EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 944-959 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Global dynamics of an Hepatitis C Virus mathematical cellular model with a logistic term Alexis Nangue1,∗, Thiery Donfack1, David Avava Ndode Yafago1 1 Department of Mathematics, Higher Teacher’s Training College, University of Maroua, Cameroon Abstract. In this paper, the aim is to analyze the global dynamics of Hepatitis C Virus (HCV) cellular mathematical model under therapy with uninfected hepatocytes proliferation. We prove that the solution of the model with positive initial values are global, positive and bounded. In addition, firstly we show that the model is locally asymptotically stable at free virus equilibrium and also at infected equilibrium. Secondly we show that the model is globally asymptotically stable at the free virus equilibrium by using an appropriate lyapunov function. 2010 Mathematics Subject Classifications: 92B99, 34D23, 92D25 Key Words and Phrases: HCV cellular model, local and global solution, invariant set, stability, basic reproduction ratio, lyapunov function 1. Introduction Viral hepatitis C is an infectious disease caused by the hepatitis C virus (HCV). It is among the causes of liver cancer, the latter being one of the biggest causes of death in the world. In particular, according to the 2018 WHO report, 10,000 people die each year from hepatitis with a prevalence rate of around 13% for hepatitis C versus 10% for hepatitis B. Mathematical and computer models have become essential tools for analyzing, predicting and controlling infectious diseases (hepatitis, HIV, ebola, dengue, chikungunya...), both at the population level and at the individual level. These models can be used to construct and test hypotheses, make predictions and evaluate effective measures to make drugs effective. Numerous mathematical models describing the temporal dynamics of hepatitis C virus (HCV) have been proposed by various authors, such as: Neumann et al. [8] in 1998; Guedj and Neumann [5] in 2010; Chatterjee et al. [1] in 2012 and J. Rong and al. [6] in 2013. Our model is inspired by the model of Guedj and Neumann [5] which considers two levels of the infection namely: extracellular infection and intracellular infection. Stability is a ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3454 Email addresses: alexnanga02@yahoo.fr (A. Nangue), thierytd@yahoo.fr (T. Donfack), avavadavid@gmail.com (D. A. Yafago) http://www.ejpam.com 944 c© 2019 EJPAM All rights reserved. A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 945 central issue in the study of the dynamics of cellular models. Our work will consist to study the global dynamics of the model (1). The mathematical properties of a certain number of models of hepatitis C virus infection have already been studied, for example in [2]. The Neumann et al. [8] model of viral dynamics, named here the cell infection(CI) model is the standard description for HCV kinetics during treatment. In this model, the change in viral load, V (t), is occurring on the level of cell infection and involves de novo infection (with constant rate β), infected cell loss (with constant rate d), virus particle production (with constant rate per infected cell p), and virus clearance from circulation (with rate constant c), in Figure 1 [8] by the following compartmental model : Figure 1: The CI model (1)comprises of de novo infection and loss of infected cells (I), production and clearance of virus in circulation (V) and target cell (T) dynamics. In the original model, the target cells T (t) are