Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Volume 3, Number 2, 2022, pp.86-105 https://doi.org/10.5206/mase/14663 GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL WITH CELL-TO-CELL TRANSMISSION UNDER THERAPY ALEXIS NANGUE, PAULIN TIOMO LEMOFOUET, SIMON NDOUVATAMA, AND EMMANUEL KENGNE Abstract. In virus dynamics, when a cell is infected, the number of virions outside the cells is reduced by one: this phenomenon is known as absorption effect. Most mathematical in intra-host models neglects this phenomenon. Virus-to-cell infection and direct cell-to-cell transmission are two fundamental modes whereby viruses can be propagated and transmitted. In this work, we propose a new virus dynamics model, which incorporates both modes and takes into account the absorption effect and treatment. First we show mathematically and biologically the well-posedness of our model preceded by the result on the existence and the uniqueness of the solutions. Also, an explicit formula for the basic reproduction number R0 of the model is determined. By analyzing the characteristic equations we establish the local stability of the uninfected equilibrium and the infected equilibrium in terms of R0. The global behaviour of the model is investigated by constructing an appropriate Lyapunov functional for uninfected equilibrium and by applying a geometric approach to the study of the infected equilibrium. Numerical simulations are carried out, to confirm the obtained theoretical result in a particular case. 1. Introduction Mathematical modeling is one of the most coveted areas in which virus infection research is under- taken. It consists of modeling the evolution of the infection using tools, mainly differential equations. The modeling of infectious diseases is a tool that has been used to study the mechanisms by which diseases spread, to predict the future course of an outbreak, and to evaluate strategies to control an epi- demic. In recent years, grandiose efforts have been devoted to the mathematical modeling of intra-host viral dynamics. Mathematical models have been developed to describe the process of in vivo infection of many viruses. Many viruses infect humans and cause different infectious diseases such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), Ebola virus, Zika virus, and nowadays new coronavirus (COVID-19 virus). They are often transmitted in the body by two fundamentally distinct modes, either by virus-to-cell infection through the extracellular space or by cell-to-cell transmission involving direct cell-to-cell contact [7, 24, 26, 35, 43]. During both infection modes, a part of infected cells returns to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus [13, 20, 10]. To model viral infection dynamics, several mathematical models have been proposed and developed [3, 9]. Most of these models are based on the assumption that healthy cells can only be infected by viruses, and so they consider only the virus-to-cell infection mode [1, 16, 29, 28, 30, 31]. Authors in [23] consider a mathematical model that describes a viral infection of HIV-1 with both virus-to-cell and cell-to-cell transmission with other features. It should be noted that the total infection rate of Received by the editors 2 February 2022; accepted 13 April 2022; published online 21 April 2022. 2010 Mathematics Subject Classification. 92B99, 34D23, 92D25, 37C75. Key words and phrases. Absorption effect, cell-to-cell transmission, global solution, global stability, treatment, well- posedness. 86 https://ojs.lib.uwo.ca/mase https://dx.doi.org/https://doi.org/10.5206/mase/14663 GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 87 uninfected cells that it has been considered is a particular case from the one we consider in this work. However, there are few virus dynamics models in literature with both modes of transmission [33, 34, 36] and taking into account the cure of infected cells. Motivated by the mentioned biological and mathe- matical considerations above, [17] propose a virus dynamics model with two transmission modes, i.e., cell-to-cell and virus-to-cell transmission modes. We note that in virus dynamics model proposed by [16], the loss of pathogens due to the absorption into uninfected cells is ignored. In biology, it is natural that, when pathogens are absorbed into susceptible cells, the number of pathogens are reduced into the blood volume : this is called absorption effect. Hence, some researchers (see, for example, [2, 27, 42, 14] and the references therein) have included the absorption effect into their models. We also find that the treatment [6] is not taken into account. So, to make this last model a little more realistic, motivated by the works in [16], in the present paper we are concerned with the effect of both virus-to-cell and cell- to-cell transmissions with absorption and antiviral treatments on the global dynamics of a generalized infection viral model. The rest of this paper is organized as follows. In Section 2, the mathematical model is constructed. Section 3 deals with the existence, the positivity and boundedness of solutions of our model. In ad- dition the threshold parameter R0 of model (2.2) is determined and the existence of the equilibria is discussed with respect to the value of R0. In Section 4, local stability of the equilibria are completely discussed. In Section 5, global stability of the equilibria is studied. The global behaviour of the model is investigated by constructing an appropriate Lyapunov functional for uninfected equilibrium and by applying Li-Muldowney global stability-criterion to the infected equilibrium. Numerical simulations are shown in Section 6. Finally, a brief discussion is presented in Section 7. 