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Discrete Dynamics in Nature and Society
VolumeΒ 2011Β (2011), Article IDΒ 201274, 19 pages
http://dx.doi.org/10.1155/2011/201274
Research Article

Global Properties of Virus Dynamics Models with Multitarget Cells and Discrete-Time Delays

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

Received 8 July 2011; Accepted 16 October 2011

Academic Editor: YongΒ Zhou

Copyright Β© 2011 A. M. Elaiw and M. A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a class of virus dynamics models with multitarget cells and multiple intracellular delays and study their global properties. The first model is a 5-dimensional system of nonlinear delay differential equations (DDEs) that describes the interaction of the virus with two classes of target cells. The second model is a (2𝑛+1)-dimensional system of nonlinear DDEs that describes the dynamics of the virus, 𝑛 classes of uninfected target cells, and 𝑛 classes of infected target cells. The third model generalizes the second one by assuming that the incidence rate of infection is given by saturation functional response. Two types of discrete time delays are incorporated into these models to describe (i) the latent period between the time the target cell is contacted by the virus particle and the time the virus enters the cell, (ii) the latent period between the time the virus has penetrated into a cell and the time of the emission of infectious (mature) virus particles. Lyapunov functionals are constructed to establish the global asymptotic stability of the uninfected and infected steady states of these models. We have proven that if the basic reproduction number 𝑅0 is less than unity, then the uninfected steady state is globally asymptotically stable, and if 𝑅0>1 (or if the infected steady state exists), then the infected steady state is globally asymptotically stable.

1. Introduction

Nowadays, various types of viruses infect the human body and cause serious and dangerous diseases. Mathematical modeling and model analysis of virus dynamics have attracted the interests of mathematicians during the recent years, due to their importance in understanding the associated characteristics of the virus dynamics and guiding in developing efficient antiviral drug therapies. Several mathematical models have been proposed in the literature to describe the interaction of the virus with the target cells [1]. Some of these models are given by a system of nonlinear ordinary differential equations (ODEs). Others are given by a system of nonlinear delay differential equations (DDEs) to account the intracellular time delays. The basic virus dynamics model with intracellular discrete time delay has been proposed in [2] and given byΜ‡π‘₯(𝑑)=πœ†βˆ’π‘‘π‘₯(𝑑)βˆ’π›½π‘₯(𝑑)𝑣(𝑑),(1.1)̇𝑦(𝑑)=π‘’βˆ’π‘šπœΜ‡π‘£π›½π‘₯(π‘‘βˆ’πœ)𝑣(π‘‘βˆ’πœ)βˆ’π‘Žπ‘¦(𝑑),(1.2)(𝑑)=𝑝𝑦(𝑑)βˆ’π‘π‘£(𝑑),(1.3) where π‘₯(𝑑),𝑦(𝑑), and 𝑣(𝑑) represent the populations of uninfected target cells, infected cells, and free virus particles at time 𝑑, respectively. Here, πœ† represents the rate of which new target cells are generated from sources within the body, 𝑑 is the death rate constant, and 𝛽 is the infection rate constant. Equation (1.2) describes the population dynamics of the infected cells and shows that they die with rate constant π‘Ž. The virus particles are produced by the infected cells with rate constant 𝑝, and are removed from the system with rate constant 𝑐. The parameter 𝜏 accounts for the time between viral entry into the target cell and the production of new virus particles. The recruitment of virus-producing cells at time 𝑑 is given by the number of cells that were newly infected cells at time π‘‘βˆ’πœ and are still alive at time 𝑑. The probability of surviving the time period from π‘‘βˆ’πœ to 𝑑 is π‘’βˆ’π‘šπœ, where π‘š is the constant death rate of infected cells but not yet virus-producing cells.

A great effort has been made in developing various mathematical models of viral infections with discrete or distributed delays and studying their basic and global properties, such as positive invariance properties, boundedness of the model solutions and stability analysis [3–19]. In [20–24], multiple inracellular delays have been incroporated into the virus dynamics model. Most of the existing models are based on the assumption that the virus attacks one class of target cells (e.g., CD4+ T cells in case of HIV or hepatic cells in case of HCV and HBV). Since the interactions of some types of viruses inside the human body is not very clear and complicated, therefore, the virus may attack more than one class of target cells. Hence, virus dynamics models describing the interaction of the virus with more than one class of target cells are needed. In case of HIV infection, Perelson et al. [25] observed that the HIV attack two classes of target cells, CD4+ T cells and macrophages. In [26, 27], an HIV model with two target cells has been proposed. In very recent works [28–30], we have proposed several HIV models with two target cells and investigated the global asymptotic stability of their steady states. In [31], we have proposed a class of virus dynamics models with multitarget cells. However, the intracellular time delay has been neglected in [26–31].

The purpose of this paper is to propose a class of virus dynamics models with multitarget cells and establish the global stability of their steady states. The first model considers the interaction of the virus with two classes of target cells. In the second model, we assume that the virus attacks 𝑛 classes of target cells. The third model generalizes the second one by assuming that the infection rate is given by saturation functional response. We incorporate two types of discrete time delays into these models describing (i) the time between the target cell is contacted by the virus particle and the contacting virus enters the cell, (ii) the time between the virus has penetrated into a cell and the emission of infectious (mature) virus particles. The global stability of these models is established using Lyapunov functionals, which are similar in nature to those used in [11, 20]. We prove that the global dynamics of these models are determined by the basic reproduction number 𝑅0. If 𝑅0≀1, then the uninfected steady state is globally asymptotically stable (GAS). If 𝑅0>1 (or if the infected steady state exists), then the infected steady state is GAS for all time delays.

2. Virus Dynamics Model with Two Target Cells and Delays

In this section, we introduce a mathematical model of virus infection with two classes of target cells. This model can describe the HIV dynamics with two classes of target cells, CD4+ T cells and macrophages [26, 27]. This model can be considered as an extension of the models given in [11, 26, 27]: Μ‡π‘₯1(𝑑)=πœ†1βˆ’π‘‘1π‘₯1(𝑑)βˆ’π›½1π‘₯1(𝑑)𝑣(𝑑),(2.1)̇𝑦1(𝑑)=π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έβˆ’π‘Ž1𝑦1(𝑑),(2.2)Μ‡π‘₯2(𝑑)=πœ†2βˆ’π‘‘2π‘₯2(𝑑)βˆ’π›½2π‘₯2(𝑑)𝑣(𝑑),(2.3)̇𝑦2(𝑑)=π‘’βˆ’π‘š2𝜏2𝛽2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έβˆ’π‘Ž2𝑦2Μ‡(𝑑),(2.4)𝑣(𝑑)=π‘’βˆ’π‘›1πœ”1𝑝1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έ+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έβˆ’π‘π‘£(𝑑),(2.5) where π‘₯1and π‘₯2 represent the populations of the two classes of uninfected target cells; 𝑦1and 𝑦2 are the populations of the infected cells. The population of the target cells are described by (2.1) and (2.3), where πœ†1 and πœ†2 represent the rates of which new target cells are generated, 𝑑1 and 𝑑2 are the death rate constants, and 𝛽1 and 𝛽2 are the infection rate constants. Equations (2.2) and (2.4) describe the population dynamics of the two classes of infected cells and show that they die with rate constants π‘Ž1 and π‘Ž2. The virus particles are produced by the two classes of infected cells with rate constants 𝑝1 and 𝑝2 and are cleared with rate constant 𝑐. Here the parameter πœπ‘– accounts for the time between the target cells of class 𝑖 are contacted by the virus particle and the contacting virus enters the cells. The recruitment of virus-producing cells at time 𝑑 is given by the number of cells that were newly infected cells at times π‘‘βˆ’πœπ‘– and are still alive at time 𝑑. Also, π‘šπ‘– is assumed to be a constant death rate for infected target cells, but not yet virus-producing cells. Thus, the probability of surviving the time period from π‘‘βˆ’πœπ‘– to 𝑑 is π‘’βˆ’π‘šπ‘–πœπ‘–, 𝑖=1,2. The time between the virus has penetrated into a target cell of class 𝑖 and the emission of infectious (matures) virus particles is represented by πœ”π‘–. The probability of survival of an immature virus is given by π‘’βˆ’π‘›π‘–πœ”π‘–, where 𝑛𝑖 is constant.

