DYNAMICS OF AN IMMUNOLOGICAL VIRAL INFECTION MODEL WITH LYTIC AND NON-LYTIC IMMUNE RESPONSE IN PRESENCE OF CELL-TO-CELL TRANSMISSION AND CURE OF INFECTED CELLS

. In this paper, we propose a mathematical model that describes the dynamics of viral infection with both modes of transmission, virus-to-cell and cell-to-cell by taking into account the non-cytolytic cure of infected cells, the lytic and non-lytic humoral immune response. We ﬁrst prove the non-negativity and boundedness of the solutions of the proposed model. Furthermore, the dynamical behaviors of the model including the local and global stability of equilibria are rigorously investigated


INTRODUCTION
The human body is exposed permanently to pathogenic agents, such as viruses which can induce viral infections. Viruses once entered the cell, they take over its machinery and metabolism to support their replication. Furthermore, viruses can spread by two different modes, either Based on the above biological and mathematical considerations, we propose a mathematical model that incorporates both modes of transmission, cure of infected cells as well as lytic and non-lytic humoral immune response. This model is formulated by the following nonlinear system (1) where the uninfected cells (U) are generated at rate λ , die at rate d U U and become infected either by free virus particles at rate β 1 UV or by direct contact with infected cells at rate β 2 UI.
The two modes of transmission are inhibited by non-lytic humoral immune response at rate 1 + q 1 A and 1 + q 2 A, respectively. The infected cells (I) die at rate d I I and return to the uninfected state by loss of cccDNA from their nucleus at rate εI. Free viruses (V) are produced by infected cells at rate kI, cleared at rate d V V and neutralized by antibodies at rate rVA. Antibodies develop in response to free virus at rate ρVA and decay at rate d A A.
It is important to note that if q 1 = q 2 = 0, then we find the model studied in [8]. Also, if β 2 = 0, then we get the model investigated by Dhar et al. in [7].
The remainder of this paper is outlined as follows. In the next section, we establish some preliminary results including the non-negativity and boundedness of solutions of system (1) as well as we discuss the existence of equilibria. In Section 3, we establish sufficient conditions for the local stability of the three equilibrium points. Section 4 is devoted to global stability of equilibria. The paper ends with a conclusion presented in Section 5.

PRELIMINARY RESULTS
Theorem 2.1. All solutions for (1) with non-negative initial conditions, remain non-negative and bounded for all t ≥ 0.
Proof. We have According to Proposition B.7 of [11], we deduce that the solutions U, I, V and A are nonnegative.
To prove the boundedness of solutions, we consider the following function Then Then U(t), I(t), V (t) and A(t) are bounded. This completes the proof.
Next, we define two threshold parameters and establish the existence of three equilibrium points for (1). It is clear that the point P 0 (U 0 , 0, 0, 0), with U 0 = λ d U , is the unique infection-free equilibrium of model (1). Then we define the first threshold parameter which represents the basic reproduction number R 0 of model (1) as follows: are the basic reproduction numbers associated to the virus-to-cell and cell-to-cell transmission modes, respectively.
In absence of humoral immune response and R 0 > 1, the model (1) has the unique equilibrium point called the immune free-equilibrium and labeled by P 1 (U 1 , I 1 ,V 1 , 0), where In presence of humoral immune response, and we define on the closed interval [0, u * ] the function h as follows When the humoral immune response has not established, we have from the last equation of Then we define the second threshold parameter which represents the antibody immune response reproduction number as follows This establishes the uniqueness of U 2 and therefore our system has another infection equilibrium called the infection equilibrium with

