STABILITY OF SIS AND SVIS MODELS WITH BEDDINGTON-DENGELIS FUNCTIONAl RESPONSES

A thesis Submitted to the Council of Faculty of Science and Science Education School of Science at the University of Sulaimani in partial fulfillment of the requirements for the degree of Master of Science in Mathematics (Dynamical Systems)

By Mediya B. Mrakhan B.Sc. Mathematics (2011), University of Garmian Supervised by Assist. Prof. Dr. Kawa A. Hasan and

Assist. Prof. Dr. Mudhafar F. Hama

March, 2016

Rashamee, 2716

SUPERVISOR CERTIFICATION

We certify that the preparation of thesis titled "Stability of SIS and SVIS Models with Beddington-Deangelis Functional Responses" accomplished by (Mediya B. Mrakhan), was prepared under my supervision in the School of Science, Faculty of Science and Science Education at the University of Sulaimani, as partial fulfillment of the requirements for the degree of Master of Science in (Mathematics).

Signatures: Name: Dr. Kawa A. Hasan Title: Assistant Professor Date: / / 2016 Signatures: Name: Dr. Mudhafar F. Hama Title: Assistant Professor Date: / / 2016

In view of the available recommendation, I forward this thesis for debate by the examining committee. Signature: Name: Prof. Dr. Karwan H. Faraj Title: Professor Date: / /2016

Linguistic Evaluation Certification

I herby certify that this thesis titled "Stability of SIS and SVIS Models with Beddington-Deangelis Functional Responses" prepared by (Mediya B. Mrakhan), has been read and checked and after indicating all the grammatical and spelling mistakes; the thesis was given again to the candidate to make the adequate corrections. After the second reading, I found that the candidate corrected the indicated mistakes. Therefore, I certify that this thesis is free from mistakes.

Signature: Name: Position: English Department, School of Languages, University of Sulaimani Date: / / 2016

EXAMINATION COMMITTEE CERTIFICATION

We certify that we have read this thesis entitled "Stability of SIS and SVIS Models with Beddington-Deangelis Functional Responses" prepared by (Mediya B. Mrakhan), and as Examining Committee, examined the student in its content and in what is connected with it, and in our opinion it meets the basic requirements toward the degree of Master of Science in Mathematic "Dynamic systam".

Signature Name: Dr. Karwan Hama Faraj Jwamer Title: Professor Date: / /2016 (Chairman)

Signature Name: Dr. Arkan N. Mustafa Title: Lecture Date: / /2016 (Member)

Signature Name: Dr. Shazad Shawki Ahmed Title: Assistant Professor Date: / /2016 (Member)

Signature Name: Dr. Mudhafar F. H. Title: Assistant Professor Date: / /2016 (Supervisor-Member)

Signature Name: Dr. Kawa A. H. Title: Assistant Professor Date: / /2016 (Supervisor-Member)

Approved by the Dean of the Faculty of Science and Science Education. Signature Name: Dr. Bakhtiar Q. Aziz Title: Professor Date: / /2016

DEDICATION

Dedicated to: • My dear Parents. • My brothers and sisters. • All my friends, special thanks Rzgar F. Mahmood.

With love and Respect.

Mediya B. Mrakhan Kurdistan- Sulaimani 2016

ACKNOWLEDGMENTS First, I am grateful to the God for the good health and wellbeing that were necessary to complete this thesis. I would like to express my gratitude to my supervisors Dr. Kawa A. Hasan and Dr. Muzafar F. Hama for the useful comments, remarks and engagement through the learning process of this thesis. I would like to extend my gratitude and thanks to the dean of the Faculty of Science and Science Education, School of Science Prof. Dr. Bakhtiar Qader Aziz. Also I would like to thanks prof. Dr. Karwan Hama Faraj Jwamer the head of Mathematics Department and Dr. Mudhafar F. Hama the dcider of Mathematics Department in the Faculty of Science and Science Education, School of Science. My very sincere thanks are due to Prof. Dr. Adil K. Jabbar for his assistance. My sincere thanks also goes to all my teachers who taught me during the first and second course of study in Masters courses. A special thanks to my family. Words cannot express how grateful I am to my mother, father, my sisters and brothers for all of the sacrifices that they’ve made on my behalf. Many thanks also for my best friend Rzgar F. Mahmood who always helped me and gave me the power for studying. I would also like to thank all of my friends who supported me in writing, and incented me to strive towards my goal.

Mediya B. Mrakhan Kurdistan- Sulaimani 2016

ABSTRACT

The objective of this thesis is to study the effect of some epidemic factors such as treatment, immigration, and vaccination on the dynamical behaviour of some epidemic models. Two types of epidemiological models are proposed and analyzed analytically as well as numerically. In the first model, the global dynamics of SIS model with saturated βSI

incidence 1+ aS+bI and saturated treatment function is investigated. A sufficient condition for the existence of equilibrium points is obtained and the dynamical behaviour of the model is discussed. Also, the global asymptotic stability of the disease - free and endemic equilibria is studied. The second model, which is the SVIS model with non-linear β I

incidence rate 1+bI2+cI 2 is studied. The effect of immigrants on the dynamical behaviour of SVIS model is considered analytically. And the sufficient condition for the existence and stability of the endemic equilibrium point is obtained .The global dynamics of the model is studied by solving it numerically for different sets of initial values and for different sets of parameter values. It has been concluded that the system is approaches either to the disease-free equilibrium point or to endemic equilibrium point.

I

Contents Abstact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Chapter One General Introduction 1.1 Introduction . . . . . . . . . . . 1.2 Basic Concepts . . . . . . . . . 1.3 Mathematical tools . . . . . . . 1.3.1 Local stability analysis . 1.3.2 Two-dimensional flow . 1.3.3 Routh-Hurwitz criterion 1.3.4 Descarte’s rule of signs . 1.3.5 Lyapunov method . . . .

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1 1 7 7 11 11 13 13

Chapter Two The Dynamics of SIS Epidemic Model with Effect of Treatment 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Existence and Stability analysis of SIS epidemic model without treatment . . . . . . . . . . . . . . . . . . . . . 2.2.2 Stability analysis of an SIS epidemic model with treatment for system (2.2) . . . . . . . . . . . . . . . . 2.2.3 Stability analysis of an SIS epidemic model with treatment for system (2.3) . . . . . . . . . . . . . . . .

16 17 19 24 28

2.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter Three The Dynamics of SVIS Epidemic Model 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . 49 II

3.3 3.4 3.5 3.6

Existence of Equilibrium point of system (3.1) Stability Analysis . . . . . . . . . . . . . . . . . Globally Stability Analysis . . . . . . . . . . . Numerical analysis . . . . . . . . . . . . . . . .

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51 54 59 64

Chapter Four Discussion and conclusions 4.1 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . 73 4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII

III

List of Figures 2.1 Block diagram of system (2.1) . . . . . . . . . . . . . . . . . . 2.2 Phase plot of system (2.4) starting from different initial points (50, 50), (150, 150) and (300, 300). . . . . . . . . . . . . 2.3 Time series of attractor given in Fig.(2.2). . . . . . . . . . . . 2.4 Phase plot of system (2.4) starting from different initial points (300, 15), (1000, 50) and (1000, 30). . . . . . . . . . . . 2.5 Time series of attractor given in Fig.(2.4). . . . . . . . . . . . 2.6 Phase plot of system (2.2) starting from different initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Time series of attractor given in Fig.(2.6). . . . . . . . . . . . 2.8 Phase plot of system (2.2) starting from different initial condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Time series of attractor given in Fig.(2.8). . . . . . . . . . . . 2.10 Phase plot system (2.3) starting from different initial points. 2.11 Time series of attractor given in Fig.(2.10). . . . . . . . . . . 2.12 Time series of the solution of system (2.3) . . . . . . . . . . . 2.13 Time series of the solution of system (2.3) . . . . . . . . . . . 3.1 Block diagram of system (3.1) . . . . . . . . . . . . . . . . . . 3.2 This figure depicts the trajectories of the model equation (3.9) with the initial conditions I0 , I1 and I2 . In this case, all solutions converges to the disease-free equilibrium point. 3.3 Time series of trajectories of system (3.1) . . . . . . . . . . . 3.4 This figure depicts the trajectories of the model equation (3.9) with initial conditions I3 , I4 and I5 . In this case, all solutions converges to the endemic equilibrium point E∗ . . 3.5 Time series of trajectories of system for data given in equation (3.9) with ρ = 0.2 . . . . . . . . . . . . . . . . . . . . . . 3.6 Time series of the solution of system (3.1) . . . . . . . . . . . IV

17 39 40 40 41 42 42 43 43 44 44 45 47 50

65 65

66 66 67

3.7 Time series of the solution of system (3.1) 3.8 Time series of the solution of system (3.1) 3.9 Time series of trajectories for system (3.1) 3.10 Time series of the solution of system (3.1)

V

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68 69 70 71

List of Tables 2.1 The effect of the varying infection rate of system (2.3). . . . 45 2.2 The effect of the varying infection rate of system (2.3). . . . 46

VI

List of Abbreviations

Symbols

Description

DF ( X, µ)

Jacobian matrix.

Rn+

{( x1 , x2 , · · · , xn ) ∈ Rn : xi ≥ 0, i = 1, 2, · · · , n}.

IntRn+

{( x1 , x2 , · · · , xn ) ∈ Rn : xi > 0, i = 1, 2, · · · , n}.

C0

Continous derivative of first order.

O (.)

Error.

T( I )

Treatment function.

VII

Chapter One General Introduction

Chapter One General Introduction

1.1

Introduction

In this chapter, various biological and epidemiological concepts, definitions and mathematical tools used in this thesis are presented.

1.2

Basic Concepts[7]

The study of the interactions of organisms (individual living creatures, either unicellular or multi-cellular) with their physical environment and with one another is referred to as an ecology. The group of organisms whose members have the same structural traits and who can interbreed with each other is known as a species. Any group of individuals, usually of a single species occupying a given area at the same time, is known as population. An ecosystem is a set of all species in a given area along with the encompassing physical environment. The application of mathematical methods to problems in ecology has emerged form a branch of ecology known as mathematical ecology. Types of interactions [21] There are three kinds of interactions that can occur between two species. They are described as follows i. Mutualism or symbiosis (+, +): Each species has an accelerating effect on the growth of the other. ii. Competition (-,-): The growth rate of each species is depressed (negatively effected) due to the existence of the other species. iii. prey-predator (-, +): The effects are such that the growth rate of one species is decreased (negatively effected), and of the other increased (positively effected).

1

Chapter 1

General Introduction

Prey-Predator model The general framework for a mathematical model that describes the dynamics between any two species, having prey -predator type of interaction, has the following structure. dx = g( x ) x − f ( x, y) dt dy = e f ( x, y) − d(y)y dt

(1.1)

Where x and y are the densities of the prey and predator respectively. Further, g( x )denotes the prey’s per-capita growth rate in the absence of predator. e is the conversion rate of the eaten prey into a new predator and d(y) is the natural mortality rate of the predator in the absence of the prey. The function f ( x, y) is called the functional response in the prey equation, while it is known as a numerical response in predator equation. Prey-dependent function The most commonly used trophic functions in ecological model are the term prey-dependent means that the consumption rate by each single predator is the only function of prey density, that is f ( x, y) = f ( x ). Functional response equations that are strictly prey-dependent such as the Holling family (type-I, II, III,IV) are predominant in the literature. Holling are identified by four basic types of functional responses which are described bellow [15]. i. Holling type-I, in which the consumption rate of individual consumers increases linearly with prey density and when the consumer satiated, then it becomes constant. It describes the wellknown Lotka-Volterra functional response with a fixed upper limit.

f (x) =

 αx

for 0 < x < α

γ

for α ≤ x

2

(1.2)

Chapter 1

General Introduction

Where α is the value of resource at which the consumer satiated at constant γ. Despite its simplicity, type-I and Lotka-Volterra functional responses [7] are still frequently used for simple ecological models. ii. Holling type-II or Holling disc equation, in which the consumption rate of individual consumers increases at a decreasing rate with prey density until it becomes constant at satiation level. It is a hyperbola, approaching the maximal value a = 1/h asymptotically.

f (x) =

ax Ax = 1 + Ahx b+x

(1.3)

Where A is a search rate, h is the time spent on handling of one prey, a is the maximum harvest rate and b is the half saturation biomass f (b) = ( 2a ) equation (1.3) remains the most popular among the ecologists, it is also known as Michaelis-Menten type, due to it is equivalent to model of enzyme kinetics developed in 1913 by Lenor Michaelis and Maude Menten [29]. iii. Holling type-III functional response, in which the consumption rate of individual consumers accelerates at first and then decelerates towards satiation. This type of functional response can be described as follows

f (x) =

Ax2 ax2 = 1 + Ahx2 b + x2

(1.4)

iv. Holling type-IV functional response, the type of functional response suggested by Andrews [2]. The Holling type-IV function and also called the Monod-Haldane function response is of the form f (x) =

Ax 1 + bx + Ahx2

3

(1.5)

Chapter 1

General Introduction

Sokol and Howell [35] proposed b = 0 to simplified Holling typeIV function of the form f (x) =

Ax ax = 1 + Ahx2 b + x2

(1.6)

Definition 1.1. [30] An Epidemic is an unusually large short-term out break of a disease, such as Cholera and Aids etc. The spread of disease depends on i. The mode of transmission, ii. Susceptibility, iii. Infections period, iv. Resistance, and many other factors. Definition 1.2. [30] The mathematical model that describes the contagious disease which spreading in the population is known as Epidemiological model. At most, the population in an epidemiological model divided into three classes: susceptible, infected, removal which can be denoted by S(t), I (t) and R(t) respectively. Here S(t) represents the number of susceptible individuals in the population who can be infected, I (t) represents the number of infected individuals in the population, that is, those with the disease who are actively transmitted it, while R(t) represent the number of removal individuals by recovery, immunization, death, or other means. Therefore, in the following number of known epidemiological models is stated those models which are formulated depending on the following assumptions i. The disease transmitted by contact between infected individuals and susceptible individuals.

4

Chapter 1

General Introduction

ii. All susceptible individuals are equally susceptible, all infected individuals are equally infections and removal individuals are equally removal. iii. The population is fixed in size that is there is no brith or immigration occurs and all deaths are taken into account. Hence if N is the population size, then

S(t) + I (t) + R(t) = N (t)

(1.7)

Surely, other assumptions can be adopted too. Epidemic Models [26, 30] By considering of what has been mentioned before, there are many epidemic models in nature. Let S(t) and I (t) represent the susceptible and the infected populations at time t. The total population is N (t) = S(t) + I (t). The most known and used models are i. SI model: SI epidemic models are used to model diseases such as influenza or the common cold, where generally no one is immune and over the course of the epidemic almost everyone eventually becomes infected. The models to be discussed are as follows (a) SI model without births and deaths dS = − Nβ SI dt dI = Nβ SI dt

(1.8a)

(b) SI model with births and deaths dS = µN − Nβ SI − µS dt dI = Nβ SI − µI dt

5

(1.8b)

Chapter 1

General Introduction

(c) SI model with immigration dS = λ − Nβ SI − µS dt dI = Nβ SI − µI dt

(1.8c)

ii. SIS model: In this model, SIS epidemic model has been applied to some sexually transmitted diseases, where there is recovery but no immunity; individuals can be infected immediately following recovery. The models to be discussed are as follows (a) SIS model without births and deaths dS = − Nβ SI + γI dt dI = Nβ SI − γI dt

(1.9a)

(b) SIS model with births and deaths dS = µN − Nβ SI + γI − µS dt dI = Nβ SI − γI − µI dt

(1.9b)

(c) SIS model with immigration dS = λ − Nβ SI + γI − µS dt dI = Nβ SI − γI − µI dt

(1.9c)

iii. SIR model: In this model, infected individuals are removed by death or hospitalization at a rate proportional to the number of infected, therefore the system which describes the dynamic of SIR model can be written as

6

Chapter 1

General Introduction

dS = −αSI dt dI = αSI − rI dt dR = rI dt

(1.10)

Where α, r are the infection rate and the removal rate respectively.