2.1. Initial Conditions

The initial conditions for system (2.1)–(2.5) take the formπ‘₯1(πœƒ)=πœ‘1(πœƒ),𝑦1(πœƒ)=πœ‘2(πœƒ),π‘₯2(πœƒ)=πœ‘3(πœƒ),𝑦2(πœƒ)=πœ‘4(πœƒ),𝑣(πœƒ)=πœ‘5πœ‘(πœƒ),π‘–ξ€Ίξ€½πœ(πœƒ)β‰₯0,πœƒβˆˆβˆ’max1,𝜏2,πœ”1,πœ”2ξ€Ύξ€»,0,𝑖=1,…,5,(2.6) where (πœ‘1(πœƒ),…,πœ‘5(πœƒ))∈𝐢([βˆ’max{𝜏1,𝜏2,πœ”1,πœ”2},0],ℝ5+), the Banach space of continuous functions mapping the interval [βˆ’max{𝜏1,𝜏2,πœ”1,πœ”2},0]into ℝ5+, where ℝ5+={(𝑧1,𝑧2,…,𝑧5)βˆΆπ‘§π‘–β‰₯0}.

By the fundamental theory of functional differential equations [32], system (2.1)–(2.5) has a unique solution(π‘₯1(𝑑),𝑦1(𝑑),π‘₯2(𝑑),𝑦2(𝑑),𝑣(𝑑)) satisfying the initial conditions (2.6).

2.2. Nonnegativity and Boundedness of Solutions

In the following, we establish the nonnegativity and boundedness of solutions of (2.1)–(2.5) with initial conditions (2.6).

Proposition 2.1. Let (π‘₯1(𝑑),𝑦1(𝑑),π‘₯2(𝑑),𝑦2(𝑑),𝑣(𝑑)) be any solution of (2.1)–(2.5) satisfying the initial conditions (2.6), then π‘₯1(𝑑), 𝑦1(𝑑), π‘₯2(𝑑), 𝑦2(𝑑),and 𝑣(𝑑) are all nonnegative for 𝑑β‰₯0 and ultimately bounded.

Proof. From (2.1) and (2.3), we have π‘₯𝑖(𝑑)=π‘₯𝑖(0)π‘’βˆ’βˆ«π‘‘0(𝑑𝑖+𝛽𝑖𝑣(πœ‰))π‘‘πœ‰+πœ†π‘–ξ€œπ‘‘0π‘’βˆ’βˆ«π‘‘πœ‚(𝑑𝑖+𝛽𝑖𝑣(πœ‰))π‘‘πœ‰π‘‘πœ‚,𝑖=1,2,(2.7) which indicates that π‘₯1(𝑑)β‰₯0, π‘₯2(𝑑)β‰₯0 for all 𝑑β‰₯0. Now from (2.2), (2.4), and (2.5), we have 𝑦𝑖(𝑑)=𝑦𝑖(0)π‘’βˆ’π‘Žπ‘–π‘‘+π‘’βˆ’π‘šπ‘–πœπ‘–ξ€œπ‘‘0𝛽𝑖π‘₯π‘–ξ€·πœ‚βˆ’πœπ‘–ξ€Έπ‘£ξ€·πœ‚βˆ’πœπ‘–ξ€Έπ‘’βˆ’π‘Žπ‘–(π‘‘βˆ’πœ‚)π‘‘πœ‚,𝑖=1,2,𝑣(𝑑)=𝑣(0)π‘’βˆ’π‘π‘‘+ξ€œπ‘‘0ξ€Ίπ‘’βˆ’π‘›1πœ”1𝑝1𝑦1ξ€·πœ‚βˆ’πœ”1ξ€Έ+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2ξ€·πœ‚βˆ’πœ”2π‘’ξ€Έξ€»βˆ’π‘(π‘‘βˆ’πœ‚)π‘‘πœ‚,(2.8) confiming that 𝑦1(𝑑)β‰₯0, 𝑦2(𝑑)β‰₯0, 𝑣(𝑑)β‰₯0 for all π‘‘βˆˆ[0,max{𝜏1,𝜏2,πœ”1,πœ”2}]. By a recursive argument, we obtain 𝑦1(𝑑)β‰₯0, 𝑦2(𝑑)β‰₯0, 𝑣(𝑑)β‰₯0 for all 𝑑β‰₯0.
To show the boundedness of the solutions, we let 𝑋1(𝑑)=π‘’βˆ’π‘š1𝜏1π‘₯1(π‘‘βˆ’πœ1βˆ’πœ”1)+𝑦1(π‘‘βˆ’πœ”1) and 𝑋2(𝑑)=π‘’βˆ’π‘š2𝜏2π‘₯2(π‘‘βˆ’πœ2βˆ’πœ”2)+𝑦2(π‘‘βˆ’πœ”2), then ̇𝑋1(𝑑)β‰€πœ†1π‘’βˆ’π‘š1𝜏1βˆ’πœŽ1𝑋1̇𝑋(𝑑),2(𝑑)β‰€πœ†2π‘’βˆ’π‘š2𝜏2βˆ’πœŽ2𝑋2(𝑑),(2.9) where 𝜎1=min{𝑑1,π‘Ž1} and 𝜎2=min{𝑑2,π‘Ž2}. Hence, limsupπ‘‘β†’βˆžπ‘‹1(𝑑)≀𝐿1, and limsupπ‘‘β†’βˆžπ‘‹2(𝑑)≀𝐿2, where 𝐿1=πœ†1π‘’βˆ’π‘š1𝜏1/𝜎1 and 𝐿2=πœ†2π‘’βˆ’π‘š2𝜏2/𝜎2. On the other hand, ̇𝑣(𝑑)β‰€π‘’βˆ’π‘›1πœ”1𝑝1𝐿1+π‘’βˆ’π‘›2πœ”2𝑝2𝐿2βˆ’π‘π‘£,(2.10) then limsupπ‘‘β†’βˆžπ‘£(𝑑)≀𝐿3, where 𝐿3=(π‘’βˆ’π‘›1πœ”1𝑝1𝐿1+π‘’βˆ’π‘›2πœ”2𝑝2𝐿2)/𝑐. It follows that the solution (π‘₯1(𝑑),𝑦1(𝑑),π‘₯2(𝑑),𝑦2(𝑑),𝑣(𝑑)) is ultimately bounded.

2.3. Steady States

It will be explained in the following that the global behavior of model (2.1)–(2.5) crucially depends on the basic reproduction number given by𝑅0=π‘’βˆ’(π‘š1𝜏1+𝑛1πœ”1)𝑝1𝛽1π‘Ž2π‘₯01+π‘’βˆ’(π‘š2𝜏2+𝑛2πœ”2)𝑝2𝛽2π‘Ž1π‘₯02π‘Ž1π‘Ž2𝑐,(2.11) where π‘₯01=πœ†1/𝑑1 and π‘₯02=πœ†2/𝑑2. We observe that 𝑅0 can be written as𝑅0=𝑅1+𝑅2,(2.12) where𝑅1=π‘’βˆ’(π‘š1𝜏1+𝑛1πœ”1)𝑝1𝛽1πœ†1π‘Ž1𝑑1𝑐,𝑅2=π‘’βˆ’(π‘š2𝜏2+𝑛2πœ”2)𝑝2𝛽2πœ†2π‘Ž2𝑑2𝑐(2.13) are the basic reproduction numbers of each class of target cell dynamics separately (see [28]).

