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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 586035, 6 pages http://dx.doi.org/10.1155/2014/586035 Research Article Dynamics Analysis of a Viral Infection Model with a General Standard Incidence Rate Yu Ji and Muxuan Zheng Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China Correspondence should be addressed to Yu Ji; Received 23 May 2014; Accepted 20 August 2014; Published 14 October 2014 Academic Editor: Peixuan Weng Copyright © 2014 Y. Ji and M. 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. 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 𝑅0 < 1, the infection-free equilibrium is globally asymptotically stable and the virus is cleared. Moreover, if 𝑅0 > 1, 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 𝑥 are assumed to become infected by free virions V at the bilinear rate 𝛽𝑥V, where 𝛽 is a positive constant rate. However, the basic reproduction number 𝑅0 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: 𝑥󸀠 = 𝜆 − 𝑑𝑥 − 𝛽 𝑦󸀠 = 𝛽 𝑥 V, 𝑥+𝑦 𝑥 V − 𝑎𝑦, 𝑥+𝑦 (1) V󸀠 = 𝑘𝑦 − 𝑢V. The basic reproduction number 𝑅0 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 𝑥𝑔(V) = 𝛽𝑥V(1 − 𝑐V) for measles outbreaks. Liu et al. [15] studied a nonlinear saturated mass action given by 𝛽𝑥(V𝑝 /(1 + 𝛼V𝑞 )), where 𝛽, 𝑝, 𝛼, and 𝑞 > 0. When 𝑝 = 𝑞 = 1, the nonlinear incidence rate becomes 𝛽𝑥(V/(1 + 𝛼V)), which has been frequently used in the viral model with saturation response. 2 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: 𝑥󸀠 = 𝜆 − 𝑑𝑥 − 𝛽 𝑦󸀠 = 𝛽 𝑥 𝑔 (V) , 𝑥+𝑦 𝑥 𝑔 (V) − 𝑎𝑦, 𝑥+𝑦 (2) V󸀠 = 𝑘𝑦 − 𝑢V, Further 𝑉1 (𝑡) ⩽ 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). Hence, 𝑉1 (𝑡) is bounded. Then we can conclude that 𝑥(𝑡), 𝑦(𝑡), and V(𝑡) are eventually bounded. Thus, there exists an 𝑀 > 0 such that 𝑥(𝑡) < 𝑀, 𝑦(𝑡) < 𝑀, V(𝑡) < 𝑀. This completes the proof. 𝐷 = {(𝑥, 𝑦, V) ∈ 𝑅+3 | 0 < 𝑥 (𝑡) ⩽ Theorem 1. There is an 𝑀 > 0, such that, for any positive solution (𝑥(𝑡), 𝑦(𝑡), V(𝑡)) of model (2), one has 𝑥(𝑡) < 𝑀, 𝑦(𝑡) < 𝑀, V(𝑡) < 𝑀. Proof. Let 𝑎 ⩽ 𝜆 − min {𝑑, , 𝑢} 𝑉1 (𝑡) . 2 2.2. Existence and Uniqueness of the Endemic Equilibrium. Obviously, 𝑄1 = (𝜆/𝑑, 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. Theorem 2. If 𝑅0 > 1, then the model (2) has a unique endemic equilibrium of the form 𝑄2 = (𝑥∗ , 𝑦∗ , V∗ ) with 0 < 𝑥∗ < 𝜆/𝑑, 𝑦∗ > 0 and V∗ > 0. Proof. At any equilibrium, the following equations hold: 𝜆 − 𝑑𝑥 − 𝛽𝑥 𝑔 (V) = 0, 𝑥+𝑦 𝛽𝑥 𝑔 (V) − 𝑎𝑦 = 0, 𝑥+𝑦 By the first and the second equations of (8), we have 𝑦 = (1/𝑎)(𝜆 − 𝑑𝑥). From the third equation, we get V = (𝑘/𝑢)𝑦. Now, we consider the following function 𝐹(𝑥) defined on the interval [0, 𝜆/𝑑]: (3) 1 𝑔 (V) − 𝑎𝑦, 𝑥+𝑦 1 where 𝑦 = (𝜆 − 𝑑𝑥) , 𝑎 𝑘 V = 𝑦. 𝑢 Therefore 𝐹󸀠 (𝑥) = 𝛽 (4) + (5) (8) 𝑘𝑦 − 𝑢V = 0. Denote ℎ = min{𝑑, 𝑎/2, 𝑢}; it follows that 𝑉1󸀠 (𝑡) ⩽ 𝜆 − ℎ𝑉1 (𝑡) . (7) If 𝑥(0) ⩽ 𝜆/𝑑, from the first equation of model (2), we have 𝑥(𝑡) ⩽ 𝜆/𝑑 when 𝑡 > 0. It is easy to see that 𝐷 is a positively invariant region for model (2). 𝐹 (𝑥) = 𝛽𝑥 Calculating the derivative of 𝑉1 along the solutions of model (2) gives 𝜆 , 𝑑 0 ⩽ 𝑦 (𝑡) , V (𝑡) ⩽ 𝑀} . 2.1. Positivity and Boundedness. The proof of positive solution is easy; we only show the boundedness of solution in the following. 𝑎 𝑎𝑢 V 𝑉1󸀠 (𝑡) = 𝜆 − 𝑑𝑥 − 𝑦 − 2 2𝑘 (6) Define where 𝑔(0) = 0, 𝑔󸀠 (V) > 0, and 𝑔󸀠󸀠 (V) ⩽ 0 when V ⩾ 0. Under this assumption, in the special case 𝑔(V) = V, the incidence rate means the standard incidence rate. If 𝑔(V) = V/(1 + 𝛼V), then that describes the model with the standard incidence rate and saturation response. The basic reproduction number of the model (2) is given by 𝑅0 = (𝛽𝑘/𝑎𝑢)𝑔󸀠 (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, 𝑅0 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. 𝑎 𝑉1 (𝑡) = 𝑥 (𝑡) + 𝑦 (𝑡) + V (𝑡) . 2𝑘 𝜆 𝜆 + (𝑉1 (0) − ) 𝑒−ℎ𝑡 . ℎ ℎ = 1 𝑑 1 (1 − ) 𝑔 (V) − 𝛽𝑥𝑔 (V) 2 𝑥+𝑦 𝑎 (𝑥 + 𝑦) 𝛽𝑥 󸀠 𝑘 𝑑 𝑑 𝑔 (V) (− ) − 𝑎 (− ) 𝑥+𝑦 𝑢 𝑎 𝑎 𝛽𝑔 (V (...truncated)