1.3

Mathematical tools

A number of mathematical tools are used throughout this thesis. The objective is to study the dynamical behavior of some epidemic models.

1.3.1

Local stability analysis

Consider the following system which consists of n-dimensional autonomous differential equations  dx1  = x1 f 1 ( x1 , x2 , . . . , xn ) = F1 ( x1 , x2 , . . . , xn )    dt       dx2  = x2 f 2 ( x1 , x2 , . . . , xn ) = F2 ( x1 , x2 , . . . , xn )  0 dt ⇒ X = F ( X ) (1.11)   ..   .       dxn  = xn f n ( x1 , x2 , . . . , xn ) = Fn ( x1 , x2 , . . . , xn ) dt Where X = ( x1 , x2 , . . . , xn ) ∈ Rn and F = ( F1 , F2 , . . . , Fn ) T ∈ Rn . The non-linear functions Fi ; i = 1, 2, . . . , n are continuously differentiable functions defined on U ⊂ Rn . Moreover, before going further to explain the linearization procedure for equation (1.11), some basic definitions needed to study the stability of system (1.11) are presented. Definition 1.3. [36] A point X ∗ = ( x1∗ , x2∗ , . . . , xn∗ ) is said to be an equilibrium point of system (1.11) if it satisfies the following equation Fi ( X ∗ ) = 0; ∀i = 1, 2, . . . , n. Definition 1.4 (Uniformly Bounded[20]). The solutions of (1.11) are uniformly bounded if, for each M∗ > 0, there is a M∗ = M∗ ( M∗) > 0 such 7

Chapter 1

General Introduction

that, for t0 ∈ R+ and ϕ ∈ C with k ϕk < M∗ , any solution x (t, t0 , ϕ) satisfies | x (t, t0 , ϕ)| < M∗ for all t = t0 . Definition 1.5. [36] An equilibrium point X ∗ of the system (1.11) is said to be stable if, for any ε > 0, there exists δ = δ(e) > 0 such that if k X (0) − X ∗ k < δ then k ϕ(t) − X ∗ k < ε for all t ≥ 0, where ϕ(t) is a solution of system (1.11) at time t. In addition if k ϕ(t) − X ∗ k → 0 as t → ∞, for all k X (0) − X ∗ k sufficiently small, then the equilibrium point X ∗ is said to be asymptotically stable. An equilibrium point which is not stable is said to be unstable. Definition 1.6. [36] An equilibrium point X ∗ of the system (1.11) is said to be globally asymptotically stable if it is asymptotically stable for any initial point X (0) ∈ U. According to the above definition, the basin of attraction of an equilibrium point X ∗ , which is defined by D ( X ∗ ) = { X (0) : ϕ(t) → X ∗ as t

→ ∞}, is equal to U if and only if it is globally asymptotically stable. Therefore, it is clear that if the equilibrium point X ∗ is globally asymptotically stable then it should be locally asymptotically stable, but the converse is not true. Definition 1.7. [31] The set of all ω −limit points of a flow given by the system (1.11) is called the ω −limit set. The α−limit set is similarly defined. Now, in order to determine the stability of X ∗ , we must understand the nature of the solutions of the system (1.11) near the equilibrium point X ∗ . Let ϕ(t) = X ∗ + E(t)

(1.12)

Where E(t) = ( E1 , E2 , . . . , En )(t) represents a small perturbation of the original solution ϕ(t) = ( ϕ1 (t), ϕ2 (t), . . . , ϕn (t)). Then depending on equation (1.12), the system (1.11) becomes dE = F ( X ∗ + E) dt 8

(1.13)

Chapter 1

General Introduction

Note that E(t) = 0 is an equilibrium point of this equation. So, by applying Tayler expansions, we obtain F ( X ∗ + E) = F ( X ∗ ) + DF ( X ∗ ) E + O(k Ek2 )

(1.14)

Here 

∂F1  ∂x  1  ∂F  2   ∂x ∗ DF ( X ) =  . 1  ..    ∂F  n ∂x1

 ∂F1 ∂xn   ∂F2    ∂xn  ..  .    ∂Fn   ··· ∂xn

∂F1 ··· ∂x2 ∂F2 ··· ∂x2 .. .. . . ∂Fn ∂x2

(1.15)

X=X∗

Represents the Variational matrix (or Jacobain matrix) of F evaluated at X ∗ . Since F ( X ∗ ) = 0, the system (1.14) simplifies to give dE = DF ( X ∗ ) E + O(k Ek2 ) dt

(1.16)

Accordingly, so as to be close to the origin, the system (1.16) can be written as follows dE ≈ DF ( X ∗ ) E = J ( E) dt

(1.17)

Note that, the equation (1.17) represents the linear approximate system of the system (1.11) at the equilibrium point X ∗ . Therefore, the stability of E(t) = 0 and hence ϕ(t) = X ∗ is given in the theorem (1.1). Definition 1.8. [31] Consider the differential equation dx = x˙ = f ( x ) dt

(1.18)

Where x = x (t) ∈ Rn is a vector valued function of an independent variable (usually time) and f : U → Rn is a smooth function defined on some subset U ⊂ Rn . System of the form (1.18), in which the vector 9

Chapter 1

General Introduction

field does not contain time explicitly, are called autonomous. Definition 1.9. [4] The system y0 = f (t, y) Is said to be asymptotically autonomous on the set Ω if and only if i. limt→∞ f (t, y) = h(y) for y ∈ Ω and this convergence is uniform for y in closed bounded subsets of Ω. ii. For every ε > 0 and every y ∈ Ω there exists a δ(ε, y) > 0 such that | f (t, x ) − f (t, y)| < ε, whenever | x − y| < δ for 0 ≤ t < ∞. Theorem 1.1. [41] Let X ∗ be an equilibrium point of the system (1.11), then X ∗ is locally asymptotically stable if all the eigenvalues of J = DF ( X ∗ ) lie strictly in the left half of a complex plane. If any of the eigenvalues lie in the strict right half of a complex plane, then the equilibrium point is unstable. Definition 1.10. [41] Let X ∗ be an equilibrium point of the system (1.11), then X ∗ is called hyperbolic equilibrium point if none of the eigenvalues of J = DF ( X ∗ ) have zero real part, otherwise, it is a non-hyperbolic equilibrium point. Note that, according to the above argument, the following classification of an equilibrium point X ∗ of the system (1.17) can be made [41]. i. A saddle point: A hyperbolic equilibrium point of the system (1.11) is called a saddle if some, but not all, of the eigenvalues of J = DF ( X ∗ ), have positive real parts and the rest of the eigenvalues have negative real parts. ii. A stable node (sink): If all the eigenvalues have negative real parts, then the hyperbolic equilibrium point is called stable node or sink. iii. An unstable node (source): If all the eigenvalues have positive real parts, then the hyperbolic equilibrium point is called an unstable node or source. 10

Chapter 1

General Introduction

iv. Center: If all the eigenvalues are purely imaginary and non-zero, then the non-hyperbolic equilibrium point is called a center.

1.3.2

Two-dimensional flow

Consider the following autonomous two-dimensional system dx = f ( x, y) dt dy = g( x, y) dt

(1.19)

Where ( x, y) ∈ Ω ⊂ Rn with f and g are sufficiently smooth functions. According to the above discussion, the stability analysis of system (1.19) is well understood. However, the following, two theorems discuss the constraints which ensure the existence and non-existence of closed orbits in the plane. Theorem 1.2 (Poincare - Bendixson theorem[14]). A non-empty compact ω or α limit set of a planer flow, which contains no equilibrium points is a closed orbit. Obviously, the Poincare-Bendixson theorem suggests that the solution of a bounded system of two-dimensional autonomous differential equations of first order converges either to a point or to closed curve. Now, the non-existence condition of a closed orbit of system (1.19) is established in the following theorem. Theorem 1.3 (Bendixson-dulac-criterion[14]). Consider the planer dynamical system given by (1.19) where f and g are at least C 0 . Let B( x, y) ∂( B f ) ∂( Bg) be C 0 function on a simply connected region Ω ⊂ R2 . If ∂x + ∂y is not identically zero and does not change sign in Ω, then the system has no periodic curve entirely in Ω.

1.3.3

Routh-Hurwitz criterion [29]

Consider the characteristic equation of J = DF ( X ∗ ) which is defined by Pn ( A) = An + c1 An−1 + c2 An−2 + . . . + cn 11

(1.20)

Chapter 1

General Introduction

Let 

c1  1  " # 0 c1 c3  D1 = c1 , D2 = det , . . . , Dk = det  0 1 c2   ... 

c3 c5 · · · c2k−1



 c2 c4 · · · c2k−2   c1 c3 . . . c2k−3    1 c2 . . . c2k−4   .. .. .. ..  . . . .  0 0 0 . . . ck

Where ci 6= 0 if i > 0, then the roots of Pn ( A) have negative real part if and only if Dk > 0 for all k = 1, 2, . . . , n. Now, applying this criterion when n = 2, the following equation can be obtained Pn ( A) = A2 + c1 A + c2 = 0 Hence " D1 = c1 , D2 = det

c1 0 1 c2

#

= c1 c2

Thus, for n = 2 the necessary and sufficient conditions for having roots with negative real parts are c1 > 0 and c2 > 0. For n = 3, the following equation can be obtained Pn ( A) = A3 + c1 A2 + c2 A + c3 = 0 And hence " D1 = c1 , D2 = det 

c1 c3

1 c2 

#

= c1 c2 − c3

c1 c3 0   D3 = det  1 c2 0  = (c1 c2 − c3 )c3 0 c1 c3 Thus, for n = 3 the necessary and sufficient conditions for having 12

Chapter 1

General Introduction

roots with negative real parts are c1 > 0, c3 > 0 and c1 c2 − c3 > 0. Therefore, an application of Routh-Hurwitz criterion gives a number of constraints on the coefficients c1 , c2 , . . . , cn which are necessary and sufficient to ensure all the eigenvalues lie in the left half of complex plane. Hence, if the Routh-Hurwitz constraints are simultaneously satisfied, then the system will be asymptotically stable at X ∗ . However, the violation of any one of the conditions implies unstable point.

1.3.4

Descarte’s rule of signs [25]

Let P( x ) define a polynomial function with real coefficients and nonzero constant term with terms in descending power of x. a. The number of positive real zeros of P either equals the number of variation in sign occurring in the coefficients of P( x ) or is less than the number of variations by a positive even integer. b. The number of negative real zeros of P either equals the number of variations in sign occurring in the coefficients of P(− x ), or is less than the number of variations by a positive even integer.

1.3.5

Lyapunov method [14]

The method of Lyapunov can be used to determine the stability and instability of the equilibrium point (especially of non-hyperbolic type) of the nonlinear systems. This method has also been used even to study the stability of the equilibrium point of the linear systems. Furthermore, it works in finite and also infinite dimensions. The basic idea of the method is given as follows: Consider the system (1.11) with an equilibrium point X ∗ , then in order to determine whether or not X ∗ is stable, it is sufficient (as shown in the definition ) to find a neighbourhood φ of X ∗ for which the orbits starting in φ remain in φ for all positive times. This condition would be satisfied if we could show that the orbit of the system (1.11) is either tangent to the boundary of φ or pointing inward toward X ∗ , also this situation should remain true even as we shrink φ down onto X ∗ . Note that the way of showing this situation is called Lyapunov method. Therefore, the general theorem for stability of an equilibrium point 13

Chapter 1

General Introduction

which makes these ideas precise is given in the following theorem. Theorem 1.4 (Lyapunov stability Theorem[14]). Let X ∗ be an equilibrium point of the system (1.11) and let L : φ → R be a C 0 function defined on some neighbourhood φ of X ∗ (φ ⊆ Rn ) such that i. L( X ∗ ) = 0 and L( X ) > 0 if X 6= X ∗ . ii.

dL dt

≤ 0 in φ − { X ∗ }.

Then X ∗ is stable. Moreover, if the second condition is replaced by the following iii.

dL dt

< 0 in φ − { X ∗ }.

Then X ∗ is asymptotically stable. Note that the function L given above is known as Lyapunov function (weak Lyapunov function) if and only if conditions (i) and (ii) holds. While it is known as strictly Lyapunov function (strong Lyapunov function) if and only if conditions (i) and (iii) holds. In addition, if φ can be chosen to be all of Rn , then X ∗ is said to be globally asymptotically stable if (i) and (iii) holds.

14

Chapter Two The Dynamics of SIS Epidemic Model with Effect of Treatment

Chapter Two The Dynamics of SIS Epidemic Model with Effect of Treatment

2.1

Introduction

The study of SIS epidemic model mainly concerns global asymptotic stability and it is one of the most basic and most important model in decreasing the spread of many diseases. In 1927, Kermack and Mckerdick [18] proposed a simple SIS model with infective immigrants. Gao and Hethcote (1995) [9] considered the SIS model with a standard disease incidence and density - Pendant demographics verbless sentence. In [23] Li and Ma studies, the SIS model with vaccination and Temporary immunity where is the verbs. Zhou and Liu [46] considered an SIS model with pule vaccination. Treatment including isolation or quarantine is an important method to prevent and control the spread of various infectious diseases. Many mathematicians [16, 22, 24, 32, 38, 39, 42–45] have begun to investigate the rule of treatment function in epidemiological models. In classical epidemic models, the treatment function of the invectives individuals is assumed to be proportional to the number of the infective individual. Because of every community should have a maximal capacity for the treatment of a disease and the resources for treatment should be every large. In (2004), Wany and Ruan [38] considered the maximal treatment capacity to cure infective individuals so that the disease can be eradicated. Recently, saturated treatment function has been widely applied in many epidemic models. For example [42], Zhang and Liu took a continuous and differentiable saturated treatment function T ( I ) = 1+rIαI where α ≥ 0, r > 0. Further, a piecewise linear treatment function was considered, that is,

T( I ) =

 KI

0 ≤ I ≤ I0

m

I > I0 16

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

where m = KI0 and K is positive constant. This means that the treatment function is proportional to the number of the infective individuals when the capacity of treatment has not been reached; otherwise it takes the maximal capacity of treatment KI0 . The treatment function has been used by some other researchers. For example Zhang and Liu [42] studied a model with a general incidence λ( I + S)n−1 SI (0 ≤ n ≤ 1) and the treatment function. Hu et al. [39] considered an epidemic model βSI with standard incidence rate N and the treatment function. Li et al. βI

[24] studied an epidemic model with nonlinear incidence rate ( 1+αI ) with the treatment function and analyzed the stability and bifurcation of the system. In this chapter, we introduce the global dynamics of SIS βSI

model with saturated incidence (Beddington-Deangelis) 1+ aS+bI and saturated treatment function. A sufficient condition for the existence of equation points is obtained and the dynamical behavior of the model is saturated. Also, the global asymptotic stability of the disease-free and endemic equilibria is studied.