Following the same line as in [28], we can show that if 𝑅0≀1, then system (2.1)–(2.5) has only one steady state 𝐸0=(π‘₯01,0,π‘₯02,0,0) which is called uninfected steady state, and if 𝑅0>1, then system (2.1)–(2.5) has two steady states 𝐸0 and infected steady state 𝐸1=(π‘₯βˆ—1,π‘¦βˆ—1,π‘₯βˆ—2,π‘¦βˆ—2,π‘£βˆ—). The coordinates of the infected steady state are given by π‘₯βˆ—1=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›Ό5𝑐𝛼1𝛼5+𝛼2𝛼3,if𝛼4βˆ’ξ€·π›Ό=0,1𝛼5+𝛼2𝛼3βˆ’π›Ό4𝑐+𝛼1𝛼5+𝛼2𝛼3βˆ’π›Ό4𝑐2+4𝛼1𝛼4𝛼5𝑐2𝛼1𝛼4,if𝛼4π‘₯β‰ 0,βˆ—2=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π›Ό3𝑐𝛼1𝛼5+𝛼2𝛼3,if𝛼4𝑐=0,𝛼2+𝛼1𝛼5+𝛼2𝛼3βˆ’π›Ό4π‘ξ€Έβˆ’ξ”ξ€·π›Ό1𝛼5+𝛼2𝛼3βˆ’π›Ό4𝑐2+4𝛼1𝛼4𝛼5𝑐2𝛼2𝛼4,if𝛼4𝑦≠0,βˆ—1=𝑑1π‘Ž1π‘’π‘š1𝜏1π‘₯01π‘₯βˆ—1ξƒͺπ‘₯βˆ’1βˆ—1,π‘¦βˆ—2=𝑑2π‘Ž2π‘’π‘š2𝜏2π‘₯02π‘₯βˆ—2ξƒͺπ‘₯βˆ’1βˆ—2,π‘£βˆ—=𝑑1𝛽1π‘₯01π‘₯βˆ—1ξƒͺ,βˆ’1(2.14) where 𝛼1=π‘’βˆ’(𝑛1πœ”1+π‘š1𝜏1)𝑝1𝛽1π‘Ž1,𝛼2=π‘’βˆ’(𝑛2πœ”2+π‘š2𝜏2)𝑝2𝛽2π‘Ž2,𝛼3=πœ†2𝛽1,𝛼4=𝛽1𝑑2βˆ’π›½2𝑑1,𝛼5=πœ†1𝛽2.(2.15)

2.4. Global Stability

In this section, we prove the global stability of the uninfected and infected steady states of system (2.1)–(2.5). The strategy of the proof is to use suitable Lyapunov functionals which are similar in nature to those used in [11, 20]. Next we will use the following notation: 𝑧=𝑧(𝑑), for any π‘§βˆˆ{π‘₯1,𝑦1,π‘₯2,𝑦2,𝑣}. We also define a function π»βˆΆβ„>0→ℝβ‰₯0 as 𝐻(𝑧)=π‘§βˆ’1βˆ’ln𝑧.(2.16) It is clear that 𝐻(𝑧)β‰₯0 for any 𝑧>0 and 𝐻 has the global minimum 𝐻(1)=0.

Theorem 2.2. (i) If 𝑅0≀1, then 𝐸0 is GAS for any 𝜏1,𝜏2,πœ”1,πœ”2β‰₯0.
(ii) If 𝑅0>1, then 𝐸1 is GAS for any 𝜏1,𝜏2,πœ”1,πœ”2β‰₯0.

