Dynamics Analysis of a Viral Infection Model with a General Standard Incidence Rate

and Applied Analysis 3


Introduction
Mathematical models of viral infection have played a significant role in the understanding of the disease in vivo [1].Analysis of the viral dynamics of a proper model can not only provide important quantitative insights into the pathogenesis, but also lead to design treatment strategies which would more effectively bring the infection under control [2].
The basic models of within-host viral infection, proposed by Nowak and May [3] and Perelson and Nelson [4], have been widely used in the studies of viral infection [5][6][7][8][9][10][11][12][13], such as HBV and HIV infection.In both of these two basic models, uninfected cells  are assumed to become infected by free virions V at the bilinear rate V, where  is a positive constant rate.However, the basic reproduction number  0 of these two models is proportional to the number of total cells of the host's organ prior to the infection.This implies that an individual with a smaller organ maybe more resistant to virus infection than an individual with a larger one.Hence, Min et al. [5] proposed the following amended Nowak and May's model with a standard incidence rate to describe the hepatitis B virus infection: The basic reproduction number  0 of model ( 1) is independent of the number of total cells of the host's organ.
In the modelling the viral infection of disease, the incidence rate, which is the rate of new infections, plays an important role in describing the viral dynamics.Bilinear and standard incidence rate are the most common incidence rates in virus infection models.However, there are still some other nonlinear incidence rates to describe disease infections.Yorke and London [14] investigated an incidence rate (V) = V(1 − V) for measles outbreaks.Liu et al. [15] studied a nonlinear saturated mass action given by (V  /(1 + V  )), where , , , and  > 0. When  =  = 1, the nonlinear incidence rate becomes (V/(1 + V)), which has been frequently used in the viral model with saturation response.

Abstract and Applied Analysis
In this paper, motivated by the above models, we formulate an amended viral infection model with a general standard incidence rate, which is described as follows: where (0) = 0,   (V) > 0, and   (V) ⩽ 0 when V ⩾ 0. Under this assumption, in the special case (V) = V, the incidence rate means the standard incidence rate.If (V) = V/(1 + V), then that describes the model with the standard incidence rate and saturation response.
The basic reproduction number of the model ( 2) is given by  0 = (/)  (0), which describes the average number of secondary infections produced by a single infected cell during the period of infection when all cells are uninfected.Clearly,  0 of model ( 2) is also independent of the number of the host's organ.The main purpose of this paper is to study the virus dynamics of model (2).
The rest of this paper is organized as follows: Section 2 studies the existence and uniqueness of equilibria of model (2).The stability of the infection-free equilibrium and the endemic equilibrium is analyzed in Section 3. Finally, concluding remarks are given in Section 4.

The Existence and Uniqueness of Equilibria
Before the analysis of the existence and uniqueness of equilibria, we will show the positivity and boundedness of solutions of model (2).

Positivity and Boundedness.
The proof of positive solution is easy; we only show the boundedness of solution in the following.

Existence and Uniqueness of the Endemic Equilibrium.
Obviously,  1 = (/, 0, 0) is the infection-free equilibrium of model ( 2), which represents the extinction of the free virus.As for the existence and uniqueness of the positive equilibrium, we have the following theorem.

Stability Analysis of Equilibria
In this section, we will analyze the stability of those two steady states.The Jacobian matrix of the vector field corresponding to model ( 2) is ) . ( 3.1.Stability of the Infection-Free Equilibrium  1 .First of all, we will study the stability of the infection-free equilibrium  1 .
Consider the Lyapunov function Calculating the derivative of  2 along the solutions of the model (2) gives Since  0 < 1, then   2 () ⩽ 0 and   2 () = 0 only if V = 0.By the Lyapunov-Lasalle Theorem, solutions in  approach the largest positively invariant subset of the set  where   2 () = 0. Thus, all solutions in the set  approach the infection-free equilibrium  1 .This completes the proof.
Proof.The Jacobian matrix of (13) at  2 becomes The characteristic equation associated with ( 2 ) is given by where From the inequality of ( 12), we get  1 > 0,  2 > 0,  3 > 0. By Routh-Hurwitz criterion, we are only to show  1  2 >  3 .In fact, all the terms of  1 ,  2 , and  3 are nonnegative, and all the three terms in  3 appear in the expansion of  1  2 .Hence,  2 is locally asymptotically stable when  0 > 1.
Furthermore, to analyze the global asymptotic stability of  2 , we introduce the results of Theorem 2.5 in [16].
Then the unique equilibrium  is globally asymptotically stable in D.
Therefore, we only need to prove that model (2) at the  2 satisfies this lemma when  0 > 1.Then we have the following conclusion.Theorem 6.If  0 > 1, then  2 is globally asymptotically stable.
Proof.Firstly, the assumption (1) of Lemma 5 is equivalent to the uniform persistence of the model ( 2) [17].Clearly, the  1 is a unique steady state at the boundary of .And the uniform persistence of ( 2) is equivalent to the instability of the  1 [18].Since the infected-free steady state  1 is unstable when  0 > 1, the assumption (1) of Lemma 5 holds.
Secondly, in order to verify assumption (2) of Lemma 5, we only need to show that model ( 2) is competitive in the convex region .Taking the diagonal matrix  = diag(1, −1, 1), it is easy to verify that the matrix  has nonpositive off-diagonal elements, where  is the Jacobian matrix (17).Hence, model ( 2) is competitive.So model ( 2) satisfies the Poincaré-Bendixson property.

Concluding Remarks
In this paper, we consider a viral infection model with a general standard incidence rate.The basic reproduction number of model ( 2) is independent of the number of the host's organ prior to the infection, which avoids the emergence of the unreasonable situation for the basic viral models of Nowak and May and Perelson and Nelson.This general incidence rate represents a variety of possible incidences such as saturation response and standard incidence.The existence and uniqueness of the positive equilibrium of model (2) have been proved in this paper.We also show the positivity and boundedness of solutions of model (2).The global stability of the infection-free equilibrium and endemic equilibrium has been analyzed, respectively.When the basic reproduction number  0 < 1, the infection-free equilibrium is globally asymptotically stable and the virus is cleared.Moreover, if  0 > 1, then the endemic equilibrium is globally asymptotically stable and the virus persists in the host.