2.2

Model formulation

In this section, we formulate SIS epidemic model with immigration βIS incidence rate 1+ aS+bI and treatment, an SIS epidemic model which consists of the susceptible individuals S(t), the infectious individuals I (t) and the total population N (t) at time t, which represented in the block diagram given by Fig.(2.1) can be represented by the following system of non-linear ordinary differential equations

Figure 2.1: Block diagram of system (2.1)

17

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

dS = A − µS − 1+βIS aS+bI + γI + T ( I ) dt dI = 1+βIS aS+bI − ( d + µ + γ ) I − T ( I ) dt

(2.1)

N (t) = S(t) + I (t) Where  δI if 0 ≤ I ≤ I 0 T( I ) = K if I > I 0 (K = δI0 ) in the rate at which infected individuals are treated, A is the recruitment rate of individuals (including newborns and immigrants) into the susceptible population, µ is the natural death rate in each class, γ is the nature recovery rate of infected individuals, d is the diseaserelated death rate, β is the infection coefficient. A, µ, γ, d, β, a and b are all positive numbers. This if 0 ≤ I ≤ I0 model (2.1) implies dS = A − µS − 1+βIS aS+bI + ( γ + δ ) I dt dI = 1+βIS aS+bI − ( d + µ + γ + δ ) I dt

(2.2)

If I > I0 model (2.1) implies dS = A − µS − 1+βIS aS+bI + γI + K dt dI = 1+βIS aS+bI − ( d + µ + γ ) I − K dt

(2.3)

Obviously, due to the biological meaning of the components S(t) and I (t) we focus on the model in the domain R2+ = {(S, I ) ∈ R2 : S ≥ 0, I ≥ 0} which is positively invariant for the system (2.1). Theorem 2.1. All solutions of system (2.1) with non-negative initial condition are uniformly bounded. Proof: Let (S(t), I (t)) be any solution of the system (2.1) with nonnegative condition (S(0), I (0)). Consider the function N (t) = S(t) +

18

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

I (t), time derivative of N (t), dN dS dI = + = A − µS − µI − dI dt dt dt dN + µN (t) ≤ A which has an integrating factor eµt and hence dt −µt where c = 1 N (0) − 1 that means a solution is N (t) = A µ + Ace µ A A A − µt − µt N (t) = µ (1 − e ) + N (0)e . Therefore, N (t) ≤ µ as t → ∞, hence So

all solutions of system (2.1) that initiate in the region R2+ are eventually confined in the region B = {(S, I ) ∈ R2+ : N =

A + ε, for any ε > 0} µ

Thus these solutions are uniformly bounded and then the proof is complete.

2.2.1

Existence and Stability analysis of SIS epidemic model without treatment

In this subsection, we study the dynamics of SIS epidemic model (2.1) without treatment, system (2.1) can be rewritten in the following form dS = A − µS − 1+βIS aS+bI + γI dt dI = 1+βIS aS+bI − ( d + µ + γ ) I dt

(2.4)

Existence of equilibrium points and Stability analysis of system (2.4) Now, we discuss the existence and stability of all positive equilibrium points of system (2.4). ˆ 0) = Obviously the system (2.4) has a disease free equilibrium point E¯ (S, E¯ ( A µ , 0) for any parameters. ˇ Iˇ) exists in the region However, the endemic equilibrium point Eˇ (S, IntR2+ if and only if there is a positive solution to the following non-

19

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

linear equations βIS + γI = 0 1 + aS + bI

(2.5a)

βIS − (d + µ + γ) I = 0 1 + aS + bI

(2.5b)

A − µS −

Now, from the equation (2.5b) we get that

(d + µ + γ)(1 + b Iˇ) Sˇ = β − a(d + µ + γ) Clearly, Sˇ is positive under the following condition β > a(d + µ + γ)

(2.6)

Put Sˇ in equation (2.5a), to obtain the following Iˇ = where P0 =

A − P0 bP0 + d + µ

µ(d+µ+γ) . β− a(d+µ+γ)

and clearly, Iˇ is positive under the following condition A > P0

(2.7)

Now, the local stability analysis for the above equilibrium points are studied. The general variational matrix of the system (2.4) at (S, I ) is given by   J (S, I ) =  

− βS(1+ aS) (1+ aS+bI )2

(1+bI ) −µ − (1βI + aS+bI )2 βI (1+bI ) (1+ aS+bI )2

βS(1+ aS) (1+ aS+bI )2

+γ

− (d + µ + γ)

   

Therefore, the variational matrix of system (2.4) at the disease free

20

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

ˆ 0) = ( A , 0) is given by equilibrium point E¯ (S, µ 

−µ

 J (E¯ ) =  

γ− βA µ+ aA

0



βA µ+ aA

− (d + µ + γ)

  

Clearly the eigenvalue of J (E¯ ) are λ1 = −µ < 0 and λ2 =

βA − (d + µ + γ) µ + aA

Therefore, E¯ is locally asymptotically stable provided that the following condition holds βA < (d + µ + γ) µ + aA

(2.8)

While, the variational matrix of system (2.4) at the endemic equilibˇ Iˇ) is given by rium point Eˇ = (S,    J (Eˇ ) =  

−µ −

− βSˇ (1+ aSˇ ) (1+ aSˇ +b Iˇ)2

β Iˇ(1+b Iˇ) (1+ aSˇ +b Iˇ)2

βSˇ (1+ aSˇ ) (1+ aSˇ +b Iˇ)2

β Iˇ(1+b Iˇ) (1+ aSˇ +b Iˇ)2

−

β Iˇ(1 + b Iˇ) (1 + aSˇ + b Iˇ)2

− (d + µ + γ)

   

βSˇ (1 + aSˇ ) − (d + µ + γ) (1 + aSˇ + b Iˇ)2 ! ˇ ˇ − β S (1 + a S ) +γ (1 + aSˇ + b Iˇ)2

β Iˇ(1 + b Iˇ) det( J (Eˇ )) = −µ − (1 + aSˇ + b Iˇ)2 

+γ





21

!

Chapter 2

=µ

The Dynamics of SIS Epidemic Model with Effect of Treatment

βSˇ (1 + aSˇ ) (d + µ + γ) − (1 + aSˇ + b Iˇ)2

!

+

β Iˇ(1 + b Iˇ) (d + µ) (1 + aSˇ + b Iˇ)2

β Iˇ(1 + b Iˇ) µbβ IˇSˇ + = (d + µ) (1 + aSˇ + b Iˇ)2 (1 + aSˇ + b Iˇ)2 from equation (2.5b)

βSˇ (1+ aSˇ +b Iˇ)

Tra( J (Eˇ )) = − µ −

= (d + µ + γ), then det( J (Eˇ )) > 0, and

β Iˇ(1 + b Iˇ) βSˇ (1 + aSˇ ) + − (d + µ + γ) (1 + aSˇ + b Iˇ)2 (1 + aSˇ + b Iˇ)2

β Iˇ(1 + b Iˇ) bβ IˇSˇ =[−µ − − ]<0 (1 + aSˇ + b Iˇ)2 (1 + aSˇ + b Iˇ)2 ˇ Iˇ are positive, So det(Eˇ ) > 0 and Tr (Eˇ ) < 0, it is If Eˇ exists then S, always locally stable. Theorem 2.2. Assume that the disease-free equilibrium point E¯ ( A µ , 0) of the system (2.4) is locally asymptotically stable in the IntR2+ of the SI −plane with Aβ < (d + µ) (2.9) µ(1 + aS + bI ) Then Eˆ is globally asymptotically stable in IntR2+ of the SI −plane. Proof: Consider the following positive definite function about E¯ ( A µ , 0) V (S, I ) = S −

A A µS − ln +I µ µ A

Clearly, V : R2+ → R is C 0 function such that V ( A µ , 0) = 0 and V (S, I ) > 0, ∀(S, I ) ∈ IntR2+ with (S, I ) 6= ( A µ , 0). Moreover, by differentiating V with respect to t along the solution of the system (2.4), the following equation can be obtained

(S − Aµ ) dS dI dV = + dt S dt dt

22

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

By substituting the value of system (2.4) in the above equation dV µ <− dt S



A S− µ

2



Aβ − d+µ− µ(1 + aS + bI )

 I<0

dV dt

is negative definite under the condition (2.9), and then V ¯ is Lyapunov function (Theorem (1.4)) with respect to E¯ ( A µ , 0). Hence E is globally asymptotically stable in IntR2+ of SI −plane. Hence

ˇ Iˇ) of Theorem 2.3. Assume that the endemic equilibrium point Eˇ (S, the system (2.4) is locally asymptotically stable in the IntR2+ of the SI −plane with b>a

(2.10)

Then Eˇ is globally asymptotically stable in IntR2+ of the SI −plane. Proof: Consider a Dulac function D =

1 SI

and assume that

βIS dS = A − µS − + γI dt 1 + aS + bI dI βIS = − (d + µ + γ) I dt 1 + aS + bI Clearly, D (S, I ) > 0 for all (S, I ) ∈ IntR2+ and it is C 0 function in the IntR2+ . Now, since D

A µ β γ dS = − − + dt SI I 1 + aS + bI S

D

dI β (d + µ + γ) = − dt 1 + aS + bI S

Hence ∆(S, I ) =

∂Dg1 ∂Dg2 1 A β(b − a) + = − 2 ( + γ) − ∂S ∂I S I (1 + aS + bI )2

Note that ∆(S, I ) dose not change the sign and is not identically zero in the IntR2+ if b > a. Then according to Bendixon-Dulac criterion 23

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

(Theorem (1.3)), there is no periodic solution in the IntR2+ . Now, since all solutions of the system (2.4) are bounded and Eˇ is a unique positive equilibrium point in the IntR2+ , hence by using the Pointcare-Bendixon theorem (Theorem (1.2)) Eˇ is globally asymptotically stable.

2.2.2

Stability analysis of an SIS epidemic model with treatment for system (2.2)

In this subsection, the existence of all possible equilibrium points of system (2.2) and their locally and globally stability analysis are discussed. It is obvious that (2.2) always has a unique disease-free equilibrium point E0 ( A µ , 0), the endemic equilibrium of system (2.2) can be obtained by solving algebraic equations A − µS −

βIS + (γ + δ) I = 0 1 + aS + bI

βIS − (d + µ + γ + δ) I = 0 1 + aS + bI By adding the first and the second equations of the system (2.2), we get that S=

A − (d + µ) I µ

(2.11)

By substituting the equation (2.11) in the second equation of system (2.2), the following equations can be obtained βA − β(d + µ) Iˆ − (d + µ + γ + δ)(µ + aA + µb Iˆ − a(d + µ) Iˆ) = 0 Which implies that Iˆ = Where T0 =

(µ + aA)(d + µ + δ + γ)(1 − T0 ) (d + µ)[−( β + µbM) + a(d + µ + δ + γ)]

βA (µ+ aA)(d+µ+δ+γ)

γ 6= 1, and M = 1 + dδ+ +µ hence

24

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

If a(d + µ + δ + γ) > β + µbM and T0 < 1 Iˆ =

(µ + aA)(d + µ + δ + γ)(1 − T0 ) >0 (d + µ)[−( β + µbM) + a(d + µ + δ + γ)]

If we put Iˆ in equation (2.11) we get 

bAM + (d + µ + δ + γ) Sˆ = − a(d + µ + δ + γ) − ( β + µbM)



<0

So this case must be omitted. If a(d + µ + δ + γ) < β + µbM and T0 > 1 holds then Iˆ =

(µ + aA)(d + µ + δ + γ)( T0 − 1) >0 (d + µ)[ β + µbM − a(d + µ + δ + γ)

With Sˆ =

bAM + (d + µ + δ + γ) >0 β + µbM − a(d + µ + δ + γ)

ˆ Iˆ) of system (2.2) Then we get a positive equilibrium point Eˆ (S, But since 0 < Iˆ ≤ I0 , then

(d + µ)[ β + µbM − a(d + µ + δ + γ)] I0 (µ + aA)(d + µ + δ + γ)

(2.12)

(d + µ)[ β + µbM − a(d + µ + δ + γ)] I0 (µ + aA)(d + µ + δ + γ)

(2.13)

T0 ≤ 1 + Since T0 > 1 1 < T0 ≤ 1 + Define N0 = 1 +

(d + µ)[ β + µbM − a(d + µ + δ + γ)] I0 (µ + aA)(d + µ + δ + γ)

(2.14)

ˆ Iˆ) when So the system (2.2) has an endemic equilibrium point Eˆ (S, 1 < T0 ≤ N0 . If T0 = 1 is disease-free equilibrium E0 ( A µ , 0). 25

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

From the above analysis, the following theorems are obtained Theorem 2.4. If T0 < 1, system (2.2) has only one disease-free equilibrium E0 ( A µ , 0), if T0 > 1 system (2.2) has a unique endemic equilibrium ˆ Iˆ) except the disease-free equilibrium E0 ( A , 0). Eˆ (S, µ

Theorem 2.5. The disease - free equilibrium point E0 of the system (2.2) is locally asymptotically stable point if T0 < 1 and it is saddle point if T0 > 1. Proof: The variational matrix of system (2.2) at the disease -free equilibrium point E0 ( A µ , 0) is given by   J ( E0 ) =  

−µ 0

δ+γ− βA µ+ aA



βA µ+ aA

− (d + µ + δ + γ)

  

Clearly the eigenvalue of J (E0 ) are λ1 = −µ < 0 and λ2 = (d + µ + δ + γ)

βA µ+ aA

−

Therefore, E0 is locally asymptotically stable if T0 < 1 and it is saddle point if T0 > 1. ˆ Iˆ) of the system (2.2) Theorem 2.6. The endemic equilibrium point Eˆ (S, is always locally asymptotically stable if it exists. Proof: The Variational matrix of system (2.2) at the endemic equilibˆ Iˆ). rium point Eˆ (S,    ˆ J (E) =  

−µ −

− βSˆ (1+ aSˆ ) (1+ aSˆ +b Iˆ)2

β Iˆ(1+b Iˆ) (1+ aSˆ +b Iˆ)2

β Iˆ(1+b Iˆ) (1+ aSˆ +b Iˆ)2

βSˆ (1+ aSˆ ) (1+ aSˆ +b Iˆ)2

26

+γ+δ

− (d + µ + δ + γ)

    

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Then βSˆ (1 + aSˆ ) − (d + µ + δ + γ) (1 + aSˆ + b Iˆ)2 ! ˆ ˆ ˆ ˆ − β S (1 + a S ) β I (1 + b I ) − +γ+δ (1 + aSˆ + b Iˆ)2 (1 + aSˆ + b Iˆ)2 ! ˆ ˆ β S (1 + a S ) β Iˆ(1 + b Iˆ)(d + µ) =µ d + µ + δ + γ − + (1 + aSˆ + b Iˆ)2 (1 + aSˆ + b Iˆ)2

β Iˆ(1 + b Iˆ) det( J (Eˆ )) = −µ − (1 + aSˆ + b Iˆ)2 



!