Proof. (i) We consider a Lyapunov functional π‘Š1=π‘Š11+π‘Š12+π‘Š13,(2.17) where π‘Š11=π‘’βˆ’π‘š1𝜏1π‘₯01𝐻π‘₯1π‘₯01ξƒͺ+𝑦1𝑒+π›Ύβˆ’π‘š2𝜏2π‘₯02𝐻π‘₯2π‘₯02ξƒͺ+𝑦2ξƒͺ+π‘Ž1𝑝1𝑒𝑛1πœ”1π‘Šπ‘£,12=π‘’βˆ’π‘š1𝜏1ξ€œπœ10𝛽1π‘₯1(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ+π›Ύπ‘’βˆ’π‘š2𝜏2ξ€œπœ20𝛽2π‘₯2π‘Š(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ,13=π‘Ž1ξ€œπœ”10𝑦1(π‘‘βˆ’πœƒ)π‘‘πœƒ+π›Ύπ‘Ž2ξ€œπœ”20𝑦2(π‘‘βˆ’πœƒ)π‘‘πœƒ,(2.18) where 𝛾=(𝑝2π‘Ž1/𝑝1π‘Ž2)𝑒𝑛1πœ”1βˆ’π‘›2πœ”2. We note that π‘Š1 is defined, continuous, and positive definite for all (π‘₯1,𝑦1,π‘₯2,𝑦2,𝑣)>0. Also, the global minimum π‘Š1=0 occurs at the uninfected steady state 𝐸0.
The time derivatives of π‘Š11 and π‘Š12 are given by π‘‘π‘Š11𝑑𝑑=π‘’βˆ’π‘š1𝜏1π‘₯1βˆ’01π‘₯1ξƒͺΜ‡π‘₯1+̇𝑦1𝑒+π›Ύβˆ’π‘š2𝜏2π‘₯1βˆ’02π‘₯2ξƒͺΜ‡π‘₯2+̇𝑦2ξƒͺ+π‘Ž1𝑝1𝑒𝑛1πœ”1̇𝑣,π‘‘π‘Š12𝑑𝑑=π‘’βˆ’π‘š1𝜏1ξ€œπœ10𝑑𝛽𝑑𝑑1π‘₯1(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ+π›Ύπ‘’βˆ’π‘š2𝜏2ξ€œπœ20𝑑𝛽𝑑𝑑2π‘₯2(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ=βˆ’π‘’βˆ’π‘š1𝜏1ξ€œπœ10π‘‘π›½π‘‘πœƒ1π‘₯1(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒβˆ’π›Ύπ‘’βˆ’π‘š2𝜏2ξ€œπœ20π‘‘π›½π‘‘πœƒ2π‘₯2(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ=π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1π‘£βˆ’π›½1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έξ€Έ+π›Ύπ‘’βˆ’π‘š2𝜏2𝛽2π‘₯2π‘£βˆ’π›½2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2.ξ€Έξ€»(2.19) Similarly, π‘‘π‘Š13/𝑑𝑑 is given by π‘‘π‘Š13𝑑𝑑=π‘Ž1𝑦1βˆ’π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έξ€Έ+π›Ύπ‘Ž2𝑦2βˆ’π‘¦2ξ€·π‘‘βˆ’πœ”2.ξ€Έξ€Έ(2.20) It follows that π‘‘π‘Š1𝑑𝑑=π‘’βˆ’π‘š1𝜏1π‘₯1βˆ’01π‘₯1ξƒͺξ€·πœ†1βˆ’π‘‘1π‘₯1βˆ’π›½1π‘₯1𝑣+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έβˆ’π‘Ž1𝑦1𝑒+π›Ύβˆ’π‘š2𝜏2π‘₯1βˆ’02π‘₯2ξƒͺξ€·πœ†2βˆ’π‘‘2π‘₯2βˆ’π›½2π‘₯2𝑣+π‘’βˆ’π‘š2𝜏2𝛽2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έβˆ’π‘Ž2𝑦2ξƒ­+π‘Ž1𝑝1𝑒𝑛1πœ”1ξ€Ίπ‘’βˆ’π‘›1πœ”1𝑝1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έ+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έξ€»βˆ’π‘π‘£+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1π‘£βˆ’π›½1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έξ€»+π›Ύπ‘’βˆ’π‘š2𝜏2𝛽2π‘₯2π‘£βˆ’π›½2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έξ€»+π‘Ž1𝑦1βˆ’π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έξ€»+π›Ύπ‘Ž2𝑦2βˆ’π‘¦2ξ€·π‘‘βˆ’πœ”2ξ€Έξ€»=π‘’βˆ’π‘š1𝜏1πœ†1π‘₯2βˆ’01π‘₯1βˆ’π‘₯1π‘₯01ξƒ­+π‘’βˆ’π‘š2𝜏2π›Ύπœ†2π‘₯2βˆ’02π‘₯2βˆ’π‘₯2π‘₯02ξƒ­+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯01𝑣+π‘’βˆ’π‘š2𝜏2𝛾𝛽2π‘₯02π‘Žπ‘£βˆ’1𝑐𝑝1𝑒𝑛1πœ”1𝑣=π‘’βˆ’π‘š1𝜏1πœ†1π‘₯2βˆ’01π‘₯1βˆ’π‘₯1π‘₯01ξƒ­+π‘’βˆ’π‘š2𝜏2π›Ύπœ†2π‘₯2βˆ’02π‘₯2βˆ’π‘₯2π‘₯02ξƒ­+π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1𝑅0ξ€Έβˆ’1𝑣.(2.21) Since the arithmetical mean is greater than or equal to the geometrical mean, then the first two terms of (2.21) are less than or equal to zero. Therefore, if 𝑅0≀1, then π‘‘π‘Š1/𝑑𝑑≀0 for all π‘₯1,π‘₯2,𝑣>0. By Theorem  5.3.1 in [32], the solutions of system (2.1)–(2.5) limit to 𝑀, the largest invariant subset of {π‘‘π‘Š1/𝑑𝑑=0}. Clearly, it follows from (2.21) that π‘‘π‘Š1/𝑑𝑑=0 if and only if π‘₯1=π‘₯01, π‘₯2=π‘₯02, 𝑣=0. Noting that 𝑀 is invariant, for each element of 𝑀 we have 𝑣=0, ̇𝑣=0. From (2.5) we drive that Μ‡0=𝑣=π‘’βˆ’π‘›1πœ”1𝑝1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έ+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έ.(2.22) Since 𝑦1(π‘‘βˆ’πœƒ)β‰₯0 and 𝑦2(π‘‘βˆ’πœƒ)β‰₯0 for all πœƒβˆˆ[0,max{𝜏1,𝜏2,πœ”1,πœ”2}], then π‘’βˆ’π‘›1πœ”1𝑝1𝑦1(π‘‘βˆ’πœ”1)+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2(π‘‘βˆ’πœ”2)=0 if and only if 𝑦1(π‘‘βˆ’πœ”1)=𝑦2(π‘‘βˆ’πœ”2)=0. Hence, π‘‘π‘Š1/𝑑𝑑=0 if and only if π‘₯1=π‘₯01, π‘₯2=π‘₯02, 𝑦1=𝑦2=𝑣=0. From LaSalle’s invariance principle, 𝐸0 is GAS for any 𝜏1,𝜏2,πœ”1,πœ”2β‰₯0.
(ii) Define a Lyapunov functional as π‘Š2=π‘’βˆ’π‘š1𝜏1π‘₯βˆ—1𝐻π‘₯1π‘₯βˆ—1ξ‚Ά+π‘¦βˆ—1𝐻𝑦1π‘¦βˆ—1𝑒+π›Ύβˆ’π‘š2𝜏2π‘₯βˆ—2𝐻π‘₯2π‘₯βˆ—2ξ‚Ά+π‘¦βˆ—2𝐻𝑦2π‘¦βˆ—2+π‘Žξ‚Άξ‚Ή1𝑝1𝑒𝑛1πœ”1π‘£βˆ—π»ξ‚€π‘£π‘£βˆ—ξ‚+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯βˆ—1π‘£βˆ—ξ€œπœ10𝐻π‘₯1(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘₯βˆ—1π‘£βˆ—ξ‚Άπ‘‘πœƒ+π‘’βˆ’π‘š2𝜏2𝛾𝛽2π‘₯βˆ—2π‘£βˆ—ξ€œπœ20𝐻π‘₯2(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘₯βˆ—2π‘£βˆ—ξ‚Άπ‘‘πœƒ+π‘Ž1π‘¦βˆ—1ξ€œπœ”10𝐻𝑦1(π‘‘βˆ’πœƒ)π‘¦βˆ—1ξ‚Άπ‘‘πœƒ+π›Ύπ‘Ž2π‘¦βˆ—2ξ€œπœ”20𝐻𝑦2(π‘‘βˆ’πœƒ)π‘¦βˆ—2ξ‚Άπ‘‘πœƒ.(2.23) Differentiating with respect to time yields π‘‘π‘Š2𝑑𝑑=π‘’βˆ’π‘š1𝜏1ξ‚΅π‘₯1βˆ’βˆ—1π‘₯1ξ‚Άξ€·πœ†1βˆ’π‘‘1π‘₯1βˆ’π›½1π‘₯1𝑣+𝑦1βˆ’βˆ—1𝑦1ξ‚Άξ€·π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έβˆ’π‘Ž1𝑦1𝑒+π›Ύβˆ’π‘š2𝜏2ξ‚΅π‘₯1βˆ’βˆ—2π‘₯2ξ‚Άξ€·πœ†2βˆ’π‘‘2π‘₯2βˆ’π›½2π‘₯2𝑣+𝑦1βˆ’βˆ—2𝑦2ξ‚Άξ€·π‘’βˆ’π‘š2𝜏2𝛽2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έβˆ’π‘Ž2𝑦2ξ€Έξ‚Ή+π‘Ž1𝑝1𝑒𝑛1πœ”1𝑣1βˆ’βˆ—π‘£ξ‚Άξ€·π‘’βˆ’π‘›1πœ”1𝑝1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έ+π‘’βˆ’π‘›2πœ”2𝑝2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έξ€Έβˆ’π‘π‘£+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯1π‘£βˆ’π›½1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έ+𝛽1π‘₯βˆ—1π‘£βˆ—ξƒ©π‘₯ln1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έπ‘₯1𝑣ξƒͺξƒ­+π›Ύπ‘’βˆ’π‘š2𝜏2𝛽2π‘₯2π‘£βˆ’π›½2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