LOCAL STABILITY
In this section, we establish sufficient conditions for the local stability of the three equilibrium points.
The Jacobian matrix, J(P), of system (1) at a point P(U, I,V, A) is given by Proof. The characteristic equation at P 0 is given by (4). Also, According to Routh-Hurwitz criterion [12,Theorem 4.4], the other two roots of the equation (4) have negative parts when R 0 < 1. Hence, P 0 is locally asymptotically stable if R 0 < 1.
Proof. The characteristic equation at the equilibrium point P 1 is given by where Assume that R A 1 < 1 < R 0 . We have From Routh-Hurwitz criterion, we deduce that all roots of the equation (5) have negative real parts. Hence, the immune-free equilibrium P 1 is locally asymptotically stable when R A 1 < 1 < R 0 .
If R A 1 > 1, then ρV 1 − d A is a positive root of the characteristic equation (5). Therfore, the immune-free equilibrium P 1 becomes unstable when R A 1 > 1.
the infection equilibrium with humoral immune response P 2 is locally asymptotically stable.
Proof. The characteristic equation at P 2 is given by Hence, It follows from the following equalities c 3 − c 4 c 5 = Based on Routh-Hurwitz criterion, we deduce that all roots of the equation (6) have negative real parts.

GLOBAL STABILITY
In this section, we focus on the global stability analysis of the equilibria P 0 , P 1 , and P 3 .
Theorem 4.1. If R 0 ≤ 1, then the infection-free equilibrium P 0 is globally asymptotically stable.
Proof. We define the Lyapunov function L 0 as where φ (x) = x − 1 − ln(x) for x > 0, and c given by the following equality 2c(d I + d U ) = ε U 0 .
Note that φ (x) ≥ 0 for all x > 0 and φ (x) = 0 if and only if x = 1. Thus, L 0 (P) ≥ 0 for all P ∈ IR * + × IR 3 + and L 0 (P) = 0 if and only if P = P 0 . Therefore, for all solution of the model system (1), we have Hence, Thus, when R 0 ≤ 1, we have dL 0 dt ≤ 0 with equality if and only if U = U 0 , I = 0, V = 0 and A = 0. Therefore, it follows from LaSalle's invariance principle that P 0 is globally asymptotically stable when R 0 ≤ 1.
To study the global stability for two infection equilibria P 1 and P 2 , we consider the following condition (H) for all i = 1, 2.
, then the immune-free equilibrium P 1 is globally asymptotically stable.
Proof. We consider the Lyapunov functional L 1 as follows Since λ = d U U 1 + β 1 U 1 V 1 + β 2 U 1 I 1 − εI 1 = d U U 1 + d I I 1 and kI 1 = d V V 1 , we get Form the condition (H) and using the equality and A = 0. From LaSalle's invariance principle, we deduce that P 1 is globally asymptotically Theorem 4.3. Assume that (H) holds for P 2 . If R A 1 > 1 and d U U 2 − εI 2 ≥ 0, then the infection equilibrium with humoral immune response P 2 is globally asymptotically stable.
Proof. To analyze the global stability of P 2 , we consider the following Lyapunov function: Hence, Thus, By the following equalities and by simple computations, we obtain Using the arithmetic-geometric inequality, we have Moreover if (H) is holds for P 2 , then Hence if d U U 2 − εI 2 ≥ 0 and R A 1 > 1, then dL 2 dt ≤ 0 with equality if and only if U = U 2 , I = I 2 , V = V 2 and A = A 2 . It follows from LaSalle's invariance principle that P 2 is globally asymptotically stable. This completes the proof.

CONCLUSION
In this work, we have proposed and analyzed the dynamics of an immunological viral infection model with lytic and non-lytic immune response in presence of cell-to-cell transmission and cure of infected cells. We proved that the proposed model is mathematically and biologically well-posed. Also, we derived two threshold parameters that are the basic reproduction number R 0 and the antibody immune response reproduction number R A 1 . Such threshold parameters parameters characterize the dynamics of the model. In addition, the local and global stability of equilibria are fully established by means of direct and indirect Lyapunov method.
Moreover, the models and results presented in [7,8] are improved and generalized.
It is known that the adaptive immunity has two important characteristics that are specificity and memory. The first refers to ability of immune system to target specific pathogens. However, the second characteristic refers to the ability of immune system to quickly remember the antigens that previously activated it and launch a more intense immune reaction when encountering the same antigen a second time. Therefore, it will be more interesting to study the effect of immunological memory on the dynamics of the proposed model by using the new generalized Hattaf fractional (GHF) derivative and its properties presented [13,14,15]. This will be the main aim of our future works.