2. Formulation and description of the model The model studied in [16] is described by the following three differential system of equations:  dT dt = λ−dT −f(T,I,V )V −g(T,I)I + ρI, dI dt = f(T,I,V )V + g(T,I)I − (a + ρ)I, dV dt = kI −µV, , (2.1) where T(t), I(t), and V (t) denote the concentrations of uninfected cells, infected cells, and free viruses at time t, respectively, λ is the recruitment rate of uninfected cells, ρ is the cure rate of infected cells, k is the production rate of free viruses by infected cells, and d, a, and µ are the death rates of uninfected cells, infected cells, and free viruses, respectively. In addition, healthy cells become infected either by free viruses at rate f(T,I,V )V or by direct contact with an infected cell at rate g(T,I)I. Hence, the term f(T,I,V )V + g(T,I)I represents the total infection rate of uninfected cells. As we mentioned above, system (2.1) does not take into account the treatment and the absorption phenomenon. we consider in this paper the following virus dynamics model with therapy and absorption effect,   dT dt = λ−dT − (1 −η)f(T,I,V )V −g(T,I)I + ρI, dI dt = (1 −η)f(T,I,V )V + g(T,I)I − (a + ρ)I, dV dt = (1 −ε)kI −µV − (1 −η)f(T,I,V )V, (2.2) where the term −(1 −η)f(T,I,V )V in the third equation represents the loss of pathogens due to the absorption into uninfected cells. In addition, the therapeutic effect in this model involved blocking virions production (referred to as drug effectiveness) and reducing new infections which, are described in fractions (1 −ε) and (1 −η), respectively. 88 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE System (2.2) is subject to the initial conditions T(0) = T0, I(0) = I0, V (0) = V0 with T0 ≥ 0, I0 ≥ 0, V0 ≥ 0. (2.3) 3. Relevant assumptions and preliminary results 3.1. Relevant assumptions. The incidence function g for direct cell-to-cell transmission mode is assumed to be continuously differentiable in the interior of R2+ and satisfies the following properties : (H01): g(0,I) = 0 for all I ≥ 0, ∂g∂T (T,I) ≥ 0 for all T ≥ 0 and I ≥ 0(or g(T,I) is a strictly increasing function with respect to T when f ≡ 0). (H02): ∂g ∂I (T,I) ≤ 0 for all T ≥ 0 and I ≥ 0. In addition, the incidence function f for virus-to-cell infection mode, which denotes the average number of cells which are infected by each virus in unit time, is assumed to be continuously differentiable in the interior of R3+ and has the properties similar to those assumed in [14, 18] : (H1): f(0,I,V ) = 0; for all I ≥ 0 and V ≥ 0. (H2): ∂f ∂T (T,I,V ) ≥ 0; for all T > 0, I ≥ 0 and V ≥ 0. (H3): ∂f ∂I (T,I,V ) ≤ 0 and ∂f ∂V (T,I,V ) ≤ 0 for all T ≥ 0, I ≥ 0 and V ≥ 0. (H4): f(T,I,V ) + V ∂f ∂V (T,I,V ) ≥ 0 for all T > 0, I ≥ 0, and V ≥ 0. The significance of these assumptions is as follows : • (H01) means that the incidence rate by cell-to-cell transmission is equal to zero if there are no susceptible cells. This incidence rate is increasing when the numbers of infected cells are constant and the number of susceptible cells increases. Biologically speaking, the greater the amount of susceptible cells, the greater the average number of cells infected by direct contact with an infected cell in the unit time. • (H02) means that the greater the amount of infected cells, the lower the average number of cells infected by direct contact in the unit time. • (H1) means that the incidence rate for virus-to-cell infection mode is equal to zero if there are no susceptible cells. • (H2) signifies that the incidence rate is increasing when the numbers of infected cells and viruses are constant and the number of susceptible cells increases. Hence, the second hypothesis means; the more the amount of susceptible cells, the more the average number of cells which are infected by each virus in the unit time will occur. • The first assumption of (H3) means the more the amount of infected cells, the less the aver- age number of cells which are infected by each virus in the unit time will be and the second assumption of (H3) means the more the amount of infected virus, the less the average number of cells which are infected by each virus in the unit time will be. • (H4) means that if the total number of cells is constant, the more the amount of virus is, then the more the number of cells which are infected in the unit time will be. Therefore, the hypotheses summarized in assumptions (H01) - (H4) are reasonable and consistent with the reality. For more informations concerning the biological significance of hypotheses (H01), (H02), (H1), (H2) and (H3) , we refer the readers to [15, 41]. Furthermore, the five assumptions (H01), (H02), (H1), (H2) and (H3) are satisfied by most incidence rates existing in the literature. 