έ+𝛽2π‘₯βˆ—2π‘£βˆ—ξƒ©π‘₯ln2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έπ‘₯2𝑣ξƒͺξƒ­+π‘Ž1𝑦1βˆ’π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έ+π‘¦βˆ—1𝑦ln1ξ€·π‘‘βˆ’πœ”1𝑦1ξƒͺξƒ­+π‘Ž2𝛾𝑦2βˆ’π‘¦2ξ€·π‘‘βˆ’πœ”2ξ€Έ+π‘¦βˆ—2𝑦ln2ξ€·π‘‘βˆ’πœ”2𝑦2ξƒͺξƒ­=π‘’βˆ’π‘š1𝜏1ξ‚΅π‘₯1βˆ’βˆ—1π‘₯1ξ‚Άξ€·πœ†1βˆ’π‘‘1π‘₯1ξ€Έ+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯βˆ—1π‘£βˆ’π‘’βˆ’π‘š1𝜏1π‘¦βˆ—1𝛽1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1+π‘Ž1π‘¦βˆ—1𝑒+π›Ύβˆ’π‘š2𝜏2ξ‚΅π‘₯1βˆ’βˆ—2π‘₯2ξ‚Άξ€·πœ†2βˆ’π‘‘2π‘₯2ξ€Έ+π‘’βˆ’π‘š2𝜏2𝛽2π‘₯βˆ—2π‘£βˆ’π‘’βˆ’π‘š2𝜏2π‘¦βˆ—2𝛽2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2+π‘Ž2π‘¦βˆ—2ξƒ­βˆ’π‘Ž1π‘£βˆ—π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£βˆ’π›Ύπ‘Ž2π‘£βˆ—π‘¦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£βˆ’π‘Ž1𝑐𝑝1𝑒𝑛1πœ”1π‘Žπ‘£+1𝑐𝑝1𝑒𝑛1πœ”1π‘£βˆ—+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯βˆ—1π‘£βˆ—ξƒ©π‘₯ln1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έπ‘₯1𝑣ξƒͺ+π›Ύπ‘’βˆ’π‘š2𝜏2𝛽2π‘₯βˆ—2π‘£βˆ—ξƒ©π‘₯ln2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έπ‘₯2𝑣ξƒͺ+π‘Ž1π‘¦βˆ—1𝑦ln1ξ€·π‘‘βˆ’πœ”1𝑦1ξƒͺ+π›Ύπ‘Ž2π‘¦βˆ—2𝑦ln2ξ€·π‘‘βˆ’πœ”2𝑦2ξƒͺ.(2.24) Using the infected steady state 𝐸1 conditions πœ†1=𝑑1π‘₯βˆ—1+𝛽1π‘₯βˆ—1π‘£βˆ—,πœ†2=𝑑2π‘₯βˆ—2+𝛽2π‘₯βˆ—2π‘£βˆ—,π‘Ž1π‘¦βˆ—1π‘’π‘š1𝜏1=𝛽1π‘₯βˆ—1π‘£βˆ—,π‘Ž2π‘¦βˆ—2π‘’π‘š2𝜏2=𝛽2π‘₯βˆ—2π‘£βˆ—,π‘π‘£βˆ—=𝑝1π‘’βˆ’π‘›1πœ”1π‘¦βˆ—1+𝑝2π‘’βˆ’π‘›2πœ”2π‘¦βˆ—2,(2.25) we obtain π‘‘π‘Š2𝑑𝑑=π‘’βˆ’π‘š1𝜏1𝑑1π‘₯βˆ—1+π‘Ž1π‘¦βˆ—1π‘’π‘š1𝜏1βˆ’π‘‘1π‘₯1βˆ’π‘₯βˆ—1π‘₯1𝑑1π‘₯βˆ—1+π‘Ž1π‘¦βˆ—1π‘’π‘š1𝜏1βˆ’π‘‘1π‘₯1ξ€Έξ‚Ή+π‘’βˆ’π‘š1𝜏1𝛽1π‘₯βˆ—1π‘£βˆ—ξ‚€π‘£π‘£βˆ—ξ‚βˆ’π‘Ž1π‘¦βˆ—1π‘¦βˆ—1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1π‘₯βˆ—1π‘£βˆ—+2π‘Ž1π‘¦βˆ—1𝑒+π›Ύβˆ’π‘š2𝜏2𝑑2π‘₯βˆ—2+π‘Ž2π‘¦βˆ—2π‘’π‘š2𝜏2βˆ’π‘‘2π‘₯2ξ€Έβˆ’π‘’βˆ’π‘š2𝜏2π‘₯βˆ—2π‘₯2𝑑2π‘₯βˆ—2+π‘Ž2π‘¦βˆ—2π‘’π‘š2𝜏2βˆ’π‘‘2π‘₯2ξ€Έ+π‘’βˆ’π‘š2𝜏2𝛽2π‘₯βˆ—2π‘£βˆ—ξ‚€π‘£π‘£βˆ—ξ‚βˆ’π‘Ž2π‘¦βˆ—2π‘¦βˆ—2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2π‘₯βˆ—2π‘£βˆ—+2π‘Ž2π‘¦βˆ—2ξƒ­βˆ’π‘Ž1π‘¦βˆ—1π‘£βˆ—π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£π‘¦βˆ—1βˆ’π›Ύπ‘Ž2π‘¦βˆ—2π‘£βˆ—π‘¦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£π‘¦βˆ—2βˆ’π‘Ž1𝑐𝑝1𝑒𝑛1πœ”1π‘£βˆ—ξ‚€π‘£π‘£βˆ—ξ‚+π‘Ž1π‘¦βˆ—1π‘₯ln1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1ξ€Έπ‘₯1𝑣ξƒͺ+π›Ύπ‘Ž2π‘¦βˆ—2π‘₯ln2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2ξ€Έπ‘₯2𝑣ξƒͺ+π‘Ž1π‘¦βˆ—1𝑦ln1ξ€·π‘‘βˆ’πœ”1𝑦1ξƒͺ+π›Ύπ‘Ž2π‘¦βˆ—2𝑦ln2ξ€·π‘‘βˆ’πœ”2𝑦2ξƒͺ=π‘’βˆ’π‘š1𝜏1𝑑1π‘₯βˆ—1ξ‚΅π‘₯2βˆ’1π‘₯βˆ—1βˆ’π‘₯βˆ—1π‘₯1ξ‚Άβˆ’π‘Ž1π‘¦βˆ—1π‘₯βˆ—1π‘₯1βˆ’π‘Ž1π‘¦βˆ—1π‘¦βˆ—1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1π‘₯βˆ—1π‘£βˆ—βˆ’π‘Ž1π‘¦βˆ—1π‘£βˆ—π‘¦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£π‘¦βˆ—1+3π‘Ž1π‘¦βˆ—1+π‘Ž1π‘¦βˆ—1π‘₯ln1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘₯1𝑣𝑦1ξƒͺ𝑒+π›Ύβˆ’π‘š2𝜏2𝑑2π‘₯βˆ—2ξ‚΅π‘₯2βˆ’2π‘₯βˆ—2βˆ’π‘₯βˆ—2π‘₯2ξ‚Άβˆ’π‘Ž2π‘¦βˆ—2π‘₯βˆ—2π‘₯2βˆ’π‘Ž2π‘¦βˆ—2π‘¦βˆ—2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2π‘₯βˆ—2π‘£βˆ—βˆ’π‘Ž2π‘¦βˆ—2π‘£βˆ—π‘¦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£π‘¦βˆ—2+3π‘Ž2π‘¦βˆ—2+π‘Ž2π‘¦βˆ—2π‘₯ln2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘₯2𝑣𝑦2+π‘Žξƒͺξƒ­1𝑝1𝑒𝑛1πœ”1ξ€·π‘’βˆ’π‘›1πœ”1𝑝1π‘¦βˆ—1+π‘’βˆ’π‘›2πœ”2𝑝2π‘¦βˆ—2βˆ’π‘π‘£βˆ—ξ€Έπ‘£π‘£βˆ—.(2.26) From (2.25), we see that the last term in (2.26) vanishes. Then, using the following equalities: π‘₯ln1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘₯1𝑣𝑦1ξƒͺξ‚΅π‘₯=lnβˆ—1π‘₯1𝑦+ln1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£βˆ—π‘¦βˆ—1𝑣ξƒͺ𝑦+lnβˆ—1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1π‘₯βˆ—1π‘£βˆ—ξƒͺ,π‘₯ln2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘₯2𝑣𝑦2ξƒͺξ‚΅π‘₯=lnβˆ—2π‘₯2𝑦+ln2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£βˆ—π‘¦βˆ—2𝑣ξƒͺ𝑦+lnβˆ—2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2π‘₯βˆ—2π‘£βˆ—ξƒͺ,(2.27) we can rewrite (2.26) as π‘‘π‘Š2𝑑𝑑=π‘’βˆ’π‘š1𝜏1𝑑1π‘₯βˆ—1ξ‚΅π‘₯2βˆ’1π‘₯βˆ—1βˆ’π‘₯βˆ—1π‘₯1ξ‚Ά+π‘’βˆ’π‘š2𝜏2𝛾𝑑2π‘₯βˆ—2ξ‚΅π‘₯2βˆ’2π‘₯βˆ—2βˆ’π‘₯βˆ—2π‘₯2ξ‚Άβˆ’π‘Ž1π‘¦βˆ—1𝐻π‘₯βˆ—1π‘₯1𝑦+𝐻1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£βˆ—π‘¦βˆ—1𝑣ξƒͺ𝑦+π»βˆ—1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1π‘₯βˆ—1π‘£βˆ—ξƒͺξƒ­βˆ’π›Ύπ‘Ž2π‘¦βˆ—2𝐻π‘₯βˆ—2π‘₯2𝑦+𝐻2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£βˆ—π‘¦βˆ—2𝑣ξƒͺ𝑦+π»βˆ—2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2π‘₯βˆ—2π‘£βˆ—.ξƒͺξƒ­(2.28) Since the arithmetical mean is greater than or equal to the geometrical mean, then the first two terms of (2.28) are less than or equal to zero. It is easy to see that if π‘₯βˆ—1,π‘¦βˆ—1,π‘₯βˆ—2,π‘¦βˆ—2,π‘£βˆ—>0, then π‘‘π‘Š2/𝑑𝑑≀0. By [32,Theorem  5.3.1], the solutions of system (2.1)–(2.5) limit to 𝑀, the largest invariant subset of {π‘‘π‘Š2/𝑑𝑑=0}. It can be seen that π‘‘π‘Š2/𝑑𝑑=0 if and only if π‘₯1=π‘₯βˆ—1,π‘₯2=π‘₯βˆ—2,𝑣=π‘£βˆ—, and 𝐻=0, that is, 𝑦1ξ€·π‘‘βˆ’πœ”1ξ€Έπ‘£βˆ—π‘¦βˆ—1𝑣=𝑦2ξ€·π‘‘βˆ’πœ”2ξ€Έπ‘£βˆ—π‘¦βˆ—2𝑣=π‘¦βˆ—1π‘₯1ξ€·π‘‘βˆ’πœ1ξ€Έπ‘£ξ€·π‘‘βˆ’πœ1𝑦1π‘₯βˆ—1π‘£βˆ—=π‘¦βˆ—2π‘₯2ξ€·π‘‘βˆ’πœ2ξ€Έπ‘£ξ€·π‘‘βˆ’πœ2𝑦2π‘₯βˆ—2π‘£βˆ—=1.(2.29) If 𝑣=π‘£βˆ—, then from (2.29), we have 𝑦1=π‘¦βˆ—1 and 𝑦2=π‘¦βˆ—2, and hence π‘‘π‘Š2/𝑑𝑑 equal to zero at 𝐸1. LaSalle’s invariance principle implies global stability of 𝐸1.

