Abstract and Applied Analysis

Volume 2014 (2014), Article ID 586035, 6 pages

http://dx.doi.org/10.1155/2014/586035

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

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China

Received 23 May 2014; Accepted 20 August 2014; Published 14 October 2014

Academic Editor: Peixuan Weng

Copyright © 2014 Yu Ji and Muxuan Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The basic viral infection models, proposed by Nowak et al. and Perelson et al., respectively, have been widely used to describe viral infection such as HBV and HIV infection. However, the basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection, which seems not to be reasonable. In this paper, we formulate an amended model with a general standard incidence rate. The basic reproduction number of the amended model is independent of total cells of the host’s organ. When the basic reproduction number , the infection-free equilibrium is globally asymptotically stable and the virus is cleared. Moreover, if , then the endemic equilibrium is globally asymptotically stable and the virus persists in the host.

#### 1. 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 [5–13], such as HBV and HIV infection. In both of these two basic models, uninfected cells are assumed to become infected by free virions at the bilinear rate , where is a positive constant rate. However, the basic reproduction number 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 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 for measles outbreaks. Liu et al. [15] studied a nonlinear saturated mass action given by , where , , , and . When , the nonlinear incidence rate becomes , which has been frequently used in the viral model with saturation response.

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 , , and when . Under this assumption, in the special case , the incidence rate means the standard incidence rate. If , then that describes the model with the standard incidence rate and saturation response.

The basic reproduction number of the model (2) is given by , 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, 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.

#### 2. 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).

##### 2.1. Positivity and Boundedness

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

Theorem 1. *There is an , such that, for any positive solution of model (2), one has , , .*

*Proof. *Let
Calculating the derivative of along the solutions of model (2) gives
Denote ; it follows that
Further
Hence, is bounded. Then we can conclude that , , and are eventually bounded. Thus, there exists an such that , , . This completes the proof.

Define If , from the first equation of model (2), we have when . It is easy to see that is a positively invariant region for model (2).

##### 2.2. Existence and Uniqueness of the Endemic Equilibrium

Obviously, 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.

Theorem 2. *If , then the model (2) has a unique endemic equilibrium of the form with , and .*

*Proof. *At any equilibrium, the following equations hold:
By the first and the second equations of (8), we have . From the third equation, we get .

Now, we consider the following function defined on the interval :
Therefore
Clearly, , . Notice that . Hence, we get when . Therefore, there exists at least one positive equilibrium with and .

Next, we will proof the uniqueness of the endemic equilibrium. Observe the terms − in . Since , at any positive equilibrium from (8), we have
Since , we get ; that is

Consequently, we deduce that at any positive equilibrium. Suppose there are at least two positive equilibria of in ; then there must be at some equilibrium, which is a contradiction. Therefore, if , there exists a unique endemic equilibrium with and . This completes the proof.

#### 3. 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

First of all, we will study the stability of the infection-free equilibrium .

Theorem 3. *If , then is globally asymptotically stable and becomes unstable when .*

*Proof. *For , the characteristic equation of (13) is given by
Note that and when ; thus all eigenvalues of (14) have negative real parts if and at least one eigenvalue becomes positive when . Hence, is locally asymptotically stable if and becomes unstable if .

Consider the Lyapunov function
Calculating the derivative of along the solutions of the model (2) gives
Since , then and only if . By the Lyapunov-Lasalle Theorem, solutions in approach the largest positively invariant subset of the set where . Thus, all solutions in the set approach the infection-free equilibrium . This completes the proof.

##### 3.2. Stability of the Endemic Equilibrium

When , there physically exists another steady state . We now consider the local stability of the endemic steady state when .

Theorem 4. *If , then is locally asymptotically stable.*

*Proof. *The Jacobian matrix of (13) at becomes
The characteristic equation associated with is given by
where
From the inequality of (12), we get , . By Routh-Hurwitz criterion, we are only to show . In fact, all the terms of , and are nonnegative, and all the three terms in appear in the expansion of . Hence, is locally asymptotically stable when .

Furthermore, to analyze the global asymptotic stability of , we introduce the results of Theorem 2.5 in [16].

Lemma 5. *Assume that*(1)*there exists a compact absorbing set K D and the system (2) has a unique equilibrium in D;*(2)*the system (2) satisfies the Poincaré-Bendixson property;*(3)*For each periodic solution to (2) with , the second additive compound matrix of the Jacobian (17) is asymptotically stable;*(4)*.**Then the unique equilibrium is globally asymptotically stable in D.*

Therefore, we only need to prove that model (2) at the satisfies this lemma when . Then we have the following conclusion.