From the second equation of system (2.2), we get µbβSˆ Iˆ + β Iˆ(1 + b Iˆ)(d + µ) >0 det( J (Eˆ )) = (1 + aSˆ + b Iˆ)2 Tra( J (Eˆ )) = − µ −

β Iˆ(1 + b Iˆ) βSˆ (1 + aSˆ ) + (1 + aSˆ + b Iˆ)2 (1 + aSˆ + b Iˆ)2

− (d + µ + δ + γ) =−µ−

bβSˆ Iˆ β Iˆ(1 + b Iˆ) − <0 (1 + aSˆ + b Iˆ)2 (1 + aSˆ + b Iˆ)2

According to the trace-determine stability criterion. ˆ Iˆ) exists then it is always locally asymptotically stable. So if Eˆ (S, Theorem 2.7. Assume that the disease-free equilibrium point E0 ( A µ , 0) of the system (2.2) is locally asymptotically stable in the IntR2+ then E0 is globally asymptotically stable in the IntR2+ of the SI −plane if Aβ < (d + µ) µ(1 + aS + bI )

(2.15)

Proof: As the same as proof of the theorem (2.2) ˆ Iˆ) of the Theorem 2.8. Assume that the endemic equilibrium point Eˆ (S, system (2.2) exists in the IntR2+ , then it is globally asymptotically stable in IntR2+ of the SI −plane. 27

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Proof: Consider a Dulac function D = P = A − µS − Q=

1 I

and assume that

βIS + (γ + δ) I 1 + aS + bI

βIS − (d + µ + γ + δ) I 1 + aS + bI

Clearly, D (S, I ) > 0 for all (S, I ) ∈ IntR2+ and it is C 0 function in the IntR2+ . Now, since DP =

βS A µS − − +γ+δ I I 1 + aS + bI

DQ =

βS − (d + µ + γ + δ) 1 + aS + bI

Hence ∆(S, I ) =

µ β(1 + bS + bI ) ∂DP ∂DQ + =− − <0 ∂S ∂I I (1 + aS + bI )2

Note that ∆(S, I ) dose not change the sign and is not identically zero in the IntR2+ . Then according to Bendixon-Dulac criterion (Theorem (1.3)), there is no periodic solution in the IntR2+ . Now, since all solutions of the system (2.2) are bounded and Eˆ is a unique positive equilibrium point in the IntR2+ , hence by using the Pointcare-Bendixon theorem (Theorem (1.2)) Eˆ is globally asymptotically stable.

2.2.3

Stability analysis of an SIS epidemic model with treatment for system (2.3)

The main goal of this subsection is to study dynamics of endemic equilibrium of system (2.3) where it can be obtained by solving alge-

28

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

braic equations A − µS −

βIS + γI + K = 0 1 + aS + bI

βIS − (d + µ + γ) I − K = 0 1 + aS + bI By substituting equation (2.11) in the second equation of system (2.3), obtained the following equation

R2 I 2 + R1 I + R0 = 0

(2.16)

Where R0 = K (µ + aA) R1 = (d + µ + γ)(µ + aA) + µbK − βA − aK (d + µ) R2 = (d + µ)[ β + µbH − a(d + µ + γ)] and H

= 1 + d+γ µ

We study the equation (2.16) as follows If β + µbH = a(d + µ + γ) then equation (2.16) has a positive root whenever R1 < 0, then I˘ =

K (µ + aA) >0 aK (d + µ) − µb( H A + K ) − µ(d + µ + γ)

bA( H A + K ) + A(d + µ + γ) + K (d + µ) <0 S˘ = − aK (d + µ) − µb( H A + K ) − µ(d + µ + γ) So this case must be omitted. If β + µbH < a(d + µ + γ), it follows from (2.16) that

(d + µ)[ a(d + µ + γ) − ( β + µbH )] I 2 + [ βA + aK (d + µ) −(d + µ + γ)(µ + aA) − µbK ] I − K (µ + aA) = 0

29

(2.17)

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Denoting two roots of equation (2.17) by I1 and I2 we have I1,2

−[ βA + aK (d + µ) − (d + µ + γ)(µ + aA) − µbK ] ∓ = 2(d + µ)[ a(d + µ + γ) − ( β + µbH )]

I1 + I2 = − I1 ∗ I2 =

√ ξ1

βA + aK (d + µ) − (d + µ + γ)(µ + aA) − µbK (d + µ)[ a(d + µ + γ) − ( β + µbH )]

−K (µ + aA) (d + µ)[ a(d + µ + γ) − ( β + µbH )]

ξ 1 =[ βA + aK (d + µ) − (d + µ + γ)(µ + aA) − µbK ]2

+ 4K (µ + aA)(d + µ)[ a(d + µ + γ) − ( β + µbH )] > 0 So the equation (2.17) has only one positive root denoted by I1 and the other is negative root,

√ R1 + ξ 1 I1 = 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] √   1 (d + µ + γ)( aA − µ) + aK (d + µ) − βA − µb(2H A + K ) − ξ 1 S1 = 2µ a(d + µ + γ) − ( β + µbH ) Then S1 > 0 holds only if

(d + µ + γ)( aA − µ) + aK (d + µ) − µb(2H A + K ) − T0 < (µ + aA)(d + µ + δ + γ)

√ ξ1

Define

(d + µ + γ)( aA − µ) + aK (d + µ) − µb(2H A + K ) − N1 = (µ + aA)(d + µ + δ + γ) The point E1 (S1 , I1 ) satisfies system (2.3) where I1 > I0

√ R1 + ξ 1 > I0 2(d + µ)[ a(d + µ + γ) − ( β + µbH )]

30

√ ξ1

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

We have p

ξ 1 > − R1 + 2(d + µ)[ a(d + µ + γ − ( β + µbH )] I0

(2.18)

Then if

− R1 +2(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 < 0 T0 < 1 −

−

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 (µ + aA)(d + µ + δ + γ)

Define N2 = 1 −

−

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 (µ + aA)(d + µ + δ + γ)

Then the equation (2.18) holds only, if

− R1 + 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 > 0

(2.19)

ξ 1 ≥ [− R1 + 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 ]2

(2.20)

Then by (2.19) we get that T0 > N2 and by (2.20) we get K (µ + aA) ≥[ βA + aK (d + µ) − (d + µ + γ)(µ + aA) − µbK ] I0

+ (d + µ)[ a(d + µ + γ) − ( β + µbH )] I02

31

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

T0 ≤1 −

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

(d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 K − (d + µ + δ + γ) I0 (µ + aA)(d + µ + δ + γ)

+ Define

N3 =1 −

+

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

K (d + µ)[ a(d + µ + γ) − ( β + µbH )] I0 − (d + µ + δ + γ) I0 (µ + aA)(d + µ + δ + γ)

Then N2 < T0 ≤ N3 . So, if T0 ≤ N3 and T0 < N1 , E1 (S1 , I1 ) is endemic equilibrium, where

√ R1 + ξ 1 I1 = 2(d + µ)[ a(d + µ + γ) − ( β + µbH )] and S1 =

1 [ A − (d + µ) I1 ] µ

If ( β + µbH ) > a(d + γ + µ), then it is easy to see that equation (2.16) has no positive root if R1 ≥ 0. If R1 < 0 then ξ 2 = R21 − 4K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)] and R1 =(d + µ + γ)(µ + aA) + µbK − βA − aK (d + µ) R1 = − T0 (d + aA)(d + µ + δ + γ) + (µ + aA)(d + µ + δ + γ)

− δ(µ + aA) + µbK − aK (d + µ) Then ξ 2 ≥ 0 implies R21 ≥ 4K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)]

32

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

and T0 ≤1 −

−

2

aK (d + µ) + δ(µ + aA) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) p

K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

Or T0 ≥1 −

+

2

µbK aK (d + µ) + δ(µ + aA) + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) p

K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

Define N4 =1 −

−

2

N5 =1 −

+

2

aK (d + µ) + δ(µ + aA) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) p

K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

aK (d + µ) + δ(µ + aA) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) p

K (µ + aA)(d + µ)[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

At the same time, R1 < 0 holds if and only if T0 > 1 −

aK (d + µ) + δ(µ + aA) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

Define N6 =1 −

aK (d + µ) + δ(µ + aA) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

Therefore, if T0 ≥ N5 , we have R1 < 0 and ξ 2 ≥ 0, then the equation 33

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

(2.16) has two positive roots I2 , I3 , where

√ − R1 − ξ 2 I2 = 2(d + µ)[ β + µbH − a(d + µ + γ)] and

√ − R1 + ξ 2 I3 = 2(d + µ)[ β + µbH − a(d + µ + γ)] Then Si = µ1 [ A − (d + µ) Ii ] > 0, (i = 2, 3) holds if T0 <1 −

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

2A[ β + µbH − a(d + µ + γ)] + + (µ + aA)(d + µ + δ + γ)

√ ξ2

and

√ δ(µ + aA) + aK (d + µ) + ξ 2 µbK T0 <1 − + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) +

2A[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

Define N7 =1 −

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

2A[ β + µbH − a(d + µ + γ)] + + (µ + aA)(d + µ + δ + γ)

√ ξ2

√ δ(µ + aA) + aK (d + µ) + ξ 2 µbK N8 =1 − + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ) +

2A[ β + µbH − a(d + µ + γ)] (µ + aA)(d + µ + δ + γ)

It is easy to see that N8 < N7 . 34

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Now from equation (2.3) Ii > I0 for (i = 2, 3). √ For − R1 − ξ 2 > 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 for I2 > I0 . If R1 + 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 < 0, then T0 >1 −

+

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

2(d + µ)[ β + µbH − a(d + µ + γ) I0 ] (µ + aA)(d + µ + δ + γ)

Define N9 =1 −

+

µbK δ(µ + aA) + aK (d + µ) + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

2(d + µ)[ β + µbH − a(d + µ + γ) I0 ] (µ + aA)(d + µ + δ + γ)

Furthermore, R1 + 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 > 0 gives T0 <1 −

+

δ(µ + aA) + aK (d + µ) µbK + (µ + aA)(d + µ + δ + γ) (µ + aA)(d + µ + δ + γ)

(d + µ)[ β + µbH − a(d + µ + γ)] I0 K + (d + µ + δ + γ) I0 (µ + aA)(d + µ + δ + γ)

Therefore, if N9 < T0 < N3 , I2 > I0 holds. Similarly, if I3 > I0 R1 + 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 < 0

(2.21)

Or R1 + 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 > 0

(2.22)

ξ 2 > R1 + 2(d + µ)[ β + µbH − a(d + µ + γ)] I0 2

(2.23)

We get that T0 < N9 or T0 > max( N3 , N9 ).

35

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

From the above analysis, obtained the following theorem Theorem 2.9. From the above discussion, the following conclusions have been drawin i. If β + µbH < a(d + µ + γ), then E1 (S1 , I1 ) is a unique endemic equilibrium of model (2.3) if T0 < N1 , E1 (S1 , I1 ) is a unique endemic equilibrium of model (2.1) if T0 < N1 and T0 < N3 . ii. If β + µbH > a(d + µ + γ), the model (2.3) has two positive equilibrium points E2 (S2 , I2 ), E3 (S3 , I3 ) if T0 < N8 ; model (2.3) has only one positive equilibrium point E2 (S2 , I2 ), if N8 < T0 < N7 ; the model (2.3) has no positive equilibrium point if T0 ≥ N7 . E2 (S2 , I2 ) is an endemic equilibrium of model (2.1) if N9 < T0 < N3 and E3 (S3 , I3 ) is an endemic equilibrium point of model (2.1) if T0 < N9 or T0 > max( N3 , N9 ). iii. If β + µbH = a(d + µ + γ), then the model (2.3) has no endemic equilibrium point. Theorem 2.10. The endemic equilibrium point E1 of the system (2.3) is locally asymptotically stable if the following condition is satisfied bβS1 I1 K < I1 (1 + aS1 + bI1 )2

(2.24)

Proof: The Variational matrix of system (2.3) at the endemic equilibrium point E1 (S1 , I1 )   J ( E1 ) =  

− βS1 (1+ aS1 ) (1+ aS1 +bI1 )2

1 (1+ bI1 ) −µ − (1βI + aS +bI )2 1

1

βI1 (1+bI1 ) (1+ aS1 +bI1 )2

βS1 (1+ aS1 ) (1+ aS1 +bI1 )2

36

+γ

− (d + µ + γ)

   

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Then 

βS1 (1 + aS1 ) det( J (E1 )) =µ d + µ + γ − (1 + aS1 + bI1 )2

+

βI1 (1 + bI1 )(d + µ) (1 + aS1 + bI1 )2



βS1 K βS1 (1 + aS1 ) − − 1 + aS1 + bS1 I1 (1 + aS1 + bI1 )2



bβS1 I1 K − (1 + aS1 + bI1 )2 I1

>µ

=µ



Tra( J (E1 )) = − µ −

−

<





βI1 (1 + bI1 ) βS1 (1 + aS1 ) + (1 + aS1 + bI1 )2 (1 + aS1 + bI1 )2

K βS1 + 1 + aS1 + bS1 I1

K bβS1 I1 − I1 (1 + aS1 + bI1 )2

So E1 (S1 , I1 ) is locally asymptotically stable if the condition (2.24) is satisfied. Otherwise it is an unstable point and periodic dynamics exists. Theorem 2.11. The endemic equilibrium points E2 and E3 of system (2.3) for I > I0 are locally asymptotically stable under the condition (2.24) with replacing E1 by E2 and E3 . Theorem 2.12. Assume that the endemic equilibrium point E1 (S1 , I1 ) of the system (2.3) is locally asymptotically stable in the IntR2+ , then it is globally asymptotically stable in IntR2+ if δ<µ

37

(2.25)

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Proof: Consider a Dulac function D = g1 = A − µS − g2 =

1 I

and assume that

βIS + γI − K 1 + aS + bI

βIS − (d + µ + γ) I − K 1 + aS + bI

Clearly, D (S, I ) > 0 for all (S, I ) ∈ IntR2+ and it is C 0 function in the IntR2+ . Now, since Dg1 = Dg2 =

βS K A µS − − +γ+ I I 1 + aS + bI I βS K − (d + µ + γ) − 1 + aS + bI I

Hence K = δI0 if I0 < I, then ∆(S, I ) =

1 β(1 + bS + bI ) ∂Dg1 ∂Dg2 + < (δ − µ) − ∂S ∂I I (1 + aS + bI )2

Thus, ∆(S, I ) dose not change the sign and is not identically zero in the IntR2+ if δ < µ. Then according to Bendixon-Dulac criterion (Theorem (1.3)), there is no periodic solution in the IntR2+ . Now, since all solutions of the system (2.3) are bounded and E1 is a unique positive equilibrium point in the IntR2+ , hence by using the Pointcare-Bendixon theorem (Theorem (1.2)) E1 is globally asymptotically stable. Theorem 2.13. The endemic equilibrium points E2 and E3 of system (2.3) for I > I0 are global asymptotically stable under the condition (2.25).