3. Basic Virus Dynamics Model with Multitarget Cells and Delays

In this section, we propose a virus dynamics model which describes the interaction of the virus with 𝑛 classes of target cells. Two types of discrete-time delays (πœπ‘–,πœ”π‘–,𝑖=1,…,𝑛) are incorporated into the model. The model is a generalization of those of one class of target cells and two classes of target cells models presented, respectively, in [26, 33]. Moreover, it can be seen that when 𝑛=1 and πœ”1=0, then the following model leads to the model presented in [11]. Μ‡π‘₯𝑖=πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π›½π‘–π‘₯𝑖𝑣,𝑖=1,…,𝑛,̇𝑦𝑖=π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆ’π‘Žπ‘–π‘¦π‘–Μ‡,𝑖=1,…,𝑛,𝑣=𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έβˆ’π‘π‘£,(3.1) where π‘₯𝑖 and 𝑦𝑖 represent the populations of the uninfected target cells and infected cells of class 𝑖, respectively, 𝑣 is the population of the virus particles. All the parameters of the model have the same biological meaning as given in the previous section.

The initial conditions for system (3.1) take the formπ‘₯𝑗(πœƒ)=πœ‘π‘—π‘¦(πœƒ),𝑗=1,…,𝑛,𝑗(πœƒ)=πœ™π‘—+𝑛(πœƒ),𝑗=1,…,𝑛,𝑣(πœƒ)=πœ‘2𝑛+1πœ‘(πœƒ),π‘—ξ€Ίξ€½πœ(πœƒ)β‰₯0,𝑗=1,…,2𝑛+1,πœƒβˆˆβˆ’max1,…,πœπ‘›,πœ”1,…,πœ”π‘›ξ€Ύξ€»,,0(3.2) where (πœ‘1(πœƒ),πœ‘2(πœƒ),…,πœ‘2𝑛+1(πœƒ))∈𝐢 and 𝐢=𝐢([βˆ’max{𝜏1,…,πœπ‘›,πœ”1,…,πœ”π‘›},0],ℝ+2𝑛+1) is the Banach space of continuous functions mapping the interval [βˆ’max{𝜏1,…,πœπ‘›,πœ”1,…,πœ”π‘›},0] into ℝ+2𝑛+1.

Similar to the previous section, the nonnegativity and the boundedness of the solutions of system (3.1) can be shown.

3.1. Steady States

It is clear that system (3.1) has an uninfected steady state 𝐸0=(π‘₯01,…,π‘₯0𝑛,𝑦01,…,𝑦0𝑛,𝑣0), where π‘₯0𝑖=πœ†π‘–/𝑑𝑖, 𝑦0𝑖=0, 𝑖=1,…,𝑛, and 𝑣0=0. The system can also have a positive infected steady state 𝐸1(π‘₯βˆ—1,…,π‘₯βˆ—π‘›,π‘¦βˆ—1,…,π‘¦βˆ—π‘›,π‘£βˆ—). The coordinates of the infected steady state, if they exist, satisfy the equalities:πœ†π‘–=𝑑𝑖π‘₯βˆ—π‘–+𝛽𝑖π‘₯βˆ—π‘–π‘£βˆ—π‘Ž,𝑖=1,…,𝑛,(3.3)π‘–π‘¦βˆ—π‘–=π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—,𝑖=1,…,𝑛,(3.4)π‘π‘£βˆ—=𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦βˆ—π‘–.(3.5) The basic reproduction number of system (3.1) is given by𝑅0=𝑛𝑖=1𝑅𝑖=𝑛𝑖=1π‘’βˆ’(π‘šπ‘–πœπ‘–+π‘›π‘–πœ”π‘–)π›½π‘–π‘π‘–πœ†π‘–π‘Žπ‘–π‘‘π‘–π‘,(3.6) where 𝑅𝑖 is the basic reproduction number for the dynamics of the interaction of the virus only with the target cells of class 𝑖.

3.2. Global Stability

In the following theorem, the global stability of the uninfected and infected steady states of system (3.1) will be established.

Theorem 3.1. (i) If 𝑅0≀1, then 𝐸0 is GAS for any πœπ‘–,πœ”π‘–β‰₯0, 𝑖=1,…,𝑛.
(ii) If 𝐸1 exists, then it is GAS for any πœπ‘–,πœ”π‘–β‰₯0, 𝑖=1,…,𝑛.