produced at a constant rate s and die with death rate constant d. In this new model we taking into account a logistic rate, r ( 1− T+I Tmax ) , that allows for a more realistic growth of hepatocytes by proliferation. It should be noted that Tmax is the maximum hepatocyte density or ’carrying capacity’ which is based on both uninfected and infected hepatocyte in the liver. It is implicitly assumed that the vast majority of hepatocytes are susceptible to infection, consistent with experi- mental findings that a large proportion of hepatocytes can be infected (Rodriguez-Inigo et al. [9] ; Stiffler et al. [10]). Initially, the model neglects the fact that when a cell is infected the number of virions outside the cells is reduced by one. For this reason the equation for dV dt should contain an extra term −(1− η)βV T . It is argued, however, that this term is small in comparison to other terms in the same equation, so that it is justified to omit it. From a mathematical point of view including this effect leads to a new system which we call the ‘modified fundamental model of virus dynamics’. The additional term can be written as −α(1− η)βV T , where the Parameter α takes the value zero or one. The phenomenon described above is governed according to J. Guedj and A. U. Neu- mann [5] by a set of three autonomous ordinary differential equations : A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 946 dT dt = rT ( 1− T + I Tmax ) − (1− η)βV T − dT ; (1a) dI dt = (1− η)βV T − δI; (1b) dV dt = (1− ε)pI − cV − α(1− η)βV T ; (1c) where the equations relate the dynamics relationship between, T as the uninfected target cells (hepatocytes), I as the infected cells and V as the viral load (amount of viruses present in the blood). The efficacy of treatment in blocking virion production and reducing new infections is described by the parameters, ε and η, respectively, which values are non- negative and less than one. For biological significance of the parameters, one assumption is employed. Infected cells have a higher turnover rate than uninfected cells, i.e. d ≤ δ and we also suppose that r ≥ d. Let T0, I0, V0 ∈ R be given real numbers. we look for solutions T, I and V of the mathematical model (1) over [t0,+∞[, T ≤ +∞ satisfying : T (t0) = T0, I(t0) = I0, and V (t0) = V0, t0 ∈ [0,+∞[. (2) (2) are called initial conditions and the given numbers T0, I0, V0 being the initial data. The aim of this paper is to analyze the global dynamics of Hepatitis C Virus (HCV) cellular mathematical model under therapy with uninfected hepatocytes proliferation described by system (1). To achieve this, we will organized the paper as follows : we study in section 1 some properties of the solutions of the studied model. Section 2 is devoted to the study of the local stability of equilibrium points and we end with the global stability of the uninfected equilibrium point in Section 3 2. Properties of solutions to the Cauchy problem (1), (2) 2.1. Existence of local and global solutions, positivity Our objective is to prove the existence of global solution T, I V defined over the whole interval [t0,+∞[ and satisfying (2). The first step in examining model (1) is to prove that local solution to the initial-value problem does, in fact, exist, and that this solution is unique. Proposition 1. Let T0, I0, V0 ∈ R be given. There exists t1 > t0 > 0 and continuously differentiable functions T , I, V : [0, t0) −→ R such that the ordered triple (T, I, V ) satisfies (1) and (T (t0), I(t0), V (t0)) = (T0, I0, V0). Proof. To prove the result, we use the classical Cauchy-Lipschitz theorem. Since the