2.3

Numerical Analysis

In this section, the global dynamical behavior of the system (2.1) is investigated. This investigation are confirming above theoretical analysis and understand the effect of treatment on the dynamics of SIS epidemic 38

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

model. The first system (2.4) is solved numerically for different sets of parameters and different sets of initial conditions. The following set of parameter values has been chosen A = 600, µ = 0.2, β = 0.7, γ = 1, a = 0.25, b = 1.5, and d = 2 (2.26) For this parametric values, it is observed that the conditions of equations (2.6), (2.7), and (2.8) are satisfied and trajectories with initial conditions (50, 50), (150, 150) and (300, 300) converge to free equilibrium ¯ 0) = (3000, 0). This indicates that the free equilibrium point point (S, is globally asymptotically stable. See Fig.(2.2) and Fig.(2.3)

Figure 2.2: Phase plot of system (2.4) starting from different initial points (50, 50), (150, 150) and (300, 300).

39

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Figure 2.3: Time series of attractor given in Fig.(2.2). Obviously Fig.(2.2) and Fig.(2.3), show the convergent of system (2.4) to the globally asymptotically stable disease-free point (3000, 0) when it confirms the theoretical analysis. However, the system (2.4) is solved for the parameter value β = 0.9 keeping other parameters fixed as given in equation (2.26), for this parameter values, it is observed that the condition of theorem (2.3) is satisfied and the phase plot of system (2.4) starting from different sets of initial data (300, 15), (1000, 50) and (1000, 30) is drawn in Fig.(2.4) and Fig.(2.5)

Figure 2.4: Phase plot of system (2.4) starting from different initial points (300, 15), (1000, 50) and (1000, 30).

40

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Figure 2.5: Time series of attractor given in Fig.(2.4). Fig.(2.4) and Fig.(2.5) show clearly the existence of a unique endemic ˇ Iˇ) = (2446.64, 50.3051) of the system (2.4) which equilibrium point (S, is globally asymptotically stable. Now, in order to discuss the dynamics of system (2.2) we choose the following set of parameter values: A = 300, µ = 0.5, β = 0.4, γ = 1.5, δ = 2, a = 0.25, b = 1.5, and d = 2

(2.27)

For this parameter values, it is observed that the condition of theorem (2.5) is satisfied and trajectories with initial conditions (30, 15), (40, 40) converge to equilibrium point E0 = (600, 0). See Fig.(2.6) and Fig.(2.7)

41

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Figure 2.6: Phase plot of system (2.2) starting from different initial conditions.

Figure 2.7: Time series of attractor given in Fig.(2.6). Clearly, Fig.(2.6) and Fig.(2.7), show the existence of disease-free point E0 and the convergent of system (2.2) to the globally asymptotically stable E0 . Now, if we choose β = 1.7, keeping other parameters as in equation (2.27), then the condition for existence of positive equilibrium point ˆ Iˆ) is satisfied and trajectories with initial values (60, 75), (50, Eˆ = (S, 50) and (150, 80) converge to Eˆ = (543, 11.4). This indicates that the equilibrium point Eˆ is globally asymptotically stable. See Fig.(2.8) and 42

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Fig.(2.9)

Figure 2.8: Phase plot of system (2.2) starting from different initial condition.

Figure 2.9: Time series of attractor given in Fig.(2.8). Finally, the dynamical behavior of the system (2.3) is discussed. To solve system (2.3) numerically, the following sets of parameter values are chosen A = 500, µ = 0.25, β = 0.02, γ = 2, δ = 0.1, a = 0.001, b = 0.01, d = 0.01, I0 = 50

(2.28)

The phase plot of system (2.3) starting from different sets of initial data is drawn in Fig.(2.10) and Fig.(2.11). 43

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Figure 2.10: Phase plot system (2.3) starting from different initial points.

Figure 2.11: Time series of attractor given in Fig.(2.10). Fig.(2.10) and Fig.(2.11) show clearly the existence of a unique endemic equilibrium point (1159.97, 807.721) of system (2.3) which is globally asymptotically stable. Now, in order to discuss the effect of varying the infection rate β on the dynamical behavior of system (2.3). The system (2.3) is solved for different values of the infection rate, as shown in the following table.

44

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

Table (2.1) The effect of the varying infection rate of system (2.3). Parameters kept fixed

Parameter

Dynamical behavior of the system (2.3)

β = 0.009

The system (2.3) approaches asymptotically stable to (1608.89, 376.07)

β = 0.01

The system (2.3) approaches asymptotically stable to (1554.13, 428.724)

β = 0.015

The system (2.3) approaches asymptotically stable to (1328.36, 645.808)

As given in equation (2.28)

and the trajectories of system (2.3) as given in table (2.1) are drawn in Fig.(2.12) (a-c).

(a) For β = 0.009.

(b) For β = 0.01.

(c) For β = 0.015.

Figure 2.12: Time series of the solution of system (2.3)

45

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

According to the above figure, it is clear that, as the infection rate increases the endemic equilibrium point still coexists and stable with increase in the value of infected individuals whereas the value of susceptible individuals, decreases. The effect of the varying treatment rate δ on the dynamics of system (2.3) is studied the system (2.3) is solved for different values of the treatment rate, shows in the following table.

Table (2.2) The effect of the varying infection rate of system (2.3). Parameters kept fixed

Parameter

Dynamical behavior of the system (2.3)

δ = 0.15

he system (2.3) approaches asymptotically stable to (1160.77, 806.949)

δ = 0.3

The system (2.3) approaches asymptotically stable to (1163.19, 804.628)

δ = 0.5

The system (2.3) approaches asymptotically stable to (1166.41, 801.529)

As given in equation (2.28)

and the trajectories of system (2.3) as given in table (2.2) are drawn in Fig.(2.13) (a-c).

46

Chapter 2

The Dynamics of SIS Epidemic Model with Effect of Treatment

(a) For δ = 0.15.

(b) For δ = 0.3.

(c) For δ = 0.5.

Figure 2.13: Time series of the solution of system (2.3) Obviously, from Fig.(2.13) (a-c) as the treatment rate increases the endemic equilibrium point of system (2.3), which is still stable and the number of susceptible individuals increases whereas the number of infected individuals decreases.

47

Chapter Three The Dynamics of SVIS Epidemic Model

Chapter Three The Dynamics of SVIS Epidemic Model

3.1

Introduction

Many infectious disease causing death in varying total population, the study of epidemic models become one of the important area in the mathematical theory of epidemiology. Epidemic model with vaccination is one of the most important models in decreasing the spread of many diseases. In [5] Kribs-Zaleta and Velasco- Hernández presented a simple two dimensional SIS model with vaccination exhibiting backward bifurcation. Farringten [6] derived relation between vaccine efficacy against transmission and analyzed the impact of vaccination program on the transmission potential of the infection in large populations. In [10] Gumel and Moghadas proposed a model for the dynamics of an infectious disease in the presence of a preventive vaccine considering non-linear incidence rate

cI 1+ I .

Liu et al. [27, 28] proposed more

realistic models that assume non-linear incidence rate given by

kI l 1+αI h

with k, l, h, α > 0. Shim E. [33] assumed that the total population is β asymptotically constant, and supposed the incidence rate N where β is the transmission rate. In this chapter, we will study the SVIS model β I with non-linear incidence rate (Holling type-IV) 1+bI2+cI 2 . The effect of immigrants on the dynamical behavior of SVIS model is considered analytically. The sufficient condition for the existence and stability of the endemic equilibrium point is obtained. The global dynamics of the model is studied by solving it numerically for different sets of initial values and for different sets of parameters values.

3.2

Model formulation

Consider an SIS model when a vaccination program is in effect and there is a constant flow of incoming immigrants. A population of size N (t) at time t is partition into three classes of individuals; susceptible, infections and vaccinated, with sizes denoted by S(t), I (t) and V (t), re49

Chapter 3

The Dynamics of SVIS Epidemic Model

spectively which represented in the block diagram given by Fig. (3.1) can be represented by the following system of non-linear ordinary differential equations.

Figure 3.1: Block diagram of system (3.1)

dS = Λ + (1 − ρ) A + β 1 V + γI − 1+βbI2 SI − µS − β 3 S +cI 2 dt dV = β 3 S − 1+σβbI2+IVcI 2 − β 1 V − µV dt dI σβ 2 IV = ρA + 1+βbI2 SI 2 + 1+ bI + cI 2 − ( α + γ + µ ) I + cI dt

(3.1)

where Λ is the constant natural birth rate, with all newborns coming into the susceptible class. There is a constant flow of A new members in to the population per a unit of time, where fraction of ρ with immigrants is infective (0 ≤ ρ ≤ 1). The rate at which the vaccine wears off is β 1 , β 2 is infection constant rate coefficient for susceptible individual and the rate at which the susceptible population is vaccinated is β 3 . There is a constant per capita natural death rate µ > 0 in each class, and fraction γ ≥ 0 of infectives recovers in unit time. The vaccine has the effect of reducing by a factor of σ, where 0 ≤ σ ≤ 1 and σ = 0 means 50

Chapter 3

The Dynamics of SVIS Epidemic Model

that the vaccine is completely effective, while σ = 1 means that the vaccine is totally infective. The disease can be fatal to some infectives and we define α to be the rate of disease related death. The parameters b, c √ are fiting parameters of the response function, where b > −2 c and c > 0. Obviously, due to the biological meaning of the variables S(t), V (t) and I (t), system (3.1) has the domain R3+ = {(S, V, I ) ∈ R3 , S ≥ 0, V ≥ 0, I ≥ 0} which is positively invariant for system (3.1) and all the solutions of system (3.1) with non-negative initial conditions are uniformly bounded as it is proved in the following theorem . Theorem 3.1. All solution of system (3.1) with non-negative initial condition are uniformly bounded. Proof: Let (S(t), V (t), I (t)) be any solution of the system (3.1) with non- negative initial condition (S(0), V (0), I (0)). Consider the function N (t) = S(t) + V (t) + I (t), time derivative of N (t), then dN = Λ + A − µ(S + V + I ) − αI dt

+ µN ≤ Λ + A which has an integrating factor eµt and hence (Λ+ A) A) a solution is N (t) = + Ce−µt where C = N (0) − (Λ+ that µ µ So

dN dt

(Λ+ A)

(Λ+ A)

−µt ) + N (0) e−µt . Therefore, N ( t ) ≤ means N (t) = µ (1 − e µ as t → ∞, hence all solutions of system (3.1) that initiate in the region

R3+ are eventually confined in the region B = {(S, V, I ) : N = S + V + I =

(Λ + A) } µ

Thus these solution are uniformly bounded and then the proof is finished.

3.3

Existence of Equilibrium point of system (3.1)

In this section, we find all possible equilibrium points of the system (3.1) and their locally stability analysis is discussed. 51

Chapter 3

The Dynamics of SVIS Epidemic Model

ˆ V, ˆ 0) Clearly, system (3.1) has a disease-free equilibrium point Eˆ (S, (Λ+ A)(µ+ β 1 ) β (Λ+ A) which it always exists, where Sˆ = and Vˆ = 3 . µ(µ+ β 1 + β 3 )

However, the endemic equilibrium point

µ(µ+ β 1 + β 3 ) ∗ ∗ ∗ E (S , V , I ∗ ) exist in

the

region IntR3+ if and only if there is a positive solution to the following non-linear equations Λ + (1 − ρ) A + β 1 V + γI −

β 2 SI − µS − β 3 S = 0 1 + bI + cI 2

(3.2a)

β3 S −

σβ 2 IV − β 1 V − µV = 0 1 + bI + cI 2

(3.2b)

ρA +

β 2 SI σβ 2 IV + − (α + γ + µ) I = 0 2 1 + bI + cI 1 + bI + cI 2

(3.2c)

Adding equation (3.2a), equation (3.2b) and equation (3.2c) we get Λ + A − µS − µV − (α + µ) I = 0 Then S∗ =

1 [(Λ + A) − µ(V ∗ + I ∗ ) − αI ∗ ] µ

(3.3)

Clearly S∗ > 0 if (Λ + A) − µ(V ∗ + I ∗ ) − αI ∗ > 0 0 < µ(V ∗ + I ∗ ) + αI ∗ < (Λ + A)

(3.4)

From the equation (3.2b), the following equation can be obtained σβ 2 I ∗ β3 S − V [ + β 1 + µ] = 0 1 + bI ∗ + cI ∗ 2 ∗

∗

V∗ =

β 3 S∗ G ∗ σβ 2 I ∗ + ( β 1 + µ) G ∗

(3.5)

where G ∗ = 1 + bI ∗ + cI ∗ 2 , D0 = Λ + A, D1 = α + µ, D2 = β 1 + β 3 + µ, and D3 = α + γ + µ 52

Chapter 3

The Dynamics of SVIS Epidemic Model

By substituting the equation (3.3) in the equation (3.5) we get β 3 G ∗ [ D0 − D1 I ∗ ] β 3 G ∗ [(Λ + A) − (µ + α) I ∗ ] = V = µ[σβ 2 I ∗ + ( β 1 + β 3 + µ) G ∗ ] µ[σβ 2 I ∗ + D2 G ∗ ] ∗

(3.6)

By substituting the equation (3.3) and the equation (3.5) in the equation (3.2c) and then simplifying the resulting term gives the following polynomial equation A5 I ∗ 5 + A4 I ∗ 4 + A3 I ∗ 3 + A2 I ∗ 2 + A1 I ∗ + A0 = 0

(3.7)

Where A5 = − c2 µD3 D2

(3.8a)

A4 =c2 µρAD2 − cβ 2 D1 K1 − cµD3 K2

(3.8b)

A3 =cµρAK2 + β 2 (cD0 − bD1 )K1 − K3

(3.8c)

A2 =K4 + K2 (µbρA − µD3 ) + β 2 K1 (bD0 − D1 ) − µb2 ρAD2

(3.8d)

A1 =µρAK2 + β 2 D0 K1 − µD3 D2

(3.8e)

A0 =µρAD2

(3.8f)

and K1 = β 1 + σβ 3 + µ K2 =2bD2 + σβ 2 K3 =µD3 D2 (2c + b2 ) + σβ 2 (µbD3 + β 2 D1 ) K4 =2cµρAD2 + σβ22 D0 A straightforward computation shows that the equation (3.7) has a positive root namely I ∗ provided that one set of the following sets of

53

Chapter 3

The Dynamics of SVIS Epidemic Model

conditions holds A1 > 0, A2 > 0, with A3 > 0

(3.9a)

A1 < 0, A2 < 0, A3 < 0, with A4 < 0

(3.9b)

A1 > 0, A3 < 0, with A4 < 0

(3.9c)

Substitution the value of I ∗ in equation (3.5) gives the value of V ∗ and then substituting the value of V ∗ and I ∗ in equation (3.3) gives the value of S∗ .

