2 Local Behavior of the Proposed Model

In this section, we find the local stability of the system (4.3.2).

Theorem 4.3.2. When R0 < 1, then the disease-free equilibrium E0 is locally asymptoti-

cally stable while for R0 > 1, E0 is unstable and the Es occurs.

Proof. The characteristic equation of the Jacobian matrix of the linearized system

corresponding to the system (4.3.2) at E0(λd , 0, 0, 0, 0) is given by

det[ξ I − J(E0)] = (b + ξ)(d + ξ)(q + ξ)

[(a + ξ)(p + ξ)−

λ

dβkN1(ξ)N2(ξ)

]= 0.

(4.3.5)

The three roots of the characteristic equation (4.3.2) are ξ1 = −b, ξ2 = −d and ξ3 =

−q, which are negative and the remaining two roots are given by the following

equation:

(a + ξ)(p + ξ)−λ

dβkN1(ξ)N2(ξ) = 0. (4.3.6)

96

Let us rewrite equation (4.3.6) as

g(ξ) = (a + ξ)(p + ξ)−λ

dβkN1(ξ)N2(ξ). (4.3.7)

Noting that |Ni(ξ)| ≤ 1, (i = 1, 2). Let us assume g(0) = ap(1 − R0) < 0 and

limξ→∞ g(ξ) = +∞. By the continuity of g(ξ) there exists at least one positive root

of g(ξ) = 0. Thus, the infection-free equilibrium E0 is unstable if R0 > 1. If we

choose the direct delta function fi(τ) = δ(t), (i = 1, 2) then we obtain Ni(ξ) =

1, (i = 1, 2). In this case equation (4.3.7) becomes

ξ2 + (a + p)ξ + ap(1 − R0) = 0. (4.3.8)

Thus, if R0 < 1, then equation (4.3.7) has two negative roots. Hence the equilib-

rium E0 is locally asymptotically stable when fi(τ) = δ(t), (i = 1, 2). If iν(ν > 0)

is a solution of equation (4.3.7), it follows that

−ν2 + (a + p)νi + ap − kβλ

dN1(iν)N2(iν) = 0,

which yields

ν4 + (a2 + p2)ξ2 + (ap)2 − (kβλ

d)2|N1(iν)|

2|N2(iν)|2 = 0. (4.3.9)

We note that for (i = 1, 2),

|Ni(iν)| =

∣∣∣∣∫ ∞

0e−aiτ

(cos(iντ)− sin(iντ)

)fi(τ)dτ

∣∣∣∣ ≤∫ ∞

0e−aiτ fi(τ)dτ. (4.3.10)

Therefore, we have

(ap)2 − (kβλ

d)2|N1(iν)|

2|N2(iν)|2 ≥ (ap)2(1 − R2

0).

Hence, if R0 < 1, then equation (4.3.8) has no positive roots. Therefore, by the

general theory on characteristic equations of delay differential equations [64], it is

97

clear that E0 is always locally asymptotically stable for fi(τ) = δ(t), (i = 1, 2), if

R0 < 1.

Theorem 4.3.3. The single infection-free equilibrium Es is locally asymptotically stable

when 1 < R0 < Rs while Es becomes unstable for R0 > Rs and recombinant virus may

persist.

Proof. The characteristic equation corresponding to the Jacobian matrix of the lin-

earized system of the system (4.3.2) is given by

det[ξ I − J(E0)] =((b + ξ)(q + ξ)− αy1c

)[(d + βv1 + ξ)kβx1N1(ξ)N2(ξ)

−(β)2x1kv1N1(ξ)N2(ξ) − (d + βv1 + ξ)(a + ξ)(p + ξ)

]

= 0. (4.3.11)

We can write the equation (4.3.11) in the form P1(ξ)P2(ξ) = 0, where

P1(ξ) = (b + ξ)(q + ξ)− αy1c,

P2(ξ) = (d + βv1 + ξ)kβx1N1(ξ)N2(ξ) − (β)2x1kv1N1(ξ)N2(ξ)

−(d + βv1 + ξ)(a + ξ)(p + ξ).

Now P1(ξ) can be written as

P1(ξ) = ξ2 + (b + q)ξ + bq(1 − Rd),

which shows that P1(ξ) = 0 has roots which have negative real parts iff Rd < 1 (i,e

R0 < Rs or one positive and one negative if Rd > 1. Similarly, P2(ξ) = 0 can be

written as

ξ3 + a2(τ)ξ2 + a1(τ)ξ + a0(τ) + (b1ξ + b2)N1(ξ)N2(ξ) = 0, (4.3.12)

98

where

a2(τ) = a + p + d + βv1,

a1(τ) = (a + p)(d + βv1) + ap,

a0(τ) = ap(d + βv1),

b1(τ) = −kβx1,

b0(τ) = −dkβx1.

When fi(τ) = δ(τ), we have Ni(ξ) = 1(i = 1, 2). In this case equation (4.3.12)

becomes

ξ3 + a2(τ)ξ2 + (a1(τ) + b1(τ))ξ + a0(τ) + b2(τ) = 0, (4.3.13)

The equation (4.3.13) have negative real parts if R0 > 1 [67], as

a2(τ) = a + p + d + d(R0 − 1) > 0,

a1(τ) + b1(τ) = (a + p)(d + d(R0 − 1) > 0,

a0(τ) + b2(τ) = apd(R0 − 1) > 0.

Finally, we have

a2(a1 + b1)− (a0 + b2) =(dR0

(a2 + (a + p)(p + dR0

)+ apd

)> 0.

Therefore, the equilibrium Es is locally asymptotically stable when fi(τ) = δ(τ)(i =

1, 2). If iν for ν > 0 is solution of equation (4.3.13), then it follows that

−iν3 + a2(τ)ν2 + a1(τ)iν + a0(τ) + (b1iν + b2)N1(iν)N2(iν) = 0. (4.3.14)

After some simplification of equation (4.3.14), we get

ν6 + (a22 − 2a1)ν

4 + (a21 − 2a0a2)ν

2 + a20 − (b2

2 + b21ν2)|N1(iν)|

2|N2(iν)|2 = 0,

(4.3.15)

99

where

a22 − 2a1 = (d + βv1)

2 + a2 + p2> 0,

a21 − 2a0a2 − b2

1|N1(iν)|2|N2(iν)|

2 = (d + βv1)2(a2 + p2) + a2p2

−(kβx1)|N1(iν)|2|N2(iν)|

2

≥ (a2 + p2)(d + βv1)2,

(a20 − b2

2)|N1(iν)|2|N2(iν)|

2 = (a2 + p2)(ap(d + βv1))2

−(dkβx1)2|N1(iν)|

2|N2(iν)|2

≥ apβv1

[ap(d + βv1) + dkβx1|N1(iν)|

2|N2(iν)|2

].

Therefore, if R0 > 1, then the equation (4.3.15) has no positive roots. The general

theory of characteristic equations of delay differential equations [64] implies that

the chronic infection equilibrium Es is locally asymptotically stable when fi(τ) =

δ(τ), (i = 1, 2).

4.3.3 Global Behavior of the Proposed Model

In this section, we study the global behavior of the system (4.3.2). To do

this we use Lyapunove functionals theory and Lasali’s invariance principle.


R0 < 1.

Proof. Let(

x(t), y(t), z(t), v(t), w(t))

be any positive solution of the system (4.3.2)

with initial conditions (4.3.3). Consider the Lyapunove functional

L0(t) = L1(t) + L2(t), (4.3.16)

100

where

L1(t) = x(t)− x0 − lnx(t)

x0+ r1y(t) + r1z(t) + r2v(t) + r1

b

cw(t),

L2(t) = r1β∫ ∞

0f1(τ)e

−a1τ∫ t

t−τx(θ)v(θ)dθdτ + r2k

∫ ∞

0f2(τ)e

−a2τ ×∫ t

t−τy(θ)dθdτ

with r1 = 1∫ ∞

0 ea1τ f1(τ)dτand r2 = 1

k∫ ∞

0 e−a1τ f1(τ)dτ∫ ∞

0 e−a2τ f2(τ)dτ.

By taking derivative of L1(t) along the positive solution of the system (4.3.16), we

have

dV1

dt= (x(t)−

x

x0)


)

+r1

(β∫ ∞

0f1(τ)e

−a1τx(t − τ)v(t − τ)dτ − ay(t) − αw(t)y(t)

)

+r1


)+ r2

(k∫ ∞

0f2(τ)e

−a2τy(t − τ)dτ − pv(t)

)

+r1b

c

(cz(t) − qw(t)

). (4.3.17)

On substituting λ = dx0 in (4.3.17) and simplifying, we get

dV1

dt=

(x(t)− x0)2


(β∫ ∞

0f1(τ)e

−a1τx(t − τ)v(t − τ)dτ


)+ r1


)+ r2k

( ∫ ∞

0f2(τ)

×e−a2τy(t − τ)dτ − pv(t)

)+ r1

b

c

(cz(t)− qw(t)

),

=−d(x(t)− x0)

2


(β∫ ∞

0f1(τ)e

−a1τx(t − τ)v(t − τ)dτ

−r1ay(t) + r2k∫ ∞

0f2(τ)e

−a2τy(t − τ)dτ − r1bq

cw(t) + r2p(R0 − 1)v(t)

−r1bq

cw(t). (4.3.18)

101

By taking the derivative of L2(t), we get

dV2

dt= r1β

∫ ∞

0f1(τ)e

−a1τ

(x(t)v(t) − x(t − τ)v(t − τ)

)dτ + r2k

∫ ∞

0f2(τ)e

−a2τ

×

(y(t)− y(t − τ

)dτ. (4.3.19)

Taking derivative of equation (4.3.16) and using equations (4.3.18) and (4.3.19) and

simplifying, we get

dV0

dt= −d

(x(t) − x0)2

x+

p(R0 − 1)v(t)

k∫ ∞

0 f1(τ)e−a1τdτ∫ ∞

0 f2(τ)e−a2τdτ

−r1bq

cw(t).

If R0 ≤ 1, it follows from equation (4.3.19) that dV1dt ≤ 0. Moreover, the equality

will be hold if x0 = λd , y(t) = 0, z(t) = 0, v(t) = 0, w(t) = 0. Hence, E0 is globally

asymptotically stable when R0 < 1 (see [70]).

Theorem 4.3.5. Then the single infection free equilibrium Es is globally asymptotically

stable if 1 < R0 < Rs.

Proof. Let (x(t), y(t), z(t), v(t), w(t)) be any positive solution of the system (4.3.2)

with initial conditions (4.3.3). Let us consider the Lyapunove functional

Ls(t) =(

x(t)− x1 ln x(t))+ r1

(y(t)− y1 ln y(t)

)+ r2

(v(t)− ln v(t)

)+ r1z(t)

+r1b

cw(t), (4.3.20)

102

where r1 and r2 have been discussed in the previous theorem. Now taking deriva-

tive of (4.3.20), we get

dVs

dt= (1 −

x1

x)


)

+r1(1 −y1

y)

(β∫ ∞

0e−a1τ f1(τ)dτx(t − τ)v(t − τ)− ay(t) − αw(t)y(t)

)

+r1


)+ r2(1 −

v1

v)

(k∫ ∞

0e−a2τ f2(τ)y(t − τ)dτ − pv(t)

)

+r1b

c

(cz(t)− qw(t)

). (4.3.21)

Using λ = dx1 − βx1v1 in equation (4.3.21), we get

dVs

dt= −

(x(t)− x1)2


(1 −

x1

x

)+ r1β

∫ ∞

0e−aτ f1(τ)dτx(t − τ)dτ

−r1βy1

y

∫ ∞

0ea1τ f1(τ)dτx(t − τ)v(t − τ)v(t − τ)dτ − r1ay(t)

+βx1v1 + r2k∫ ∞

0e−a2τ f2(τ)dτy(t − τ)− r2k

v1

v(t)

∫ ∞

0ea2τ f2(τ)y(t − τ)dτ

+βx1v1 + r1(αy1 −bq

c)w(t). (4.3.22)

Let us define

Lss(t) = Ls(t) + r1β∫ ∞

0e−a1τ f1(τ)

∫ t

t−τ

[x(ξ)v(ξ) − x1v1 − xv ln

x(ξ)v(ξ)

x1v1

]dξdτ

+r2k∫ ∞

0e−a2τ f2(τ)

∫ t

t−τ

[y(ξ) − y1 − y1 ln

y(ξ)

y1

]dξdτ. (4.3.23)

103

Taking derivative of equation (4.3.24) and using equation (4.3.22), we get

dVss

dt= −

(x(t) − x1)2


(1 −

x1

x

)+ r1β

∫ ∞

0e−a1τ f1(τ)x(t − τ)dτ

−r1βy1

y

∫ ∞

0ea1τ f1(τ)x(t − τ)v(t − τ)dτ − r1ay(t) + βx1v1

+r2k∫ ∞

0e−τ f2(τ)dτy(t − τ)− r2k

v1

v(t)

∫ ∞

0e−a2τ f2(τ)y(t − τ)dτ

+βx1v1 + r1(αy1 −bq

c)w(t) + r1β

∫ ∞

0e−a1τ f1(τ)

[x(t)v(t) − x(t − τ)v(t − τ)

+x1v1 lnx(t − τ)v(t − τ)

x(t)v(t)

]dξdτ + r2k

∫ ∞

0e−a2τ f2(τ)

[y(t)− y(t − τ)

+y1 lny(t − τ)

y(t)

]

= −(x(t) − x1)

2

x+ βx1v1

(1 −

x1

x

)− r1β

∫ ∞

0e−a1τ f1(τ) ln

x(t − τ)v(t − τ)

x1v1y(t)dτ

+βx1v1 − r2ky1

∫ ∞

0e−a2τ f2(τ) ln

v1y(t − τ)

y1v(t)dτ + βx1v1 − r1β

∫ ∞

0e−a1τ f1(τ)

× lnx(t − τ)v(t − τ)

x(t)v(t)dτ + r2ky1

∫ ∞

0e−a2τ f2(τ) ln

y(t − τ)

y(t)dτ

+r1(αy1 −bq

c)w(t),

= −(x(t) − x1)

2

x+ βx1v1

(1 −

x1

x−

x

x1

)− r1βx1v1

∫ ∞

0e−a1τ f1(τ)

×

[y1x(t − τ)v(t − τ)

x1v1y(t)− 1 − ln

y1x(t − τ)v(t − τ)

x1v1y(t)

]dτ

−r2ky1

∫ ∞

0e−a2τ f2(τ)

[v1y(t − τ)

y1v(t)− 1 − ln

v1y(t − τ)

y1v(t)

]dτ

−αr1pd

βk(Rs − R0)w(t). (4.3.24)

If R0 ≤ Rs, it follows from equation (4.3.24) that dVssdt for x1, y1, v1 > 0. Moreover,

the equality will be hold if x = x1 and y = y1, v = v1, z = 0 and w = 0. Thus, the

solutions limit to the largest invariant subset of ˙Lss(t) = 0. Then, we can conclude

that Es is globally asymptotically stable. This completes the proof.

