Dynamics of a Viral Infection Model with General Contact Rate between Susceptible Cells and Virus Particles

and Applied Analysis 3 3. Local Stability In this section, we study the local stability of each of feasible equilibria of system (2) by analyzing the corresponding characteristic equations, respectively. The Jacobian matrix J of (2) at (x, y, V) is


Introduction
Viral infection within-host, such as hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV) infections, is a complicated kinetic process, and mathematical model is always important, which can give a hand to understand the complexity between the responses of the body and variant conditions [1][2][3][4][5][6].
The basic viral infection model contains three variables, susceptible host cells (), infected host cells (), and free virus particles (V), which can be formulated by the following differential equations [7,8]: in which susceptible host cells are produced at a constant rate, , die at the rate of , and become infected with the rate of V.Infected host cells are produced at the rate of V and die at the rate of .Free virus particles are released from infected host cells at the rate of  and die at the rate of V.It is assumed that parameters , , , , , and  are all positive constants.
Note that there is an assumption that the infection term is based on the mass-action principle, which means that there is a constant contact rate () between susceptible host cells and virus particles in (1).However, many experiments of microparasitic infections suggest the infection rate may be a nonlinear relationship [3,[9][10][11], such as dose-dependent infection rate.Thus, to meet more biological practice, we replace the constant contact rate () with a general contact rate ((V)) between susceptible cells and virus particles and obtain the following modified viral infection model: where the contact rate function (V) satisfy the following assumption (H1): (H1) (V) : R + → R + , continuous and differentiable, (0) = ,   (V) < 0 and (∞) = 0.
The primary goal of this paper is to carry out a mathematical analysis of system (2) and predict whether the infection 2 Abstract and Applied Analysis disappears or survives.The organization of this paper is as follows.In the next section, some preliminary results are given, including the dissipativity of system (2), the definition of basic reproduction number of the virus, and the existence of the disease-free equilibrium and endemic equilibrium.In Section 3, by analyzing the corresponding characteristic equations, we study the local stability of the equilibria.In Section 4, by using suitable Lyapunov function and LaSalle's invariance principle [12], we first prove that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable.Then using Theorem 4.6 in [13], we obtain the uniform persistence of (2) if the basic reproduction number is greater than unity.A brief discussion is given in Section 5 to conclude this work.

Preliminary Results
In this section, we first show that all solutions of system (2) are positive and ultimately bounded.Then the existence of feasible equilibria is given under the condition of basic reproduction number of the virus.
Because of the biological meaning of the components ((), (), V()), we focus on the model in the first octant of R 3 and consider system (2) with initial conditions The following result shows that system (2) is dissipative.
Theorem 1.Under the initial conditions (3), all solutions of system (2) are positive for  > 0 and there exists a constant  > 0, such that all solutions satisfy () < , () < , and V() <  for all sufficiently large .
Note that a free virus particle has an average lifetime of 1/ and parameter  is the burst size, which means the total number of virions produced by an infected cell during its life span.Thus, at the beginning of the infectious process, the average number of newly virus particles generated from one virus particle, which is the basic reproduction number of virus by [14,15], can be defined as Now, we begin to find the equilibria of model ( 2) by the following algebraic system Solving the third algebraic equation of (7), we can obtain  = V/.By combining this equality with the second equation of (7), we have  = /(V) or V = 0.When V = 0, it is easy to have  = 0 and  = / by the third and first equations of (7); that is, system (2) always has a diseasefree equilibrium state, denoted as  0 = (/, 0, 0).If V ̸ = 0, substituting  = /(V) in the first equation of (7), we have Note that Thus, if  1 (0) >  2 (0), that is,  0 > 1, there is a unique positive root for (8).We summarize the above analyses in the following result.

Local Stability
In this section, we study the local stability of each of feasible equilibria of system (2) by analyzing the corresponding characteristic equations, respectively.The Jacobian matrix  of (2) at (, , V) is At disease-free equilibrium  0 , Clearly, the determinant of the lower right-hand 2 × 2 matrix is positive and its trace is negative only if  0 < 1, so its eigenvalues have negative real parts in this case.Thus,  0 is locally asymptotically stable if and only if  0 < 1.
When  0 > 1, the endemic equilibrium  1 exists, and the Jacobian matrix at  1 is The characteristic equation of ( 12) is given by in which Here, we used  * (V * ) = / and the assumption (H1); that is,   (V) < 0.
Because  1 and  3 are both positive, by Routh-Hurwitz criterion,  1 is locally asymptotically stable if and only if  1  2 −  3 > 0. After a simple algebraic calculation, we have that is positive because   (V) < 0. Thus,  1 is locally asymptotically stable if and only if  0 > 1.
We summarize the above results and Proposition 2 in the following theorem.Theorem 3. If  0 < 1, then only disease-free equilibrium  0 exists and is locally asymptotically stable.When  0 > 1,  0 is unstable and the endemic equilibrium  1 appears and is locally asymptotically stable.

Global Stability and Disease Persistence
For the global stability of the equilibria, we first have the following.
Theorem 4. The disease-free equilibrium  0 is globally asymptotically stable if only  0 exists; that is,  0 < 1.
Proof.The result follows from an application of Theorem 4.6 in [13], with  1 = int(R 3 + ) and  2 = bd(R 3 + ).Since the proof is similar to that of Lemma 3.5 in [16], here we only sketch the modifications that  0 is a weak repeller for  1 .