Proof. (i) Define a Lyapunov functional π‘Š1 as follows: π‘Š1=𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–π‘₯0𝑖𝐻π‘₯𝑖π‘₯0𝑖ξƒͺ+𝑦𝑖+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–ξ€œπœπ‘–0π‘₯𝑖(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ+π‘Žπ‘–ξ€œπœ”π‘–0𝑦𝑖+π‘Ž(π‘‘βˆ’πœƒ)π‘‘πœƒ1𝑝1𝑒𝑛1πœ”1𝑣,(3.7) where 𝛾𝑖=(π‘Ž1𝑝𝑖/π‘Žπ‘–π‘1)𝑒𝑛1πœ”1βˆ’π‘›π‘–πœ”π‘–. The time derivative of π‘Š1 along the solution of system (3.1) satisfies π‘‘π‘Š1=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–ξƒ©π‘₯1βˆ’0𝑖π‘₯𝑖ξƒͺξ€·πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π›½π‘–π‘₯𝑖𝑣+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆ’π‘Žπ‘–π‘¦π‘–+π‘’βˆ’π‘šπ‘–πœπ‘–ξ€·π›½π‘–π‘₯π‘–π‘£βˆ’π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξ€Έ+π‘Žπ‘–ξ€·π‘¦π‘–βˆ’π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξƒ­+π‘Žξ€Έξ€Έ1𝑝1𝑒𝑛1πœ”1𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξƒͺ=βˆ’π‘π‘£π‘›ξ“π‘–=1π‘’βˆ’π‘šπ‘–πœπ‘–π›Ύπ‘–πœ†π‘–ξƒ¬π‘₯2βˆ’π‘–π‘₯0π‘–βˆ’π‘₯0𝑖π‘₯𝑖+π‘Ž1𝑐𝑝1𝑒𝑛1πœ”1𝑛𝑖=1π‘’βˆ’(π‘šπ‘–πœπ‘–+π‘›π‘–πœ”π‘–)𝑝𝑖𝛽𝑖π‘₯0π‘–π‘Žπ‘–π‘ξƒͺ𝑣=βˆ’1𝑛𝑖=1π‘’βˆ’π‘šπ‘–πœπ‘–π›Ύπ‘–πœ†π‘–ξƒ¬π‘₯2βˆ’π‘–π‘₯0π‘–βˆ’π‘₯0𝑖π‘₯𝑖+π‘Ž1𝑐𝑝1𝑒𝑛1πœ”1𝑅0ξ€Έβˆ’1𝑣.(3.8) Since the arithmetical mean is greater than or equal to the geometrical mean, then the first term of (3.8) is less than or equal to zero. Therefore, if 𝑅0≀1, then π‘‘π‘Š1/𝑑𝑑≀0 for all π‘₯𝑖,𝑦𝑖,𝑣>0. Similar to the previous section, one can show that the maximal compact invariant set in {π‘‘π‘Š1/𝑑𝑑=0} is the singleton {𝐸0} when 𝑅0≀1. The global stability of 𝐸0 follows from LaSalle’s invariance principle.
To prove (ii), we consider the Lyapunov functional π‘Š2=𝑛𝑖=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–π‘₯βˆ—π‘–π»ξ‚΅π‘₯𝑖π‘₯βˆ—π‘–ξ‚Ά+π‘¦βˆ—π‘–π»ξ‚΅π‘¦π‘–π‘¦βˆ—π‘–ξ‚Ά+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—ξ€œπœπ‘–0𝐻π‘₯𝑖(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)π‘₯βˆ—π‘–π‘£βˆ—ξ‚Άπ‘‘πœƒ+π‘Žπ‘–π‘¦βˆ—π‘–ξ€œπœ”π‘–0𝐻𝑦𝑖(π‘‘βˆ’πœƒ)π‘¦βˆ—π‘–ξ‚Άξ‚Ή+π‘Žπ‘‘πœƒ1𝑝1𝑒𝑛1πœ”1π‘£βˆ—π»ξ‚€π‘£π‘£βˆ—ξ‚.(3.9) Differentiating with respect to time yields π‘‘π‘Š2=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–ξ‚΅π‘₯1βˆ’βˆ—π‘–π‘₯π‘–ξ‚Άξ€·πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π›½π‘–π‘₯𝑖𝑣+𝑦1βˆ’βˆ—π‘–π‘¦π‘–ξ‚Άξ€·π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆ’π‘Žπ‘–π‘¦π‘–ξ€Έ+π‘’βˆ’π‘šπ‘–πœπ‘–ξ€·π›½π‘–π‘₯π‘–π‘£βˆ’π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξ€Έ+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘₯𝑖𝑣ξƒͺ+π‘Žπ‘–ξ€·π‘¦π‘–βˆ’π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξ€Έ+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘¦lnπ‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘¦π‘–+π‘Žξƒͺξƒ­1𝑒𝑛1πœ”1𝑝1𝑣1βˆ’βˆ—π‘£ξ‚Άξƒ©π‘›ξ“π‘–=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξƒͺ=βˆ’π‘π‘£π‘›ξ“π‘–=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–ξ‚΅πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’πœ†π‘–π‘₯βˆ—π‘–π‘₯𝑖+𝑑𝑖π‘₯βˆ—π‘–+𝛽𝑖π‘₯βˆ—π‘–π‘£ξ‚Άβˆ’π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦βˆ—π‘–π‘¦π‘–+π‘Žπ‘–π‘¦βˆ—π‘–+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘₯𝑖𝑣ξƒͺ+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘¦lnπ‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘¦π‘–βˆ’π‘Žξƒͺξƒ­1𝑐𝑒𝑛1πœ”1𝑝1π‘£π‘£βˆ’βˆ—π‘£π‘›ξ“π‘–=1π›Ύπ‘–π‘Žπ‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έ+π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1π‘£βˆ—.(3.10) Using the infected steady state conditions (3.3)–(3.5), and the following equality: π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1π‘£βˆ—=π‘Ž1𝑒𝑛1πœ”1𝑝1𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦βˆ—π‘–=𝑛𝑖=1π›Ύπ‘–π‘Žπ‘–π‘¦βˆ—π‘–,(3.11) we obtain π‘‘π‘Š2=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–ξ‚΅π‘‘π‘–π‘₯βˆ—π‘–+π‘’π‘šπ‘–πœπ‘–π‘Žπ‘–π‘¦βˆ—π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π‘₯βˆ—π‘–π‘₯𝑖𝑑𝑖π‘₯βˆ—π‘–+π‘’π‘šπ‘–πœπ‘–π‘Žπ‘–π‘¦βˆ—π‘–ξ€Έ+𝑑𝑖π‘₯βˆ—π‘–ξ‚Άβˆ’π‘Žπ‘–π‘¦βˆ—π‘–π‘¦βˆ—π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–π‘₯βˆ—π‘–π‘£βˆ—+2π‘Žπ‘–π‘¦βˆ—π‘–βˆ’π‘Žπ‘–π‘¦βˆ—π‘–π‘£βˆ—π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘£π‘¦βˆ—π‘–+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘₯𝑖𝑣ξƒͺ+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘¦lnπ‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘¦π‘–+ξƒͺ𝑛𝑖=1π‘’βˆ’π‘šπ‘–πœπ‘–π›Ύπ‘–π›½π‘–π‘₯βˆ—π‘–βˆ’π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1ξƒͺ𝑣=𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–π‘‘π‘–π‘₯βˆ—π‘–ξ‚΅π‘₯2βˆ’βˆ—π‘–π‘₯π‘–βˆ’π‘₯𝑖π‘₯βˆ—π‘–ξ‚Άβˆ’π‘Žπ‘–π‘¦βˆ—π‘–π‘₯βˆ—π‘–π‘₯π‘–βˆ’π‘Žπ‘–π‘¦βˆ—π‘–π‘£βˆ—π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘£π‘¦βˆ—π‘–βˆ’π‘Žπ‘–π‘¦βˆ—π‘–π‘¦βˆ—π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–π‘₯βˆ—π‘–π‘£βˆ—+3π‘Žπ‘–π‘¦βˆ—π‘–+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘₯𝑖𝑣𝑦𝑖+π‘Žξƒͺξƒ­1𝑒𝑛1πœ”1𝑝1𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦βˆ—π‘–βˆ’π‘π‘£βˆ—ξƒͺπ‘£π‘£βˆ—.(3.12) From (3.5), we can see that the last term in (3.12) vanishes. Then, by using the following equality: π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘₯𝑖𝑣𝑦𝑖ξƒͺξ‚΅π‘₯=lnβˆ—π‘–π‘₯𝑖𝑣+lnβˆ—π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘£π‘¦βˆ—π‘–ξƒͺ𝑦+lnβˆ—π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–π‘₯βˆ—π‘–π‘£βˆ—ξƒͺ,(3.13) we can rewrite (3.12) as π‘‘π‘Š2=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–π‘‘π‘–π‘₯βˆ—π‘–ξ‚΅π‘₯2βˆ’βˆ—π‘–π‘₯π‘–βˆ’π‘₯𝑖π‘₯βˆ—π‘–ξ‚Άβˆ’π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π»ξ‚΅π‘₯βˆ—π‘–π‘₯𝑖𝑣+π»βˆ—π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘£π‘¦βˆ—π‘–ξƒͺ𝑦+π»βˆ—π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦π‘–π‘₯βˆ—π‘–π‘£βˆ—.ξƒͺξƒͺξƒ­(3.14) It is easy to see that if π‘₯βˆ—π‘–,π‘¦βˆ—π‘–,π‘£βˆ—>0, 𝑖=1,…,𝑛, then π‘‘π‘Š2/𝑑𝑑≀0 for all (π‘₯𝑖,𝑦𝑖,𝑣)>0 (the arithmetical mean is greater than or equal to the geometrical mean and 𝐻β‰₯0). Clearly, the singleton {𝐸1} is the only invariant set in {π‘‘π‘Š2/𝑑𝑑=0}. LaSalle's invariance principle implies global stability of 𝐸1.

4. Virus Dynamics Model with Saturation Infection Rate

In this section, we proposed a virus dynamics model which describes the interaction of the virus with 𝑛 classes of target cells taking into account the saturation infection rate and multiple intracellular delays: Μ‡π‘₯𝑖(𝑑)=πœ†π‘–βˆ’π‘‘π‘–π‘₯𝑖𝛽(𝑑)βˆ’π‘–π‘₯𝑖(𝑑)𝑣(𝑑)1+𝛼𝑖𝑣(𝑑),𝑖=1,…,𝑛,̇𝑦𝑖𝑒(𝑑)=βˆ’π‘šπ‘‘πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+𝛼𝑖𝑣(𝑑)βˆ’π‘Žπ‘–π‘¦π‘–Μ‡π‘£(𝑑),𝑖=1,…,𝑛,(𝑑)=𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έβˆ’π‘π‘£(𝑑),(4.1) where 𝛼𝑖,𝑖=1,…,𝑛 are positive constants. The variables and parameters of the model have the same definitions as given in Section 2. We mention that if 𝑛=1 and πœ”1=0, then model (4.1) leads to the model presented in [9], and if 𝑛=2 and πœ”1=πœ”2=0, 𝛼1=𝛼2=1, then model (4.1) leads to the model presented in [34].

4.1. Steady States

It is clear that system (4.1) has an uninfected steady state 𝐸0=(π‘₯01,…,π‘₯0𝑛,𝑦01,…,𝑦0𝑛,𝑣0), where π‘₯0𝑖=πœ†π‘–/𝑑𝑖, 𝑦0𝑖=0, and 𝑣0=0. The system can also have a positive infected steady state 𝐸1(π‘₯βˆ—1,…,π‘₯βˆ—π‘›,π‘¦βˆ—1,…,π‘¦βˆ—π‘›,π‘£βˆ—). The coordinates of the infected steady state, if they exist, satisfy the equalities:πœ†π‘–=𝑑𝑖π‘₯βˆ—π‘–+𝛽𝑖π‘₯βˆ—π‘–π‘£βˆ—1+π›Όπ‘–π‘£βˆ—π‘Ž,𝑖=1,...,𝑛,π‘–π‘¦βˆ—π‘–=π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—1+π›Όπ‘–π‘£βˆ—,𝑖=1,...,𝑛,π‘π‘£βˆ—=𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦βˆ—π‘–.(4.2) The basic reproduction number 𝑅0 for system (4.1) is the same as given by (3.6).

4.2. Global Stability

In this section, we study the global stability of the uninfected and infected steady states of system (4.1).

