A Viral Infection Model with a Nonlinear Infection Rate

  • Yumei Yu1,

    Affiliated with

    • JuanJ Nieto2,

      Affiliated with

      • Angela Torres3 and

        Affiliated with

        • Kaifa Wang4Email author

          Affiliated with

          Boundary Value Problems20092009:958016

          DOI: 10.1155/2009/958016

          Received: 28 February 2009

          Accepted: 27 May 2009

          Published: 29 June 2009

          Abstract

          A viral infection model with a nonlinear infection rate is constructed based on empirical evidences. Qualitative analysis shows that there is a degenerate singular infection equilibrium. Furthermore, bifurcation of cusp-type with codimension two (i.e., Bogdanov-Takens bifurcation) is confirmed under appropriate conditions. As a result, the rich dynamical behaviors indicate that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus.

          1. Introduction

          Mathematical models can provide insights into the dynamics of viral load in vivo. A basic viral infection model [1] has been widely used for studying the dynamics of infectious agents such as hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV), which has the following forms:
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ1_HTML.gif
          (1.1)

          where susceptible cells ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq1_HTML.gif ) are produced at a constant rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq2_HTML.gif , die at a density-dependent rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq3_HTML.gif , and become infected with a rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq4_HTML.gif ; infected cells ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq5_HTML.gif ) are produced at rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq6_HTML.gif and die at a density-dependent rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq7_HTML.gif ; free virus particles ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq8_HTML.gif ) are released from infected cells at the rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq9_HTML.gif and die at a rate http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq10_HTML.gif . Recently, there have been many papers on virus dynamics within-host in different aspects based on the (1.1). For example, the influences of spatial structures on virus dynamics have been considered, and the existence of traveling waves is established via the geometric singular perturbation method [2]. For more literature, we list [3, 4] and references cited therein.

          Usually, there is a plausible assumption that the amount of free virus is simply proportional to the number of infected cells because the dynamics of the virus is substantially faster than that of the infected cells, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq11_HTML.gif . Thus, the number of infected cells http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq12_HTML.gif can also be considered as a measure of virus load http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq13_HTML.gif (e.g., see [57]). As a result, the model (1.1) is reduced to
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ2_HTML.gif
          (1.2)

          As for this model, it is easy to see that the basic reproduction number of virus is given by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq14_HTML.gif , which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process. Furthermore, we know that the infection-free equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq15_HTML.gif is globally asymptotically stable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq16_HTML.gif , and so is the infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq17_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq18_HTML.gif .

          Note that both infection terms in (1.1) and (1.2) are based on the mass-action principle (Perelson and Nelson [8]); that is, the infection rate per susceptible cell and per virus is a constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq19_HTML.gif . However, infection experiments of Ebert et al. [9] and McLean and Bostock [10] suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose and is usually sigmoidal in shape. Thus, as Regoes et al. [11], we take the nonlinear infection rate into account by relaxing the mass-action assumption that is made in (1.2) and obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ3_HTML.gif
          (1.3)
          where the infection rate per susceptible cell, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq20_HTML.gif , is a sigmoidal function of the virus (parasite) concentration because the number of infected cells http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq21_HTML.gif can also be considered as a measure of virus load (e.g., see [57]), which is represented in the following form:
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ4_HTML.gif
          (1.4)

          Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq22_HTML.gif denotes the infectious dose at which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq23_HTML.gif of the susceptible cells are infected, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq24_HTML.gif measures the slope of the sigmoidal curve at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq25_HTML.gif and approximates the average number of virus that enters a single host cell at the begin stage of invasion, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq26_HTML.gif measures the infection force of the virus, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq27_HTML.gif measures the inhibition effect from the behavioral change of the susceptible cells when their number increases or from the production of immune response which depends on the infected cells.

          In fact, many investigators have introduced different functional responses into related equations for epidemiological modeling, of which we list [1217] and references cited therein. However, a few studies have considered the influences of nonlinear infection rate on virus dynamics. When the parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq28_HTML.gif , [18, 19] considered a viral mathematical model with the nonlinear infection rate and time delay. Furthermore, some different types of nonlinear functional responses, in particular of the form http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq29_HTML.gif or Holling-type functional response, were investigated in [2023].

          Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq30_HTML.gif in (1.4). To simplify the study, we fix the slope http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq31_HTML.gif in the present paper, and system (1.3) becomes
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ5_HTML.gif
          (1.5)
          To be concise in notations, rescale (1.5) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq32_HTML.gif . For simplicity, we still use variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq33_HTML.gif instead of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq34_HTML.gif and obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ6_HTML.gif
          (1.6)

          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq35_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq36_HTML.gif is the average life time of susceptible cells and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq37_HTML.gif is the average life-time of infected cells. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq38_HTML.gif is always valid by means of biological detection. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq39_HTML.gif , the virus does not kill infected cells. Therefore, the virus is non cytopathic in vivo. However, when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq40_HTML.gif , which means that the virus kills infected cells before its average life time, the virus is cytopathic in vivo.

          The main purpose of this paper is to study the effect of the nonlinear infection rate on the dynamics of (1.6). We will perform a qualitative analysis and derive the Allee-type dynamics which result from the appearance of bistable states or saddle-node state in (1.6). The bifurcation analysis indicates that (1.6) undergoes a Bogdanov-Takens bifurcation at the degenerate singular infection equilibrium which includes a saddle-node bifurcation, a Hopf bifurcation, and a homoclinic bifurcation. Thus, the nonlinear infection rate can induce the complex dynamic behaviors in the viral infection model.

          The organization of the paper is as follows. In Section 2, the qualitative analysis of system (1.6) is performed, and the stability of the equilibria is obtained. The results indicate that (1.6) can display an Allee effect. Section 3 gives the bifurcation analysis, which indicates that the dynamics of (1.6) is more complex than that of (1.1) and (1.2). Finally, a brief discussion on the direct biological implications of the results is given in Section 4.

          2. Qualitative Analysis

          Since we are interested in virus pathogenesis and not initial processes of infection, we assume that the initial data for the system (1.6) are such that
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ7_HTML.gif
          (2.1)
          The objective of this section is to perform a qualitative analysis of system (1.6) and derive the Allee-type dynamics. Clearly, the solutions of system (1.6) with positive initial values are positive and bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq41_HTML.gif , and note that (1.6) has one and only one infection-free equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq42_HTML.gif . Then by using the formula of a basic reproduction number for the compartmental models in van den Driessche and Watmough [24], we know that the basic reproduction number of virus of (1.6) is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ8_HTML.gif
          (2.2)

          which describes the average number of newly infected cells generated from one infected cell at the beginning of the infectious process as zero. Although it is zero, we will show that the virus can still persist in host.

          We start by studying the equilibria of (1.6). Obviously, the infection-free equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq43_HTML.gif always exists and is a stable hyperbolic node because the corresponding characteristic equation is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq44_HTML.gif .

          In order to find the positive (infection) equilibria, set
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ9_HTML.gif
          (2.3)
          then we have the equation
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ10_HTML.gif
          (2.4)

          Based on (2.4), we can obtain that

          (i) there is no infection equilibria if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq45_HTML.gif ;

          (ii) there is a unique infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq46_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq47_HTML.gif ;

          (iii) there are two infection equilibria http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq49_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq50_HTML.gif .

          Here,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ11_HTML.gif
          (2.5)
          Thus, the surface
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ12_HTML.gif
          (2.6)

          is a Saddle-Node bifurcation surface, that is, on one side of the surface http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq51_HTML.gif system (1.6) has not any positive equilibria; on the surface http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq52_HTML.gif system (1.6) has only one positive equilibrium; on the other side of the surface http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq53_HTML.gif system (1.6) has two positive equilibria. The detailed results will follow.

          Next, we determine the stability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq55_HTML.gif . The Jacobian matrix at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq56_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ13_HTML.gif
          (2.7)
          After some calculations, we have
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ14_HTML.gif
          (2.8)

          Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq57_HTML.gif in this case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq58_HTML.gif is valid. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq59_HTML.gif and the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq60_HTML.gif is a saddle.

          The Jacobian matrix at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq61_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ15_HTML.gif
          (2.9)

          By a similar argument as above, we can obtain that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq62_HTML.gif . Thus, the equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq63_HTML.gif is a node, or a focus, or a center.

          For the sake of simplicity, we denote
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ16_HTML.gif
          (2.10)

          We have the following results on the stability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq64_HTML.gif .

          Theorem 2.1.

          Suppose that equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq65_HTML.gif exists; that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq66_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq67_HTML.gif is always stable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq68_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq69_HTML.gif , we have

          (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq70_HTML.gif is stable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq71_HTML.gif ;

          (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq72_HTML.gif is unstable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq73_HTML.gif ;

          (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq74_HTML.gif is a linear center if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq75_HTML.gif .

          Proof.

