Effect of Time Delay and Antibodies on HCV Dynamics with Cure Rate and Two Routes of Infection

In this paper we propose and analyze an HCV dynamics model taking into consideration the cure of infected hepatocytes and antibody immune response. We incorporate both virus-to-cell and cell-to-cell transmissions into the model. We incorporate a distributed-time delay to describe the time between the HCV or infected cell contacts an uninfected hepatocyte and the emission of new active HCV. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive two threshold parameters which fully determine the existence and stability of the three steady states of the model. Using Lyapunov functionals, we established the global stability of the steady states. The theoretical results are confirmed by numerical simulations.


Introduction
Hepatitis C virus is considered one of the dangerous human viruses that infects the liver and causes the lever cirrhosis.Mathematical modeling and analysis of within-host HCV dynamics have been studied by many authors (see e.g. [1]- [12]).These works can help researchers for better understanding the HCV dynamical behavior and providing new suggestions for clinical treatment.Immune response plays an important role in controlling the dynamics of several viruses (see e.g.[13] [14] [15] [16] [17]).Cytotoxic T Lymphocyte (CTL) and antibodies play a central role of immune response.CTL cells attack and kill the where, s, y, p and z represent the concentration of uninfected hepatocytes, infected hepatocytes, HCV particles and antibodies, respectively.The uninfected hepatocytes are generated at a constant rate β, die at rate ˆs δ , where δ is the natural death rate constant.The infection rate due to both virus-to-cell and cell-to-cell transmissions is given by 1 2 sp sy α α

+
, where 1 α and 2 α are con- stants.The infected hepatocytes die at rate εy and cure at rate ρy, where ε and ρ are constants.Constant m is the generation rate of the HCV from infected hepatocytes.Antibodies attack the HCV at rate qzp , proliferate at rate rzp and die at rate μz, where q, r and μ are constants.
It is assumed in model ( 1)-( 4) that, the hepatocytes can produce HCV particles once they are contacted by HCV or infected cells.However, there is a time period from the moment of the uninfected hepatocytes that are contacted by the HCV or infected cells and the moment of producing new active HCV particles [10] [11].
The aim of this paper is to study the qualitative behavior of an HCV dynamics model with antibody immune response.We have incorporated distributed time delay and both virus-to-cell and cell-to-cell transmissions.We derive two threshold parameters and establish the global stability of the three steady states of the model using Lyapunov method.

The Model
We propose the following HCV dynamics model with distributed time delay:

ˆ, s t s t s t p t s t y t y t
We assume that, the HCV or infected cell contacts an uninfected hepatocyte at time t τ − , the cell becomes infected at time t, where τ is a distributed para- ( ) ( ) where ς and h are positive constants.Let us denote ( ) ( ) Let the initial conditions for system (5)-( 8) be given as: where C is the Banach space of continuous functions mapping the interval [ ] Then, the uniqueness of the solution for 0 t > is guaranteed [42].

Basic Properties
In this subsection, we investigate the nonnegativity and boundedness of solutions.
Next, we establish the boundedness of the model's solutions.The nonnegativity of the model's solution implies that is positively invariant with respect to system (5)-(8).

Global Stability
The following theorems investigate the global stability of the steady states of system ( 5)-( 8).Let us define the function , , , , , , s y p z s t y t p t z t = .Theorem 1. Suppose that 0 1 R ≤ , then the infection-free steady state 0 Π is globally asymptotically stable (GAS).
Proof.Constructing a Lyapunov functional .
For each element of 0 Γ we have ( ) 0 y t = , then ( ) and Equation ( 6) we get ( ) ( ) Proof.Let us define a function , , , L s y p z as: along the trajectories of system ( 5)-( 8), we get Collecting terms of Equation ( 19), we get Applying condition of equilibrum 1 Π : Simplify Equation ( 20) and let 1 i = , in Equation( 21 Equation ( 22) can be rewrite as: We note that From Lemma 2 we have 1 2 p p ≤ , then, , s y and 1 0 p > , where , , s s y y p p = = = and 0 z = .Thus, the global asymptotic stability of 1 Π follows from LIP when 1 1 z R ≤ , and , , , L s y p z as: Collecting terms of Equation ( 24) and applying the equilibrium conditions for

. y p qp z mF
Equation ( 25) can be simplified as: We note that,

Numerical Simulations
This section is devoted to performing some numerical simulations for model ( 5)-( 8).Let us choose ( ) ( ) Π exists and it is globally asymptotically stable.From Figure 3, we find that the numerical results agree with the theoretical one presented in Theorem 3.For all initial conditions the states reach the steady state .This case corresponds to a chronic HCV infection with active antibody immune response.
Case II: Effect of the time delays on the free HCV particles dynamics: Let us take the initial conditions (Initial-2).We choose the values 1 0.001 α = and 0.01 r = . we assume that 1 τ τ * = .Table 2 contains the values of all threshold parameters and equilibria of system ( 26)-( 29) with different values of τ * .
From Table 2 we can see that, the values of 0 R , and 1 z R are decreased as τ * is increased.Moreover, τ * has a significant effect on the stability of steady states of the system.Table 2 and Figure 4 show that a high value of τ * Table 2.The values of the threshold parameters and the equilibria of system ( 26)-( 29) with different values of τ * .decreases the concentration of infected hepatocytes, free HCV particles, antibodies, and increases the population of uninfected hepatocytes.Therefore, the steady states of the system will eventually stabilized around the healthy state 0 Π .

1 µ
hepatocyte during the time delay period, where is a con- stant.( ) ρ τ is a probability distribution function satisfying ( ) 0 ρ τ > and define the basic reproduction number for the humoral immune response Hum R which comes from the limiting (linearized)

17 )
Journal of Applied Mathematics and Physics

1 ΠFigure 1 .Figure 2 .
Figure 1.The simulation of trajectories of system (26)-(29) in case of R 0 ≤ 1.(a) The concentration of uninfected hepatocytes; (b) The concentration of infected hepatocytes; (c) The concentration of free HCV particles; (d) The concentration of antibodies.

Figure 3 .
Figure 3.The simulation of trajectories of system (26)-(29) in case of 1 1 z R > .(a) The concentration of uninfected hepatocytes; (b) The concentration of infected hepatocytes; (c) The concentration of free HCV particles; (d) The concentration of antibodies.

Figure 4 .
Figure 4.The effect of delays on the behaviour of all trajectories of system (26)-(29).(a) The concentration of uninfected hepatocytes; (b) The concentration of infected hepatocytes; (c) The concentration of free HCV particles; (d) The concentration of antibodies. )