104


Here, we present numerical simulation. We use Runge-Kutta order

four method to find numerical results. For our numerical simulation we used pa-

rameters values β = 0.004 (estimated), λ = 2 [77], d = 1/10 [77], α = 0.004,

(estimated), a = 1/2 (estimated), p = 2 [77], k = 50 [78], b = 2 [74] ,c = 2000

[74], m1 = 1/2 = m2 = 1/2 (assumed), q = 2 (assumed), with initial conditions

x(0) = 13, y(0) = 6, z(0) = 3, v(0) = 149, w(0) = 1. Our numerical results show

that by using continuous delays in latent and virus production periods, the con-

centration of healthy cells increases and the virus load reduces as shown in figures

(a) and (d), respectively. Figure (b) shows that the concentration of infected cells

decreases due to reducing the viral load by postponing the production period of

infected cells. Figure (d) shows that the concentration of double infected cells in-

creases which release the recombinant virus to fight with pathogens virus. Figure

(e) the shows that recombinant virus are decreasing with the passage of time.

105

1 2 3 4 5t

12

14

16

18

xHtL

0.2 0.4 0.6 0.8 1.0t

1

2

3

4

5

6

yHtL

(a) (b)

1 2 3 4 5t

500

1000

1500

2000

2500

zHtL

0.5 1.0 1.5 2.0 2.5 3.0t

20

40

60

80

100

120

140

vHtL

(c) (d)

0.5 1.0 1.5 2.0 2.5 3.0t

200 000

400 000

600 000

800 000

1´ 106

wHtL

(e)Figure

Figure 4.1: The dynamics of the system (4.3.2) showing concentration of cells andviruses with continuous time delay

106

4.4 Conclusion and Discussion

In this chapter, the asymptotic analysis of delayed HIV-1 epidemic models with

two continuous time delays have been presented. The first model representing dis-

tributed intracellular delay in the laten period. For this model, the corresponding

characteristic equations was found and it was shown that if the basic reproduc-

tion ratio is less than unity, the infection-free equilibrium is locally asymptotically

stable. It is also proved that the chronic-infection equilibrium exists and is locally

asymptotically stable if the basic reproduction ratio is greater than one. Similarly

the global stability of the infection-free equilibrium and the chronic-infection equi-

librium of the proposed model has been completely established under certain con-

ditions. It is clear from these results that intracellular delay describing the latent

period has great effect on the stability of feasible equilibria and therefore do not

induce periodic oscillations. The second model represents the time delays in the

latent period and the virus production period. It was shown that if the basic re-

production ratio is less than unity, the infection-free equilibrium is locally asymp-

totically stable. We also proved that the single infection equilibrium exists and

is locally asymptotically stable if the basic reproduction ratio is greater than one.

Similarly, the global stability of the infection-free equilibrium and the single infec-

tion equilibrium of the proposed model have been completely established under

certain conditions. It is clear from these results that intracellular delays describing

the latent period and viral production period have great effect on the stability of

feasible equilibria and never be missed for modeling HIV-1 infection.

Chapter 5

Global dynamics of HIV-1 modelswith cure rate and saturation response

5.1 Overview

This chapter is devoted to study of two HIV-1 infection models, including

recovery rate of unproductively infected cells to uninfected cells and the saturation

mass actio. First model describes the effect of incorporation of recovery rate. The

dynamical properties of this model is discussed by using basic reproductive num-

bers. Next, the second model is proposed to study the effect of saturation mass

action principle. Then, this model will be analyzed for local and global stability.

Numerical simulation will be carried out to support the derived analytical results.

5.2 Study of HIV-1 Model with Effect of Cure Rate

In this section, the effect of the incorporation of recovery rate of unproduc-

tively infected CD+4 cells to uninfected cells is studied. The recovery of these cells

107

108

to uninfected cells is due to loss of all DNA from their nucleus by using drugs ther-

apy [79]. The following model is obtained after incorporating the above mentioned

terms:

dx

dt= λ − dx(t)− βe−aτx(t − τ)v(t − τ) + γy(t),

dy

dt= βe−aτx(t − τ)v(t − τ)− (a + γ)y(t) − αw(t)y(t),

dz

dt= αw(t)y(t) − bz(t), (5.2.1)

dv


dw

dt= cz(t)− qw(t),

where γ is the rate of conversion of infected cells. τ denotes time lag in contact

and infection processes. The stability analysis of this model will be discussed in

the coming sections.

5.2.1 Preliminaries and Assumptions

This section discusses the non-negativity and well-posedness of the system

(5.2.1) will be presented. Let B = C([−τ, 0]; R5) be the Banach space of continuous

mapping. These are the mapping from [−τ, 0] to R5 with the sup-norm. For the

system (4.2.1), consider (x(φ), y(φ), z(φ), v(φ), w(φ)) ∈ X, satisfying

x(φ) ≥ 0, y(φ) ≥ 0, z(φ) ≥ 0, v(φ) ≥ 0, w(φ) ≥ 0, φ ∈ [−τ, 0]. (5.2.2)

We have unique solution of the equation(4.2.1) under the given initial conditions.

Theorem 5.2.1. The solutions of the system (5.2.1) are non-negative when the initial

conditions are bounded and non-negative.

109

Proof. The following solution of the system (5.2.1) is obtained by using constant of

variation formula

x(t) = x(0)e−∫ t


0βe−aτx(t − τ)v(t − τ)e

−∫ t


y(t) = y(0)e−∫ t

0 (a+αz(ζ))dζ +∫ t

0


)e−∫ t

η (a+αv(ζ))dζdη.

It can be seen that x(t) and y(t) are positive. The solutions z(t), v(t), and w(t) have

been shown to be positive in chapter 3. Here, we have proved the positivity by

using constant of variation formula instead of Fundamental theorem of delay dif-

ferential equation [69]. For boundedness of the solution (x(t), y(t), z(t), v(t), w(t)),

Let us consider

F(t) = ckx(t) + cky(t) + ckz(t) +ac

2v(t) +

bk

2w(t). (5.2.3)

The derivative of the equation (5.2.3) yields

dF

dt= ck


)

+ck(

βe−aτx(t − τ)v(t − τ)− ay(t) − γy(t) − αw(t)v(t))

+ck(

aw(t)y(t) − bz(t))+

ac

2

(ky(t) − pv(t)

)+

bk

2

(cz(t) − qw(t))

= ckλ −(

dckx(t) +a

2cky(t) +

b

2ckz(t) + q

bk

2w(t) + p

ac

2v(t)

)

≤ ckλe−aτ − A⋆F(t),

where A⋆ = min{d, a2 , b

2 , q, p}. This means that F(t) is bounded.

5.2.2 Existence of the Equilibria

The system (4.2.1) has the following three possible biologically meaningful

equilibria, disease-free equilibrium E0(x0, y0, z0, v0, w0) (found in section (3.2.2) ),

110

single infection equilibrium Es(x1, y1, z1, v1, w1) and double infection equilibrium

¯E(x2, y2, z2, v2, w2) are given by

E =

((a + γ)p

βke−aτ,

λkβe−aτ − dp(a + γ)

kaβe−aτ, 0,

λkβe−aτ − dp(a + γ)

paβe−aτ, 0

),

¯E =

((αλc + γbq)p

αcdp + βkqbe−aτ,

bq

αc,

q

αc(

ckλαβe−a(τ) − αcdp(a + γ)− abqkβe−aτ

αcdp + bkqβe−aτ),

kqb

αcp

,αckβλe−a(τ) − αcdp(a + γ)− abqkβe−aτ

α(αcdp + bkqβe−aτ)

).

Each equilibrium point can be interpreted as; E0 is an infection-free equilibrium

corresponding to maximal levels of healthy CD+4 T cells. The second equilibrium

Es correspond to positive levels of healthy CD+4 T cells, infected cells, virus, but

free from recombinant viruses. The third equilibrium ¯E correspond to positive lev-

els of healthy CD+4 T cells, infected cells, virus, and recombinant virus. The basic

reproduction number (see [80]) is obtained from the proposed model as follows

R0 =kβλe−aτ

dp(a + γ).

Here R0 depends on recovery rate γ which is different from the work of Tian et al.

[42]. For R0 < 1, E0 is the only equilibrium which is biologically meaningful. If

R0 > 1, there is another equilibrium point Es. But ¯E exists if and only if Rd > 1 ,

where

Rd =αβλcke−aτ − αcdp(a + γ)

βbkq(a + γ)e−aτ=

(a + γ)αcdp

βbkqe−aτ(R0 − 1).

Suppose that Rs = 1 +βbkqe−aτ

αcdp , and Rd > 1 if and only if R0 > Rs.

5.2.3 Stability Results of Infection Free Equilibrium

Local dynamical behavior of the system (5.2.1) is discussed in this section.

111

Theorem 5.2.2. For R0 < 1, the disease-free equilibrium E0 is locally asymptotically

stable while for R0 > 1, E0 becomes unstable and the single infection equilibrium E occurs.

Proof. After linearlization around E0 the system (4.2.1) becomes

dx

dt= −dx(t)− βe−aτ λ

dv(t − τ) + γy(t),

dy

dt= βe−aτ λ

dv(t − τ)− (a + γ)y(t),

dz

dt= −bz(t), (5.2.4)

dv


dw

dt= cz(t)− qw(t).

The characteristic equation corresponding to the Jacobian matrix of the linearized

system (5.2.4) is given by

(b + ρ)(d + ρ)(q + ρ)

[(a + γ + ρ)(p + ρ)−

λ

dβke−τ(ρ+a)

]. (5.2.5)

Where ρ stands for eigenvalue. The first factor of the equation (5.2.5) has three

negative roots and the second factor of equation (5.2.5) can be determined as

(a + γ + ρ)(p + ρ) =λ

dβke−τ(ρ+a). (5.2.6)

The modulus of the left hand side of equation (5.2.6) satisfies

|(a + γ + ρ)(p + ρ)| ≥ (a + γ)p,

provided that ρ has non- negative real part. The modulus of the right hand side of

(5.2.6) gives

λ

dβk|e−τ(ρ+a)| = |(a + γ)pR0 | < (a + γ)p.

112

But this is contradiction to equation (5.2.6). Thus, when R0 < 1, then all the eigne

value have negative real part. Thus, the infection free state E0 is locally asymptot-

ically stable. For R0 > 1, we have

g(ρ) = (a + γ + ρ)(p + ρ)−λ

dβke−τ(ρ+a).

Now g(0) = (a + γ)p(1 − R0) < 0 and limρ→∞ g(ρ) = +∞. There exists at least

one positive root of g(ρ) = 0. Therefore, the infection-free equilibrium E0 is unsta-

ble if R0 > 1 (see [81]).


R0 < 1.

Proof. Let us consider

V0(t) =1

2(x(t)−

λ

d)2 +

λ

dy(t) +

λ

dz(t) +

(a + γ)λ

kdv(t) +

bλ

cdw(t)

+βλ

de−aτ

∫ t

t−τx(ζ)v(ζ)d(ζ). (5.2.7)

Here V0 stands for Lyapunov functional. The derivative of equation (5.2.7) and the

use of the system (5.2.1) yields

dV0

dt= (x(t)−

λ

d)


)

+λ

d

(βe−aτx(t − τ)v(t − τ)− ay(t) − γy(t) + αw(t)y(t))

+λ

d(w(t)y(t) − bz(t)

)+

(a + γ)λ

kd

(ky(t) − pv(t)

)

+bλ

cd

(cz(t)− qw(t)

)+

βλ

de−aτ

∫ t

t−τx(ζ)v(ζ)d(ζ). (5.2.8)

113

After further simplification, equation (5.2.8) becomes

dV0

dt= −(x(t)−

λ

d)

((x(t) −

λ


)−

(λ

d− x(t)

)γy

−qbλ

cdw(t)−

(a + γ)pλ

dk(

kβλe−aτ

(a + γ)dp− 1)v(t),

= −(x(t)−λ

d)

((x(t) −

λ


)−

(λ

d− x(t)

)γy

−(a + γ)pλ

dk(1 − R0)v(t)−

qbλ

cdw(t). (5.2.9)

Thus, when R0 < 1, then equation (5.2.9) implies that dV0dt < 0 and equality will

be satisfied if only if x0 = λd , y(t) = 0, z(t) = 0, v(t) = 0, w(t) = 0. Thus, using

LaSalle’s invariance principle (see [70]), we can see that E0 is globally asymptoti-

cally stable when R0 < 1.

5.2.4 Stability Results of Single Infection Equilibrium

This section is devoted to the analysis of E.

Theorem 5.2.4. The single infection equilibrium E is locally asymptotically stable if 1 <

R0 < Rs, while E becomes unstable for R0 > Rs.

Proof. The linearized form of the model (4.2.1) at E(x1, y1, z1, v1, w1) is given as

follows

x(t) = −dx(t)− βe−aτ(x1v(t − τ) + v1x(t − τ)) + γy(t),

y(t) = βe−aτ(x1v(t − τ) + v1x(t − τ)− (a + γ)y(t) − αy1w(t)],

z(t) = αy1w(t)− bz(t), (5.2.10)

v(t) = ky(t) − pv(t),

w(t) = cz(t)− qw(t).ρ

114

Let f1(ρ) f2(ρ) = 0 be the characteristic equation of the Jacobian matrix of the sys-

tem (5.2.10), where

f1(ρ) = ρ2 + (b + q)ρ + bq −cα(λkβe−aτ − dp(a + γ))

akβe−aτ,

f2(ρ) = ρ3 +

(a + γ + p +

kβλ

(a + γ)pe−aτ

)ρ2

+

[kβλ

(a + γ)pe−aτ(a + γ + p) + (a + γ)p

]ρ

+ kβλe−aτ − (a + γ)(ρ + d)pe−ρτ.

Now f1(ρ) can be simplified as

f1(ρ) = ρ2 + (b + q)ρ + bq(1 − Rd),

which represents that f1(ρ) = 0 has negative real part roots if and only if Rd < 1

. Therefore, if R0 > R1, E is unstable. After some simplification f2(ρ) = 0 can be

written as

ρ3 + a2(τ)ρ2 + a1(τ)ρ + a0(τ)− (c1ρ + c2)e

−ρτ = 0, (5.2.11)

where

a2(τ) = a + γ + p +kβλ

(a + γ)pe−aτ

and

a1(τ) =kβλ

(a + γ)pe−aτ(a + γ + p) + (a + γ)p,

a0(τ) = kβλe−aτ , c1 = (a + γ)p, c2 = (a + γ)pd.

It is clear that ρ = 0 is not root of (5.2.11) if R0 > 1. When τ = 0, then (5.2.11)

becomes

ρ3 + a2(0)ρ2 + (a1(0)− c1)ρ + a0(0)− c2 = 0, (5.2.12)

115

Using Routh-Hurwitz criterion (see [67]), we can prove that

a2(0) = a + γ + p +kβλ

(a + γ)p> 0,

a1(0)− c1 =kβλ

(a + γ)p(a + γ + p) > 0,

a0(0)− c2 = (a + γ)pd(R0 |τ=0 − 1) > 0.

Similarly,

a2(0)(a1(0)− c1)− (a0(0)− c2) =k2β2λ2

(a + γ)2p2(a + γ + p)

+kβλ

(a + γ)p(a + γ + p)2 + (a + γ)pd > 0.

Thus, any root of equation (5.2.11) has negative real part when τ = 0. Now, we

consider the distribution of the roots when τ > 0. Let ρ = iν(ν > 0) be the purely

imaginary root of (5.2.11), then,

−iν3 − a2(τ)ν2 + ia1(τ)ν + a0(τ)− (ic1ν + c2)e

−iντ = 0.