Theorem 6. *If , then 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 is a unique steady state at the boundary of . And the uniform persistence of (2) is equivalent to the instability of the [18]. Since the infected-free steady state is unstable when , 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 , 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.

Next, we will show that model (2) satisfies assumption (3) of Lemma 5. The second additive compound matrix of the Jacobian matrix (13) is
The second compound system (2) along a periodic solution is

To show that the model (21) is asymptotically stable, consider a Lyapunov function
From the uniform persistence, there exists a constant such that

The right derivative of along a solution to (21) and can be estimated as follows:

Then, we will get
Therefore,
where
Notice that , , where . We get
Denote ; we get
By inequality (26) and Gronwall inequality, we have
This implies that when . And from inequality (23), we get when . As a result, the second additive compound matrix of the Jacobian (17) is asymptotically stable.

At last, we will prove that model (2) satisfies assumption (4) of Lemma 5. Let be the Jacobian matrix of (2) at . Then

From the inequality of (12), we can conclude that all the terms of are negative, so . This verifies assumption (4) of Lemma 5. Hence, is globally asymptotically stable if by Lemma 5. This completes the proof.

#### 4. 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 , the infection-free equilibrium is globally asymptotically stable and the virus is cleared. Moreover, if , then the endemic equilibrium is globally asymptotically stable and the virus persists in the host.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research is jointly supported by the Research Foundation for Youth Scholars of Beijing Technology and Business University (QNJJ2013-24) and the General Program of Science and Technology Development Project of Beijing Municipal Education Commission (KM201310011006).

#### References

- P. de Leenheer and H. L. Smith, “Virus dynamics: a global analysis,”
*SIAM Journal on Applied Mathematics*, vol. 63, no. 4, pp. 1313–1327, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Lewin, T. Walters, and S. Locarnini, “Hepatitis B treatment: rational combination chemotherapy based on viral kinetic and animal model studies,”
*Antiviral Research*, vol. 55, no. 3, pp. 381–396, 2002. View at Publisher · View at Google Scholar · View at Scopus - M. A. Nowak and R. M. May,
*Virus Dynamics*, Oxford University Press, Oxford, UK, 2000. View at MathSciNet - A. S. Perelson and P. W. Nelson, “Mathematical analysis of HIV-1 dynamics in vivo,”
*SIAM Review*, vol. 41, no. 1, pp. 3–44, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Min, Y. Su, and Y. Kuang, “Mathematical analysis of a basic virus infection model with application to HBV infection,”
*The Rocky Mountain Journal of Mathematics*, vol. 38, no. 5, pp. 1573–1585, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. A. Gourley, Y. Kuang, and J. D. Nagy, “Dynamics of a delay differential equation model of hepatitis B virus infection,”
*Journal of Biological Dynamics*, vol. 2, no. 2, pp. 140–153, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - T. Wang, Z. Hu, F. Liao, and W. Ma, “Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity,”
*Mathematics and Computers in Simulation*, vol. 89, pp. 13–22, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. M. Cai and X. Z. Li, “Analysis of a SEIV epidemic model with a nonlinear incidence rate,”
*Applied Mathematical Modelling*, vol. 33, no. 7, pp. 2919–2926, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. Hattaf, N. Yousfi, and A. Tridane, “Mathematical analysis of a virus dynamics model with general incidence rate and cure rate,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 4, pp. 1866–1872, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. P. Wang and R. Xu, “Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 2, pp. 964–978, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. Wang, A. Fan, and A. Torres, “Global properties of an improved hepatitis B virus model,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 4, pp. 3131–3138, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Song and A. U. Neumann, “Global stability and periodic solution of the viral dynamics,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 1, pp. 281–297, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. X. Hu, J. J. Zhang, H. Wang, W. Ma, and F. Liao, “Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment,”
*Applied Mathematical Modelling*, vol. 38, no. 2, pp. 524–534, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. A. Yorke and W. P. London, “Recurrent outbreaks of measles, chickenpox and mumps. II. Systematic differences in contact rates and stochastic effects,”
*The American Journal of Epidemiology*, vol. 98, no. 6, pp. 469–482, 1973. View at Google Scholar · View at Scopus - W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,”
*Journal of Mathematical Biology*, vol. 25, no. 4, pp. 359–380, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Y. Li and L. C. Wang, “Global stability in some SEIR epidemic models,” in
*Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory*, vol. 126 of*The IMA Volumes in Mathematics and Its Applications*, pp. 295–311, Springer, New York, NY, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - G. Butler and P. Waltman, “Persistence in dynamical systems,”
*Journal of Differential Equations*, vol. 63, no. 2, pp. 255–263, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “Global dynamics of a SEIR model with varying total population size,”
*Mathematical Biosciences*, vol. 160, no. 2, pp. 191–213, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus