Global stability of a distributed delayed viral model with general incidence rate

In this paper, we discussed an infinitely distributed delayed viral infection model with nonlinear immune response and general incidence rate. We proved the existence and uniqueness of the equilibria. By using the Lyapunov functional and LaSalle invariance principle, we obtained the conditions of global stabilities of the infection-free equilibrium, the immune-exhausted equilibrium and the endemic equilibrium. Numerical simulations are given to verify the analytical results.


Introduction
During recent decades there has been a lot of research regarding mathematical modelling of viruses dynamics via models of ordinary differential equations (ODE). Advances in immunology have lead us to better understand the interactions between populations of virus and the immune system, therefore several nonlinear sytems of ordinary differential equations (ODE's) has been proposed. Nowak and Bengham [1] study the model (1) y ′ (t) = βx(t)v(t) − ay(t) − py(t)z(t), (2) v ′ (t) = ky(t) − uv(t), (3) z ′ (t) = cy(t)z(t) − bz(t).
Where x(t) denotes the number of healthy cells, y(t) denotes the infected cells, v(t) denotes the number of matures viruses and z(t) denotes the number of CTL (cytotoxic T lymphocyte response) cells. Uninfected target cells are assumed to be generated at a constant rate s and die at rate d. Infection of target cells by free virus is assumed to occur at rate β. Infected cells die at rate a and are removed at rate p by the CTL immune response. New virus is produced from infected cells at rate k and dies at rate u. The average lifetime of uninfected cells, infected cells and free virus is thus given by 1/d, 1/a and 1/u, respectively. c denotes rate at which the CTL response is produced, and b denotes death rate of the CTL response, respectively, with all given constants positive.
Generally, the type of incidence function used in a model has an important role in modeling the dynamics of viruses. The most common, is the bilinear incidence rate βxv. However, this rate is not useful all the time. For instance, using bilinear incidence suggests that model can not describe the infection process of hepatitis B, where individuals with small liver are more resistant to infection than the ones with a bigger liver. Recently, works about models of infections by viruses have used the incidence function of type Beddington-DeAngelis and Crowley-Martin. In [2] the authors propose a model of infection by virus with Crowley-Martin functional response (1+ax(t))(1+bv(t)) . Li and Fu in [3] study the following system: They construct a Lyapunov functional to establish the global dynamics of the system. More recently, in [4], the authors consider the system They give also results of global stability as well as Hopf bifurcation results. Yang and Wei in [5] consider a more general incidence rate, they study the system giving some results about global stability in terms of the basic reproduction number and the immune response reproduction number. Here the f (v) is assumed to be a continuous function on v that belongs to (0, ∞) and satisfies f (0) = 0, f ′ (v) > 0 for all v greater or equal to 0 and f ′′ (v) < 0 for all v greater or equal to 0.
In [6] the authors consider a more general incidence rate f (x, y, v)v, where f is assumed to be continuously differentiable in the interior of R 3 + and satisfies the following hypotheses iii) ∂f ∂y (x, y, v) ≤ 0 and ∂f ∂v (x, y, v) ≤ 0 for all x ≥ 0, y ≥ 0, v ≥ 0. Finally, in [7] analyse an infection model by virus , with general incidence and immune response, which generalizes the systems by [2,6,8]: where f (x, y, v) is a continuous and differentiable function in the interior of R 3 + and satisfies the conditions Another viral infection model is the studied in [9] given by: Where x, y, v, z denotes the non infected cells, infected cells, virus and specific virus CTL at time t, respectively. Conditions on functions f i ,h, w, and g i are specified in [9]. A related work is [10].
Based on the discussion above, we will study a delayed viral infection model with general incidence rate and CTL immune response given bẏ The dynamics of uninfected cells, x, in absence of infection is governed by x ′ = n(x), where n(x) is the intrinsic growth rate of uninfected cells accounting for both productions and natural mortality, which is assumed to satisfy the following: is continuously differentiable, and existx > 0 such as that n(x) = 0, n(x) > 0 for x ∈ [0,x), and n(x) < 0 for x <x. Typical functions appearing in the literature are n(x) = s − dx and n(x) = s − dx + rx (1 − x/x max ).
All parameters are nonnegative and the distributions f i for i = 1, 2 are assumed to satisfy the following (see [9] and [11] ) : However, the uniqueness and Global stability results on the positive equilibrium require the following assumption.
Motivated by the work mentioned above we consider the following model In this paper, we will study the global dynamics model of (21), organization is as follows: in section 2, we prove the existence and uniqueness of the infection free equilibrium, the CTLinactivated infection equilibrium and the CTL-activated infection equilibrium. In section 3, the conditions that allow the global stability of each equilibrium are determined and proved. Section 4 provide several numerical simulations that shows the results obtained in section 3 and 4. Finally, in section 5 we summarize the results obtained, comparing them with the previous models studied in literature, and setting the guidelines for possible future work.

Positivity and Boundedness
For system (21) the suitable space is C 4 = C × C × C × C, where C is the Banach space of fading memory type ( [12]): where α > 0 is a constant and the norm of a φ ∈ C is defined as φ = sup θ≤0 |φ(θ)|e αθ . The nonnegative cone of C is defined as C + = C((−∞, 0], R + ). Proof. To see that x(t) is positive, we proceed by contradiction. Let t 1 the first value of time such that x(t 1 ) = 0. From the first equation of (21) we see that x ′ (t 1 ) = n(0) > 0 and x(t 1 ) = 0, therefore there exists ǫ > 0 such that x(t) < 0 for t ∈ (t 1 − ǫ, t 1 ), this leads to a contradiction. It follows that x(t) is always positive. With a similar argument we see that y(t), v(t) and z(t) are positive for t ≥ 0.
Theorem (1) implies that omega limit set of system (21) are contained in the following bounded feasible region: It can be verified that the region Γ is positively invariant with respect to model (21) and that the model is well posed.

Existence and uniqueness of equilibria of system
At any equilibrium we have The system (21) always has an infection free equilibrium E 0 = (x, 0, 0, 0). In addition to E 0 the system could have two types of Chronic infection Equilibria E 1 = (x 1 , y 1 , v 1 , 0) and E 2 = (x 2 , y 2 , v 2 , z 2 ) in Γ where the entries of E 1 and E 2 are strictly positives. The equilibria E 1 and E 2 are called CTL-inactivated infection equilibrium (CTL-IE) and CTL-activated infection equilibrium (CTL-AE), respectively.
We define the general reproduction number as which is the ratio of the per capita production and decay rates of mature viruses at an equilibrium (x, y, v, z) with z = 0. In particular, at the infection free equilibrium, E 0 , we denote R(x, 0, 0) by R 0 , representing the basic production number for viral infection: From assumption (H2) we have that ϕ 1 is invertible, so we can define from equation (29): Define also H(x) = n(x) − f (x,ŷ,v)v, we have H(0) = n(0) > 0 and H(x) = −f (x,ŷ,v)v, with f (x,ŷ,v) > f (0,ŷ,v) = 0 by (ii), so there exists ax ∈ (0,x) such that H(x) = 0. We denote: and refer it as the viral reproduction number. From assumptions on f it is easy to see that f (x, 0, 0) > f (x, y, v), for all y, v > 0, moreover for all x ∈ [0,x) we have f (x, y, v) < f (x, y, v), so: Particularly, R 0 > R 1 . The basic reproduction number for the CTL response is given by: For proof of the existence and uniqueness of equilibria, we require two additional assumption. First, we define the following sets: The following conditions are used to guarantee the uniqueness of the equilibria.
Theorem 2. Assume that i) − iii) and H 1 − H 4 are satisfied.
When x =x and y = v = z = 0 the equations (26)-(29)are satisfied, therefore E 0 = (x 0 , 0, 0, 0) is a steady state called the infection free equilibrium. To proof that it is unique when R 0 < 1, we look for the existence of a positive equilibrium.

Global stability
Let R 0 and R 1 be defined as in previous section.
Proof. Define a Lyapunov functional Calculating the derivative of V 1 along the positive solution of system (21), it follows thaṫ Using n(x 0 ) = 0 and simplifying, we get Using the following inequalities: We have that Since R 0 ≤ 1, we haveV 1 ≤ 0. Therefore the disease free E 1 is stable,V 1 = 0 if and only if x = x 0 , y = 0, v = 0, z = 0. So, the largest compact invariant set in {(x, y, v, z) :V 1 = 0} is just the singleton E 1 . From LaSalle invariance principle , we conclude that E 1 is globally asymptotically stable.
Using the history function from previous example we can plot the solution with Matlab. By theorem (4), E 1 is globally asymptotically stable as we can see in figure (2).
Finally, for β = 1, we have R 0 > 1 and R 1 = 6.088529, so there exists an infection free equilibrium E 0 and two equilibria By theorem (5) we have global stability of equilibrium E 2 . The solutions are showed in figure (3).
Example 3. In order to show that our model, generalizes the previous articles, we propose the following incidence function to show our results: this function satisfies Set the other functions and parameters as in example 1, then we obtain model: The infection free equilibrium is E 0 = (666.6666, 0, 0, 0) as in previous cases, with R 0 = 70.0722745β, so R 0 > 1 iff β > 0.01427097. Therefore, if we set β = 0.1 then we have R 0 > 1, R 1 = 4.9234560677357286281 and there exists two more equilibria points, 15