The modula of the above equation result in

Hs(ν2) = ν6 + [a2

2(τ)− 2a1(τ)]ν4 + [a2

1(τ)− 2a0(τ)a2(τ)− c21]ν

2

+a20(τ)− c2

2 = 0. (5.2.13)

Since

a22(τ)− 2a1(τ) = (a + γ)2 + p2 + d2R2

0 > 0,

a21(τ)− 2a0(τ)a2(τ)− c2

1 = d2[(a + γ)2 + p2]R20 > 0,

a20(τ)− c2

2 = (a + γ)2 p2d2(R20 − 1) > 0.

We see that all the coefficients of the above equation are positive which implies that

the function Hs(ν2) is monotonically increasing for 0 ≤ ν2

< ∞ with Hs(0) > 0.

116

Therefore, equation (5.2.13) has no positive roots if R0 > 1. Thus all the roots of

the equation (5.2.13) have negative real part if τ > 0 and R0 > 1.

Theorem 5.2.5. If R0 > 1 and R0 < Rs, then the infected equilibrium Es is global stable.

Proof. Let us construct the Lyapunove functional

Vs(t) = (x − x1 ln x) + (y − y1 ln y) + z +a + γ

k(v − v1 ln v) +

b

cw

+x1v1βe−aτ

t∫

t−τ

(x(θ)v(θ)

v1x(θ + τ)− ln x(θ)v(θ)

)dθ. (5.2.14)

The derivative of equation (5.2.14) yields

Vs(t) = (1 −x1

x)x + (1 −

y1

y)y + z +

a + γ

k(1 −

v1

v)v +

b

cw

+ x1v1βe−aτ

×

(x(t)v(t)

x(t + τ)v1−

x(t − τ)v(t − τ)

x(t)v1− ln(x(t)v(t)) + ln(x(t − τ)v(t − τ))

).

= (1 −x1

x)(λ − dx(t)− βe−aτx(t − τ)v(t − τ) + γy(t)) (5.2.15)

+(1 −y1

y)(βe−aτ x(t − τ)v(t − τ)− (a + γ)y(t) − αw(t)y(t))

+αw(t)y(t) − bz(t) +a + γ

k(1 −

v1

v)(ky(t) − pv(t)) +

b

c(cz(t) − qw(t))

+x1v1βe−aτ

(x(t)v(t)

x(τ + t)v1−

x(t − τ)v(t − τ)

x(t)v1+ ln

x(t − τ)v(t − τ))

(x(t)v(t)

).

The proposed system (5.2.1) at single infection equilibrium E(x1, y1, z1, v1, w1) yields

the following identities: λ = dx1 + βe−aτx1v1 + γy1, βe−aτx1v1 = (a + γ)y1,

ky1 = pv1. If τ is very large, that is, when the time delay in the contact of un-

infected targeted cells and pathogen virus and the latent period is very large, then

the rate of infection will be very small and contrariety if τ is very small, then the

infection will spread more rapidly. Therefore, we suppose that delay is very large,

117

Parameters Definition Value(day−1)

λ cell production rate 2 cell/mm3

d death rate of cells 0.01β infection rate 0.004 mm3/vira Death rate infected cell 0.5α double infection production Assumed α = βb Death rate of double-infected cell 2k virus production rate 50 vir/cellp Removal virus 3c Production rate of recombinant

by a double-infected cell 2000 vir/cellq Rate of removal of recombinant Assumed q = pτ Time Delay 1.0 ∼ 1.5 days

Table 5.1: Parameters Values used for Numerical Simulation

and by taking limit we get

limτ→∞

(x(t + τ)) = x(t). (5.2.16)

Using the above identities and the assumption (5.2.16) in equation (5.2.15), we get

V1(t) = dx1

(2 −

x

x1−

x1

x

)+ βe−aτx1v1

(3 −

x1

x−

yv1

y1v−

y1x(t − τ)v(t − τ)

yx1v1

+ lnx(t − τ)v(t − τ)

xv

)− (

x1

x− 1)γy +

αdp

βk(R0 − Rs)w(t). (5.2.17)

The following inequalities presented in [42], equation (5.2.17) results in dV1dt < 0

provided that R0 < R1. Thus, we have Es is globally asymptotically stable.

5.2.5 Numerical Discussion

This section is related to the numerical discussion and plots in the support of our

derived theoretical results. The drugs therapy can control HIV-1 infection. Using

118

drugs therapy, the infected cells revert to the uninfected cells. For numerical sim-

ulation, we have taken the values of the parameters from [25] and presented in

table (5.1) Figures 5.1 to 5.3 are the oscillations of uninfected cells, infected cells,

double infected cells, pathogen virus and recombinant virus. Figure 5.1 shows the

dynamical behavior of HIV-1 infection for the delay term τ = 1.5 and for different

recovery rates γ = 0.01, 0.1, 0.3, 0.5, 0.7, 0.9. It represents that as the value of the

recovery rate increases, the density of of uninfected cells increases and the concen-

tration of infected cells decreases. Figure 5.2 shows that varying time delay τ = 0.7

and keeping the values of γ constant, the amplitude of oscillation increases and the

rate towards stability decreases. Figure 5.3 shows that if we further reduce the de-

lay time τ = 0.4, then amplitudes of oscillations increases. The amplitudes of the

oscillations in Figure 5.3 is almost twice of that in Figure 5.1 though their frequen-

cies are almost not changed. These figures show that introducing even very small

time delay in the model can produce significant quantitative changes in solutions,

which cannot be observed from the model without time delay. Also, as the value

of recovery rate increases the infected cells revert to the healthy cells more rapidly

and converging to stable equilibrium. Therefore, we can also claim that the cure

rate γ is very important parameter and by improving the cure rate, we can reduce

this infection. Moreover, the significant qualitative changes due to existence of de-

lay can be observed. These results also suggests that time delay is important tool

which must be incorporated in HIV-1 infection models.

119

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

time t

x(t)

Virus−free host cells

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(a) x(t) (Density of infected cells) verses t

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

7

8

9

10

time t

y(t)

Infected cells

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(b) y(t) (Density of uninfected cells) verses t)

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

7

time t

z(t)

Double−infected cells

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(c) z(t) (Density of double infected cells) versest

0 100 200 300 400 500 600 700 8000

50

100

150

time t

v(t)

Pathogen virus

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(d) v(t) (Density of pathogen virus) verses t

0 100 200 300 400 500 600 700 8000

500

1000

1500

2000

2500

3000

3500

time t

w(t

)

Recombinant (genetically modified) virus

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(e) w(t) (Density of recombinant virus) verses t

Figure 5.1: The dynamics of the system (5.2.1) for τ = 1.5, showing convergence tothe stable equilibrium E.

120

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

120

time t

x(t)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(a) x(t) (density of infected cells) verses t

0 100 200 300 400 500 600 700 800 9000

1

2

3

4

5

6

7

8

time t

y(t)

Infected cells

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(b) y(t) (density of uninfected cells) verses t)

0 100 200 300 400 500 600 700 800 9000

1

2

3

4

5

6

time t

z(t)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(c) z(t) (density of double infected cells) verses t

0 100 200 300 400 500 600 700 800 9000

20

40

60

80

100

120

time t

v(t)

Pathogen virus

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(d) v(t) (density of pathogen virus) verses t

0 100 200 300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

time t

w(t

)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(e) w(t) (density of recombinant virus) verses t

Figure 5.2: The dynamics of the system (5.2.1) for τ = 0.7 showing convergence to

the stable equilibrium ¯E.

121

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120

time t

x(t)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(a) x(t) (density of infected cells) verses t

0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

7

time t

y(t)

Infected cells

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(b) y(t) (density of uninfected cells) verses t)

0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

time t

z(t)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(c) z(t) (density of double infected cells) verses t

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120

time t

v(t)

Pathogen virus

γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(d) v(t) (density of pathogen virus) verses t

0 200 400 600 800 1000 1200 14000

500

1000

1500

2000

2500

3000

time t

w(t

)


γ=0.01γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9

(e) w(t) (density of recombinant virus) verses t

Figure 5.3: The dynamics of the system (5.2.1) for τ = 0.4 showing oscillating be-havior.

122

5.3 Dynamics of Dose Dependent Infection Rate in HIV-

1 Model

In this section, we develop HIV-1 infection model with Holling type-II func-

tional response and two time delays. The Holling type-II functional response is

represented by 11+σv(t)

. The proposed HIV-1 infection model leads to the following

system of equations:

dx

dt= λ − dx(t)−

βx(t)v(t)

1 + σv(t),

dy

dt=

βe−a1τ1 x(t − τ1)v(t − τ1)

1 + σv(t − τ1)− ay(t) − αw(t)y(t),

dz

dt= αw(t)y(t) − bz(t),

dv

dt= ke−a2τ2y(t − τ2)− pv(t),

dw

dt= cz(t) − qw(t), (5.3.1)

where the variables and parameters in equation (5.3.1) have been defined in section

(3.4.1). The positivity and boundedness of the proposed model will be introduced

[82]. Then, the reproduction number will be investigated to prove the global be-

havior of the proposed model. The global stability of the system 5.3.1 at E0 and E

will be found. Numerical simulation will be used to justify the analytical results.

123

5.3.1 Preliminarily Results

In this section, we discuss some basic properties of the model. The following

theorem gives boundedness and positivity of the solutions.

Theorem 5.3.1. The solutions of the model (4.2.17) are non-negative and bounded pro-

vided the given initial conditions are non-negative and bounded.

Proof. Suppose that X = C[(−max(τ1, τ2), 0); R5] is the Banach space of contin-

uous mapping from [(−max(τ1, τ2), 0); R5] to R5 with the spermium norm. We

further suppose that x(t) =(

x(t), y(t), z(t), v(t), w(t))

and xt(ν) = x(t + ν) for

ν ∈ [(−max(τ1, τ2), 0]. By using fundamental theory of FDEs [69], for any initial

condition ϕ ∈ X with ϕ ≥ 0 we know that there exists a unique solution x(t, ϕ)

satisfying x(ν, ϕ) = ϕ(υ). Now the system (4.2.17) can be written as ˙x(t) = f (xt),

where

f (xt) =

λ − dx(0)−βx(0)v(0)1+σv(0)

βe−a1τ1 x(−τ1)v(−τ1)1+σv(−τ1)

− ay(0)− αw(0)y(0)

αw(0)y(0) − bz(0)

ky(0)− pv(0)

cz(0)− qw(0)

.

It can be shown that if any ϕ ∈ X satisfies ϕ ≥ 0, ϕi(0) = 0 for some i, then

f (ϕi) ≥ 0. Therefore, x(t, ϕ) ≥ 0 for all t ≥ 0 in its maximal interval of existence

124

if ϕ ≥ 0 [17]. Next, we show the boundedness of the solution. To do this let us

consider

M⋆(t) = cke−a1τ1 x(t − τ1) +ac

2ea2τ2v(t + τ2) +

bk

2w(t) (5.3.2)

+cky(t) + ckz(t). (5.3.3)

Calculating the derivative of (4.2.18), and using the system (4.2.17), we have

dM⋆

dt= cke−a1τ1

(λ − dx(t − τ1)− βx(t − τ1)v(t − τ1)

)

+ck

(βe−a1τ1 x(t − τ1)v(t − τ1)− (a + αw(t))y(t)

)

+ck


)+

ac

2ea2τ2

(ke−a2τ2y(t)

−pv(t + τ2)

)+

bk

2

(cz(t)− qw(t)

). (5.3.4)

After some re-arrangement of equation (4.2.19), we get

dM⋆

dt= ckλe−a1τ1 −

(cdke−aτ1 x(t − τ1) +

1

2acky(t)

+1

2bckz(t) +

bkq

2w(t) +

ac

2pea2τ2v(t + τ2)

)

≤ ckλe−a1τ1 − ǫM⋆(t),

where ǫ = min{d, a2 , b

2 , q, p}. Which shows that M⋆(t) is bounded.

125

5.3.2 Existence of the Equilibria and Basic Reproductive Num-

bers

The system (4.2.17) has three equilibria [83], E0 and E and ¯E as follows

E =

(apea1τ1+a2τ2 + σλk

k(σd + β),

pe−a1τ1

ak

(λβk − apdea1τ1+a2τ2

σd + β

), 0,

λβk − apdea1τ1+a2τ2

apea1τ1+a2τ2(σd + β)0

),

¯E =

(A

B,

qb

αc,

1

Bαc

(αcβke−(a1τ1+a2τ2)A − a(αcp + kqbσe−a2τ2))B

(αcp + kqbσe−a2τ2)

),

kqbe−a2τ2

αcp,

1

Bα

(αcβke−(a1τ1+a2τ2)S − a(αcp + kqbσe−a2τ2)B

(αcp + kqbσe−a2τ2)

)),

where, A = αcpλ + σkqbde−a2τ2λ and B = αcpd + σkqbde−a2τ2 + βkqbe−a2 τ2 .

In epidemiological models the threshold quantity R0 is called the basic reproduc-

tion number of the disease which is a key concept [38]. The basic reproductive

number for our proposed model is given by

R0 =λβk

apde−(a1τ1+a2τ2).

For the third equilibrium to exist, the density of the recombinant virus must be

exist and should be greater than zero, which determine the other reproductive

number

Rd =αcdp

βbkqe−a2τ2(R0 − 1).

Hence, Rd > 1 if and only if R0 > Rs, where Rs = 1 +βbkqe−a2τ2

αcdp .

5.3.3 Global Dynamics of Proposed Model for the Case R0 < 1

This section is devoted to the global analysis of the model.

126

Theorem 5.3.2. If R0 < 1, then the infection free equilibrium is E0 is globally asymptoti-

cally stable.

Proof. Let us consider

W0(t) =e−a1τ1

2(x(t)−

λ

d)2 +

λ

d(y(t) + z(t)) + ea2τ2

aλ

kdv(t)

+bλ

cdw(t) +

βλ

de−a1τ1

∫ t

t−τ1

x(φ)v(φ)

1 + σv(φ)d(φ)

+aλ

d

∫ t

t−τ2

y(φ)dφ. (5.3.5)

Differentiating equation (5.3.5) with respect to t, we get

dV0

dt= −e−a1τ1

(x(t)−

λ

d

)dx

dt+

λ

d

dy

dt+

λ

d

dz

dt+ ea2τ2

aλ

kd

dv

dt

+βλ

de−a1τ1

(Ψ(t − τ1)−

x(t)v(t)

1 + σv(t)

)

+bλ

cd

dw

dt+

aλ

d

(y(t − τ2)− y(t)

), (5.3.6)

where Ψ(t − τ1) =x(t−τ1)v(t−τ1)

1+σv(t−τ1). Using system (5.3.1) in equation (5.3.6), we get

dV0

dt= −e−a1τ1(x(t)−

λ

d)

(λ − dx(t)−

βx(t)v(t)

1 + σv(t)

)

+λ

d

(βe−a1τ1Ψ(t − τ1)− (a + αw(t))y(t)

)(5.3.7)

+λ

d


)+ ea2τ2

aλ

kd

(ky(t) − pv(t)

)

+βλ

de−a1τ1

(Ψ(t − τ1)−

x(t)v(t)

1 + σv(t)

)

+bλ

cd

(cz(t) − qw(t)

)+

aλ

d

(y(t − τ2)− y(t)

).