first order system of ordinary differential equations (1) is autonomous, it is enough to show A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 947 that the function f : R3 −→ R3 defined by : f(x, y, z) =  f1(x, y, z) f2(x, y, z) f3(x, y, z)  =  rx(1− x+y Tmax )− dx− (1− η)βzx (1− η)βzx− δy (1− ε)py − cz − α(1− η)βzx  is locally Lipschitz in its u = (x, y, z) argument. In fact, it suffices to notice that the jacobian matrix ∇f(x, y, z) =  r(1− 2x+y Tmax ) − rx Tmax −(1− η)βx (1− η)βz −δ (1− η)βx −α(1− η)βx (1− ε)p −c− α(1− η)βx  is locally bounded for every u = (x, y, z) ∈ R3. Hence, f has a continuous, bounded derivative on any compact subset of R3 and though f is locally Lipschitz in u = (x, y, z); In addition f is continuous. By Cauchy-Lipschitz theorem, there exists a unique solution, x(t), to the ordinary differential equation u′(t) = f(x(t)) with initial value u(t0) = u0 = (x0, y0, z0) on [t0, t1] for some time t1 > t0 ≥ 0. This completes the proof of the proposition. Remark 1. Since f is a continuously differentiable function, we deduce a unique maximal solution of initial value problem (1), (2). In addition, f , being indefinitely continuously differentiable, we can also deduce that this solution is indefinitely continuously differen- tiable. Additionally, we may show that for positive initial data, solutions of Cauchy problem (1), (2) remain positive as long as they exist. Theorem 1. Let (T, I, V ) be a solution of the Cauchy problem (1), (2) on an interval [t0, t1[. Assume the initial data of (1), (2) satisfy T0 > 0, I0 > 0, and V0 > 0 then T (t), T (t) and V (t) remain positive for all t ∈ [t0, t1[. Proof. Call the variables xi. If there is an index i and a time t for which xi(t) = 0, let 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 equation for xi in the system (1) can be written in the form : dxi(t) dt = −xif(x) + g(x), where g(x) is non-negative. As a consequence dxi(t) dt ≥ −xif(x) and xi(t) > 0, ∀t ∈ [t0, t∗]. A contradiction. This completes the proof of the theorem. A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 948 Remark 2. With this, we have a general idea that the model is sustainable, and can say with assurance that it remains biologically valid as long as it began with biologically- reasonable (i.e, positive) data. This also shows that once infected, it is entirely possible that the virus may continue to exist beneath a detectable threshold without doing any damage. Remark 3. One reason why we choose the strict inequalities for the initial data is that often in biological (or chemical) applications we are interested in the case of solutions where all unknowns are positive. This means intuitively that all elements of the model are ’active’. On the other hand it is sometimes relevant to consider solutions with non-strict inequalities. The fact the statement of the theorem with strict inequalities implies the corresponding statement with non-strict inequalities is based on continuous dependence of the solution on initial data. It will now be shown, with the help of the continuation criterion the existence of global solutions. Theorem 2. If (1 − ε)p − δ > 0 the solution of the initial value problem (1), (2), with positive initial data, exists globally in time in the future, i.e. on [t0,+∞[. Proof. To prove this, it is enough to show that all variables are bounded on an arbitrary finite interval [t0; t). Using the positivity of the solutions, is suffices to show that all variables are bounded above. The sum of equations (1a), (1b) and (1c) leads to : d dt (T + I + V ) = rT ( 1− T + I Tmax ) − dT + ((1− ε)p− δ)I − cV − α(1− η)βV T. Hence d dt (T + I + V ) ≤ rTmax 4 − ( √ rT√ Tmax − √ rTmax 2 )2 +A (T + I + V ) where A = max{c, d, (1− ε)p− δ}. It follows that : d dt (T + I + V ) ≤ rTmax 4 +A (T + I + V ) . (3) From differential inequation (3), T (t) + I(t) + V (t) should verify : T (t) + I(t) + V (t) ≤ N(t), where the function N is the solution of the following initial value problem (4), (5): dN dt = AN + rTmax 4 (4) N(t0) = N0 = T0 + I0 + V0. (5) A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 949 The solution of the initial value problem (4), (5) is given by : N(t) = −rTmax 4A + ( N0 + rTmax 4A ) eA(t−t0). It follows that, T (t) + I(t) + V (t) ≤ −rTmax 4A + ( T0 + I0 + V0 + rTmax 4A ) eA(t−t0). This last equation leads to : T (t) + I(t) + V (t) ≤ ( T0 + I0 + V0 + rTmax 4A ) eA(t−t0). Therefore, T (t) + I(t) + V (t) ≤ g(t) (6) where : g(t) = ( T0 + I0 + V0 + rTmax 4A ) eA(t−t0). g is a continuous function thus g is bounded function on [t0, t]. Furthermore according to the positivity result and inequation (6), we have : 0 < N(t) ≤ g(t) on [t0, t]. Therefore, since N is bounded on any finite interval, T , I ,V are also bounded. This completes the proof of the theorem. 2.2. Asymptotic behaviour and invariant set The following theorem shows that all solutions of model (1) in R3 + are ultimately bounded and that solutions with positive initial value conditions are positive, which indi- cates that model (1) is well-posed. Otherwise for biological reasons only initial data are considered for which at the initial time T + I ≤ Tmax. Then we will prove that if the initial data satisfy this previous inequality the solution also does so. Theorem 3. For any positive initial data T0, I0 and V0 of the Cauchy problem (1), we have : T (t) ≤ C1, I(t) ≤ C1 and V (t) ≤ C2, where C1 = (r − d)Tmax r , C2 = max ( (1− ε)p c C1, V0 ) when t tends to +∞. A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 950 Proof. Adding equations (1a) and (1b) implies, since d ≤ δ : d dt (T + I) = r ( 1− T + I Tmax ) T − dT − δI ≤ r ( 1− T + I Tmax ) T − d(T + I) ≤ r ( 1− T + I Tmax ) T + r ( 1− T + I Tmax ) I − d(T + I) d dt (T + I) ≤ r ( 1− T + I Tmax ) (T + I)− d(T + I). (7) We know that T (t) + I(t) ≤M(t), where M satisfies : dM dt = (r − d)M − r Tmax M2 (8) M(t0) = T (t0) + I(t0) = T0 + I0. (9) (8) is a first order differential equation of Bernoulli type. Hence, by direct calculation, using (9), we obtain the following expression of M : M(t) = (r − d)Tmax r + (r − d)Tmax ( z0 − r (r−d)Tmax ) e−(r−d)(t−t0) which shows that M(t) −→ (r−d)Tmax r as t −→ +∞ and hence, M is bounded. Since 0 ≤ T (t) + I(t) ≤M(t), T + I is bounded and therefore T (t) ≤ (r−d)Tmax r , I(t) ≤ (r−d)Tmax r . Moreover, as far as V is concerned, according to 1c, we have : dV dt = (1− ε)pI − cV − α(1− η)βV T ≤ (1− ε)pI − cV ≤ (1− ε)pC1 − cV i.e. dV dt ≤ −cV + (1− ε)pC1 (10) We know, by differential calculus, that V (t) ≤ N(t), where N satisfies : dN dt ≤ −cV + (1− ε)pC1, (11) N(t0) = V (t0)) = V0. (12) A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 951 (11) is a non homogeneous first order differential equation. Hence, by direct calculation, using (12), we obtain the following expression of N : N(t) = (1− ε)pC1 c + e−c(t−t0) ( V0 − (1− ε)pC1 c ) ; = max ( (1− ε)p c C1, V0 )( 1 + e−c(t−t0) − e−c(t−t0) ) , which shows that: N(t) ≤ max ( (1− ε)p c C1, V0 ) and therefore, N is bounded as t tends to +∞. Since 0 ≤ V (t) ≤ N(t), V is bounded and hence V (t) ≤ max ( (1−ε)p c C1, V0 ) . At the end we achieve the proof of theorem 3 . As consequences of Theorem 3 we have the followings : Remark 4. Let S be a solution