3.2. Positivity and boundedness. First of all we show that the solutions of system 2.2 with non- negative initial conditions remain nonnegative and bounded for all t ≥ 0. Let R3+ = {(T,I,V ) ∈ R3 : T ≥ 0, I ≥ 0, V ≥ 0}. It is well known by the fundamental theory of ordinary differential equations (uniqueness of solutions GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 89 of Cauchy problem), that system (2.2) has a unique local solution (T(t),I(t),V (t)) satisfying the initial conditions (2.3). We have the following results. Theorem 3.1. Let (T,I,V ) be a solution of the initial value problem (2.2), (2.3) on an interval [t0, t1) with t1 > t0 ≥ 0. Assume that the initial data of the initial value problem (2.2), (2.3) satisfy T0 ≥ 0, I0 > 0, and V0 > 0. Then T , I and V remain positive for all t ∈ [t0, t1). Proof. We first prove that T(t) is positive for all t > 0. Suppose T(t) is not always positive. Let τ > 0 be the first time such that T(τ) = 0. By the first equation of (2.2) we have dT dt (τ) = λ + ρI(τ) > 0, provided T(τ) > 0, which implies T(t) < 0 for t ∈ (τ−�,τ), for sufficiently small � > 0. A contradiction. Therefore T(t) is positive for all t > 0. Now let us show that I(t) and V (t) are positive. Call the variables xi. If there is an index i and a time t (t0 ≤ t < t1) for which xi(t) = 0, we consider t∗ be the infimum of all such t for any i. Then the restriction of the solution to the interval [t0, t∗) is positive and xi(t∗) = 0 for a certain value of i. The second and third equation of system (2.2) can be written in the form: dxi(t) dt = −xiFi(x1,x2,x3) + Gi(x1,x2,x3), i ∈{2, 3} where F2(x1,x2,x3) = a + ρ, F3(x1,x2,x3) = µ + (1 −η)f(x1,x2,x3), G2(x1,x2,x3) = (1 −η)f(x1,x2,x3)x3 + g(x1,x2), G3(x1,x2,x3) = (1 −η)kx2 are. non-negative. As a consequence dxi(t) dt ≥−xiFi(x1,x2,x3) and d dt (log xi) ≥−Fi(x1,x2,x3) ≥−C, i ∈ 2, 3, for a positive constant C. To show this, we use the fact that the solution remains in a compact set. It follows that xi(t∗) ≥ xi(t0)e−C(t∗−t0) > 0, which is a contradiction and the Theorem 3.1 is proven. � This means that the first quadrant R3+ is positively invariant with respect to system (2.2). The boundedness of the solutions is guaranteed by the following theorem. Theorem 3.2. All solutions of (2.2) are uniformly bounded in the compact subset Ω = ß (T,I,V ) ∈ R3+ : T ≤ λ δ , I ≤ λ δ , V ≤ (1 −ε)kλ δ ™ , where δ = min{a,d}. Proof. Let (T(t),I(t),V (t)) be any solution with nonnegative initial condition (T0,I0,V0). Adding the first two equations of system (2.2), we obtain d dt (T + I) = λ−dT −aI, ≤ λ−δ(T + I). Hence lim sup t→+∞ (T(t) + I(t)) ≤ λ δ . Similarly, from the third equation of system (2.2) one has : dV dt ≤(1 −ε)k λ δ −µV −f(T,I,V )V, ≤(1 −ε)k λ δ −µV. 90 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE Hence, lim sup t→∞ (V (t)) ≤ (1 −ε)kλ δ . Hence, all solutions of system (2.2) starting in R3+ are eventually confined in the region Ω. This completes the proof. � Remark 3.1. We would like to make the following remarks. (i) These two previous theorems show mathematically and biologically the well-posedness [22] of our model (2.2). (ii) From these two results, if the initial data satisfy the inequalities T0 + I0 ≤ λδ and V0 ≤ (1−ε)kλ δ then the whole solution most satisfy these inequalities. This means that we have identified an invariant subset and for biological considerations, we will study system (2.2) in the subset Ω. 3.3. Equilibra. 3.3.1. Basic reproduction numbers and disease-free equilibrium. Now, we discuss the existence of equi- libria. By simple computation system (2.2) always has one uninfected equilibrium of the form E0 = (T0, 0, 0) with T0 = λ d . This will allow us to determine a threshold parameter to discuss the dynamic behaviour of the epidemic model. This later will be decisive for the rest of the work. This parameter is called the basic reproduction number and it measures the expected average number of new infected cells generated by a single virion in a completely healthy cell. According to the concept of next generation matrix in [8] and the computation of the basic reproduction number presented in [39], we can compute the basic reproduction number of system (2.2). We have the following result : Proposition 3.3. The basic reproduction number of the model (2.2) is given as : R0 = (1 −ε)(1 −η)kf(λ d , 0, 0) + (µ + (1 −η)f(λ d , 0, 0))g(λ d , 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) which can be rewritten as R0 = R01 + R02 where R01 = (1 −ε)k(1 −η)f(λ d , 0, 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) and R02 = g(λ d , 0) a + ρ . Proof. Based on notations in [39], the nonnegative matrix F and the non-singular M-matrix V are given by : F = Ç g(T0, 0) (1 −η)f(λ d , 0, 0) 0 0 å and V = Ñ −(a + ρ) 0 (1 −ε)k −µ− (1 −η)f(λ d , 0, 0) é . The next generation matrix is given by F.V −1 = á − g(λ d , 0) a + ρ − (1 −ε)(1 −η)kf(λ d , 0, 0) (a + ρ) ( µ + (1 −η)f(λ d , 0, 0) ) − (1 −µ)f(λd , 0, 0) µ + (1 −µ)f(λ d , 0, 0) 0 0 ë , GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 91 and the basic reproduction number of system (2.2) is defined as R0 = ρ(−F.V −1) where ρ(A) denotes the spectral radius of a matrix A. It follows that : R0 = (1 −ε)(1 −η)kf(λ d , 0, 0) (a + ρ) ( µ + (1 −η)f(λ d , 0, 0) ) + g(T0, 0) a + ρ , = (1 −ε)(1 −η)kf(λ d , 0, 0) + (µ + (1 −η)f(λ d , 0, 0))g(λ d , 0) (a + ρ)(µ + (1 −η)f(λ d , 0, 0)) . This completes the proof of theorem 3.3. � Remark 3.2. According to [16], R01 is the basic reproduction number corresponding to virus-to-cell infection mode, whereas R02 is the basic reproduction number corresponding to cell-to-cell transmission mode. 3.3.2. Infected equilibrium. Proposition 3.4. If R0 > 1, then system (2.2) has a unique chronic infection equilibrium of the form E∗ = (T∗,I∗,V ∗) with T∗ ∈ (0,T0), I∗ > 0 and V ∗ > 0. Proof. To find the other equilibrium of system (2.2), which is named the infected equilibrium, we solve the algebraic system   λ−dT − (1 −η)f(T,I,V )V −g(T,I)I + ρI = 0, (3.1) (1 −η)f(T,I,V )V + g(T,I)I − (a + ρ)I = 0, (3.2) (1 −ε)kI −µV − (1 −η)f(T,I,V )V = 0. (3.3) Adding (3.2) and (3.3) one has : V = 1 µ [(1 −ε)k + g(T,I) − (a + ρ)]I. (3.4) Reporting (3.4) into (3.2) yields : [(1 −ε)k + g(T,I) − (a + ρ)](1 −η)f(T,I,V ) + µg(T,I) −µ(a + ρ) = 0. (3.5) Since I = 1 a (λ−dT) ≥ 0, we have T ≤ λ d . Hence, there is no biological equilibrium when T > λ d . We define the function ψ on the interval [ 0,T0 ] by : ψ(T) = [(1 −ε)k + g(T,I) − (a + ρ)](1 −η)f(T,I,V ) + µg(T,I) −µ(a + ρ). We have ψ(0) = −µ(a + ρ) < 0. Moreover, ψ(T0) =(1 −ε)k(1 −η)f(T0, 0, 0) + [µ + (1 −η)f(T0, 0, 0)]g(T0, 0), − (a + ρ) [ µ + (1 −η)f(T0, 0, 0) ] =(a + ρ) ( µ + (1 −η)f(T0, 0, 0) )ß (1 −ε)k(1 −η)f(T0, 0, 0) + [µ + (1 −η)f(T0, 0, 0)]g(T0, 0) (a + ρ)(µ + (1 −η)f(T0, 0, 0)) − 1 ™ , =(a + ρ) ( µ + (1 −η)f(T0, 0, 0) ) (R0 − 1) 92 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE and dψ dT (T) =(1 −η) Å ∂g ∂T (T,I) − d a . ∂g ∂I (T,I) ã f(T,I,V ) + (1 −η){ ∂f ∂T (T,I,V ) − d a ∂f ∂I (T,I,V ) + 1 µ [( ∂g ∂T (T,I) − d a . ∂g ∂I (T,I))I − d a ((1 −ε)k + g(T,I) − (a + ρ)]. ∂f ∂V (T,I,V )} [(1 −ε)k + g(T,I) − (a + ρ)] + µ Å ∂g ∂T (T,I) − d a ∂g ∂I (T,I) ã , = Å ∂g ∂T (T,I) − d a . ∂g ∂I (T,I) ã [µ + (1 −η)f(T,I,V )] + (1 −η) Å ∂f ∂T (T,I,V ) − d a ∂f ∂I (T,I,V ) ã [(1 −ε)k + g(T,I) − (a + ρ)] + (1 −η) I µ Å ( ∂g ∂T (T,I) − d a . ∂g ∂I (T,I)) ã [(1 −ε)k + g(T,I) − (a + ρ)] − (1 −η) d a 1 µ ∂f ∂V (T,I,V ) [(1 −ε)k + g(T,I) − (a + ρ)]2 . We deduce that dψ dT (T) > 0. Hence, for R0 > 1, there exists a unique infected equilibrium E∗ = (T∗,I∗,V ∗) with T∗ ∈ ( 0, λ d ) , I∗ > 0 and V ∗ > 0. This completes the proof of proposition 3.4. � 4. Local stability In this section, we discuss the local stability of the two equilibria of system (2.2). Theorem 4.1. The disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and becomes unstable if R0 > 1. Proof. The Jacobian matrix of system (2.2) at the disease-free equilibrium E0 is given by : JE0 = Ñ −d −g(T0, 0) + ρ −(1 −η)f(T0, 0, 0) 0 g(T0, 0) − (a + ρ) (1 −η)f(T0, 0, 0) 0 (1 −ε)k −µ− (1 −η)f(T0, 0, 0) é . Computing the characteristic equation of JE0 , one has − (X + d)(X2 + a1X + a0) = 0 (4.1) where a1 = −g(T0, 0) + (a + ρ) + µ + (1 −η)f(T0, 0, 0), a0 = −g(T0, 0) [ µ + (1 −η)f(T0, 0, 0) ] + (a + ρ) [ µ + (1 −η)f(T0, 0, 0) ] −(1 −ε)(1 −η)kf(T0, 0, 0). a0 and a1 can also be written in the form : a1 = µ + (1 −η)f(T0, 0, 0) + (a + ρ)(1 −R02), and a0 = (a + ρ)(µ + (1 −η)f(T0, 0, 0))(1 −R0). GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 93 Since R0 < 1, it follows that a0 and a1 are positive. From the Routh-Hurwitz criteria [12] we know that all roots of X2 + a1X + a0 = 0 have negative real parts. Thus all roots of (4.1) have negative real parts. Therefore, the disease-free equilibrium E0 is locally asymptotically stable for R0 < 1 and unstable if R0 < 1. � Next, we study the local stability of the chronic infection equilibrium E∗. Note that the equilibrium E∗ does not exist if R0 < 1 and E∗ = E0 when R0 = 1. Theorem 4.2. The chronic infection equilibrium E∗ is locally asymptotically stable if R0 > 1, aI∗ ∂g ∂T (T∗,I∗) > (1 −ε)kd and (H4) are satisfied. Proof. The Jacobian matrix of system (2.2) at the chronic infection equilibrium E∗ is given by JE∗ = Ñ a11 a12 a13 a21 a22 a23 a31 a32 a33 é where a11 = −d− (1 −η)V ∗ ∂f ∂T (E∗) − I∗ ∂g ∂T (T∗,I∗), a12 = −(1 −η)V ∗ ∂f ∂I (E∗) − I∗ ∂g ∂I (T∗,I∗) −g(T∗,I∗) + ρ, a13 = −(1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ã , a21 = (1 −η)V ∗ ∂f ∂T (E∗) + I∗ ∂g ∂T (T∗,I∗), a22 = (1 −η)V ∗ ∂f ∂I (E∗) + I∗ ∂g ∂I (T∗,I∗) + g(T∗,I∗) − (a + ρ), a23 = (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ã , a31 = (1 −η)V ∗ ∂f ∂T (E∗), a32 = (1 −ε)k − (1 −η)V ∗ ∂f ∂T (E∗), a33 = −µ− (1 −η) Å V ∗ ∂f ∂T (E∗) + f(E∗) ã . Computing the characteristic equation of JE∗, one has X3 + a2X + a1X + a0 = 0, (4.2) where a2 =d + a + ρ−g(T∗,I∗) + µ + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ã + (1 −η)V ∗ ∂f ∂T (E∗) − (1 −η)V ∗ ∂f ∂I (E∗) + I∗ ∂g ∂T (T∗,I∗) − I∗ ∂g ∂I (T∗,I∗), a1 =d ï a + ρ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗) + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãò + ï µ + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãòï a + ρ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗) ò + (1 −η)(a + µ) ∂f ∂T (E∗) + (1 −η)aI∗ ∂f ∂T (E∗) − (1 −η)(1 + µ)V ∗ ∂f ∂I (E∗). 