Substituting λ = dx0 in (5.3.7), we obtain

dV0

dt= −e−a1τ1(x(t)−

λ

d)2 −

apea2τ2λ

dk(1 − R0)v(t)

−qbλ

cdw(t). (5.3.8)

127

Equation (5.3.8) implies that dV0dt ≤ 0 provided that R0 ≤ 1. But the equality holds

only if x0 = λd , y(t) = 0, z(t) = 0, v(t) = 0, w(t) = 0. Thus, E0 is globally

asymptotically stable when R0 < 1.

5.3.4 Global behavior of the model for the case R0 > 1

This section is related to the study of the global character of the model.

Theorem 5.3.3. When R0 > 1 and R0 < Rs, then E is globally asymptotically stable.

Proof. Let us construct the following Lyapunov functional

Ws(t) = W11(t) + βx1v1e−a1τ1W12(t) + aW13(t), (5.3.9)

where

W11(t) = e−a1τ1(x(t)− x1 ln x(t)) + (y(t) − y1 ln y(t)) + z(t)

+aea2τ2

k(v(t) − v1 ln v(t)) +

b

cw(t),

W12(t) =∫ t

t−τ1

(x(ρ)v(ρ)

x1v1(1 + σv(t))−

1

(1 + σv1)ln(

x(ρ)v(ρ)

x1v1(1 + σv(t)))

)dρ,

W13(t) =∫ t

t−τ2y(ρ)dρ.

The derivative of equation (5.3.9) yields:

dVs

dt= e−a1τ1(1 −

x1

x(t))x(t) + (1 −

y1

y(t))y(t) +

b

cw(t)

+z(t) +aea2τ2

k(1 −

v1

v)v(t) + βx1v1e−a1τ1

(Ψ(t)

x1v1

−1

1 + σv1ln

( x(t)v(t)

x1v1(1 + σv(t))

)−

x(t − τ1)v(t − τ1)

x1v1(1 + σv(t − τ1))

+1

1 + σv1ln(

x(t − τ1)v(t − τ1)(1 + σv1)

x1v1(1 + σv(t − τ1))

)

+a(

y(t)− y(t − τ2)).

128

Using E in the system (5.3.1), we obtain the following identities:

λ = dx1 +βe−aτx1v1

1 + σv1,

βe−aτx1v1 = (a + γ)y1,

ky1 = pv1.

The use of these identities in (4.3.3) yields the following equations

dVs

dt= e−a1τ1(2 −

x1

x−

x

x1) +

βx1v1e−aτ1

1 + σv1

×

(3 −

x1

x−

v1y(t − τ2)

vy1−

(1 + σv(t))y1x(t − τ1)v(t − τ1)

(1 + σv(t − τ1))x1v1y

+ ln(x(t − τ1)v(t − τ1)(1 + σv(t))

x(t)v(t)(1 + σv(t − τ1)))

)

+αdp

aβke−a2τ2(R0 − Rs)w(t). (5.3.10)

The following inequities hold [42],

e−a1τ1(2 −x1

x−

x

x1) ≤ 0,

(3 −

x1

x−

v1y(t − τ2)

vy1−

(1 + σv(t))y1x(t − τ1)v(t − τ1)

(1 + σv(t − τ1))x1v1y

+ ln( x(t − τ1)v(t − τ1)(1 + σv(t))

x(t)v(t)(1 + σv(t − τ1))

)−

v1y(t − τ2)

vy1

)≤ 0.

Therefore, with the help of these inequalities equation, (5.3.10) can give the result

dVsdt < 0, provided that R0 ≤ Rs but the equality holds, when x = x1 and y = y1

v = v1 and w = 0. We conclude that E0 is globally asymptotically stable (see

[70]).

129

5.3.5 Numerical Discussion

In this section, we give some numerical examples to support the derived

theoretical results. For numerical simulation, we use the values of the parameters

from [84, 33] with initial conditions x(0) = 5.0, y(0) = 1.0, z(0) = 2.0, v(0) =

0.5, w(0) = 4.0.

Example 5.3.1. In the system (5.3.1), we choose the parameters values as λ = 4, d =

0.21, a = 0.33, c = 40, b = 5.6, p = q = 5.6, τ1 = 10, τ2 = 10, k = 50, α = β =

0.004, σ = 0.000001, a1 = a2 = 0.1. The results of numerical simulation are represented

in Fig (1 − 5). It gives that R0 = 0.0272889842985 < 1 and the system (5.3.1) has

infection free equilibrium at E0(19, 0, 0, 0, 0). By theorem (5.4.2), we obtain infection-free

equilibrium E0 of the system (5.4.1) is globally asymptotically stable. In 5.4, the densities

of uninfected cells, infected cells, double infected cells, pathogen viruses and recombinant

viruses are given. It can be seen that there is no effect of time delay in infection free equilib-

rium. Similarly, Fig (5.5) and (5.5) are the numerical representation of infected equilibria.

Example 5.3.2. In the system (5.3.1), we set λ = 2, d = 0.10, a = 0.5, c = 40, b = p =

q = 5.6, σ = 0.0005, α = β = 0.002, a1 = a2 = 0.2, τ1 = τ2 = 5 with the above initial

conditions. It shows that 1 < R0 = 1.34 < R1 = 13.84 and the system (5.3.1) has single

infection equilibrium E0(2.94, 140, 3.696, 170, 0, 0). Thus by theorem (5.4.3), we prove

that the system (5.3.1) is globally asymptotically stable.

Example 5.3.3. In the system (5.3.1), we take λ = 2, α = β = 0.002, d = 0.10, a =

0.5, c = 40, b = 2, p = q = 5.6, k = 70, a1 = a2 = 0.2, τ1 = τ2 = 5, σ = 0.0009 with

the above initial conditions. It shows that R1 = 1.34 > 1 and thus the system (5.3.1) is

globally asymptotically stable.

130

Figure 5.4: The plot supports example (5.3.1) and shows the stability of E0

131

Figure 5.5: The plot supports example (5.3.2) and shows the stability of E

132

Figure 5.6: The plot supports example (5.3.2)

133

5.4 Conclusion

In this chapter, two delayed HIV-1 models with cure rate and saturation

response are presented. It is proved that the first proposed model, included cure

rate, has three equilibrium solutions E0, E and ¯E. It has shown that E0 is locally

as well as globally asymptotically stable for R0 ∈ (0, 1), which loses its stability

at R0 = 1. Then, E0 bifurcates into E. Then, it is shown that E is also locally and

asymptotically stable. Further, as the value of recovery rate is increased, the in-

fected cells revert to the uninfected cells more rapidly which resulting decrease in

infected cells and increase in healthy cells. And this infection can easily be reduced

if we improve the cure rate. This work is more productive for giving the direction

of controlling the infection of HIV-1. Our proposed work suggests that by incor-

porating the recovery rate HIV-1 infection can be controlled. In the second HIV-1

infection model, we have incorporated the saturation rate has been proposed. The

basic reproduction number R0 is found and two others reproduction numbers Rs

and Rd are also investigated which are different from the basic reproduction num-

ber R0. It has been shown that the infection-free equilibrium is globally asymp-

totically stable if R0 < 1. While if R0 > 1, then the infection-free equilibrium E0

becomes unstable and there occurs a single-infection equilibrium E which is glob-

ally asymptotically stable if R0 < Rs. Furthermore, if Rs < Rd, then Es is unstable

and there exists Ed. It can be seen that R0 is independent of the parameters of the

activation of CTLs. It means that CTLs do not permit to eliminate the virus.

Chapter 6

Optimal Control Strategies of HIV-1Models

6.1 Overview

In this chapter, the applications of optimal control theory to HIV-1 infec-

tion models with time delay and without time delay are are discussed. First, the

optimal control theory is applied to HIV-1 infection model (without time delay) of

fighting a virus with another virus, without delay. To develop the control strate-

gies, two control variables u1 and u2 are incorporated. The control u1 denotes the

efficaciousness of drug therapy in blocking off the infection of new cells, and the

control u2 denotes the efficacy of drug therapy in decreasing the production of new

viruses. The physical meaning of our control problem is to minimize the infected

cells measure and free virus particles and maximize the healthy cells density in

blood. Next, the single discrete delayed HIV-1 infection model with two control

variables is introduced. To do this, first the controlled HIV-1 infection model with

delay is formulated. Then, optimal control strategy is developed to minimize the

density of infected cells, the cost of treatment and the density of pathogen virus.

134

135

The existence of an optimal control for the control problem is proved and the opti-

mality system is derived. Numerical simulations are carried out. In the last section,

the double delayed- HIV-1 model with optimal controls is presented. The optimal

controls are introduced into the proposed model in order to discuss efficacy of drug

therapies to block the infection of new cells and prevent the production of new free

virions. The aim of the present study is to apply optimal control strategies to max-

imize the density of uninfected CD+4 cells in the body by using minimum drug

therapies. Finally, the numerical simulation results are carried out which confirms

the effectiveness of the proposed model.

6.2 Control Strategy of HIV-1 Model

In this section, the HIV-1 epidemic model [25] is investigated to develop

control strategy for controlling HIV-1 infection. Two control variables u1(t) and

u2(t) are introduced for the efficaciousness of drug therapy in blocking off the

infection of new cells and the efficacy of drug therapy in decreasing the production

of new virus. Then, the proposed model becomes

dx(t)

dt= λ − dx(t)− βx(t)v(t) + u1(t)x(t),

dy(t)

dt= βx(t)v(t) − ay(t) − αw(t)y(t),

dz(t)

dt= aw(t)y(t) − bz(t), (6.2.1)

dv(t)

dt= ku2(t)y(t) − pv(t),

dw(t)

dt= cz(t)− qw(t).

136

with initial conditions x(0) ≥ 0, y(0) ≥ 0, z(0) ≥ 0, v(0) ≥ 0 and w(0) ≥ 0. Here,

all the parameters and variables are discussed in the chapter 3 and u1 and u2 are

weights constants. Let the objective functional [85] be defined to maximize the

concentration of uninfected cells as follows

J(u1(t), u2(t)) =∫ t

0

[Bx(t)− (S1u2

1(t) + S2u22(t))

](6.2.2)

Then, we have to maximize this objective functional. Here, Bx(t) represents the

benefits of T cells and the other terms S1u21(t) + S2u2

2(t) are systemic costs of the

drug treatments. S1 and S2 are positive constants Our goal is to increase the num-

ber of the uninfected CD4+T cells, reducing the viral load (the number of free

virions), and minimizing the cost of treatment. We are looking to find an optimal

control pair u1(t) and u2(t) such that

J(u1(t), u2(t)) = max{J(u1(t), u2(t)) \ (u1(t), u2(t)) ∈ U}, (6.2.3)

where U = {(u1(t), u2(t)) \ ui(t) is Lebesgue measurable on [0, 1], and

0 ≤ ui(t) ≤ 1, i = 1, 2} is the control set.

6.2.1 Existence of the optimal Control Problem

In this section, we prove the existence of control problem. For the existence

of the optimal pair, we will follow [86, 87, 88, 89, 85].

Theorem 6.2.1. For the control problem (6.2.1), there exists u(t) = (u1(t), u2)(t) ∈ U

such that

max(u1(t),u2(t))∈U

J(u1(t), u2(t)) = J(u1(t), u2(t)).

Proof. To prove the existence of optimal control, we will follow [88].

137

(H1) We see that the set of controls and comparable state variables is nonempty

[87].

(H2) We also note that the solutions are bounded. Therefore, the control set U is

convex and closed.

(H3) Using the boundedness of the solution, we see that the right hand sides of

the state system are bounded by a linear function in the state and control

variables. Our state system is bilinear in u1(t) and u2(t).

(H4) Moreover, the integrand of the objective functional is concave on U. Finally,

we can prove that there exists constants h1, h2 > 0, and ρ > 1 such that the

integrand L(x(t), u1(t), u2(t)) of the objective functional satisfies

L(x(t), u1(t), u2(t)) = h2 − h1(|u1(t)|2 + |u2(t)|

2)ρ/2.

Pontryagin’s Maximum Principle given in [88] provides necessary conditions

for an optimal control problem. This principle changes the equations (6.2.1), (6.2.2)

and (6.2.3) into a problem of maximizing an Hamiltonian H point wisely with

respect to u1 and u2:

H(t) = Bx(t)−1

2

(S1u2

1(t) + S2u22(t)

)+

5

∑m=1

λm(t)gm(x, y, z, v, w) (6.2.4)

where gm(x(t), y(t), z(t), v(t), w(t) is the right hand side of the differential equa-

tion of state variables x(t), y(t), z(t), v(t), w(t). By applying Pontryagins Maxi-

mum Principle [88], we state and prove the following results.

138

Theorem 6.2.2. Given optimal controls u1(t), u2(t) and solutions x(t), y(t), z(t), v(t), w(t)

of the corresponding state system (6.2.1), there exists adjoint variables λm(t), m = 1, ..., 5

such that

λ′1(t) = (λ1(t)− λ2(t))βv(t)− B + λ1(t)d − λ1(t)u1(t),

λ′2(t) = aλ2(t) + (λ2(t)− λ3(t))αw(t)− λ5(t)ku2(t),

λ′3(t) = bλ3(t)− cλ5(t),

λ′4(t) = (λ1(t)− λ2(t))βx(t) + λ4(t)p,

λ′5(t) = (λ2(t)− λ3(t))αy(t) + λ5(t)q,

with transversality conditions λm(t) = 0, m = 1, 2, ..., 5.

Proof. By setting x = x, y = y, z = z, v = v and w = w and differentiating the

Hamiltonian with respect to states variable x(t), y(t), z(t), v(t) and w(t), respec-

tively, we obtain the following adjoint system

λ′1(t) = (λ1(t)− λ2(t))β ˜v(t) − B + λ1(t)d − λ1(t)u1(t),

λ′2(t) = aλ2(t) + (λ2(t)− λ3(t))α ˜w(t) − λ5(t)ku2(t),

λ′3(t) = bλ3(t)− cλ5(t),

λ′4(t) = (λ1(t)− λ2(t))βx(t) + λ4(t)p,

λ′5(t) = (λ2(t)− λ3(t))α ˜y(t) + λ5(t)q,

satisfying transversality conditions λm = 0, m = 1, 2...5.

Theorem 6.2.3. The control pair (u1(t), u2(t)), which maximizes the objective functional

139

over the region U are given by

u1(t) = max{min{λ1(t)x(t)

S1, 1}, 0},

u2(t) = max{min{λ4(t)ky(t)

S2, 1}, 0}

Proof. The optimality conditions yield the following

∂H

∂u1= λ1(t)x(t)− S1u1(t), (6.2.5)

and

∂H

∂u2= λ4(t)ky(t)− S2u2(t). (6.2.6)

Solving equations (6.2.5) and (6.2.6) simultaneously for the optimal control vari-

ables u1(t) and u2(t), we get

u1(t) =x(t)λ1(t)

S1, (6.2.7)

and

u2(t) =ky(t)λ4(t)

S2. (6.2.8)

By using the property of control space, equations (6.2.7) and (6.2.8) can be written

as

u1(t) =

0 if x(t)λ1(t)S1

≤ 0,

x(t)λ1(t)S1

if 0 <x(t)λ1

S1< 1,

1 ifx(t)λ1(t)

S1≥ 1.

u2(t) =

0 ifky(t)λ4(t)

S2≤ 0,

k ˜y(t)λ4(t)S2

if 0 <ky(t)λ4(t)

S2< 1,

1 ifky(t)λ4(t)

S2≥ 1.