3.4

Stability Analysis

In the following, the local stability analysis for the above equilibrium points is studied, the general variational matrix of the system (3.1) is given by 

a11 a12 a13



  J (S, V, I ) =  a21 a22 a23  a31 a32 a33 Where β2 I − µ − β 3 < 0, a12 = β 1 > 0, 1 + bI + cI 2 β 2 S(1 − cI 2 ) γ− (1 + bI + cI 2 )2 σβ 2 I β 3 > 0, a22 = −[ + µ + β 1 ] < 0, 1 + bI + cI 2 σβ 2 V (1 − cI 2 ) − (1 + bI + cI 2 )2 β2 I σβ 2 I > 0, a = > 0, 32 1 + bI + cI 2 1 + bI + cI 2 β 2 (1 − cI 2 )(S + σV ) − (α + γ + µ) (1 + bI + cI 2 )2

a11 = − a13 = a21 = a23 = a31 = a33 =

Theorem 3.2. The disease-free equilibrium point Eˆ of system (3.1) is

54

Chapter 3

The Dynamics of SVIS Epidemic Model

locally asymptotically stable in the IntR3+ if β2 <

µ(α + γ + µ) (Λ + A)

(3.10)

Proof: Therefore, the variational matrix about the disease equilibrium ˆ V, ˆ 0) is given below point Eˆ = (S,    J ( Eˆ ) =   

−µ − β 3

β1

γ − β 2 Sˆ

     

β3

− β1 − µ

−σβ 2 Vˆ

0

0

β 2 Sˆ + σβ 2 Vˆ − (α + γ + µ)

Clearly, the eigenvalue of J ( Eˆ ) are λ1 = −µ < 0, λ2 = −(µ + β 1 + β 3 ) < 0 and λ3 =

(µ + β 1 + β 3 )[ β 2 (Λ + A) − µ(α + γ + µ)] + β 2 β 3 (Λ + A)(σ − 1) µ(µ + β 1 + β 3 )

<0 Since, 0 < σ < 1, by condition (3.10), then λ3 < 0. The disease-free equilibrium point Eˆ is locally asymptotically stable µ(α+γ+µ) in the IntR3+ if β 2 < (Λ+ A) . Theorem 3.3. Assume that the positive equilibrium point E∗ (S∗ , V ∗ , I ∗ ) of the system (3.1) exists and let the following inequalities hold cI ∗ 2 > 1

(3.11a)

2 θ12 < θ11 θ22

(3.11b)

2 θ13 < θ11 θ33

(3.11c)

2 θ23 < θ22 θ33

(3.11d)

55

Chapter 3

The Dynamics of SVIS Epidemic Model

Here we have θ11

β2 I ∗ = + µ + β 3 ; θ12 = β 1 + β 3 ; 1 + bI ∗ + cI ∗ 2

θ22 =

σβ 2 I ∗ + µ + β1 1 + bI ∗ + cI ∗ 2

θ13 = [

β2 I ∗ β 2 S∗ (1 − cI 2 ) + γ − ]; 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2

θ23

σβ 2 V ∗ (1 − cI 2 ) σβ 2 I ∗ − ; = 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2

θ33

β 2 (1 − cI 2 )(S∗ + σV ∗ ) = (α + γ + µ) − (1 + bI ∗ + cI ∗ 2 )2

Then it is locally asymptotically stable in the IntR3+ . Proof: The linearized system of the system (3.1) can be written as dU dW = = J ( E ∗ )U dt dt Here, W = (S, V, I )t and U = (u1 , u2 , u3 )t with u1 = S − S∗ , u2 = V − V ∗ , u3 = I − I ∗ . Moreover, E∗ = ( aij )3×3 ; i, j = 1, 2, 3 is the variational matrix of the system (3.1) at E∗ in which a11 = −

β 2 S∗ (1 − cI 2 ) β2 I ∗ − µ − β , a = β , a = γ − 3 12 1 13 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2

a21 = β 3 , a22

σβ 2 I ∗ σβ 2 V ∗ (1 − cI 2 ) = −[ + µ + β 1 ], a23 = − 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2

a31

β2 I ∗ σβ 2 I ∗ = , a32 = , 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2

a33

β 2 (1 − cI 2 )(S∗ + σV ∗ ) = − (α + γ + µ) (1 + bI ∗ + cI ∗ 2 )2

56

Chapter 3

The Dynamics of SVIS Epidemic Model

and β2 I ∗ β 2 S∗ (1 − cI 2 ) du1 =[− − µ − β ] u + β u + [ γ − ] u3 3 1 1 2 dt 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2 σβ 2 I ∗ du2 σβ 2 V ∗ (1 − cI 2 ) = β 3 u1 + [− − µ − β 1 ]u2 + [− ] u3 dt 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2 β 2 I ∗ u1 σβ 2 I ∗ u2 β 2 (1 − cI 2 )(S∗ + σV ∗ ) du3 = + +[ dt 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2

− (α + γ + µ)]u3 Now, consider the following function 1 1 1 P = u21 + u22 + u23 2 2 2 By differentiating P with respect to t, we get dP du du du = u1 1 + u2 2 + u3 3 dt dt dt dt

=[−

β2 I ∗ − µ − β 3 ]u21 + β 1 u1 u2 + [γ 2 ∗ ∗ 1 + bI + cI

β 2 S∗ (1 − cI 2 ) σβ 2 I ∗ − ]u1 u3 + β 3 u1 u2 + [− (1 + bI ∗ + cI ∗ 2 )2 1 + bI ∗ + cI ∗ 2

57

Chapter 3

−µ−

The Dynamics of SVIS Epidemic Model

β 1 ]u22

σβ 2 V ∗ (1 − cI 2 ) β 2 I ∗ u1 u3 + [− ] u2 u3 + (1 + bI ∗ + cI ∗ 2 )2 1 + bI ∗ + cI ∗ 2

+

σβ 2 I ∗ u2 u3 β 2 (1 − cI 2 )(S∗ + σV ∗ ) + [ − (α + γ + µ)]u23 2 2 2 ∗ ∗ ∗ ∗ 1 + bI + cI (1 + bI + cI )

=[−

β2 I ∗ σβ 2 I ∗ 2 − µ − β ] u + [− − µ − β 1 ]u22 3 1 2 2 ∗ ∗ ∗ ∗ 1 + bI + cI 1 + bI + cI

+ [ β 1 + β 3 ] u1 u2 + [

β 2 (1 − cI 2 )(S∗ + σV ∗ ) − (α + γ + µ)]u23 2 2 ∗ ∗ (1 + bI + cI )

β 2 S∗ (1 − cI 2 ) σβ 2 I ∗ β2 I ∗ +γ− ] u1 u3 + [ +[ 1 + bI ∗ + cI ∗ 2 (1 + bI ∗ + cI ∗ 2 )2 1 + bI ∗ + cI ∗ 2 σβ 2 V ∗ (1 − cI 2 ) ] u2 u3 − (1 + bI ∗ + cI ∗ 2 )2 1 β2 I ∗ 1 σβ 2 I ∗ 2 =− [ + µ + β 3 ] u1 + [ β 1 + β 3 ] u1 u2 − [ 2 1 + bI ∗ + cI ∗ 2 2 1 + bI ∗ + cI ∗ 2 β2 I ∗ β2 I ∗ 1 2 + µ + β ] u + [ +γ + µ + β 1 ]u22 − [ 3 1 2 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2

−

β 2 S∗ (1 − cI 2 ) 1 ] u u − [(α + γ + µ) 3 1 2 (1 + bI ∗ + cI ∗ 2 )2

β 2 (1 − cI 2 )(S∗ + σV ∗ ) 2 1 σβ 2 I ∗ − ] u3 − [ + µ + β 1 ]u22 2 2 2 ∗ ∗ ∗ ∗ 2 (1 + bI + cI ) 1 + bI + cI σβ 2 I ∗ σβ 2 I ∗ 1 2 + µ + β 1 ] u2 + [ − [ 2 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2 σβ 2 V ∗ (1 − cI 2 ) 1 − ] u u − [(α + γ + µ) 2 3 2 (1 + bI ∗ + cI ∗ 2 )2 β 2 (1 − cI 2 )(S∗ + σV ∗ ) 2 − ] u3 (1 + bI ∗ + cI ∗ 2 )2

58

Chapter 3

The Dynamics of SVIS Epidemic Model

1 1 1 1 = − θ11 u21 + θ12 u1 u2 − θ22 u22 − θ11 u21 + θ13 u1 u3 − θ33 u23 2 2 2 2 1 1 − θ22 u22 + θ23 u2 u3 − θ33 u23 2 2 p p 1 p 1 p = − ( θ11 u1 − θ22 u2 )2 − ( θ11 u1 − θ33 u3 )2 2 2 p 1 p − ( θ22 u2 − θ33 u3 )2 2 dP dt

<0 by Lyapunov function (Theorem (1.4)), therefore the origin and then E∗ is locally asymptotically stable point in the IntR3+ . Obviously, due to condition (3.11a)-(3.11d), it is obtained that

3.5

Globally Stability Analysis

In this section, the global dynamics of the system (3.1) is carried out and the obtained result are shown in the following theorems. Theorem 3.4. The disease-free equilibrium point Eˆ of the system (3.1) is globally asymptotically stable in the subregion ( φ=

β β (S, V, I ) : 0 < I < C0 , 0 < I < K0 , 1 + 3 ≤ 2 S V

where C0 =

−b+

q

r β S b2 −4c(1− γ3 )

2c

−b+

, and K0 =

r

(µ + β 1 )(µ + β 3 ) SV σβ V

b2 −4c(1− (α+2µ)µ ) 2c

.

Proof: Consider the function L(S, V, I ) =

Z S u1 − Sˆ Sˆ

u1

du1 +

Z V u2 − Vˆ Vˆ

u2

du2 + I

By differentiating L with respect to t along the solution of the system (3.1), we will get dL S − Sˆ dS V − Vˆ dV dI = . + . + dt S dt V dt dt

59

)

Chapter 3

By substituting the value of

The Dynamics of SVIS Epidemic Model dS dV dt , dt

and

dI dt

in the above equation

dL S − Sˆ β 2 SI − µS − β 3 S] = [Λ + (1 − ρ) A + β 1 V + γI − dt S 1 + bI + cI 2 V − Vˆ σβ 2 IV β 2 SI + [ β3 S − − β V − µV ] + [ ρA + 1 V 1 + bI + cI 2 1 + bI + cI 2 σβ 2 IV − (α + γ + µ) I ] + 1 + bI + cI 2 here Λ + (1 − ρ) A = µSˆ + β 3 Sˆ − β 1 Vˆ β 1 Vˆ + µVˆ − β 3 Sˆ = 0, ρA = 0 β 2 SI dL S − Sˆ ˆ = [µS + β 3 Sˆ − β 1 Vˆ + β 1 V + γI − − µS dt S 1 + bI + cI 2 V − Vˆ σβ 2 IV − β3 S] + [ β 1 Vˆ + µVˆ − β 3 Sˆ + β 3 S − V 1 + bI + cI 2 β 2 SI σβ 2 IV − β 1 V − µV ] + + − (α + γ + µ) I 1 + bI + cI 2 1 + bI + cI 2 dL (µ + β 3 ) (µ + β 1 ) β β =− (S − Sˆ )2 − (V − Vˆ )2 + ( 1 + 3 )(S − Sˆ )(V dt S V S V ˆ ˆ γSI σβ 2 IV β 2 SI β 2 SI + γI − − − Vˆ ) − + S 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2 σβ 2 I Vˆ β 2 SI σβ 2 IV + + + − (α + µ) I − γI 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2

dL (µ + β 3 ) (µ + β 1 ) β β =− (S − Sˆ )2 − (V − Vˆ )2 + ( 1 + 3 )(S − Sˆ )(V dt S V S V σβ V β γ 2 2 ˆ +( − Vˆ ) + ( − )SI − (α + µ)) I 2 S 1 + bI + cI 1 + bI + cI 2 (3.12) Now for any (S, V, I ) in φ and by equation (3.12) we get dL (µ + β 3 ) (µ + β 1 ) β β <− (S − Sˆ )2 − (V − Vˆ )2 + ( 1 + 3 )(S − Sˆ )(V − Vˆ ) dt S V S V

60

Chapter 3

The Dynamics of SVIS Epidemic Model

(µ + β 3 ) (µ + β 1 ) dL <− (S − Sˆ )2 − (V − Vˆ )2 + 2 dt S V

r

(µ + β 1 )(µ + β 3 ) (S SV

− Sˆ )(V − Vˆ ) dL <−[ dt

r

(µ + β 3 ) (S − Sˆ ) − S

r

(µ + β 1 ) (V − Vˆ )]2 < 0 V

dL dt

is a negative definite and hence L is a Lyapunov function (Theorem (1.4)) with respect to Eˆ hence, Eˆ is globally asymptotically stable in the subregion φ. Theorem 3.5. Assume that the endemic equilibrium point E∗ (S∗ , V ∗ , I ∗ ) of the system (3.1) is locally asymptotically stable, then it is globally asymptotically stable in the subregion Ω that satisfies the following conditions:

[( [

β1 β3 1 + ) G ]2 < [(µ + β 3 ) G + β 2 I ][(µ + β 1 ) G + σβ 2 I ] S V SV

(3.13)

1 β 2 r2 (S∗ + σV ∗ ) γG β 2 S∗ r2 2 − + β ] < [( µ + β ) G + β I ][ r G − ] 2 3 2 1 S SG ∗ SI G∗ (3.14)

σβ 2 V ∗ r2 2 1 β 2 r2 (S∗ + σV ∗ ) [σβ 2 − ] < [(µ + β 1 ) G + σβ 2 I ][r1 G − ] VG ∗ VI G∗ (3.15) Where G = 1 + bI + cI 2 , G ∗ = 1 + bI ∗ + cI ∗ 2 , r1 (α + γ + µ) and r2 = 1 − cI I ∗ . Proof: Consider the function ∗

∗

∗

P(S , V , I ) =

Z S τ1 − S∗ S∗

τ1

dτ1 +

Z V τ2 − V ∗ V∗

τ2

dτ2 +

Z I τ3 − I ∗ I∗

τ3

dτ3

By differentiating L with respect to t along the solution of the system

61

Chapter 3

The Dynamics of SVIS Epidemic Model

(3.1), we get dP S − S∗ dS V − V ∗ dV I − I ∗ dI = . + . + . dt S dt V dt I dt By substituting the value of

dS dV dt , dt

and

dI dt

in the above equation

dP S − S∗ β 2 SI = [Λ + (1 − ρ) A + β 1 V + γI − − µS − β 3 S] dt S 1 + bI + cI 2

+

V − V∗ σβ 2 V I I − I∗ − β V − µV ] + [ β3 S − [ρA 1 V I 1 + bI + cI 2

+

β 2 SI σβ 2 V I + − (α + γ + µ) I ] 1 + bI + cI 2 1 + bI + cI 2

Where Λ + (1 − ρ ) A =

β 2 S∗ I ∗ + µS∗ + β 3 S∗ − β 1 V ∗ − γI ∗ 2 ∗ ∗ 1 + bI + cI

σβ 2 V I ∗ + β 1 V ∗ + µV ∗ − β 3 S∗ = 0 2 ∗ ∗ 1 + bI + cI ρA = (α + γ + µ) I ∗ −