of system (1). If S0 ∈ R × R3 + then, the limit of S(t) exits when t −→ +∞ . In other words the solution is globally bounded in the future. In particular, S is periodic if and only if S is stationary under the condition that S(t) admits a finite limit when t tends to infinity. Theorem 4. Let (t0, S0 = (T0, I0, V0)) ∈ R×R3 + and ([t0, T [, S = (T, I, V )) be a maximal solution of the Cauchy problem (1), (2) (T ∈]t0,+∞[). If T (t0) + I(t0) ≤ C1 and V (t0) ≤ C2 then the set : Ω = { (T (t), I(t), V (t)) ∈ R3 + : T (t) + I(t) ≤ C1, V (t) ≤ C2 } , where C1 = (r − d)Tmax r , C2 = max ( (1− ε)p c C1, V0 ) , is a positively invariant set by system (1). 3. Equilibria and Basic reproduction number R0 In this section, we will derive the basic reproduction number, and compute the equi- libria of model (1). When there is no viral infection, the uninfected hepatocytes dynamics is determined by : dT dt = rT ( 1− T Tmax ) − dT. (13) A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 952 Thus, in the absence of viral infection, the amount of susceptible cells will attend to a positive constant level T 0, which is : T 0 = r − d r Tmax ≤ Tmax. (14) Now, using the idea of next generation matrix for a general compartmental disease trans- mission model in [4], we can obtain the basic reproduction ratio of (1). Proposition 2. The basic reproduction ratio R0 of model (1) is given by : R0 = (1− ε)(1− η)pβT 0 δ(c+ (1− η)αβT 0) . Proof. Using (13) and (14), we know that E0 = (T 0, 0, 0) is the virus-free equilibrium or uninfected equilibrium, which exists for all positive parameter values. Based on the notations in [4], we have DF =  0 (1− η)βT 0 (1− ε)p 0  , DV =  δ 0 0 c+ α(1− η)βT 0  and DV −1 =  1 δ 0 0 1 c+α(1−η)βT 0  . and the next generation matrix is : DF.DV −1 =  0 (1− η)βT 0 (1− ε)p 0  1 δ 0 0 1 c+α(1−η)βT 0  =  0 (1−η)βT 0 c+α(1−η)βT 0 (1−ε)p δ 0  . According to [4, Theorem 2], the basic reproduction number of (1) is defined by R0 = ρ(DF.DV −1) = (1− θ)pβT 0 δ(c+ (1− η)αβT 0) , where ρ(A) denotes the spectral radius of a matrix A and 1− θ = (1− ε)(1− η). Remark 5. Henceforth (1−ε)(1−η) = (1−θ) and θ denotes the overall drug effectiveness [3]. Besides the virus-free equilibrium point E0, we now discuss the existence of the infected equilibrium point E+ = (T ∗, I∗, V ∗), in which X∗ means a positive constant. To determine T ∗, I∗ and V ∗ we should solve the following algebraic system : rT ( 1− T + I Tmax ) − (1− η)βV T − dT = 0; (1− η)βV T − δI = 0; (15) (1− ε)pI − cV − α(1− η)βV T = 0. We have the following result by easy way : A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 953 Proposition 3. If R0 > 1 and (1− ε)p−αδ > 0 then the mathematical model (1) admits an infected equilibrium point E+ = (T ∗, I∗, V ∗) where T ∗ = cδ (1− η)β[(1− ε)p− αδ] ; V ∗ = c (1− ε)p− αδ { (1− η)kβ[(1− ε)p− αδ](r − d)− crδ (1− η)βcr + (1− η)2kβ2[(1− ε)p− αδ] } = c (1− ε)p− αδ I∗ and I∗ = (1− η)kβ[(1− ε)p− αδ](r − d)− crδ (1− η)βcr + (1− η)2kβ2[(1− ε)p− αδ] . Remark 6. (i) The point (0, 0, 0) is also an equilibrium of the model (1). (ii) T ∗ can be expressed with respect to R0, thus we have: T ∗ = cδT0 R0δc+ (1− η)αβT0(R0 − 1) This means that T ∗ exists if and only if R0 > 1. 