140

In compact form, we can re-write u1(t) and u2(t) in the following form

u1(t) = max{min{λ1(t)x(t)

S1, 1}, 0}, (6.2.9)

u2(t) = max{min{kλ4(t)y(t)

S2, 1}, 0}. (6.2.10)

Using equations (6.2.9) and (6.2.10), we can write the following optimality system

dx(t)dt = λ − dx(t)− βx(t)v(t) + max{min{λ1(t)x(t)

S1, 1}, 0}y(t),

dy(t)dt = βx(t)v(t)− ay(t)− αw(t)y(t),

dz(t)dt = αw(t)y(t)− bz(t),

dv(t)dt = k(1 − max{min{

λ4(t)ky(t)S2

, 1}, 0})y(t)− pv(t),

dw(t)dt = cz(t)− qw(t).

(6.2.11)

The optimal controls variables and state variables are found by solving the opti-

mality system (6.2.11), the adjoint system dλidt with initial and boundary conditions,

equations of the control variables (6.2.9) and (6.2.10). Once we obtain an optimal

control problem, our Hamiltonian becomes

H⋆ = Bx⋆(t)−1

2

(S1u2

1(t) + S2u22(t)

)+

5

∑m=1

λm(t)gm(x⋆, y⋆, z⋆, v⋆, w⋆).

141

6.2.2 Numerical Solution of the Optimal Control Problem

In this section, we show the numerical simulation of optimal control

problem. To do this, we use Runge-Kutta four order scheme forward in time for

the state system and backward in time for the adjoint system. For numerical sim-

ulation we use the parameter values given in table 6.1. The numerical results are

given in Figures 1 − 5 which show potency of drug therapies based on the den-

sities of uninfected cells, infected cells, free viruses and recombinant viruses with

and without control. Fig 6.1 shows that without treatment, the number of unin-

fected x(t) cells decreases drastically. But after treatments the concentration of

CD4+T cells are increasing up to some time. The concentration of infected cells

y(t) decrease rapidly right from the very beginning of treatment and throughout

the period of investigation given in Fig 6.2, while the concentration of infected

cells without treatment arise at the beginning and become stable toward the end

of the period. The density of double infected cells z(t), shows in Fig 6.3. This Fig

shows that after introducing therapy, the density z(t) decays towards zero. Simi-

larly, viral load v(t) increases drastically is given in Fig 6.4 with out treatment but

with treatment there is no increase in the concentration of free virus. Moreover,

the intensity of recombinant viruses w(t) decrease with the passage of time after

therapy as these viruses are used to infect the infected cells. This can be seen in

Fig 6.5. Also, we see that optimal treatments u1 and u2 required with the change

in time to block new infection of cells and prevent viral production with minimum

side effects given in Fig 6.6. For numerical simulation we use the following data.

142

Parameters Definition Values with Sources

λ Production rate of host cell 2 cell/mm3, [25]d Death rate of host cell 0.6 Assumedβ Infection rate of host cell by virus 0.04 mm3/vir, [25]a Death rate of HIV-1 infected cell 0.1, [25]α Infection rate by recombinant 0.2, Assumedb double-infected cells death rate 0.3, Assumedk virus production rate 0.2, vir/cell Assumedp death rate of virus 0.3, Assumedc Recombinant virus production rate 0.5, vir/cell Assumedq Recombinant removal rate 0.3, Assumed

S1 Weight Constant 40 AssumedS2 Weight Constant 20, AssumedB Constant 1, [90]

Table 6.1: Numerical Values of the Parameters of the Model with Sources

0 5 10 15 20 251

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time(week)

x(t)

Control in the uninfected target cells

w/o controlwith control

Figure 6.1: The graph represents the uninfected target cells x(t) with and withoutcontrol. The concentration of uninfected target cells increases after the optimalcontrol strategy.

143

0 5 10 15 20 25−1

0

1

2

3

4

5

6

Time(week)

y(t)

Control in the infected cells


Figure 6.2: The graph represents the density of infected cells y(t) with and withoutcontrol. The concentration of infected cells approaches to a small number due tooptimal control.

0 5 10 15 20 250

1

2

3

4

5

6

7

8

Time(week)

z(t)

Control in the double infected cells


Figure 6.3: The graph represents the density of double infected cells z(t) with andwithout control. The concentration of double infected cells approaches to a smallnumber due to optimal control.

144

0 5 10 15 20 25−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time(week)

v(t)

Control in the pathogen virus


Figure 6.4: The figure shows the density of free virus v(t), with and without con-trol. The concentration of pathogen viruses reaches to a small number due to opti-mal control strategy.

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

Time(week)

w(t

)

Control in the recombinant virus


Figure 6.5: The graph shows the density of recombinant virus, with and withoutcontrol.

145

6.3 Control Strategy of the Single delayed HIV-1 model

In the section, optimal control theory is applied to single delayed HIV-1

model. The proposed optimal strategy is based on the delayed HIV-1 model [42].

To develop the control strategy, two control variables u1(t) and u2(t) are proposed.

The control u1(t) denotes the efficaciousness of drug therapy in blocking off the

infection of new cells, while the control u2(t) denotes the efficacy of drug therapy

in reducing the production of new virus. For instance, u1(t) = 1, the obstruction is

100 percent efficient. On the other hand, if u2(t) = 0, there is no blockage. These

controls are measurable functions satisfy 0 ≤ uj(t) < 1 for j = 1; 2, and represent

two different treatment strategies. These two control variables can be adjusted in

the proposed model to obtain

dx(t)

dt= Λ − dx(t)− (1 − u1(t))βx(t)v(t),

dy(t)

dt= (1 − u1(t))βe−aτ x(t − τ)v(t − τ)− ay(t) − αw(t)y(t),

dz(t)

dt= αw(t)y(t) − bz(t), (6.3.1)

dv(t)

dt= k(1 − u2(t))y(t) − pv(t),

dw(t)


with initial conditions

x(φ) ≥ 0, y(φ) ≥ 0, i(φ) ≥ 0, v(φ) ≥ 0, φ ∈ [−τ, 0], (6.3.2)

where(

x(φ), y(φ), v(φ), i(φ))

∈ X and τ = max{τ1, τ2}. All the variables and

parameters in equation (6.3.1) have been discussed in section (3.4.1). Our target is

to find the control functions u1(t) and u2(t) to maximized the density of uninfected

146

cells and recombinant viruses during the time period [0, T]. Further, to minimized

the cost of treatment, the density of pathogen virus and the density of infected cells.

In order to achieve our target, we construct to maximized the following objective

functional

J(u1(t), u2(t)) =∫ T

0

(ρ1x(t) + ρ2w(t)−

1

2(ξ1u2

1(t) + ξ2u22(t))

)dt, (6.3.3)

where ρ1 and ρ2 are weight constants which balance the size of the optimal con-

ditions and the parameters ξ1 ≥ 0 and ξ1 ≥ 0 are established on the benefits and

costs of the treatment. These parameters equalizes the size of the terms u1(t) and

u2(t) which reflect the severeness of the side effects of the drugs. We have to maxi-

mize the functional defined in (6.3.3) by raising the number of the uninfected cells

and recombinant viruses. Moreover, our achievement is the decreasing viral load

and minimizing the cost of treatment. We find optimal controls functions u⋆

1(t)

and u⋆

2(t) such that

J(u⋆

1(t), u⋆

2(t)) = max{J(u1(t), u2(t)) \ (u1(t), u2(t)) ∈ U},

where U = {(u1(t), u2(t)) \ ui is lebesgue measurable on [0, 1], 0 ≤ ui(t) ≤ 1,

i = 1, 2}, is the control set.

6.3.1 Existence of the Control Pair

In this section, we show the existence of the control strategy. To do this,

first we find the lagrangian of the optimal control problem which can be found as

follows:

L(t) = ρ1x(t) + ρ2w(t)−1

2(ξ1u2

1(t) + ξ2u22(t)).

147

The corresponding Hamiltonian [91] can be defined as

H(x, y, z, v, w, xτ, vτ, u1, u2, λ(t)

)=

1

2

(ξ1u2

1 + ξ2u22

)− ρ1x(t)− ρ2w(t)

+λ1(t)(

Λ − dx(t)− (1 − u1(t))βx(t)v(t))

+λ2(t)((1 − u1(t))βe−aτ xτvτ − ay(t)

−αw(t)y(t))+ λ3(t)


)

+λ4(t)(

k(1 − u2(t))y(t) − pv(t))

+λ5(t)(

cz(t)− qw(t))

, (6.3.4)

where xτ = x(t − τ) and vτ = v(t − τ). This Hamiltonian determines the con-

trol functions for the proposed optimal control problem. To show the existence of

optimal pair, we use the idea of Fleming and Rishel presented in [86].

Theorem 6.3.1. For the control problem with the model (6.3.1), there always exists

u⋆(t) = (u⋆

1(t), u⋆

2(t)) ∈ U such that


J(u1(t), u2(t)) = J(u⋆

1(t), u⋆

2(t)).

Proof. The existence of optimal control, we will follow [86].

(H1) We see that the set of controls and comparable state variables are nonnegative

and nonempty [92].

(H2) The admissible set U is closed and compact.

(H3) We see that the right hand side of the integrand of the objective functional is

concave on U. Also, is continuous and bounded above by sum of bounded

control and state variables and can be written as a linear function of u1 and

u2 with coefficients depending on state variables and time.

148

(H4) Finally, we can prove that there exist constants h1, h2 > 0, and η > 1 such that

the integrand L(x(t), w(t), u1(t), u2(t)) of the objective functional satisfies

L(

x(t), w(t), u1(t), u2(t))= h2 − h1(|u1|

2 + |u2|2)η/2.

We conclude that there exists an optimal control.

Next, we use Pontryagin’s Maximum Principle [88] to discuss the following

theorem.

Theorem 6.3.2. For the optimal control u⋆

1(t), u⋆

2(t) and solutions x⋆(t), y⋆(t), z⋆(t), v⋆(t),

and w⋆(t) of the corresponding state system (6.3.1), there are adjoint variables λi(t), i =

1, 2, ..., 5, satisfying the equations

dλ1

dt= ρ1 + λ1(t)

(d + (1 − u⋆

1(t))βv⋆(t))+ λ1(t + τ)λ2(t)βe−aτv⋆(t − τ)(u⋆

1(t)− 1),

dλ2

dt= aλ2(t) + (λ2(t)− λ3(t))αw⋆(t) + λ4k(u⋆

1(t)− 1),

dλ3

dt= bλ3(t)− cλ5(t), (6.3.5)

dλ4

dt= λ1(t)(1 − u⋆

1(t))βx⋆(t − τ) + λ4(t + τ)λ2(t)βe−aτ x⋆(t)(u⋆

1 (t)− 1) + λ4(t)p,

dλ5

dt= ρ2 + (λ2(t)− λ3(t))αy⋆(t) + λ5(t)q,

with transversality conditions

λj(T) = 0, j = 1, 2, ..., 5. (6.3.6)

149

Proof. Using Pontryagin’s Minimum Principle presented in [88], we get the follow-

ing system of adjoint variables,

dλ1

dt= −

∂H(t)

∂x− λ1(t + τ)

∂H

∂xτ, λ1(T) = 0,

dλ2

dt= −

∂H(t)

∂y, λ2(T) = 0,

dλ3

dt= −

∂H(t)

∂z, λ3(T) = 0,

dλ4

dt= −

∂H(t)

∂v(t)− λ4(t + τ)

∂H(t

∂vτ, λ4(T) = 0,

dλ5

dt= −

∂H(t)

∂w, λ5(T) = 0.

Further, adjusting x(t) = x⋆(t), y(t) = y⋆(t), z(t) = z⋆(t), v(t) = v⋆(t) and w(t) =

w⋆(t), we get the adjoint system (6.3.5) satisfying tranversality conditions λj(T) =

0, j = 1, 2, ..., 5.

Theorem 6.3.3. The control pair (u⋆

1(t), u⋆

2(t)), which maximizes the objective functional

J over the region U are given by

u⋆

1(t) = max

{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), 1}, 0

},

u⋆

2(t) = max

{min{

λ4(t)ky⋆(t)

ξ2, 1}, 0

}.

Proof. The optimality conditions yield the following

∂H

∂u1= ξ1u⋆

1(t)+ λ1(t)βx⋆(t)v⋆(t)− λ2(t)βe−aτ x⋆(t − τ)v⋆(t − τ), (6.3.7)

and

∂H

∂u2= ξ2u⋆

2(t)− λ4(t)ky⋆(t). (6.3.8)

150

Solving equations (6.3.7) and (6.3.8) simultaneously for the optimal control pair

u⋆

1(t) and u⋆

2(t), we get

u⋆

1(t) =β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), (6.3.9)

u⋆

2(t) =λ4(t)ky⋆(t)

ξ2. (6.3.10)

By using the property of control space, equations (6.3.9) and (6.3.10) can be written

as

u⋆

1(t) =

0 ifβξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

)≤ 0,

βξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

)if

0 <βξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ)))− λ1(t)x⋆(t)v⋆(t)

)< 1,

1 ifβξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

)≥ 1.

u⋆

2(t) =

0 ifλ4(t)ky⋆(t)

ξ2≤ 0,

λ4(t)ky⋆

ξ2if 0 <

λ4(t)ky⋆(t)ξ2

< 1,

1 ifλ4(t)ky⋆(t)

ξ2≥ 1.

The above two equations ,that is, the values of u⋆

1(t) and u⋆

2(t) can be written in

compact form as follows

u⋆

1(t) = max

{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))

−λ1(t)x⋆(t)v⋆(t)

), 1}, 0

}(6.3.11)

and

u⋆

2(t) = max

{min{

λ4(t)ky⋆(t)

ξ2, 1}, 0

}. (6.3.12)

Here, we call formula (6.3.11) and (6.3.12) for u⋆

1(t) and u⋆

2(t), the characterization

of the optimal control.

151

Therefore, we get the following optimality system:

dx⋆(t)

dt= Λ − dx⋆(t)− βx⋆(t)v⋆(t)

(1 − max

{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0

}),

dy⋆(t)

dt=

(1 − max

{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), 1}, 0

})

× βx⋆(t − τ)v⋆(t − τ)− ay⋆(t)− αw⋆(t)y⋆(t),

dz⋆(t)

dt= αy⋆(t)w⋆(t)− bz⋆(t),

dv⋆(t)

dt= k(1 − max

{min{

λ4y⋆(t)

ξ2, 1

}, 0})y⋆(t)− pv⋆(t),

dw⋆(t)

dt= cz⋆(t)− qw⋆(t).