94 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE and a0 =d Å (a + ρ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗) ãï µ + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãò + (1 −η)aV ∗I∗ ∂f ∂V (E∗) ∂g ∂T (T∗,I∗) −d(1 −ε)(1 −η)kV ∗ ∂f ∂V (E∗) −d(1 −ε)(1 −η)kf(E∗) + aµ Å V ∗ ∂f ∂T (E∗) + I∗ ∂g ∂T (T∗,I∗) ã + aI∗ ∂g ∂T (T∗,I∗)f(E∗) −dµV ∗ ∂f ∂I (E∗). =d Å a + ρ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗) ãï µ + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãò + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãÅ aI∗ ∂g ∂T (T∗,I∗) −d(1 −ε)k ã − (1 −η)dµV ∗ ∂f ∂I (E∗) + aµ Å (1 −η)V ∗ ∂f ∂T (E∗) + I∗ ∂g ∂T (T∗,I∗) ã . Since R0 > 1 and a + ρ−g(T∗,I∗) = 1 µ [(1 −ε)k + g(T∗,I∗) − (a + ρ)](1 −η)f(T∗,I∗,V ∗) = V ∗ I∗ (1 −η)f(T∗,I∗,V ∗) > 0, (4.3) we deduce that a0, a1, and a2 are positive. In fact, firstly, we deduce from (4.3) that a + ρ − g(T∗,I∗) > 0. Furthermore, using assumptions (H01), (H01), (H2), (H3) and (H4), we obtain re- spectively, ∂g ∂T (T∗,I∗) ≥ 0, ∂g ∂I (T∗,I∗) ≤ 0, ∂f ∂T (E∗) ≥ 0, ∂f ∂I (E∗) ≤ 0 and V ∗ ∂f ∂V (E∗) + f(E∗) ≥ 0. This shows that a2 is positive. Secondly, from the same arguments used to prove the positivity of a2, we deduce that a1 is positive. Finally, in addition to assumptions (H01), (H01), (H2), (H3) and (H4), using the fact that aI∗ ∂g ∂T (T∗,I∗) −d(1 −ε)k > 0, we deduce that a0 is positive. Moreover, a1a2 −a0 =da1 + (a + ρ + µ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗))a1 + (1 −η)Å V ∗ ∂f ∂T (E∗)f(E∗) ã a1 + Å (1 −η)(V ∗ ∂f ∂T (E∗)) + I∗ ∂g ∂T (T∗,I∗) ã a1 − (1 −η)V ∗ ∂f ∂I (E∗)a1 − (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãÅ da + dρ−dg(T∗,I∗) −dI∗ ∂g ∂I (T∗,I∗) + aI∗ ∂g ∂T (T∗,I∗) −d(1 −ε)k ã −dµ Å a + ρ−g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗) − (1 −η)V ∗ ∂f ∂I (E∗) ã −aµ ï (1 −η)V ∗ ∂f ∂T (E∗) + I∗ ∂g ∂T (T∗,I∗) ò , GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 95 =(d + µ)a1 + (1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ãÅ a1 −aI∗ ∂g ∂T (T∗,I∗) ã (a + ρ −g(T∗,I∗) − I∗ ∂g ∂I (T∗,I∗)) Å a1 −d(1 −η) Å V ∗ ∂f ∂V (E∗) + f(E∗) ã −dµ ã − (1 −η)(a1 −dµ)V ∗ ∂f ∂I (E∗) + Å (1 −η)V ∗ ∂f ∂I (E∗) + I∗ ∂g ∂T (T∗,I∗) ã (a1 −aµ) + d(1 −ε)(1 −η)k Å V ∗ ∂f ∂V (E∗) + f(E∗) ã . One deduces that a1a2−a0 > 0. From the Routh-Hurwitz criteria in [12] we know that all roots of (4.2) have negative real parts. Thus, the chronic infection equilibrium E∗ is locally asymptotically stable for R0 > 1. � 5. Global stability In this section, we investigate the global stability of the disease-free equilibrium E0 and the chronic infection equilibrium E∗. For the global stability, we assume that a ≥ d. Biologically, this assumption is often satisfied because a represents the death rate of infected cells and includes the possibility of death by bursting of infected cells. Furthermore, this assumption is considered by many authors; see, for example, [16, 37, 32, 40]. Particularly in [4], this condition means that the average life-time of infected cells 1 a is shorter than the average life-time of infected cells 1 d . Therefore, we have the following result. Theorem 5.1. If R0 < 1 and a ≥ d then the uninfected equilibrium E0 is globally asymptotically stable. Proof. Construct the following Lyapunov functional L(T(t),I(t),V (t)) ≡ L(t) = I(t) + (1 −η)f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) V (t). Calculating the time derivative of L(t) along the positive solution of system (2.2), we have dL dt = dL dT dT dt + dL dI dI dt + dL dV dV dt , = dI dt + (1 −η)f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) dV dt , =(1 −η)f(T,I,V )V + g(T,I)I − (a + ρ)I + (1 −η)(1 −ε)kf(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) I −µ (1 −η)f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) V − (1 −η)2f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) f(T,I,V )V, = ï f(T,I,V ) − (1 −η)f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) (µ + (1 −η)f(T,I,V )) ò V + (a + ρ)I ï (1 −η)(1 −ε)kf(T0, 0, 0) (a + ρ)(µ + (1 −η)f(T0, 0, 0)) + g(T,I) (a + ρ) − 1 ò . 96 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE According to Remark 3.1, we consider solutions for which T(t) ≤ λ d . Using the expression of R0, one has dL dt ≤ ï f(T, 0, 0) − (1 −η)f(T0, 0, 0) µ + (1 −η)f(T0, 0, 0) (µ + (1 −η)f(T,I,V )) ò V + (a + ρ)I ï (1 −η)(1 −ε)kf(T0, 0, 0) + (µ + (1 −η)f(T0, 0, 0))g(T0, 0) (a + ρ)(µ + (1 −η)f(T0, 0, 0)) − 1 ò , ≤ µf(T, 0, 0) + f(T, 0, 0)f(T0, 0, 0) −µf(T0, 0, 0) −f(T0, 0, 0)f(T, 0, 0) µ + (1 −η)f(T0, 0, 0) + (a + ρ)I[R0 − 1], ≤ µ ( f(T, 0, 0) −f(T0, 0, 0) ) µ + (1 −η)f(T0, 0, 0) + (a + ρ)I[R0 − 1], ≤(a + ρ)I[R0 − 1]. Consequently, dL dt ≤ 0 for R0 < 1. Moreover, it is easy to show that the largest compact invariant set I in V ={(T,I,V ) / dL dt = 0}, ={(T,I,V ) / I = V = 0} is the singleton {E0}. By the LaSalle invariance principle[19], the disease-free equilibrium