β 2 S∗ I ∗ σβ 2 V ∗ I ∗ − 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2

dP S − S∗ β 2 S∗ I ∗ = [ + µS∗ + β 3 S∗ − β 1 V ∗ − γI ∗ + β 1 V + γI 2 ∗ ∗ dt S 1 + bI + cI β 2 SI V − V∗ σβ 2 V I ∗ − − µS − β 3 S] + [ + β 1 V ∗ + µV ∗ 2 ∗ ∗ V 1 + bI + cI 2 1 + bI + cI I − I∗ σβ 2 V I − β 1 V − µV ] + [(α + γ + µ) I ∗ − β3 S + β3 S − 2 I 1 + bI + cI ∗

−

β 2 S∗ I ∗ σβ 2 V ∗ I ∗ β 2 SI σβ 2 V I − + + 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2 1 + bI + cI 2 1 + bI + cI 2

− (α + γ + µ) I ]

62

Chapter 3

The Dynamics of SVIS Epidemic Model

dP S − S∗ = [−(µ + β 3 )(S − S∗ ) + β 1 (V − V ∗ ) + γ( I − I ∗ ) dt S β 2 SI β 2 S∗ I β 2 S∗ I β 2 S∗ I ∗ − + − ] + 1 + bI ∗ + cI ∗ 2 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2

+

V − V∗ σβ 2 V ∗ I ∗ [−(µ + β 1 )(V − V ∗ ) + β 3 (S − S∗ ) + V 1 + bI ∗ + cI ∗ 2

σβ 2 V I σβ 2 V ∗ I σβ 2 V ∗ I I − I∗ − + − ]+ [−(α + γ I 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2 σβ 2 V ∗ I ∗ β 2 SI β 2 S∗ I ∗ − + + µ)( I − I ) − 1 + bI ∗ + cI ∗ 2 1 + bI ∗ + cI ∗ 2 1 + bI + cI 2 ∗

σβ 2 V I β 2 S∗ I β 2 S∗ I σβ 2 V ∗ I + + − + 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2 1 + bI + cI 2 β2 I (V − V ∗ )2 dP −(S − S∗ )2 = [(µ + β 3 ) + ] − [(µ + β 1 ) dt S V 1 + bI + cI 2

( I − I ∗ )2 σβ 2 I ]− [(α + γ + µ) + I 1 + bI + cI 2 −

β 2 (1 − cI I ∗ )(S∗ + σV ∗ ) β3 ∗ ∗ β1 + ] ] + ( S − S )( V − V )[ S V (1 + bI + cI 2 )(1 + bI ∗ + cI ∗ 2 )

γ β 2 S∗ (1 − cI I ∗ ) + (S − S∗ )( I − I ∗ )[ − S S(1 + bI + cI 2 )(1 + bI ∗ + cI ∗ 2 )

+

β2 σβ 2 ∗ ∗ ] + ( V − V )( I − I )[ 1 + bI + cI 2 1 + bI + cI 2

σβ 2 V ∗ (1 − cI I ∗ ) ] − V (1 + bI + cI 2 )(1 + bI ∗ + cI ∗ 2 ) dP −(S − S∗ )2 β I (V − V ∗ )2 σβ I = [(µ + β 3 ) + 2 ] − [(µ + β 1 ) + 2 ] dt 2S G 2V G β1 β3 ( S − S ∗ )2 β I + (S − S )(V − V )[ + ] − [(µ + β 3 ) + 2 ] S V 2S G ∗

−

∗

( I − I ∗ )2 β r (S∗ + σV ∗ ) β 2 S ∗ r2 ∗ ∗ γ [r1 − 2 2 ] + ( S − S )( I − I )[ − 2I GG ∗ S SGG ∗

63

Chapter 3

The Dynamics of SVIS Epidemic Model

β2 (V − V ∗ )2 σβ 2 I ( I − I ∗ )2 + ]− [(µ + β 1 ) + ]− [r1 G 2V G 2I β 2 r2 (S∗ + σV ∗ ) σβ 2 V ∗ r2 ∗ ∗ σβ 2 − ] + (V − V )( I − I )[ − ] GG ∗ G VGG ∗

dP −1 = [ dt G

r

(µ + β 3 ) G + β 2 I (S − S∗ ) − 2S

r

(µ + β 1 ) G + σβ 2 I (V − V ∗ )]2 2V s r β 2 r2 (S∗ +σV ∗ ) r G − (µ + β 3 ) G + β 2 I 1 1 G∗ − [ (S − S∗ ) − ( I − I ∗ )]2 G 2S 2I s r β 2 r2 (S∗ +σV ∗ ) r G − 1 (µ + β 1 ) G + σβ 2 I 1 G∗ (V − V ∗ ) − ( I − I ∗ )]2 − [ G 2V 2I

dP <0 dt

So

dP dt

is a negative definite and P is a Lyapunov function (Theorem

(1.4)) with respect to E∗ hence, E∗ is globally asymptotically stable in the subregion Ω.

3.6

Numerical analysis

In this section, the global dynamics of the system (3.1) is studied. The system (3.1) is solved numerically for different sets of initial conditions and different sets of parameters. The following set of parameter values is chosen Λ = 500, ρ = 0, A = 100, β 1 = 0.12, β 2 = 0.00001, β 3 = 0.2,

(3.16)

b = 2.5, c = 1.5, γ = 0.14, µ = 0.12, σ = 0.6, α = 0.2 For this parameter values, it is observed that the condition of the theorem (3.2), and the theorem (3.4) is satisfied and trajectories with initial conditions I0 = (150, 200, 800), I1 = (300, 400, 200), I2 = (200, 700, 800) converge to the disease-free equilibrium point Eˆ = (2727.2727, 2272.727 272, 0) where all three populations coexist in the form of a stable equi64

Chapter 3

The Dynamics of SVIS Epidemic Model

librium point. This indicates that the free equilibrium point Eˆ is globally asymptotically stable. See Fig.(3.2)

Figure 3.2: This figure depicts the trajectories of the model equation (3.9) with the initial conditions I0 , I1 and I2 . In this case, all solutions converges to the disease-free equilibrium point. Note: In the following figures, we will use the following representations: solid line for S; dashed line for V and dotted line for I.

(a) Time series of trajectories of system (3.1) for data given in equation (3.9) starting at I0

(b) Time series of trajectories of system (3.1) for data given in equation (3.9) starting at I1 .

Figure 3.3: Time series of trajectories of system (3.1) 65

Chapter 3

The Dynamics of SVIS Epidemic Model

However, for the set of parameters values given the equation (3.9) with ρ = 0.2 system (3.1) approaches asymptotically to endemic equilibrium point E∗ = (2664.03, 2220.02, 43.4795) in the IntR3+ starting from different sets of initial conditions I3 = (400, 2500, 8), I4 = (50, 500, 10), and I5 = (2500, 700, 20). Thus, our simulation results show that the equilibrium E∗ = (2664.0302022, 2220.024349, 43.47954323) is globally asymptotically stable. See Fig.(3.4)

Figure 3.4: This figure depicts the trajectories of the model equation (3.9) with initial conditions I3 , I4 and I5 . In this case, all solutions converges to the endemic equilibrium point E∗

(a) trajectories starting at I3

(b) trajectories starting at I4

Figure 3.5: Time series of trajectories of system for data given in equation (3.9) with ρ = 0.2 66

Chapter 3

The Dynamics of SVIS Epidemic Model

Now the effect of varying the fraction of immigrant individuals, which arrive infected on the dynamics of the system (3.1) is studied system (3.1) is solved for parameters values ρ = 0.05, ρ = 0.4 and ρ = 0.6 respectively keeping other parameters fixed as given in equation (3.16) with ρ = 0.2 and then the trajectories of system (3.1) are drawn in Fig.(3.6).

(a) For ρ = 0.05 the system approaches asymptotically to (2711.457277,2259.54474,10.87424254)

(b) For ρ = 0.4 the system approaches asymptotically to (2600.789808,2167.324432,86.95715964)

(c) For ρ = 0.6 the system approaches asymptotically to (2537.548945,2114.623854,130.4352001)

Figure 3.6: Time series of the solution of system (3.1) According to Fig.(3.6) the disease-free equilibrium point Eˆ of system (3.1) becomes unstable point and the solution the system (3.1) approaches asymptotically to the endemic equilibrium point E∗ (where ρ increases). Now, in order to discuss the effect variation of rate on the dynamical behavior of the system (3.1), the system is solved for different values of infection rate β 2 = 0.01, β 2 = 0.05 and β 2 = 0.08 respectively, keeping other parameters fixed as given in equation (3.16) with ρ = 0.2 and then the solution of system (3.1) is drawn in Fig.(3.7) (a-c). 67

Chapter 3

The Dynamics of SVIS Epidemic Model

(a) For β 2 = 0.01 the system approaches asymptotically to (2662.652085,2218.08013,44.7254181)

(b) For β 2 = 0.05 the system approaches asymptotically to (2657.74771,2211.16522,49.1576489)

(c) For β 2 = 0.08 the system approaches asymptotically to (2654.533001,2206.63609,52.6615874)

Figure 3.7: Time series of the solution of system (3.1) From Fig.(3.7) (a-c) we see that, if the infection rate increases the endemic equilibrium point of system (3.1) still coexists and stable but number of susceptible and vaccinated individuals decrease while the number of the infected individuals increases. Also, in order to discuss the effect of varying the vaccination converge rate on the dynamical behavior of system (3.1) is studied too. The system is solved numerically for different value of β 3 = 0.4, β 3 = 0.6 and β 3 = 0.8 keeping the rest of parameters fixed as given in equation (3.16) with ρ = 0.2 and time series of the solution of system (3.1) are drawn in Fig.(3.8) (a-c).

68

Chapter 3

The Dynamics of SVIS Epidemic Model

(a) For β 3 = 0.04 the system approaches asymptotically to (1838.520986,3052.53385,43.4794363)

(b) For β 3 == 0.6 the system approaches asymptotically to (1396.44464,3488.61033,43.47938)

(c) For β 3 = 0.8 the system approaches asymptotically to (1127.089953,3756.965124,43.4793459)

Figure 3.8: Time series of the solution of system (3.1) From Fig.(3.8) (a-c), it can be roted that the system (3.1) still approaches to endemic equilibrium point. Similarly, the effect of variation of the number of individuals who lose vaccine immunity and return to susceptible on the dynamical behavior of system (3.1) is studied and the system is solved for the value β 1 = 0.14, β 1 = 0.16 and β 1 = 0.18 keeping other parameters as given in equation (3.16) with ρ = 0.2 and then the solution of system (3.1) are drawn in Fig.(3.9) (a-c) respectively.

69

Chapter 3

The Dynamics of SVIS Epidemic Model

(a) For β 1 = 0.14 the system approaches asymptotically to (2760.55296,2123.50155,43.4795556)

(b) For β 1 = 0.16 the system approaches asymptotically to ( 2849.03216,2035.02232,43.4795669)

(c) For β 1 = 0.18 the system approaches asymptotically to (2930.43302,1953.62143,43.4795774)

Figure 3.9: Time series of trajectories for system (3.1) From Fig.(3.9) (a-c) it is observed that as β 1 increases the system (3.1) still approaches to endemic equilibrium point and increasing β 1 causes increasing in the susceptible and infected but the number of vaccinated decreases. Finally, the effect of vaccine efficiency against the disease on the dynamical behavior of system (3.1) is investigated. The system is solved for different values of σ = 0.8, σ = 0.9 and σ = 0.95 keeping other parameters as given in the equation (3.16) with ρ = 0.2 and then the solution of the system (3.1) are drawn in Fig.(3.10) (a-c) respectively.

70

Chapter 3

The Dynamics of SVIS Epidemic Model

(a) For σ = 0.8 the system approaches asymptotically to (2664.0301439,2220.024027,43.4796857)

(b) For σ = 0.9 the system approaches asymptotically to (2664.0301148,2220.023866,43.4797569)

(c) For σ = 0.95 the system approaches asymptotically to (2664.0301002,2220.02378,43.4797925)

Figure 3.10: Time series of the solution of system (3.1) From Fig.(3.10) (a-c) we conclude that as the vaccine efficiency increases the endemic equilibrium point of system (3.1) still coexists and stable.

71

Chapter Four Discussion, Conclusions and Future Work

Chapter Four Discussion, Conclusions and Future Work In this chapter, we discussed about the main goal in this thesis. it have section discussion and conclusions and section Future work, as follows:

4.1

Discussion and Conclusions

In chapter two, we have analyzed a SIS epidemic model with treatment. The global dynamics of SIS model with saturated incidence βSI 1+ aS+bI

and saturated treatment function is investigated. A sufficient condition for the existence of equilibrium points is obtained and the dynamical behaviour of the model is discussed. Also, the global asymptotic stability of the disease-free and endemic equilibria is studied analytically as well as numerically . From numerically simulation (subsection 2.3) the following results are obtained the SIS system (2.1) is approaches either to the disease-free equilibrium point or to endemic equilibrium point. Also, the infection rate β of system (2.3) increases the endemic equilibrium point still coexists and stable with increase in the value of infected individuals whereas the value of susceptible individuals, decreases and the effect treatment rate δ increases the endemic equilibrium point of system (2.3), which is still stable and the number of susceptible individuals increases whereas the number of infected individuals decreases. In chapter three, we have analyzed a SIS epidemic model with the effect of vaccine and immigrants on the dynamical behavior on it. The local as well as global stability analysis of each possible equilibrium point are studied analytically as well as numerically. From numerically simulation (subsection 3.6) the following results are obtained the SVIS system (3.1) is approaches either to the disease-free equilibrium point or to endemic equilibrium point. As the fraction of infected immigrant individuals ρ increases then

73

Chapter 4

Discussion and conclusions

the number of susceptible and vaccinated individuals increase but the number of infected individual’s increases. If the infection rate β 2 increases then the number of susceptible and vaccinated individuals decrease but the number of infected individual’s increases increasing of β 3 causes increasing in the vaccinated but the number of susceptible infected decrease (very slowly). As β 1 increases then the number of susceptible and infected individuals increase, however the number of vaccinated individuals decreases. Finally as the vaccine efficiency increases then the number of susceptible and vaccinated individuals decrease but the number of infected individuals increases.

4.2

Future work

In this thesis, using

βIS 1+ aS+bI

system (3.1), but can be change

in the system (2.1) and βIS 1+ aS+bI

and

β2 I 1+bI +cI 2

β2 I 1+bI +cI 2

in the

in the system (2.1)

and (3.1) respectively make a new system and worked by the same way to find the existence, stability and global dynamics of the model of the disease free equilibrium point and endemic equilibrium point.