(6.3.13)

Now the equations (6.3.11) and (6.3.12), and Hamiltonian H⋆ at

152

(x⋆, y⋆, z⋆, v⋆, w⋆, x⋆τ1, y⋆τ2

, v⋆τ1, u⋆

1 , u⋆

2 , λ1, λ1, λ2, λ3, λ4, λ5

), can be written as

H⋆(t) =1

2

(ξ1(max{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0})2 + ξ2(max{min{

λ4(t)ky⋆(t)

ξ2, 1}, 0})2

)

− ρ1x⋆(t)− ρ2w⋆(t) + λ1

[Λ − dx⋆(t)−

(1 − max{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

)βx⋆(t)v⋆(t)

]

+ λ2

[(1 − max{min{

β

ξ1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

)βe−aτx⋆(t − τ)v⋆(t − τ)− ay⋆(t)− αw⋆(t)y⋆(t)

]

+ λ3

[αw⋆(t)y⋆(t)− bz⋆(t)

]

+ λ4

[k(1 − max{min{

λ4(t)ky⋆(t)

ξ2, 1}, 0}

)y⋆(t)− pv⋆(t)

]+ λ5

[cz⋆(t)− qw⋆(t)

].

(6.3.14)

To find out the optimal control and state variables, we will numerically solve the

above systems (6.3.13) and (6.2.2).

6.3.2 Numerical Simulation of the Optimality System

In this section, we present numerical results of the optimal control problem.

For numerical simulation, we use Runge-Kutta method. We solve the state sys-

tem forward in time by Runge-Kutta fourth order scheme and the adjoint system

by backward fourth order scheme. Some of the values of the parameters are con-

sidered from real data (see for more detail references [33, 24]). The given figures

1 −−6 are the simulation results of our theoretical results. We can make some

decisions on the potency of drug therapies based on the densities of uninfected

153

Parameters Meaning Value Reference

λ uninfected cells 2 cell/mm3 [25]d death rate of cells 0.01 day−1 [25]α infection of double infected cells 0.0012 mm3/vir [25]β Rate of infection of cell 0.004mm3/vir [25]b Death rate of double infected cells 2 day−1 [25]k Virous production per infected cell 50 vir/cell [25]a Death rate of infected cells 0.5 day−1 [93, 94, 95, 96]p Clearance rate of virous 3 day−1 [93, 94, 95, 96]q clearances rate of recombinant 3 day−1 [93, 94, 95, 96]m Death rate for infected but not yet

virous producing cells 0.5 day−1 Assumedτ time delay 1 Assumed.

Table 6.2: Values of the Parameters with Different Sources

cells, infected cells, double infected cells, free virus and recombinant virus. Fig 6.6

shows that the density of uninfected target cells increases after the optimal control

strategy. Fig 6.7 represents that the number of infected cells approaches to a very

small number after treatment, while without treatment the number of infected cells

increase. Similarly, Fig 6.8, shows that the concentration of double infected cells re-

duced to a small number after treatment. While, Fig 6.9 justify the decrease in the

number of pathogen viruses after applying optimal control, due to which the in-

fection is controlled up to some time. Fig 6.10 represents that after treatment the

density of recombinant virus increases with the passage of time and as a result

these viruses may causes the control of infected cells. We see in Fig 6.11 optimal

treatments u1 and u2 are required with the change in time to reduce new infection

of cells and keep viral production with minimum side effects.

154

0 2 4 6 8 100

1

2

3

4

5

6

Time(weeks)

Den

sity

of u

ninf

ecte

d ce

lls

Control in the uninfected cells


Figure 6.6: The graph represents control in the density of uninfected cells versestime t in weeks.

0 2 4 6 8 100

1

2

3

4

5

6

Time(weeks)

Den

sity

of i

nfec

ted

cells

Control in the infected cells


Figure 6.7: The graph represents control in the density of infected cells verses timet in weeks.

155

0 2 4 6 8 101

2

3

4

5

6

7

8

Time(weeks)

Den

sity

of d

oubl

e in

fect

ed c

ells

Control in the double infected cells


Figure 6.8: The graph represents control in the density of double infected cellsverses time t in weeks.

0 2 4 6 8 10−2

−1

0

1

2

3

4

Time(weeks)

Den

sity

of p

atho

gen

viru

s

Control in the density of pathogen viruses


Figure 6.9: The graph represents control in the density of pathogen virus versestime t in weeks.

156

0 2 4 6 8 100

5

10

15

20

25

30

Time(weeks)

Rec

ombi

nant

viru

s

Control in the recombinant viruses


Figure 6.10: The graph represents control in the density of recombinant virusverses time t in weeks.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time(weeks)

Con

trol

var

iabl

es

Control variables in the epidemic model

u1u2

Figure 6.11: The graph shows the control pair in the optimal control problem.

157

6.4 Optimal Control Problem of Double Delayed HIV-

1 Model

In this section, we introduce our third optimal strategies in the in the

double delayed HIV-1 model [80]. To do this, we use two control variables u1 and

u2. After adjusting these control variables the control problem becomes

dx(t)

dt= Λ − dx(t)− (1 − u1(t))βx(t)v(t),

dy(t)

dt= (1 − u1(t))βe−a1 τ1 x(t − τ1)v(t − τ1)− ay(t) − αw(t)y(t),

dz(t)

dt= αw(t)y(t) − bz(t), (6.4.1)

dv(t)

dt= ke−a2τ2(1 − u2(t))y(t − τ2)− pv(t),

dw(t)

dt= cz(t)− qw(t),

with initial conditions

x(φ) ≥ 0, y(φ) ≥ 0, i(φ) ≥ 0, v(φ) ≥ 0, φ ∈ [−τ, 0], (6.4.2)

where(

x(φ), y(φ), v(φ), i(φ))∈ X and τ = max{τ1, τ2} where the variables u1 and

u2 are used to reduce the density of infected cells and pathogen virus and increase

the number of uninfected cells and recombinant virus. To proceed further, let the

objective functional be defined by

J(u1, u2) =∫ t f

0

[Ax(t)− (ν1u2

1 + ν2u22)

]. (6.4.3)

Then, we have to maximized the above objective functional. Here, Ax(t) repre-

sents the benefits of uninfected CD+4 cells and the other terms ν1u21 + ν2u2

2 are

systemic costs of the drug treatments, where ν1 and ν2 are positive constants rep-

resenting the relative weights attached to the drug therapies which balance the size

158

of the terms u1 and u2. Our aim is to increase the density of the uninfected CD4+T

cells, reducing the viral load (the number of free virion) and minimizing the cost

of treatment. Next, we find an optimal controls functions u⋆

1(t) and u⋆

2(t) such that

J(u⋆

1(t), u⋆

2(t)) = max{J(u1(t), u2(t)) \ (u1(t), u2(t)) ∈ U}, (6.4.4)

where U = {(u1(t), u2(t)) \ ui is the control set which is Lebesgue measurable on

[0, 1], 0 ≤ ui(t) ≤ 1, i = 1, 2}.

6.4.1 Existence of the Optimal Control Problem

Let us define the lagrangian of the optimal control problem as

L(t) = Ax(t)−1

2(ξ1u2

1(t) + ξ2u22(t)). (6.4.5)

Then, the corresponding Hamiltonian becomes

H(

x, y, z, v, w, xτ1, vτ1

, yτ2 , u1, u2, λ(t))=

1

2

(ν1u2

1 + ν2u22

)− Ax(t)

+ λ1(t)(

Λ − dx(t)− (1 − u1(t))βx(t)v(t))

+ λ2(t)((1 − u1(t))βe−a1 τ1 xτ1

vτ1− ay(t) − αw(t)y(t)

)

+ λ3(t)(

αw(t)y(t) − bz(t))+ λ4(t)

(ke−a2τ2(1 − u2(t))yτ2

− pv(t))+ λ5(t)

(cz(t)− qw(t)

),

(6.4.6)

where xτ1= x(t − τ1), yτ1

= y(t − τ1) and vτ2 = v(t − τ2). This Hamiltonian is

used to find the control functions for the proposed optimal control problem. To

check the existence of optimal pair, we use Fleming and Rishel in [97].

Theorem 6.4.1. There exists u⋆ = (u⋆

1 , u⋆

2) ∈ U, for the control problem with model

159

(6.4.1), such that


J(u1(t), u2(t)) = J(u⋆

1(t), u⋆

2(t)).

Proof. We show the existence of the solution through a well known result in [97],

and the solution exists if the following hypotheses are satisfied: (H1) see that the

set of controls and comparable state variables is nonempty. (H2) The admissible

control set U is closed and convex. (H3) Each right hand side of (6.4.1) is contin-

uous, is bounded above by a sum of the bounded control and state variables, and

can be written as a linear functions of u1 and u2 with coefficients depending on

time and state. (H4) Finally, we can prove that there exist constants h1, h2 > 0, and

κ1 such that the integrand L(x(t), u1(t), u2(t)) of the objective functional satisfies

L(

x(t), u1(t), u2(t))= h2 − h1(|u1|

2 + |u2|2)κ1/2.

Next, we use Pontryagin’s Minimum Principle [88] to discuss the following

theorem.

Theorem 6.4.2. There exists the following adjoint variables λi(t), i = 1, 2, ..., 5, for the

optimal controls u⋆

1(t), u⋆

2(t) and state variables x⋆(t), y⋆(t), z⋆(t), v⋆(t), and w⋆(t) of

the corresponding state system (6.4.1), which satisfy the following equations

dλ1

dt= A + λ1(t)

(d + (1 − u⋆

1(t))βv⋆(t))+ λ1(t + τ1)λ2(t)βe−a1τ1v⋆(t − τ)(u⋆

1(t)− 1),

dλ2

dt= aλ2(t) + (λ2(t)− λ3(t))αw⋆(t)− λ2(t + τ2)λ4(t)e

−a2τ2k(1 − u∗2(t)),

dλ3

dt= bλ3(t)− cλ5(t), (6.4.7)

dλ4

dt= λ1(t)(1 − u⋆

1(t))βx⋆(t) + λ4(t + τ1)λ2(t)βe−a1τ1 x⋆(t − τ1)(u⋆

1(t)− 1) + λ4(t)p,

dλ5

dt= (λ2(t)− λ3(t))αy⋆(t) + λ5(t)q,

160

with transversality conditions

λj(t f ) = 0, j = 1, 2, ..., 5. (6.4.8)

Proof. Using Pontryagin’s Minimum principe, we get the following system of equa-

tions of adjoint variables,

dλ1

dt= −

∂H(t)

∂x− λ1(t + τ1)

∂H(t)

∂xτ1

, λ1(t f ) = 0,

dλ2

dt= −

∂H(t)

∂y− λ(t + τ2)

∂H(t)

∂yτ2

, λ2(t f ) = 0,

dλ3

dt= −

∂H(t)

∂z, λ3(t f ) = 0,

dλ4

dt= −

∂H(t)

∂v(t)− λ4(t + τ1)

∂H(t)

∂vτ1

, λ4(t f ) = 0,

dλ5

dt= −

∂H(t)

∂w, λ5(t f ) = 0.

Further, adjusting x(t) = x⋆(t), y(t) = y⋆(t), z(t) = z⋆(t), v(t) = v⋆(t) and w(t) =

w⋆(t), we get the adjoint system (6.4.8) satisfying tranversality conditions λj(t f ) =

0, j = 1, 2, ..., 5.

λ′1(t) = −

∂H(t)

∂x− χ[0,T−τ](t)

∂H(t + τ)

∂xτ= ρ1 + λ1(t)

(d + (1 − u⋆

1(t))βv⋆(t))

+χ[0,T−τ](t)λ2(t + τ)βe−aτv⋆(t)(u⋆

1 (t + τ)− 1),

λ′2(t) = −

∂H

∂y= aλ2(t) + (λ2(t)− λ3(t))αw⋆(t) + λ4k(u⋆

1(t)− 1),

λ′3(t) = −

∂H

∂z= bλ3(t)− cλ5(t),

λ′4(t) = −

∂H(t)

∂v(t)− χ[0,T−τ](t)

∂H(t + τ)

∂vτ= λ1(t)(1 − u⋆

1(t))βx⋆(t) + χ[0,T−τ](t)λ2(t + τ)βe−a

λ′5(t) = −

∂H

∂w= ρ2 + (λ2(t)− λ3(t))αy⋆(t) + λ5(t)q,

161

Theorem 6.4.3. The control pair (u⋆

1(t), u⋆

2(t)), which maximizes the objective functional

J over the region U can be written as

u⋆

1(t) = max{min{β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), 1}, 0},

u⋆

2(t) = max{min{λ4(t)e

−a2τ2ky⋆(t − τ2)

ν2, 1}, 0}.

Proof. : By using the optimality conditions, we get the following values,

∂H

∂u1= ν1u⋆

1(t)+ λ1(t)βx⋆(t)v⋆(t)− λ2(t)βx⋆(t − τ)v⋆(t − τ), (6.4.9)

and

∂H

∂u2= ν2u⋆

2(t)− λ4(t)e−a2τ2ky⋆(t − τ2). (6.4.10)

Solving equations (6.4.9) and (6.4.10) simultaneously for the optimal control pair

u⋆

1(t) and u⋆

2(t), we get

u⋆

1(t) =β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), (6.4.11)

u⋆

2(t) =λ4(t)e

−a2τ2ky⋆(t − τ2)

ν2. (6.4.12)

By using the property of control space, equations (6.4.11) and (6.4.12) can be writ-

ten as

u⋆

1(t) =

0, ifβν1

(λ2(t)e

−a1τ1 x⋆(t − τ1)v⋆(t − τ1))− λ1(t)x

⋆(t)v⋆(t))≤ 0,

βν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), if

0 <βν1

(λ2(t)e

−aτ x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

)< 1,

1, ifβν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

)≥ 1.

162

u⋆

2(t) =

0, ife−a2τ2 λ4(t)ky⋆(t−τ2)

ξ2≤ 0,

e−a2τ2 λ4(t)ky⋆

ν2, if 0 <

λ4(t)ky⋆(t)ξ2

< 1,

1, ife−a2τ2 λ4(t)ky⋆(t)

ν2≥ 1.