E0 is globally asymptotically stable for R0 < 1. � We now investigate the global dynamics of system (2.2) when R0 > 1. Firstly, we need the following lemma. Lemma 5.2. If R0 > 1, then differential system (2.2) is uniformly persistent. Proof. Considering the notations as in theorem 4.5 of [11], we denote by X1 = Int(R 3 +) the interior of R3+ and by X2 = Bd(R 3 +) the boundary of R 3 +. Since (T,I,V ) is bounded, there exists a compact set B of R3+ in which all the solutions of the differential system (2.2) initiated in R 3 + finally enter and stay there forever. Let us denote by ω(x0) the omega limit set of the solution x = x(t,x0) of the system (2.2) (By the criterion of Poincaré-Bendixson, and the fact that the solutions of (2.2) remain in a compact, and the omega limit set always exists). We need to determine Ω2 defined as in Theorem 4.5 in [11] by Ω2 = ⋃ y∈Y2 ω(y) (5.1) with, Y2 = {x ∈ X2 / φt(x) ∈ X2; ∀t > 0}. Setting Y2 = {(T,I,V )T ∈ Bd(R3+) / I = V = 0}, one has Ω2 = {E0 = (T0, 0, 0)} where T0 = λ d . Thus, solutions initiated on the T-axis converge to E0, then E0 is an isolated recovering of Ω2 (since E0 is an equilibrium of (2.2)) and secondly acyclic (because there is no non-trivial solution in Bd(R3+) which links E0 to itself). Finally, if it is shown that E0 is a weak repeller for X1, the proof will be achieved. By definition, E0 is a weak repeller for X1 if for each solution with initial data (t0,x0) ∈ J×X we have : lim t→+∞ sup d(x(t,x0),E 0) > 0. (5.2) GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 97 Inequality (5.2) holds if V s(E0) ∩ Int(R3+) = ∅, (5.3) where V s(E0) denotes the stable manifold of E0. Suppose that (5.2) does not hold for a solution x = x(t,x0) with initial data x0 ∈ X1. Considering the fact that the closed positive orthant is positively invariant with respect to the system (2.2), then, lim t→+∞ sup d(x(t,x0),E0)) = lim t→+∞ inf d(x(t,x0),E0)) = 0. Therefore, we will have lim t→+∞ x(t,x0) = E 0 which is clearly impossible if (5.3) is verified. It remains to show that (5.3) is valid in order to achieve a contradiction. For this, let us recall that the Jacobian matrix associated with the system (2.2) in E0 is given by ∇F(E0) = Ñ −d −g(T0, 0) + ρ −(1 −η)f(T0, 0, 0) 0 g(T0, 0) − (a + ρ) (1 −η)f(T0, 0, 0) 0 (1 −ε)k −µ− (1 −η)f(T0, 0, 0) é . The characteristic polynomial of ∇F(E0) is given by P∇F(E0)(X) = −(X + d) ( X2 + a1X + a0 ) . Since the product of the real parts of the roots of the polynomial T(X) = X2 + a1X + a0 worth a0 = (a+ρ)(µ+ (1−η)f(T0, 0, 0))(1−R0) ≤ 0, then the point E0 is unstable for the system (2.2). This implies that the matrix ∇F(E0) defined previously has an eigenvalue with a positive real part denoted x+ and two others with negative real parts x 1 − and x 2 − which may or may not coincide with x 1 −. The eigenspace associated with the eigenvalue x1− is the vector space generated by the vector (1.0.0). If x1− 6= x2−, then the eigenspace associated with x2− has the structure (0,v2,v3) with v2 and v3 verifying the following equation :Ç g(T0, 0) − (a + ρ) (1 −η)f(T0, 0, 0) (1 −ε)k −µ− (1 −η)f(T0, 0, 0) å . Ç v2 v3 å = x2− Ç v2 v3 å . (5.4) In the case of equality between x2− and x 1 −, we note that the squared matrix at the left side of (5.4) will be not diagonalizable. Indeed if it was diagonalizable, it would be similar to the matrix D = Ç x2− 0 0 x2− å , and this would imply thatÇ x2− 0 0 x2− å = Ç g(T0, 0) − (a + ρ) (1 −η)f(T0, 0, 0) (1 −ε)k −µ− (1 −η)f(T0, 0, 0) å which is absurd. Therefore, the structure of the eigenvector associated with x2− will have the structure (v1,v2,v3), where v1 satisfies the equation ∇F(E0). Ñ v1 v2 v3 é = x2− Ñ v1 v2 v3 é . In both cases (i.e., the cases x2− = x 1 − and x 2 − 6= x1−), we will always have (v2,v3) /∈ R2+. Indeed, the matrix defined in (5.4) is a Metzler matrix and irreducible. Hence, the stability modulus of x+ of this Metzler matrix will be an eigenvalue to which will correspond an eigenvector u of the positive orthant. The vector u being unique, this implies that (v1,v2,v3) is not contained in this positive orthant, i.e., (v1,v2,v3) /∈ R3+. Therefore, V s(E0) ∩ int(R3+) = ∅ which completes the proof.. � 98 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE Remark 5.1. According to [5], we can deduce from lemma 5.2 the existence of a compact absorbing set in Ω. Next, we focus on the global stability of the chronic infection equilibrium E∗ by assuming that R0 > 1 and the incidence function f satisfies the hypothesis (H4). To prove the global stability of E∗, we apply the geometrical approach developed by Li and Muldowney [21]. Theorem 5.3. Assume that R0 > 1, (1−η) q1 < δ and (H4) hold, then the chronic infection equilibrium E∗ is globally asymptotically stable where q1 = 1 t ∫ t 0 Å I(s) ∂f ∂T − I(s) ∂f ∂I −V (s) ∂f ∂V ã ds Proof. The second additive compound matrix of the Jacobian matrix J, of the system (2.2) is defined by J[2] = Ñ J11 + J22 J23 −J13 J32 J11 + J33 J12 −J31 J21 J22 + J33 é = Ñ c11 c12 c13 c21 c22 c23 c31 c32 c33 é where c11 = −(a + d + ρ) − I ∂g ∂T − (1 −η) Å V ∂f ∂T + V ∂f ∂I ã + I ∂g ∂I + g, c12 = (1 −η) Å V ∂f ∂V + f ã , c13 = (1 −η) Å V ∂f ∂V + f ã , c21 = (1 −ε)k − (1 −η)V ∂f ∂I , c22 = − ï d + µ + (1 −η)V ∂f ∂T + I ∂g ∂T + (1 −η) Å V ∂f ∂V + f ãò , c23 = ρ− (1 −η)V ∂f ∂I − I ∂g ∂I −g, c31 = (1 −η)V ∂f ∂T , c32 = (1 −η)V ∂f ∂T + I ∂g ∂T , c33 = − ï a + ρ + µ + (1 −η) Å V ∂f ∂V + f ãò + (1 −η)V ∂f ∂I + I ∂g ∂I + g, and Jij; i,j = 1, 2, 3 is the (k,l)th entry of the matrix J. We consider the matrix P(T,I,V ) = diag Å 1, I V , I V ã which has inverse given by P−1 = diag Å 1, V I , V I ã and the matrix Pf which is obtained by replacing each entry pij of P by its derivative in the direction of the solution of system (2.2). Thus Pf = diag Å 0, I′V −V ′I V 2 , I′V −V ′I V 2 ã GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 99 with I′ = dI dt and V ′ = dV dt . It follows that, PfP −1 = diag Å 0, I′ I − V ′ V , I′ I − V ′ V ã . Furthermore, one has B = PfP −1 + P(J[2]P−1 = Ç B11 B12 B21 B22 å , where B11 = −(a + d + ρ) − I ∂g ∂T − (1 −η) Å V ∂f ∂T + V ∂f ∂I ã + I ∂g ∂I + g, B12 = Ä (1 −η)V I Ä V ∂f ∂V + f ä (1 −η)V I Ä V ∂f ∂V + f ää , B21 = ÖÄ (1 −ε)k − (1 −η)V ∂f ∂I ä I VÄ (1 −η)V ∂f ∂T ä I V è , B22 = Ö − Ä d + µ + (1 −η)V ∂f ∂T + I ∂g ∂T + (1 −η) Ä V ∂f ∂V + f ää + Ä I′ I − V ′ V ä ρ− (1 −η)V ∂f ∂I − I ∂g ∂I −g (1 −η)V ∂f ∂T + I ∂g ∂T − Ä a + ρ + µ + (1 −η) Ä V ∂f ∂V + f ää + (1 −η)V ∂f ∂I + I ∂g ∂I + g + Ä I′ I − V ′ V äè . We define the norm on R3 as ‖(u,v,w)‖ = max{|u|, |v| + |w|} for all (u,v,w) ∈ R3 Then the Lozinskii measure µ with respect to the norm ‖ ·‖ can be estimated as follows (see [25]): µ(B) ≤ sup{g1,g2} (5.5) where g1 = µ1(B11) + ‖B12‖ et g2 = µ1(B22) + ‖B21‖. Here, µ1 denotes the Lozinskii measure with respect to the l1 vector norm, and ‖B12‖ and ‖B21‖ are matrix norms with respect to the l1 norm. Moreover, we have µ1(B11) = −(a + d + ρ) − I ∂g ∂T − (1 −η) Å V ∂f ∂T + V ∂f ∂I ã + I ∂g ∂I + g. (5.6) ‖B12‖ = (1 −η) V I Å V ∂f ∂V + f ã . (5.7) According to the second equation of system (2.2), (5.7) becomes: ‖B12‖ = I′ I + (1 −η) V 2 I ∂f ∂V + a + ρ−g. µ1(B22) = I′ I − V ′ V −µ− (1 −η) Å V ∂f ∂V + f ã + max{−d;−a}, (5.8) = I′ I − V ′ V −µ− (1 −η) Å V ∂f ∂V + f ã − δ (5.9) and ‖B21‖ = ï (1 −ε)k − (1 −η)V ∂f ∂I + (1 −η)V ∂f ∂T ò I V . (5.10) Hence, we obtain : g1 = I′ I −d + (1 −η)V 2 I ∂f ∂V − (1 −η) Ä V ∂f ∂T + V ∂f ∂I ä + I ∂g ∂I − I ∂g ∂T , ≤ I ′ I − δ, since a ≥ d. (5.11) 100 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE By using the first equation of (2.2), (5.8) and (5.10), one also has : g2 = I′ I − V ′ V −µ− (1 −η) Å V ∂f ∂V + f ã − δ + ï (1 −ε)k − (1 −η)V ∂f ∂I + (1 −η)V ∂f ∂T ò I V , = I′ I − δ + (1 −η) Å I ∂f ∂T − I ∂f ∂I −V ∂f ∂V ã . (5.12) From (5.5), (5.11) and (5.12), we get : µ(B) ≤ I′ I − δ + (1 −η) Å I ∂f ∂T − I ∂f ∂I −V ∂f ∂V ã . From Lemma 5.2 we know that the system (2.2) is uniformly persistent when R0 > 1. Then there exists a compact absorbing set K ⊂ Ω [5]. Along each solution (T(t),I(t),V (t)) of (2.2) with X0 = (T(0),I(0),V (0)), we have 1 t ∫ t 0 (µ(B(X(s),X0))) ds ≤ (1 −η) 1 t ln Å I(t) I0 ã −δ + 1 t ∫ t 0 Å I(s) ∂f ∂T − I(s) ∂f ∂I −V (s) ∂f ∂V ã ds, which implies that lim sup t→∞ sup X0∈K 1 t ∫ t 0 (µ(B(X(s),X0))) ds ≤ −δ + lim sup t→∞ sup X0∈K (1 −η) 1 t ∫ t 0 Å I(s) ∂f ∂T − I(s) ∂f ∂I −V (s) ∂f ∂V ã ds, ≤ −δ + (1 −η) q1, < 0. Then, based on Theorem 3.5 in [21], we deduce that the chronic infection equilibrium E∗ is globally asymptotically stable. This completes the proof of the theorem 5.3. � 6. Applications and numerical simulations As an application of our theoretical results, we consider the system   dT dt = λ−dT − (1 −η)β1TV α0 + α1T + α2V + α3TV −β2TI + ρI, dI dt = (1 −η)β1TV α0 + α1T + α2V + α3TV + β2TI − (a + ρ)I, dV dt = (1 −ε)kI −µV − (1 −η)β1TV α0 + α1T + α2V + α3TV , (6.1) which is a particular case of system (2.2) by letting f(T,I,V ) = β1T α0 + α1T + α2V + α3TV and g(T,I) = β2T where β1, β2, α1, α2, α3 and α4 are non negative constants. The functions g and f satisfy (H01), (H02) conditions and (H1), (H2), (H3) conditions, respectively. We have f(T,I,V ) + V ∂f ∂V (T,I,V ) = β1T(α0 + α1T) (α0 + α1T + α2V + α3TV )2 ≥ 0 GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 101 and we conclude that assumption (H4) is also satisfied. Other state variables and parameters are the same as in model (2.2). The threshold number, R0 takes the following form R0 = ((1 −η)(1 −ε)kβ1λd + (µ(α0d + α1λ) + (1 −η)β1λ) β2λ d(a + ρ) (µ(α0d + α1λ) + (1 −η)β1λ) . Firstly, we simulate the model (6.1) by using the following parameter values : λ = 10 cells mm−3 day−1 [15], η = 0.4 , β1 = 0.000024 mm 3virion−1day−1 [15], ρ = 0.01 [38], d = 0.02day−1[15]; ε = 0.5, k = 600 virions cell−1day−1, µ = 3 day−1 [15], α0 = 1, α1 = 0.1, α2 = 0.01, α3 = 0.00001, β2 = 0.0001 [38], a = 0.5 day−1 [15]. According to the values of these parameters, R0 = 0.1257 < 1, which means that R0 satisfies the conditions mentioned in Theorem 4.1 and Theorem 5.1. This implies that the disease- free equilibrium E0 = (500, 0, 0) is globally asymptotically stable. Furthermore, numerical simulation shown in Figure 1 confirms the result. 