4. Local stability analysis In this subsection, we investigate the local stability of the equilibria E0 and E+ by finding the eigenvalues of the associated Jacobian matrices. The jacobian matrix J (T, I, V ) of model (1) is given by : J (T, I, V ) =  p1 rT Tmax −(1− η)βT (1− η)βV −δ (1− η)βT −α(1− η)βV (1− ε)p p2  (16) where p1 = r ( 1− 2T + I Tmax ) − d− (1− η)βV and p2 = − (c+ α(1− η))βT. First, we have Theorem 5. For model (1), the virus-free equilibrium point E0 = (T 0, 0, 0) is locally asymptotically stable if R0 < 1 and unstable if R0 > 1 A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 954 Proof. The local stability of the uninfected steady state E0 = (T 0, 0, 0) is governed by the eigenvalues of the matrix J (E0) =  −(r − d) 0 − (1−η)(r−d)β r 0 −δ (1−η)(r−d)β r 0 (1− ε)p b2  where b2 = − ( c+ α (1− η)(r − d)β r ) . The characteristic equation of the linearised system is given by the following equation : [−λ− (r−d)][λ2 + λ r [cr+ δr+αβ(1− η)(r−d)]− β r (1− η)(r−d)[(1− ε)p− δα] + crδ] = 0 i.e. [−λ− (r − d)][λ2 + λa1 + a2] = 0 where coefficients are given by : a1 = 1 r [cr + δr + αβ(1− η)(r − d)] a2 = −β r (1− η)(r − d)[(1− ε)p− δα] + cδ. λ = −(r−d) is already a negative eigenvalue of the jacobian matrix J (E0) and to achieve the study we will use the Routh-Hurwitz criterion. If a1 and a2 are all positive, then ap- plying the Routh-Hurwitz criterion to the quadratic equation guarantees the eigenvalues to have negative real part, and that two conditions must be satisfied for local asymptotic stability of the uninfected steady state : a1 > 0 i.e. 1 r [cr + δr + αβ(1− η)(r − d)] > 0 is satisfied. a2 > 0 i.e. −β r (1− η)(r − d)[(1− ε)p− δα] + cδ > 0 i.e. β r (1− η)(r − d)[(1− ε)p+ αδ] < cδ it follows that : β r (1− η)(r − d)(1− ε)p < cδ + αδβ r (1− η)(r − d) A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 955 which yields pβ(1− η)(r − d)(1− ε) < δ(cr + αβ(1− η)(r − d)). that leads to : pβ(1− η)(r − d)(1− ε) δ(cr + αβ(1− η)(r − d)) < 1 ⇒ R0 < 1. All conditions are satisfied if and only if R0 < 1 for local asymptotic stability of the uninfected steady state. This completes the proof. Next we consider the local stability of the unique infected equilibrium pont E+ when R0 > 1 and (1− ε)p− αδ > 0. Using (16) and the following equation : rT ∗ ( 1− T ∗ + I∗ Tmax ) − (1− η)βV ∗T ∗ − dT ∗ = 0, the jacobian matrix of model (1) at infected equilibrium point E+ is : J(E+) =  − rT Tmax − rT Tmax −(1− η)βT (1− η)βV −δ (1− η)βT −α(1− η)βV (1− ε)p −c− α(1− η)βT.  . We have the following characteristic equation associated with the above jacobian matrix J(E+) : |J(E+)− λI| = λ3 +A1λ 2 +A2λ+A3 = 0, Where : A1 = α (1− η)βT ∗ + c+ δ + T ∗r Tmax , A2 = α (−η + 1)T ∗V ∗β2η − α (−η + 1)T ∗V ∗β2 − T ∗β εηp+ α (1− η)T ∗βδ + T ∗βε p +T ∗βηp− T ∗βp+ cδ + 1 Tmax ( T ∗2βrα (1− η)− T ∗V ∗βηr + T ∗V ∗βr + T ∗cr + T ∗δ r ) , i.e. A2 = α(1− η)βT ∗ (δ − (1− η)βV ∗) + rT ∗ Tmax (1− η)βV ∗ + (1− η)βT ∗ ( α rT ∗ Tmax − (1− ε)p ) + rT ∗ Tmax (c+ δ) + δc A3 = ( −V ∗β2εη2p+ α (1− η)V ∗β2δη + 2V ∗β2εηp+ V ∗β2η2p− α (1− η)V ∗β2δ − V ∗β2εp ) T ∗ A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 956 + T ∗ Tmax (−T ∗βεηpr + T ∗βδrα (1− η) + T ∗βεpr + T ∗βηpr − V ∗βcηr − T ∗βpr + V ∗βcr + cδr) − ( 2V ∗β2ηp− V ∗β2p ) T ∗. Obviously we have : A1 > 0 and A2 > 0 if and only if α rT ∗ Tmax − (1− ε)p > 0 and δ − (1− η)βV ∗ > 0. Let ∆ = ∣∣∣∣ A1 1 A3 A2 ∣∣∣∣ = A1A2 −A3. Then, by Routh-Hurwitz criterion, we have the following result. Theorem 6. For model (1), when α rT ∗ Tmax − (1 − ε)p > 0 and δ − (1 − η)βV ∗ > 0 are valid, then the unique infected equilibrium E+ is locally asymptotically stable if ∆ > 0 and unstable if ∆ < 0. Especially we have : Corollary 1. Suppose that rT ∗ Tmax c− (1− η)βT ∗(αδ + (1− ε)p) > 0, then ∆ > 0 is always valid, i.e. E+ is locally asymptotically stable only if it exists in this case. Proof. We have : ∆ = ∣∣∣∣ A1 1 A3 A2 ∣∣∣∣ = A1A2 −A3 = T ∗βδηp− T ∗V ∗β2η2p+ T ∗V ∗β2εp+ 2T ∗V ∗β2ηp+ T ∗βcεp+ T ∗β cηp+ T ∗βδεp+ c2δ + cδ2 + T ∗ ( α (1− η)T ∗β2ε p+ α (1− η)T ∗β2ηp− α (1− η)V ∗β2c+ 2α (1− η)βcδ + (α (1− η))2 T ∗V ∗β3η ) + T ∗ ( α (1− η)V ∗β2cη + V ∗β2εη2p− 2V β2εηp− βcεηp− βδεηp− α (1− η)T ∗β2εηp ) + T ∗ Tmax ( (α (1− η))2 T ∗2β2r + 2α (1− η)T ∗βcr + 2α (1− η)T ∗βδr − V ∗βδηr + V ∗βδr + c2r ) + T ∗ Tmax ( α (1− η)T ∗2βr2 − T ∗V ∗βηr2 + T ∗V ∗βr2 + T ∗cr2 + T ∗δr2 + 2cδr + δ2r ) − (α (1− η))2 T ∗2V ∗β3 + (α (1− η))2 T ∗2β2δ − α (1− η)T ∗2β2p+ α (1− η)T ∗βδ2 − T ∗V ∗β2p − T ∗βcp− T ∗βδp, i.e, ∆ = (1− η)βT ∗ ( α rT ∗ Tmax − (1− ε)p ) (c+ α(1− η)βT ∗ + δ) + δc2 + δ2c+ (1− η)βT ∗αδc +(c+ δ) rT ∗ Tmax ( rT ∗ Tmax + c+ α(1− η)βT ∗ + δ ) + (1− η)βV ∗ ( rT ∗ Tmax c− (1− η)βT ∗(αδ + (1− ε)p) ) +α(1− η)βT ∗(δ − (1− η)βV ∗) ( rT ∗ Tmax + c+ α(1− η)βT ∗ + δ ) . Clearly ∆ > 0 if and only if rT ∗ Tmax c− (1− η)βT ∗(αδ + (1− ε)p) > 0. As a result, ∆ > 0 is always valid. This completes the proof of this corollary. A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 957 5. Global stability analysis of the model (1) at uninfected steady state For the global stability of the equilibria, we have : Theorem 7. The infection-free steady state E0 of model (1) is globally asymptotically stable if the basic reproduction number R0 < c c+α(1−η)βT 0 . Remark 7. Since R0 = (1− ε)(1− η)pβT 0 δ(c+ (1− η)αβT 0) = (1− θ)pβT 0 δ(c+ (1− η)αβT 0) and (1− ε)(1− η) = (1− θ). Then (1− θ)pβT 0 = R0δ(c+ (1− η)αβT 0)). therefore (1− θ)pβT 0I c − δ < 0 ⇐⇒ R0δ(c+ α(1− η)βT 0) c − δ < 0 ⇐⇒ R0 < c c+ α(1− η)βT 0 . Proof. Consider the Lyapunov function : L1(T, I, V ) = T − T 0 − T 0 ln T T 0 + I + (1− η)βT 0 c V L1 is defined, continuous and positive definite for all T > 0, I > 0, V > 0. Also, the global minimum L1 = 0 occurs at the infection free equilibrium E0. Further, function L1, along the solutions of system (1) at E0, satisfies : dL1 dt = ∂L1 ∂T dT dt + ∂L1 ∂I dI dt + ∂L1 ∂V dV dt = dT dt − dT 0 dt dT dt + dI dt + (1− η)βT0 c dV dt = ( 1− dT 0 dt ) Ṫ + İ + (1− η)βT0 c V̇ . Further collecting terms, we have dL1 dt = (T − T0) [ r − r(T + I) Tmax − d− (1− η)βV ] + (1− η)βV T − δI + (1− η)βT0 c [(1− ε)pI − cV − α(1− η)βV T ] . A. Nangue, T. Donfack, D. A. Ndode Yafago / Eur. J. Pure Appl. Math, 12 (3) (2019), 944-959 958 Hence: dL1 dt = (T − T 0) [ − r Tmax (T − T 0)− r Tmax I ] − δI + (1− η)(1− ε)pβT 0I c −α(1− η)2β2V TT 0 c = − r Tmax [(T − T 0)2 − (T − T 0)I] + ( (1− η)(1− ε)pβT 0I c − δ ) I −α(1− η)2β2V TT 0 c This leads to : dL1 dt = − r Tmax [(T − T 0)(T + I − T 0)] + ( (1− θ)pβT 0I c − δ ) I − α(1− η)2β2V TT 0 c , since 1− θ = (1− η)(1− ε). Furthermore, we have : dL1 dt ≤ − r Tmax [(T − T 0)(T + I − T 0)] + I(R0 − c c+ α(1− η)βT 0 )− α(1− η)2β2V TT 0 c . R0 ≤ c c+α(1−η)βT 0 and theorem 4 ensure dL1 dt ≤ 0 for all T > 0, I > 0 , V > 0. The equality dL1 dt = 0 holds only at the free equilibrium E0 . Therefore, the largest compact invariant subset of the set M = {(T, I, V ) ∈ Ω : dL1 dt = 0} is the singleton {E0}. By the Lasalle invariance principle, the infection-free equilibrium is globally asymptotically stable if R0 ≤ c c+α(1−η)βT 0 . This completes the proof of theorem 7. 6. Concluding remark It is clear that this paper is a starting point for further investigations. In a very near future we will attempt to solve the case of the global asymptotic stability of the infected equilibrium point. Constructing a Lyapunov function for this infected equilibrium model appears to be very complex. We will think about Li-Muldowney global-stability criterion [7] for example. Acknowledgements We are grateful to Professor Alan Rendall for valuable and tremendous discussions about this paper. 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