The above two equations for u⋆

1(t) and u⋆

2(t) can be written as

u⋆

1(t) = max{min{β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))

−λ1(t)x⋆(t)v⋆(t)

), 1}, 0}, (6.4.13)

and

u⋆

2(t) = max{min{λ4(t)e

−a2τ2ky⋆(t − τ2)

ν2, 1}, 0}. (6.4.14)

Here, we call formula (6.4.13) and (6.4.14) for u⋆

1(t) and u⋆

2(t) the characteriza-

tion of the optimal control. By using these control variables, we get the following

optimality system.

dx⋆(t)

dt= Λ − dx⋆(t)− βx⋆(t)v⋆(t)

(1 − max{min{

β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

),

dy⋆(t)

dt=

(1 − max{min{

β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ)v⋆(t − τ))− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

)

× βx⋆(t − τ)v⋆(t − τ)− ay⋆(t)− αw⋆(t)y⋆(t),

dz⋆(t)

dt= αy⋆(t)w⋆(t)− bz⋆(t),

dv⋆(t)

dt= k(1 − max{min{

λ4e−a2τ2y⋆(t − τ2)

ν2, 1}, 0})y⋆(t)− pv⋆(t),

dw⋆(t)

dt= cz⋆(t)− qw⋆(t)ρ2 + (λ2(t)− λ3(t))αy⋆(t) + λ5(t)q,

(6.4.15)

163

along with equations (6.4.13) and (6.4.14), and initial conditions (6.4.2). The Hamil-

tonian H⋆ at

(x⋆, y⋆, z⋆, v⋆, w⋆, x⋆τ1, y⋆τ2

, v⋆τ1, u⋆

1 , u⋆

2 , λ1, λ1, λ2, λ3, λ4, λ5

),

is given by

H⋆(t) =1

2

(ν1(max{min{

β

ξ1

(λ2(t)e

−a1τ1 x⋆(t − τ1)v⋆(t − τ1))− λ1(t)x

⋆(t)v⋆(t)), 1}, 0})2

+ ν2(max{min{λ4(t)ky⋆(t − τ2)

ν2, 1}, 0})2

)− Ax⋆(t)

+ λ1

[Λ − dx⋆(t)−

(1 − max{min{

β

ξ1

(λ2(t)e

−a1τ1 x⋆(t − τ1)v⋆(t − τ1))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

)βx⋆(t)v⋆(t)

]

+ λ2

[(1 − max{min{

β

ν1

(λ2(t)e

−a1τ1 x⋆(t − τ1)v⋆(t − τ1))

− λ1(t)x⋆(t)v⋆(t)

), 1}, 0}

)βe−a1τ1 x⋆(t − τ1)v

⋆(t − τ1)− ay⋆(t)− αw⋆(t)y⋆(t)

]

+ λ3

[αw⋆(t)y⋆(t)− bz⋆(t)

]

+ λ4

[k(1 − max{min{

λ4(t)ke−a2τ2y⋆(t − τ2)

ν2, 1}, 0}

)y⋆(t − τ2)− pv⋆(t)

]

+ λ5

[cz⋆(t)− qw⋆(t)

].

(6.4.16)

To find out the optimal control and state variables, we will solve numerically

(6.4.15) and(6.4.16).

164

6.4.2 Numerical Results of the Optimal Strategy

In this section, we consider the numerical solution of our control problem. For this

purpose we consider the the values given in table 6.2. Figures 1− 5 are the simula-

tion results from which we can conclude the effectiveness of drug therapies based

on the densities of uninfected cells, infected ells,free virus, double infected cells

and recombinant virus. Fig 6.12 shows that the density of uninfected target cells

with and without control. We see that without treatment, the density of uninfected

cells decreases drastically. But after treatment the number of cells is maintained

from the beginning to the end of the period. Fig 6.13 represents the concentration

of infected CD4+ T cells with and without control. The concentration of infected

cells decreases rapidly right from the very beginning of treatment and throughout

the period of investigation, while the concentration of infected cells without treat-

ment increases. Similarly, Fig 6.14 shows that the concentration of double infected

cells with treatment and without treatment. The number of double infected cells

decreases after treatment as wee see in Fig 6.14 that number of single infected cells

decreases. Moreover, there is no direct effect of (RTIs) and (PIs) on the density of

double infected cells. From fig 6.15, we see that the viral load increases drastically

without treatments but with treatments there is no increase in the concentration of

free virus. Fig 6.6 represents that there in no effect of our optimal control strategies

on the density of recombinant virus as our focus is only to use the drugs which

have the effect only on the reducing the number of pathogen virus.

165

Table 6.3: The Values of Parameters Used for Numerical Simulation

Parameters Definition Values with sources

λ Production rate of host cell 4 cell/mm3 [13]d Death rate of host cell 0.01 [16]β Infection rate of host cell by virus 0.004 mm3/vir [17]a Death rate of HIV-1 infected cell 0.09[18]α Infection rate by recombinant 0.004 [17]b Death rate of double-infected cell 1 (assumed)k HIV-1 production rate by a cell 0.02 vir/cell [13]p Removal rate of HIV-1 0.004 (assumed)c Production rate of recombinant

by a double-infected cell 0.05 vir/cell (assumed)q Removal rate of recombinant 1 (assumed)a1 rate of death of cells before infection 0.05 [13]a2 rate of clearness of virus

before attachment to cells 0.01 [13]τ1 latent period 1 [13]τ2 virus production period 1 [13]ξ1 Weight Constant 9000 (assumed)ξ2 Weight Constant 770 (assumed)A Constant 100 (assumed)

166

0 5 10 155

10

15

20

25

30

Time(weeks)

Sus

cept

ible

cel

ls

Control in the susceptible cells


Figure 6.12: The graph represents the density of uninfected cells verses time t inweeks.

0 5 10 152

4

6

8

10

12

14

Time(weeks)

dens

ity o

f inf

ecte

d ce

lls

Control in infected cells


Figure 6.13: The graph represents the density of infected cells verses time t inweeks.

167

0 5 10 15

2.7

2.75

2.8

2.85

2.9

2.95

3

3.05

Time(weeks)

dens

ity o

f dou

ble

infe

cted

cel

ls

Control in double infected cells


Figure 6.14: The graph represents the density of double infected cells verses time tin weeks.

0 5 10 150

5

10

15

20

25

30

35

40

45

Time(weeks)

dens

ity o

f Pat

hoge

n vi

ruse

s

Control in the pathogen viruses


Figure 6.15: The graph represents the density of pathogen virus verses time t inweeks.

168

0 5 10 150

0.5

1

1.5

2

2.5

3

Time(weeks)

dens

ity o

f rec

ombi

nant

viru

ses

Control in the recombinant viruses


Figure 6.16: The graph represents the density of recombinant virus verses time t inweeks.

0 5 10 15−0.05

0

0.05

0.1

0.15

0.2

Time (Weeks)

Con

trol

var

iabl

es u

1 an

d u2

Control variables in HIV−1 model

u1u2

Figure 6.17: The graph shows the behavior of control pair in the optimal controlproblem.

169

6.5 Conclusion and Discussion

In this chapter, optimal control strategies of three mathematical HIV-1

infection models were discussed. Presently, there is no effective therapy for HIV-1

infection in order to cure this infection. Different treatments are used to block the

evolution of the viruses in the body. Moreover, the cost of treatment is beyond

the reach of many infected patients. Hence, an optimal therapy was applied to

HIV-1 infection model ( with time delay without time delay) of fighting a virus

with another virus, in order to minimize the cost of treatment, reduce the viral

load and boost the immune response. Therefore, the first HIV-1 model is proposed

without time delay and the optimal control strategy is applied to this model. This

strategy aims to block new infection and prevent viral production by using drug

therapy with minimum side effects. Our numerical results shown that the viral

load reducing after treatment and the concentration of uninfected CD4+ T popu-

lation increases. Next, the single delayed HIV-1 model with two controls variables

was presented in detail. Here the optimal therapy is used in order to minimize the

cost of treatment, reduce the viral load and improve the immune response. Two

controls variables which measure the efficacy of reverse transcriptase and protease

inhibitors, respectively are used. Also, an efficient numerical method is devel-

oped to identify the best treatment strategy of HIV-1 infection in order to block the

new infection, increase the density of recombinant viruses and keep the number of

pathogen minimum by practicing drug therapy with minimum side effects. The

numerical simulations shown that the viral load drops off after treatment while the

densities of uninfected cells CD4+ cells and recombinant virus increases. Further,

170

the double delayed HIV-1 infection mathematical model with two controls vari-

ables was taken into account. The mathematical analysis of the proposed model

shows the effectiveness of the model in increasing the density of uninfected CD4+

T cells, reducing the density of infected cells and free virions in the body with a

minimum side effects. Also, our proposed strategy suggests that how one can min-

imize the cost of treatment. Certainly, these results could be useful in developing

improved treatment regimen for addressing the challenge of HIV-1/AIDS.

Chapter 7

Analytical Study of Fractional OrderHIV-1 Model

7.1 Overview

In this chapter, HIV-1 model of ordinary differential equation is extended

to fractional order model. Then, the Laplace-Adomian decomposition method is

used to find an analytical solution of the system of nonlinear fractional differential

equations. The convergence of the proposed method is presented. The obtained

results are compared with the available results of integer-order derivatives. Nu-

merical results shown that the Laplace-Adomian decomposition method (L-ADM)

is very simple and accurate for solving fractional-order HIV-1 infection model.

7.1.1 Formulation of the Model

The following HIV-1 infection model for the dynamics of HIV infection

which consists of five the following subclasses: the density of uninfected CD4+

T cells x(t), the productively infected CD4+ T cells y(t), the double infected cells

z(t), the concentration of pathogen virus v(t) and the concentration of recombinant

171

172

virus (w(t)).

dx


dy

dt= βx(t)v(t) − ay(t) − αw(t)y(t),

dz

dt= aw(t)y(t) − bz(t), (7.1.1)

dv


dt


with given initial conditions x(0) = m1, y(0) = m2, z(0) = m3, v(0) = m4 and

w(0) = m5.

The rate of generation of uninfected cells is denoted by λ. d is the rate of death

of infected cells. β is the constant rate at which a T-cell is contacted by the virus

and a is the natural death rate of cells. The rate of production of pathogen from

the infected cells is denoted by k. α is the constant rate of infected CD4+ T cells

with recombinant virus. b, p and q are the death rates of infected cells, pathogen

virus and the recombinant virus, respectively. The new system is described by the

following system of fractional differential equations

cDσx(t) = λ − dx(t)− βx(t)v(t),

cDσy(t) = βx(t)v(t) − ay(t) − αw(t)y(t),

cDσz(t) = αw(t)y(t) − bz(t), (7.1.2)

cDσv(t) = ky(t) − pv(t),

cDσw(t) = cz(t) − qw(t),

where cDσ represents Caputo fractional derivative.

173

7.1.2 Uniqueness of Solution

In this section, we prove the uniqueness of the solution.

Theorem 7.1.1. There is a unique solution for the initial value problem (7.1.2) under

given initial conditions and the solution remains in R+5 .

Proof. We know that the solution on (0, ∞) of the initial value problem exists and

is unique by using the Theorem 3.1 and Remark 3.2 of [98]. Next, we will prove

that the nonnegative octant R+5 is a positively invariant region. For this, we have to

show that on each hyperplane bounding the nonnegative orthant the vector field

points to R+5 . From the system (2), we obtain dx

dt |x=0 = λ ≥ 0,dydt |y=0 = βxv ≥ 0,

dvdt |z=0 = αwy ≥ 0, dv

dt |v=0 = ky ≥ 0, dwdt |v=0 = cz.

Therefore, the solution will remain in R+5 .

7.1.3 The Laplace Adomian Decomposition Method (L-ADM)

In 1980, Adomian decomposition method (ADM) was identified by Ado-

mian. This method is effective for finding numerical and explicit solution of a sys-

tem of differential equations which represent physical problems. This numerical

method works efficiently for a system of deterministic as well as stochastic differ-

ential equations. The system of linear and nonlinear boundary value problems or

Cauchy problems can also be solved by this method. This section is devoted to the

general procedure for the solution of the model (7.1.2) with given initial conditions.

174

By applying Laplace transform on both sides of the model (7.1.2), we get

sσL{x(t)} − sσ−1x(0) = L{λ − dx(t)− βx(t)v(t)},

sσL{y(t)} − sσ−1y(0) = L{βx(t)v(t) − ay(t) − αw(t)y(t)},

sσL{z(t)} − sσ−1z(0) = L{αw(t)y(t) − bz(t)}, (7.1.3)

sσL{v(t)} − sσ−1v(0) = L{ky(t) − pv(t)},

sσL{w(t)} − sσ−1w(0) = L{cz(t) − qw(t)}.

After using the initial conditions (7.1.3), we get

L{x(t)} =m1

s+

1

sσL{λ − dx(t)− βx(t)v(t)},

L{y(t)} =m2

s+

1

sσL{βx(t)v(t) − ay(t) − αw(t)y(t)},

L{z(t)} =m3

s+

1

sσL{αy(t)w(t) − bz(t)}

L{v(t)} =m4

s+

1

sσL{ky(t) − pv(t)},

L{w(t)} =m5

s+

1

sσL{cz(t)− qw(t)}.

(7.1.4)

The solutions x(t), y(t), z(t), v(t) and w(t) in the form of infinite series can be writ-

ten as

x(t) =∞

∑i=0

xn,

y(t) =∞

∑i=0

yn,

z(t) =∞

∑i=0

zn,

v(t) =∞

∑i=0

vn,

w(t) =∞

∑i=0

wn.

(7.1.5)

175

The nonlinear terms involved in the model are x(t)y(t) and y(t)w(t) are decom-

posed by Adomian polynomial as

x(t)y(t) =∞

∑i=0

Gn, y(t)z(t) =∞

∑i=0

Hn, (7.1.6)

where Gn and Hn are Adomian polynomials defined as

Gn =1

Γ(n + 1)

dn

dηn

[n

∑k=0

ηkxk

n

∑k=0

ηkyk

] ∣∣∣∣η=0

,

Hn =1

Γ(n + 1)

dn

dηn

[n

∑k=0

ηkyk

n

∑k=0

ηkzk

] ∣∣∣∣η=0

.

(7.1.7)

By using the system (7.1.6) and the system (7.1.5) in the model (7.1.4), we get

L(x0) =m1

s, L(y0) =

m2

s, L(z0) =

m3

s,L(v0) =

m4

s, L(w0) =

m5

s, (7.1.8)

L(x1) =1

sσ

(λ − dx0 − βG0

),

L(y1) =1

sσ

(βG0 − ay0 − αH0

),

L(z1) =1

sσ

(αH0 − bz0

),

L(v1) =1

sσ

(ky0 − pv0

),

L(w1) =1

sσ

(cz0 − qw0

),

(7.1.9)

...

L(xn+1) =1

sσ

(λ − dxn − βGn

),

L(yn+1) =1

sσ

(βGn − ayn − αHn

),

L(zn+1) =1

sσ

(αHn − bzn

),

L(vn+1) =1

sσ

(kyn − pvn

),

L(wn+1) =1

sσ

(czn − qwn

).

(7.1.10)

176

Next, we discuss the mathematical behavior of the solution x(t), y(t), z(t), v(t) and

w(t) for the different values of n. By applying the inverse Laplace transform to

both sides of the system (7.1.8) the values of x0, y0, z0, v0 and w0 are obtained. Sub-

stituting these values in the system (10), the first components x1, y1, z1, v1 and w1

can be found. The other terms of x2, y2, z2, v2 and w2 can be calculated recursively

in a similar way and we can write the solution in the form

x(t) = x0 + x1 + x2 + x3...,

y(t) = y0 + y1 + y2 + y3...,

z(t) = z0 + z1 + z2 + z3...,

v(t) = v0 + v1 + v2 + v3...,

w(t) = w0 + w1 + w2 + w3....

(7.1.11)

In the next section, we find numerical solution of the above system.


An analytical approximate solution in terms of an infinite power se-

ries can be obtained by using L-ADM. For numerical simulation, we consider the

values of parameters from [13]. The first few components of L-ADM solution

x(t), y(t), z(t), v(t) and w(t) are calculated. In order to do this, we computed the

first three terms of the L-ADM series solution for the system (7.1.2) which can be

written after simplification

x0 = 3, y0 = 6, z0 = 3, v0 = 149, w0 = 1.(7.1.12)

177

By using the initial values, the first approximation becomes

x1 = 3 + (1.4936)tσ

Γ(σ + 1),

y1 = 6 − (2.476)tσ

Γ(σ + 1),

z1 = 3 − (5.9928)tσ

Γ(σ + 1),

v1 = 149 + (299.89)tσ5

Γ(σ + 1),

w1 = 1 + (117)tσ

Γ(σ + 1).