74

References

[1] Anderson R. M. and May R. M., (1979), “Population Biology of Infectious Disease: Part I”, Nature 280, pp. 361-376. [2] Andrews J. F., (1968), “A Mathematical Model for the Continuous Culture of Microorganisms Utilizing Inhibitory Substrates”, Biotechnol. Bioeng. 10, pp. 707-723. [3] Blanchard P., et. al., (2002), Differential Equations, Boston University. [4] Brauer F. and Nohel J. A., (1989), The Qualitative Theory of Ordinary Differential Equations: An Introduction”, Dover Publications, New York. [5] Christopher M. Kribs-Zaleta and Jorge X. Velasco- Hernández, (2000), “A Simple Vaccination Model with Multiple Endemic States”, Mathematical Biosciences, 164, pp. 183-201. [6] Farrington C. P., (2003), “On Vaccine Efficacy and Reproduction Numbers”, Mathematical Biosciences, 185, pp. 89-109. [7] Freedman H. I., (1980), “Deterministic Mathematical Models in Population Ecology”, Marcel Dekker, Inc, New York, USA. [8] Gao L. Q. and Hethcote H. W., (1992), “Disease Models of with Density-Dependent Emographics”, J. Math. Biol., 30, pp. 717-731. [9] Gao L. Q. and Hethcote H. W., (1995), “Disease Models of with Density-Dependent Emographics”, J. Math. Biol., 50, pp. 17-31. [10] Gumel A. B. and Moghadas S. M., (2003), “A Qualitative Study of a Vaccination Model with Nonlinear Incidence”, Appl. Math. Comput., 143, pp. 409-419. [11] Halloran M.E. et. al., (1991), “Direct and Indirect Effects in Vaccine Efficacy and Effectiveness”, Am. J. Epidemiol., 133, pp. 323-31. VIII

[12] Hasan K., (2013), “Stability of SVIS Model where the Vaccine is Utterly Infective”, J. Math. Comput. Sci., no., 6, pp. 1601-1616. [13] Hethcote H. W. and Tudor D. W., (1980), “Integral Equations Models for Endemic Infectious Disease”, J. Math. Biol., 9, pp. 37-47. [14] Hirsch M. W. and Smale S., (1974), Differential Equation, Dynamical System, and Linear Algebra, New York, Academic Press. [15] Holling C. S., (1959), “The Components of Predation as Revealed by a Study of Small-Mammal Predation of the European Pine Sawfly”, The Canadian Entomologist, 91, pp. 293-320. [16] Hu Z. X.; Liu S. and Wang H., (2008), “Backward Bifurcation of an Epidemic Model with Standard Incidence Rate and Treatment Rate”, Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2302-2312. [17] Zhou J. and Hethcote H.W., (1994), “Population Size Dependent Incidence in Models for Diseases without Immunity”, J. Math. Biol., 32, pp. 809-834. [18] Kermack W. O. and Mckendrick A. G., (1927), “Contributions to Mathematical Theory of Epidemics”, Proc. R. Soc. Lond. A., 115, pp. 700-721. [19] Kermack W. O. and McKendrick A. G., (1932), “Contributions to the Mathematical Theory of Epidemics, part II”, Proc. Roy. Soc. London, 138, pp. 55-83. [20] Kobayashi K., (1994), “Uniformly Bounded Solutions of Functional Differential Equations”, Tokyo J. Math. vol. 17, No. 1. [21] Kundo K. and Chattopadhyay J., (2006), “A Ratio-Dependent EcoEpidemiological Model of the Salton Sea”, Mathematical methods in Applied Sci.ences, vol.29, pp. 191-207. [22] Lacitignola D., (2013), “Saturated Treatments and Measles Resurgence Episodes in South Africa: a Possible Linkage”, Mathematical Biosciences and Engineering: MBE, vol. 10, no. 4, pp. 1135-1157. IX

[23] Li J. and Ma Z., (2002), “Qualitative Analysis of SIS-Epidemic Model with Vaccination and Varying Total Population Size”, Math. Comput. Model, 20, pp. 1235-1243. [24] Li X. Z.; Li W. S. and Ghosh M., (2009), “Stability and Bifurcation of an SIS Epidemic Model with Treatment, Chaos”, Solitons amp & Fractals, vol. 42, no. 5, pp. 2822-2832. [25] Lial M. L., Hornspy J. and Schneider D. I., (2001), Pre-calculus, Addison- Wesley, Inc. New York, USA. [26] Lih-Ing W. Roeger, (2013), “Dynamically Consistent Discrete-Time SI and SIS Epidemic Models”, Website: www.aimSciences.org, pp. 653-662. [27] Liu W. M.; Levin S. A., and Iwasa Y., (1986), “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models”, Journal of Mathematical Biology, vol. 23, no. 2, pp. 187-204. [28] Liu W. M.; Hethcote H. W., and Levin S. A., (1987), “Dynamical behavior of epidemiological models with nonlinear incidence rates”, Journal of Mathematical Biology, vol. 25, no. 4, pp. 359-380. [29] May R. M., (1973), Stability and Complexity in Model Ecosystems, Princeton, New Jersey: Princeton University Press. [30] Murray J. D., (1993), Mathematical Biology, Springer-verlag, New York. [31] Perko L., (1983), Differential Equations and Dynamical Systems, Texts in Applied Mathematics, Vol.7, Springer-Verlag, New York. [32] Safi M. A.;Gumel A. B. and Elbasha E. H., (2013), “Qualitative Analysis of an Age-Structured SEIR Epidemic Model with Treatment”, Applied Mathematics and Computation, vol. 219, no. 22, pp. 10627-10642. [33] Shim E., (2006), “A Note on Epidemic Models with Infective Immigrants and Vaccination”, Mathematical Biosciences and Engineering, vol. 3, pp. 557-566. X

[34] Shulin Sun, (2012), “Global Dynamics of a SEIR Model with a Varying Total Population Size and Vaccination”, Int. J. of Math. Analysis, no. 40, pp. 1985-1995. [35] Sokol W. and Howell J. A., (1980), “Kinetics of Phenol Oxidation by Washed Cells”, Biotechnol. Bioeng. 23, pp. 2039-2049. [36] Stuart A. M. and Humphries A. R., (1996), Dynamical Systems and Numerical Analysis, Cambridge University Press. [37] Wang J. and Jiang Q., (2014,), “Analysis of an SIS Epidemic Model with Treatment”, Advances in Difference Equations 2014:246. [38] Wang W. D. and Ruan S. G., (2004), “Bifurcation in an Epidemic Model with Constant Removal Rate of the Infectives”, Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 775-793. [39] Wang W. D., (2006), “Backward Bifurcation of an Epidemic Model with Treatment”, Mathematical Biosciences, vol. 201, no. 1-2, pp. 5871. [40] Wensheng Y., Xuepeng Li and Zijun Bai, (2008), “Permanence of Periodic Holling type-IV predator-Prey System with Stage Structure for Prey”, Mathematical and Computer Modelling, 48, pp. 677-684. [41] Wiggins S., (1990), Introduction to Applied Non-Linear Dynamical System and Chaos, Spring-Verlag, New York, Inc. [42] Zhang X. and Liu X. N., (2008), “Backward Bifurcation of an Epidemic Model with Saturated Treatment Function”, Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 433-443. [43] Zhang X. and Liu X. N., (2009), “Backward Bifurcation and Global Dynamics of an SIS Epidemic Model with General Incidence Rate and Treatment”, Nonlinear Analysis: RealWorld Applications, vol. 10, no. 2, pp. 565-575. [44] Zhou L. H. and Fan M., (2012), “Dynamics of an SIR Epidemic Model with Limited Medical Resources Revisited”, Nonlinear Analysis: RealWorld Applications, vol. 13, no. 1, pp. 312-324. XI

[45] Zhou X. Y. and Cui J. A., (2011), “Analysis of Stability and Bifurcation for an SEIR Epidemic Model with Saturated Recovery Rate”, Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4438-4450. [46] Zhou Y. and Liu H., (2003), “Stability of Periodic Solutions for an SIS Model with Pulse Vaccination”, Math. Comput. Model, 38, pp. 299-308.

XII

‫الـخالصـة‬ ‫اهلدف من هذا البحث هو دراسة تأثري بعض املفاهيم الوبائية ‪ ,‬مثل عوامل العالج واهلجرة والتلقيح على‬ ‫السلوك الديناميكي لبعض النماذج الوبائية حيث عرضنا نوعني من هذه النماذج الوبائية ومتت دراستها حتليليا‬ ‫وعدديا‪ .‬يف النموذج األول‪ ،‬مت التحمص يف الديناميكية العامة لنموذج )‪ (SIS‬مع معدل االصابة املشبعة (‬

‫𝐼𝑆𝛽‬ ‫𝐼𝑏‪1+𝑎𝑆+‬‬

‫)‬

‫ودالة العالج املشبعة حيث حددنا شرطا كافيا عند توفره يتم احلصول على نقطة للتوازن ومتت ايضا مناقشة‬ ‫السلوك الديناميكى للنموذج‪ .‬كذلك متت دراسة استقرار احملاذيات العامة لعامل اخللو من املرض كليا من جهة‬ ‫والتوازن يف تزامن املرض من جهة اخرى‪.‬‬ ‫اما النموذج الثانى وقد كان منوذج (‪ )SVIS‬مع معدل االصابة الالخطي (‬

‫𝐼 ‪𝛽2‬‬

‫) حيث مت اعتبار‬

‫‪1+𝑏𝐼+𝑐𝐼2‬‬

‫تاثريعامل اهلجرة ( املهاجرين) على السلوك الديناميكى لنموذج (‪ )SVIS‬حتليليا كما واعطيت شرطا كافيا يضمن‬ ‫وجود واستقرارية نقطة التوازن لعامل التزامن املرضي‪.‬‬ ‫واخريا متت دراسة الديناميكية العامة للنموذج حبله عدديا جملموعات خمتلفة من القيم البدائية و جمموعات‬ ‫خمتلفة من قيم معلمية حيث توصلنا اىل االستنتاج الذي يقودنا اما اىل نقطة التوازن يف عامل اخللو املرضى او اىل‬ ‫نقطة التوازن يف عامل التزامن املرضىي‪.‬‬

‫ديـنـاميـكـيـة النـظـم ‪ SIS‬و ‪ SVIS‬مـع دالــة االستـجابـة مـن‬ ‫النـمـط ‪Beddington-Deangelis‬‬ ‫رسـالـة‬ ‫مـقدمـة اىل جملـس فـاكـليت العـلـوم و تربـيـة العـلـوم‬ ‫سـكول العـلـوم يف جـامـعـة السليمـانـيـة‬ ‫كـجزء مـن مـتـطلبـات نيـل شـهـادة‬ ‫مـاجستري يف عـلـوم الرياضيـــات‬ ‫(النظام الديناميكي)‬ ‫من قبـل‬

‫ميديا باوخان مراخان‬

‫بكالوريوس يف الرياضيات (‪ ،)2011‬يف اجلامعة طةرميان‬ ‫بــإشــرافات‬

‫أ‪ .‬م‪ .‬د‪ .‬كاوه أمحد حسن‬ ‫أ‪ .‬م‪ .‬د‪ .‬مظفر فتاح محه‬

‫مارس‪2016 ،‬‬

‫جـمـادى األول‪1437 ،‬‬

‫ثـوخـتـة‬ ‫ئامانج لةم تويَذينةوةية ليكوَلينةوةية لة كاريطةري هةنديَك لة ضةمكةثةتاييةكان وةك فاكتةرى‬ ‫ضارةسةر وكوَضبةرى وكوتان لة سةر هةلَسوكةوتى ديناميكى هةنديَك لة موَديلة ثةتاييةكان‪ .‬دوو جوَر لةو‬ ‫موَديالنة ثيَشكةش كران بوَ ليَكولَينةوة لة رووى شيكارى و ذمارةيي‪ .‬لة موَديَلى يةكةم ديناميكيةتى طشتى‬ ‫موَديَلى (‪ )SIS‬ليَكوَلينةوةى لة سةر كرا لة طةلَ تيَكراى بةركةوتى تيَر (‬

‫𝐼𝑆𝛽‬ ‫𝐼𝑏‪1+𝑎𝑆+‬‬

‫) و نةخشةى ضارةسةرى‬

‫تيَر و مةرجيَك دياريكرا بة روودانى خالَى هاوسةنطى بةدةست ديَت و هةروةها ديناميكيةتى موَديَلةكة‬ ‫تاوتويَى لةبارةوة كرا‪ .‬هةروةها ليَكوَلينةوة لةسةر جيَطريبوونى كةنارة طشتيةكانى بوَ فاكتةرى بةدةربوون لة‬ ‫نةخوَشى لة اليَكةوة و هاوسةنطى لة فاكتةرى دريَذة ثيَدانى بة نةخوَشى لة اليَكى ترةوة‪.‬‬ ‫موَديَلى دووةم موَديَلى (‪ )SVIS‬بوو لة طةلَ تيَكراى بةركةوتى ناهيَلَى (‬

‫𝐼 ‪𝛽2‬‬

‫) وكاريطةرى فاكتةرى‬

‫‪1+𝑏𝐼+𝑐𝐼2‬‬

‫كوَضبةرى لة سةر هةلَسوكةوتى ديناميكى بوَ موَديَلى (‪ )SVIS‬لة رووى شيكاريةوة بة هةند وةرطريا و‬ ‫مةرجيَك دةستنيشان كرا كة بوون و جيَطريبوونى خالَى هاوسةنطى فاكتةرى دريَذةثيَدانى بة نةخوَشى‬ ‫مسوَطةر دةكات ‪.‬‬ ‫لة كوَتاييدا ليَكوَلينةوة ى ذمارةيي لة ديناميكيةتى طشتى موَديَلةكة كرا بوَ ضةند كوَمة َليَكى جياواز لة‬ ‫نرخة سةرةتاييةكان و ضةند كوَمة َليَكى جياواز لة نرخى ثارةميرتى و طةيشتينة ئةو دةرئةجنامة كة يان‬ ‫بةرةو خالَى هاوسةنطى لة فاكتةرى بةدةربوون لة نةخوَشىمان دةبات يان بةرةوخالَى هاوسةنطى لة فاكتةرى‬ ‫دريَذة ثيَدان بة نةخوَشيمان دةبات‪.‬‬

‫ديـنـامـيـكـيـةتـى سـيـستةمـى ‪ SIS‬و ‪ SVIS‬لـةطـة َل نـةخـشـةى‬ ‫وةآلم دانـةوة لـة جـؤرى ‪Beddington-Deangelis‬‬ ‫نامةيةكة‬ ‫ثيَشكةشى كراوة بة ئةجنومةنى فاكةلَتى زانست و ثةروةردة زانستةكان‬ ‫سكوىل زانست لة زانكؤى سليَمانى‬ ‫وةك بةشيَك لة ثيَداويستيةكانى بةدةستهيَنانى برِوانامةى‬ ‫ماستةرلة زانستى مامتاتيك‬ ‫(سيستةمى ديناميكةكان)‬ ‫لــةاليةن‬

‫ميديا باوةخان مراخان‬

‫بةكالوريس لة مامتاتيك (‪ ،)2011‬لة زانكؤى طةرميان‬

‫بة سةرثةرشتياريان‬

‫ث‪.‬ى‪.‬د‪ .‬كاوة ئةمحد حسن‬ ‫ث‪.‬ى‪.‬د‪ .‬مظفر فتاح محة‬

‫ئادار‪2016 ،‬‬

‫رِةشـةمـآ‪2716 ،‬‬