Theorem 4.1. (i) If 𝑅0≀1, then 𝐸0 is GAS for any πœπ‘–,πœ”π‘–β‰₯0, 𝑖=1,…,𝑛.
(ii) If 𝐸1 exists, then it is GAS for any πœπ‘–,πœ”π‘–β‰₯0, 𝑖=1,…,𝑛.

Proof. (i) Define a Lyapunov functional π‘Š1 as follows: π‘Š1=𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–π‘₯0𝑖𝐻π‘₯𝑖π‘₯0𝑖ξƒͺ+𝑦𝑖+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–ξ€œπœπ‘–0π‘₯𝑖(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)1+𝛼𝑖𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ+π‘Žπ‘–ξ€œπœ”π‘–0𝑦𝑖+π‘Ž(π‘‘βˆ’πœƒ)π‘‘πœƒ1𝑒𝑛1πœ”1𝑝1𝑣,(4.3) The time derivative of π‘Š1 along the trajectories of (4.1) satisfies π‘‘π‘Š1=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–ξƒ©π‘₯1βˆ’0𝑖π‘₯𝑖ξƒͺξ‚΅πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π›½π‘–π‘₯𝑖𝑣1+𝛼𝑖𝑣+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆ’π‘Žπ‘–π‘¦π‘–+π‘’βˆ’π‘šπ‘–πœπ‘–ξƒ©π›½π‘–π‘₯𝑖𝑣1+π›Όπ‘–π‘£βˆ’π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξƒͺ+π‘Žπ‘–ξ€·π‘¦π‘–βˆ’π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξƒ­+π‘Žξ€Έξ€Έ1𝑒𝑛1πœ”1𝑝1𝑛𝑖=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξƒͺ=βˆ’π‘π‘£π‘›ξ“π‘–=1π›Ύπ‘–π‘’βˆ’π‘šπ‘–πœπ‘–ξƒ¬πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’πœ†π‘–π‘₯0𝑖π‘₯𝑖+𝑑𝑖π‘₯0𝑖+𝛽𝑖π‘₯0𝑖𝑣1+π›Όπ‘–π‘£ξƒ­βˆ’π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1𝑣=𝑛𝑖=1π›Ύπ‘–π‘’βˆ’π‘šπ‘–πœπ‘–πœ†π‘–ξƒ¬π‘₯2βˆ’π‘–π‘₯0π‘–βˆ’π‘₯0𝑖π‘₯π‘–ξƒ­βˆ’π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1π‘Žπ‘£+1𝑐𝑒𝑛1πœ”1𝑝1𝑛𝑖=1π‘’βˆ’(π‘šπ‘–πœπ‘–+π‘›π‘–πœ”π‘–)𝑝𝑖𝛽𝑖π‘₯0π‘–π‘£π‘Žπ‘–π‘ξ€·1+𝛼𝑖𝑣=𝑛𝑖=1π›Ύπ‘–π‘’βˆ’π‘šπ‘–πœπ‘–πœ†π‘–ξƒ¬π‘₯2βˆ’π‘–π‘₯0π‘–βˆ’π‘₯0𝑖π‘₯π‘–ξƒ­βˆ’π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1π‘Žπ‘£+1𝑐𝑒𝑛1πœ”1𝑝1𝑛𝑖=1𝑅𝑖𝑣1+𝛼𝑖𝑣=𝑛𝑖=1π›Ύπ‘–π‘’βˆ’π‘šπ‘–πœπ‘–πœ†π‘–ξƒ¬π‘₯2βˆ’π‘–π‘₯0π‘–βˆ’π‘₯0𝑖π‘₯𝑖+π‘Ž1𝑐𝑒𝑛1πœ”1𝑝1𝑅0ξ€Έπ‘Žβˆ’1π‘£βˆ’1𝑐𝑒𝑛1πœ”1𝑝1𝑛𝑖=1𝑅𝑖𝛼𝑖𝑣21+𝛼𝑖𝑣.(4.4) It is clear that if 𝑅0≀1, then π‘‘π‘Š1/𝑑𝑑≀0 for all π‘₯𝑖,𝑦𝑖,𝑣>0, where equality occurs at 𝐸0. The global stability of 𝐸0 follows from LaSalle’s invariance principle.
To prove (ii), we consider the Lyapunov functional: π‘Š2=𝑛𝑖=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–π‘₯βˆ—π‘–π»ξ‚΅π‘₯𝑖π‘₯βˆ—π‘–ξ‚Ά+π‘¦βˆ—π‘–π»ξ‚΅π‘¦π‘–π‘¦βˆ—π‘–ξ‚Ά+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—1+π›Όπ‘–π‘£βˆ—ξ€œπœπ‘–0𝐻π‘₯𝑖(π‘‘βˆ’πœƒ)𝑣(π‘‘βˆ’πœƒ)1+π›Όπ‘–π‘£βˆ—ξ€Έπ‘₯βˆ—π‘–π‘£βˆ—ξ€·1+𝛼𝑖ξƒͺ𝑣(π‘‘βˆ’πœƒ)π‘‘πœƒ+π‘Žπ‘–π‘¦βˆ—π‘–ξ€œπœ”π‘–0𝐻𝑦𝑖(π‘‘βˆ’πœƒ)π‘¦βˆ—π‘–ξ‚Άξ‚Ή+π‘Žπ‘‘πœƒ1𝑒𝑛1πœ”1𝑝1π‘£βˆ—π»ξ‚€π‘£π‘£βˆ—ξ‚.(4.5) Differentiating with respect to time yields π‘‘π‘Š2=𝑑𝑑𝑛𝑖=1π›Ύπ‘–ξ‚Έπ‘’βˆ’π‘šπ‘–πœπ‘–ξ‚΅π‘₯1βˆ’βˆ—π‘–π‘₯π‘–πœ†ξ‚Άξ‚΅π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’π›½π‘–π‘₯𝑖𝑣1+𝛼𝑖𝑣+𝑦1βˆ’βˆ—π‘–π‘¦π‘–ξ‚Άξƒ©π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έβˆ’π‘Žπ‘–π‘¦π‘–ξƒͺ+π‘’βˆ’π‘šπ‘–πœπ‘–ξƒ©π›½π‘–π‘₯𝑖𝑣1+π›Όπ‘–π‘£βˆ’π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ+𝛽𝑖π‘₯βˆ—π‘–π‘£βˆ—1+π›Όπ‘–π‘£βˆ—ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξ€·1+𝛼𝑖𝑣π‘₯𝑖𝑣1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξ€Έξƒͺξƒͺ+π‘Žπ‘–ξ€·π‘¦π‘–βˆ’π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξ€Έ+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘¦lnπ‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έπ‘¦π‘–+π‘Žξƒͺξƒ­1𝑒𝑛1πœ”1𝑝1𝑣1βˆ’βˆ—π‘£ξ‚Άξƒ©π‘›ξ“π‘–=1π‘’βˆ’π‘›π‘–πœ”π‘–π‘π‘–π‘¦π‘–ξ€·π‘‘βˆ’πœ”π‘–ξ€Έξƒͺ=βˆ’π‘π‘£π‘›ξ“π‘–=1π›Ύπ‘–ξƒ¬π‘’βˆ’π‘šπ‘–πœπ‘–ξ‚΅πœ†π‘–βˆ’π‘‘π‘–π‘₯π‘–βˆ’πœ†π‘–π‘₯βˆ—π‘–π‘₯𝑖+𝑑𝑖π‘₯βˆ—π‘–+𝛽𝑖π‘₯βˆ—π‘–π‘£1+π›Όπ‘–π‘£ξ‚Άβˆ’π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯π‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έ1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘¦βˆ—π‘–π‘¦π‘–+π‘Žπ‘–π‘¦βˆ—π‘–+π‘’βˆ’π‘šπ‘–πœπ‘–π›½π‘–π‘₯βˆ—π‘–π‘£βˆ—1+π›Όπ‘–π‘£βˆ—ξƒ©π‘₯lnπ‘–ξ€·π‘‘βˆ’πœπ‘–ξ€Έπ‘£ξ€·π‘‘βˆ’πœπ‘–ξ€Έξ€·1+𝛼𝑖𝑣π‘₯𝑖𝑣1+π›Όπ‘–π‘£ξ€·π‘‘βˆ’πœπ‘–ξƒͺξ€Έξ€Έ+π‘Žπ‘–π‘¦βˆ—π‘–ξƒ©π‘¦lnπ‘–ξ€·π‘‘βˆ’πœ”