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] 300 320 340 360 380 400 420 440 460 480 500 U n in fe ct e d c e lls T (t ) (a) 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] -10 0 10 20 30 40 50 60 70 80 In fe ct e d c e lls I (t ) (b) 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] -1000 0 1000 2000 3000 4000 5000 6000 V ir u s lo a d V (t ) (c) 0 80 2000 4000 60 500 V (t ) 6000 400 Phase diagram I(t) 8000 40 300 T(t) 10000 20020 100 0 0 (d) Figure 1. Time evolutions of model (2.2) with initial values (300; 100; 15). E0 is globally asymptotically stable In Figure 1 (a), the proliferation of uninfected cells reaches its equilibrium value at λ d and T(t) converges to λ d = 500 whereas in Figure 1 (b) and in Figure 1 (c), Infected cells I(t) and viral load V (t) 102 A. NANGUE, P. T. LEMOFOUET, S. NDOUVATAMA, AND E. KENGNE converge to zero. Secondly, we choose β1 = 0.000024 mm 3virion−1day−1 [15] and the other parameter values are the same as above. The reason to just modify the parameter β1 is based on the fact that R0 is an increasing function with respect to β1. By calculating, we R0 = 1.2571 > 1, which means that it satisfies the conditions mentioned in Theorem 4.1 and Theorem 5.3. This implies that the chronic infection equilibrium E∗ = (405, 12.10, 1275) is globally asymptotically stable. In addition to this, numerical simulations shown in Figure 2 confirms the result. 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] 240 260 280 300 320 340 360 380 400 420 440 U n in fe ct e d c e lls T (t ) (a) 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] 0 10 20 30 40 50 60 70 80 90 100 In fe ct e d c e lls I (t ) (b) 0 100 200 300 400 500 600 700 800 900 1000 Time t [in days] 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 V ir u s lo a d V (t ) (c) 0 100 2000 4000 450 V (t ) 6000 Phase diagram 400 I(t) 8000 50 T(t) 10000 350 300 0 250 (d) Figure 2. Time evolutions of model (2.2) with initial values (300; 75; 15). E∗ is globally asymptotically stable In Figure 2, we observe damped oscillations at the beginning of the simulation. Infection will not become totally extinct, but a considerable reduction of the viral load and infected cells will be observed. Particularly, in Figure 2 (a), the number of uninfected cells T(t) decrease rapidly and then increase slightly until equilibrium is reached whereas in Figure 2 (b), infected cells I(t) do not tend to zero as t increases and in Figure 2 (c) viruses persist in the presence of treatment leading to the system going to an endemic equilibrium. GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 103 7. Conclusion In this study, we have proposed and studied a virus dynamic model with a generalized functional response, treatment and absorption effect. By analyzing the characteristic equations of model (2.2) at the disease-free equilibrium point, it has been completely established that the disease-free equilibrium is locally asymptotically stable if the basic reproduction number R0 is less than or equal to one (R0 ≤ 1). The local stability result of the chronic infection equilibrium of model (2.2) is shown in Theorem 4.2. If the basic reproduction number is greater than one (R0 > 1), then the disease-free equilibrium is unstable and the chronic infection equilibrium is locally asymptotically stable. The global behaviour of the model is investigated by constructing an appropriate Lyapunov functional for disease-free equilibrium and by applying Li-Muldowney global stability-criterion to the chronic infection equilibrium. Numerical simulations are carried out, performed in MATLAB, to confirm obtained theoretical result in a particular case. Furthermore, for model (2.2), we found that the basic reproduction number is less than that of a model without absorption effect. From the above discussion, it can be seen that there is a positive effect on eliminating viruses from the blood vessel than the model without absorption effect. Finally, it should be noted that most of the results contained in this work extend and complete the results of the works in [14] and [17]. 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Virol. 3 (2013), no. 1, 44–50. https://doi.org/10.1007/978-981-13-6581-22 https://doi.org/10.1007/978-981-13-6581-22 https://doi.org/10.1007/s10884-017-9622-2 https://doi.org/10.1142/S1793524512600121 GLOBAL ANALYSIS OF A GENERALIZED VIRAL INFECTION CELLULAR MODEL 105 Higher Teachers’ Training College, University of Maroua, P.O.Box 55, Maroua, Cameroon Email address: alexnanga02@gmail.com Higher Teachers’ Training College, University of Maroua, P.O.Box 55, Maroua, Cameroon Email address: plemofouettiomo@yahoo.fr Higher Teachers’ Training College, University of Maroua, P.O.Box 55, Maroua, Cameroon Email address: simonvotsomafils@gmail.com Corresponding author, School of Physics and Electronic Information Engineering, Zhejiang Normal Uni- versity, Jinhua 321004, China Email address: ekengne6@zjnu.edu.cn 1. Introduction 2. Formulation and description of the model 3. Relevant assumptions and preliminary results 3.1. Relevant assumptions 3.2. Positivity and boundedness 3.3. Equilibra 4. Local stability 5. Global stability 6. Applications and numerical simulations 7. Conclusion References