(7.1.13)

For the second approximation, we use the initial and first approximate values to

get

x2 = 0.3t2σ+1

Γ(2σ + 2)− 014336

t2σ

Γ(2σ + 1)− 0.72238

tσ

Γ(σ + 1),

y2 = 3tσ

Γ(σ + 1)− 0.3242

tσ+1

Γ(σ + 2)− 0.508912

t2σ+1

Γ(2σ + 2),

z2 = 0.0144tσ+1

Γ(σ + 2)− 5.41856

t2σ+1

Γ(2σ + 2),

v2 = 299.8212tσ

Γ(σ + 1)+ 296

t2σ

Γ(2σ + 1),

w2 = 117tσ

Γ(σ + 1)− 287.88

t2σ

Γ(2σ + 1).

(7.1.14)

Similarly, we can get other approximations. Thus, the L-ADM series solution of

the system (7.1.5) at σ = 1, is identical to the results presented in [33]. By using

178

L-ADM, the solution after three terms becomes

x(t) = 3 + (4.5136)t − (1.0728)t2 − 0.5600t3,

y(t) = 6 + (8.9928)t − (34.42)t2 − (0.5089121)t3,

z(t) = 3 − (5.9928)t + (0.0014)t2 − (5.41856)t3,

v(t) = 149 + (598.8212)t + (296)t2,

w(t) = 1 + (234)t − (287.88)t2.

(7.1.15)

For σ = 0.95, we get the following system of series

x(t) = 6 + 0.7870550345t0.95 − 0.7845218562t1.9 + 0.5661093291t2.90,

y(t) = 12 + 1.773297592t0.95 − 0.7845218562t1.90 + 0.5661093291t2.90,

z(t) = 6 − 6.115846854t0.95 − 1.022499122t2.90 + 0.7536239617t1.95,

v(t) = 298 + 612.0247390t0.95 + 162.4321420t1.90,

w(t) = 2 + 238.8045928t0.95 − 157.5391685t1.90.

(7.1.16)

Next, we choose σ = 0.85, to get the following series after some modification

x(t) = 6 + 0.8155783471t0.85 − 0.9280851532t1.70 + 0.7193120298t2.70,

y(t) = 12 + 1.837562884t0.85 − 0.9280851532t1.70 + 0.7193120298t2.70,

z(t) = 6 − 6.337488549t0.85 + 0.8231484547t1.85 − 1.299211797t2.70,

v(t) = 298+ 634.2048563t0.85 + 192.1563538t1.70,

w(t) = 2 + 247.4590042t0.85 − 186.3679924t1.70.

(7.1.17)

179

Similarly, the solution after three terms for σ = 0.75 is given by

x(t) = 6 + 0.8391376836t0.75 − 0.1078429583t1.50 + 0.9027033339et2.50,

y(t) = 12 + 1.890643943t0.75 − 0.1078429583t1.50 + 0.9027033339t2.50,

z(t) = 6 − 6.520557442t0.75 + 0.8953222645t1.75 − 1.630450726t2.50,

v(t) = 298 + 652.5249180t0.75 + 223.2845723t1.50,

w(t) = 2 + 254.6072690t0.75 − 216.5585298t1.50.

(7.1.18)

0 10 20 30 40 50

t

1

2

3

4

xHtL

Figure 7.1: Solution behavior of the uninfected CD4+ T cells x for σ = 1 (solid line)σ = 0.95 (dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line).

180

0 10 20 30 40 50

t

2

4

6

8

10

yHtL

Figure 7.2: Solution behavior of the productively infected CD4+ T cells y for σ = 1(solid line) σ = 0.95 (dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line).

0 10 20 30 40 50t

2

4

6

8

10

zHtL

Figure 7.3: Solution behavior of double infected cells z for σ = 1(solid line) σ =0.95 (dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line).

181

0 10 20 30 40 50t

50

100

150

200

ΝHtL

Figure 7.4: Solution behavior of pathogen virus v for σ = 1(solid line) σ = 0.95(dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line).

0 10 20 30 40 50t

2

4

6

8

10

12

�HtL

Figure 7.5: Solution behavior of the recombinant virus w for σ = 1 (solid line)σ = 0.95(dashed line) σ = 0.85 (dot-dashed line) σ = 0.75 (dotted line).

182

Figures (7.1) to (7.5) represent the numerical solutions of the fractional order

model of the uninfected CD4+ T cells x(t), the productively infected CD4+ T cells

y(t), double infected cells z(t), pathogen virus v(t) and the recombinant virus w(t),

respectively at time t. From the graphical results in Figure (7.1) to (7.5), it can be

seen that the results obtained by using Laplace-Adomian decomposition method

are accurate representation of HIV-1 infection model. Therefore, the presented

method can predict the behavior of these variables accurately for the region un-

der consideration. All these figures shows the approximate solutions for different

values of σ = 1, 0.95, 0.85, 0.75 using Laplace-Adomian decomposition method.

All these figures show that the approximate solutions depend continuously on the

time-fractional derivative. Also, it is clear that the efficiency of this access can be

dramatically increased by decreasing the step size.

7.1.5 Convergence Analysis

The above solution is a series solution, which is rapidly and uniformly converge

to the exact solution. To check the convergence of the series, we use classical tech-

niques [99]). For sufficient conditions of convergence of this method, we discuss

the following theorem by using the idea presented in [100].

Theorem 7.1.2. Let X be a Banach spaces and T : X → X be a contractive nonlinear

operator such that for all x∗, x′∈ X, ||T(x∗) − T(x∗

′)|| ≤ k||x∗ − x∗

′||, 0 < k < 1.

Then by Banach contraction principle T has a unique point x⋆ such that Tx⋆ = x⋆, where

x⋆ = (x, y, z, v, w). The given sereies can be written by applying Adomian decomposition

method as follows

x⋆n = Tx⋆n−1, x⋆n−1 =n−1

∑i=1

x⋆i , n = 1, 2, 3, ...,

183

and assume that x⋆0 = x⋆0 ∈ Br(x) where Br(x⋆) = x⋆′∈ X : ||x⋆

′− x⋆|| < r, then, we

have

(i) x⋆n ∈ Br(x⋆);

(ii) limn→∞

x⋆n = x⋆.

Proof. To prove (i), we use mathematical induction. For n = 1, we have

||x⋆0 − x|| = ||T(x⋆0)− T(x⋆)|| ≤ k||x⋆0 − x⋆||.

Let the result is true for n − 1, then

||x⋆0 − x⋆|| ≤ kn−1||x⋆0 − x⋆||.

We have

||x⋆n − x⋆|| = ||T(x⋆n−1)− T(x⋆)|| ≤ k||x⋆n−1 − x⋆|| ≤ kn||x⋆0 − x⋆||.

Hence, by using the above inequality, we have

||x⋆n − x⋆|| ≤ kn||x⋆0 − x⋆|| ≤ knr < r,

which implies that x⋆n ∈ Br(x⋆).

(ii) Since ||x⋆n − x⋆|| ≤ kn||x⋆0 − x⋆|| and as limn→∞ kn = 0.

So, we have limn→∞ ||x⋆n − x⋆|| = 0 ⇒ limn→∞ x⋆n = x⋆.

7.2 Conclusion

We have developed in this work a fractional order HIV-1 epidemic model.

The present work describes the spread of an HIV-1 infection in body. We used the

184

Laplace-Adomian decomposition method to obtain the solution of the fractional

epidemic model. The comparison for different values of σ has been presented

and numerical simulations are discussed. The convergence of the method is also

presented.

Chapter 8

Summary and Suggestions for furtherDirections

The work presented in this thesis focuses on mathematical analysis and

control strategies of HIV-1 infection model. Time delays models are introduced.

These time delays are introduced in latent period, rate of contact between virus

and uninfected cells and virus production periods. These time delays have main

contribution in prolonging the time duration of HIV-1 to AIDS. The other pro-

posed control strategies are the incorporation of saturation mass action principle

and the recovery of unproductively infected cells to uninfected cells. Two con-

trols which measure the efficacy of reverse transcriptase and protease inhibitors,

respectively are also used. Then, optimal control strategies are applied to HIV-1 in-

fection model to develop strategies for reducing the concentration of infected cells

and virus in the body. Next, another model is formulated by converting the inte-

ger order model to fractional order epidemic model of HIV-1. Laplace-Adomian

decomposition method is used to find an analytical solution of the system of non-

linear fractional differential equations. For completing this research work, It is

divided into 8 chapters.

185

186

The first chapter is related to the history of HIV-1 infection. In this chapter origin

and transmission of HIV-1 are briefly introduced. The history mathematical mod-

eling of HIV-1 infection is also discussed.

In the second chapter, basic definitions related to mathematical modeling are pre-

sented. The basic material about definitions of equilibrium points, local and global

stability, Lyapunove functions and invariance principle are presented. Delay dif-

ferential equations and stability techniques of their solutions are also discussed.

Next, optimal control theory, that is, Hamiltonian, Lagrangien, optimality condi-

tions and Pontryagin’s maximum (or minimum) principle are also presented. The

last section of this chapter is devoted to the basic definitions of Fractional calculus.

In this section, Caputo’s approach to fractional calculus, fractional integral and Ca-

puto’s definition of fractional derivative are discussed.

The third chapter is devoted to the formulation and mathematical analysis of the

three delayed HIV-1 infection models. The first model describes the delayed HIV-1

infection model which interacts between uninfected cells, infected cells, pathogen

virus and CTLs response. It can be seen that R0 is independent of the parame-

ters of the activation of CTLs. All the possible equilibrium points of this model

are found. Further the local behavior of this model is found using Routh Hurwits

criterion and global behavior is determined by using Lyapunov functional the-

ory. The numerical analysis of the proposed model is also presented. The second

proposed HIV-1 infection model of fighting a virus with another virus is formu-

lated for time delays in contact process and time delay in latent period. Then,

the uniqueness, positivity and boundedness of the solution of the proposed model

are found. The equilibria points and basic reproductive numbers are determined.

187

Then, using certain conditions on these reproductive numbers, local and global

stability of the equilibria points are found. It is shown that large time delays in

contact and infection processes can reduce this infection in the body. Numerical

solution are carried out to support the derived analytical results. Then, this study

is further extended to another delayed HIV-1 infection model by using two times

delays i.e., time delay in latent period and time delay in virus production periods.

The theory of stability analysis is used to prove the local and global stability of the

proposed model. Numerical simulation is carried out and it is shown that time can

prolong the time from HIV-1 to AIDS.

The fourth chapter is devoted to the applications of continuous time delays. Two

continues delayed HIV-1 models are formulated. The first HIV-1 model is single

(continuous) delayed model and the second proposed model is double (continues)

delayed HIV-1 model. For the first proposed model, all the basic properties are pre-

sented. The basic reproduction number R0 is found. It is proved that if R0 < 1, the

infection-free steady state is globally asymptotically stable and if R0 > 1, then the

infection-free steady state becomes unstable and there occurs an infected steady

state which is asymptotically stable. From our derived results, it has been deduced

that to control the concentration of the virus and infected cells, a strategy should

be plan to bring the value of the basic reproduction number below one. The sec-

ond delayed HIV-1 infection with time delays in the latent period and the virus

production period, are considered. The global stability of the infection-free equi-

librium and the single infection equilibrium of the proposed model are completely

established under certain conditions on R0, Rs and Rd. It is clear from these results

that intracellular delays describing the latent period and viral production period

188

have great effect on the stability of feasible equilibria. Numerical results of this

model have shown that continuous delays in latent period and virus production

period can help in reducing the load of pathogen virus. Thus the concentration of

infected cells are reduced and CD4+ cells are increased.

The fifth chapter is the formulation of two new models. The first model is related

to the incorporation of cure rate in HIV-1 infection model. The existence, non-

negativity and boundedness of the solution of this model are presented. Then,

the basic reproduction number R0 is found and the stability analysis of the pro-

posed model is discussed. The numerical results have shown that depending on

the amount of drug administered, the viral load decreases and the CD4+ T popu-

lation grows significantly. It is also shown that the concentration of healthy cells,

infected cells and viral load stabilize around equilibrium points. Then, this model

is further extended to another HIV-1 infection model by incorporating saturation

response action. It is proved that this model has at most three equilibria. It is also

shown that when R0 < 1 , then the infected cells dies out and the infection free

equilibrium is globally stable. When R0 > 1, then it is proved that the infection is

persistent. Numerical simulations have indicated that depending on the fraction

of cells surviving the incubation period, the solutions approach either an infected

steady state or a periodic orbit.

The sixth chapter is the applications of optimal control strategies to three HIV-1

infection models. First, the HIV-1 infection model of fighting a virus with another

virus is considered and optimal therapy is applied. These control strategies aims

to minimize the cost of treatment, reduce the viral load and infected cells and boost

189

the immune response. The optimal controls are incorporated to represent the ef-

ficiency of drug treatment in inhibiting viral production and preventing new in-

fections. Existence for the optimal control pair is established and the Pontryagins

maximum principle is used to characterize these optimal controls. The optimal-

ity system is derived and solved numerically. Our numerical results have shown

that the viral load reducing after treatment and the CD4+ T population increases.

Morover, the optimal control theory is applied to single delayed HIV-1 mathe-

matical model with two controls variables. These controls variables are used to

measure the efficacy of reverse transcriptase and protease inhibitors, respectively.

Existence for the optimal control pair is established and the Pontryagins maximum

principle is used to characterize the optimal controls pair. An efficient numerical

method is developed to identify the best treatment strategy of HIV infection in or-

der to block new infection, increase the density of recombinant viruses and keep

the number of pathogen viruses minimum by practicing drug therapy with min-

imum side effects.Finally , the double delayed HIV-1 infection model with two

controls variables is presented. The optimal problem has shown the effectiveness

of the model in increasing the density of uninfected CD4+T cells, reducing the den-

sity of infected cells and free virions in the body with a minimum side effects. It is

concluded that using this strategy, one can minimize the cost of treatment. These

results could be useful in developing improved treatment regimen for addressing

the challenge of HIV/AIDS.

In chapter 7, the model is formulated by converting the integer order model to frac-

tional order epidemic model. Then, the Laplace-Adomian decomposition method

190

is used to find an analytical solution of the system of nonlinear fractional differ-

ential equations. The convergence of the proposed method is discussed. The new

results are compared with the available results of integer-order derivatives. Nu-

merical results show that the Laplace-Adomian decomposition method (L-ADM)

is very simple and accurate method for solving fractional-order HIV-1 infection

model.

8.1 Future Direction

In this research work, all the possible efforts have been made to study the

control strategies for reducing of HIV-1 infection in human body. However, this

study can be elaborated by incorporating some of the following assumptions: One

may consider the situation when the recombinant virus also infects susceptible

cells but at a lower rate. Moreover, the double hopf bifurcation of HIV-1 infection

model fighting a virus with an other virus may be studied. Further, the delayed

HIV-1 model of fighting a virus with another virus with waning term will be taken

in account and it will be proved that viral load differs that without virus waning.

The study of chaos and periodic solutions in HIV-1 model with time delay in cel-

lular immune response. Analysis for HIV-1 infection by fractional differential time

delay model with cure rate can be considered

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