          After some calculations, the matrix trace of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq76_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ17_HTML.gif
          (2.11)
          and its sign is determined by
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ18_HTML.gif
          (2.12)
          Note that
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ19_HTML.gif
          (2.13)

          which means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq77_HTML.gif is a monotone decreasing function of variable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq78_HTML.gif .

          Clearly,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ20_HTML.gif
          (2.14)
          Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq79_HTML.gif implies that
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ21_HTML.gif
          (2.15)
          Squaring (2.15) we find that
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ22_HTML.gif
          (2.16)
          Thus,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ23_HTML.gif
          (2.17)

          This means that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq80_HTML.gif . Thus, under the condition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq81_HTML.gif and the sign of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq83_HTML.gif is always valid if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq84_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq86_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq87_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq88_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq89_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq90_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq91_HTML.gif .

          For (1.6), its asymptotic behavior is determined by the stability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq92_HTML.gif if it does not have a limit cycle. Next, we begin to consider the nonexistence of limit cycle in (1.6).

          Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq93_HTML.gif is a saddle and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq94_HTML.gif is a node, a focus, or a center. A limit cycle of (1.6) must include http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq95_HTML.gif and does not include http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq96_HTML.gif . Since the flow of (1.6) moves toward down on the line where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq98_HTML.gif and moves towards up on the line where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq100_HTML.gif , it is easy to see that any potential limit cycle of (1.6) must lie in the region where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq101_HTML.gif . Take a Dulac function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq102_HTML.gif , and denote the right-hand sides of (1.6) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq104_HTML.gif , respectively. We have
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ24_HTML.gif
          (2.18)

          which is negative if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq105_HTML.gif . Hence , we can obtain the following result.

          Theorem 2.2.

          There is no limit cycle in (1.6) if
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ25_HTML.gif
          (2.19)
          Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq106_HTML.gif as long as it exists. Thus, inequality (2.19) is always valid if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq107_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq108_HTML.gif , using the expression of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq109_HTML.gif in (2.5), we have that inequality (2.19) that is equivalent to
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ26_HTML.gif
          (2.20)
          Indeed, since
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ27_HTML.gif
          (2.21)
          we have (2.19) that is equivalent to
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ28_HTML.gif
          (2.22)
          that is,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ29_HTML.gif
          (2.23)
          Thus,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ30_HTML.gif
          (2.24)
          On the other hand, squaring (2.23) we find that
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ31_HTML.gif
          (2.25)
          which is equivalent to
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ32_HTML.gif
          (2.26)

          The combination of (2.24) and (2.26) yields (2.20).

          Furthermore,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ33_HTML.gif
          (2.27)
          is equivalent to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq110_HTML.gif , both
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ34_HTML.gif
          (2.28)

          are equivalent to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq111_HTML.gif . Consequently, we have the following.

          Corollary 2.3.

          There is no limit cycle in (1.6) if either of the following conditions hold:

          (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq113_HTML.gif ;

          (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq115_HTML.gif .

          When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq116_HTML.gif , system (1.6) has a unique infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq117_HTML.gif . The Jacobian matrix at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq118_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ35_HTML.gif
          (2.29)
          The determinant of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq119_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ36_HTML.gif
          (2.30)
          and the trace of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq120_HTML.gif is
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ37_HTML.gif
          (2.31)

          Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq121_HTML.gif is a degenerate singular point. Since its singularity, complex dynamic behaviors may occur, which will be studied in the next section.

          3. Bifurcation Analysis

          In this section, the Bogdanov-Takens bifurcation (for short, BT bifurcation) of system (1.6) is studied when there is a unique degenerate infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq122_HTML.gif .

          For simplicity of computation, we introduce the new time http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq123_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq124_HTML.gif , rewrite http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq125_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq126_HTML.gif , and obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ38_HTML.gif
          (3.1)

          Note that (3.1) and (1.6) are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq127_HTML.gif -equivalent; both systems have the same dynamics (only the time changes).

          As the above mentioned, assume that

          (H1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq128_HTML.gif

          Then (3.1) admits a unique positive equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq129_HTML.gif , where
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ39_HTML.gif
          (3.2)
          In order to translate the positive equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq130_HTML.gif to origin, we set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq131_HTML.gif and obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ40_HTML.gif
          (3.3)

          Since we are interested in codimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq132_HTML.gif bifurcation, we assume further that

          (H2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq133_HTML.gif

          Then, after some transformations, we have the following result.

          Theorem 3.1.

          The equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq134_HTML.gif of (1.6) is a cusp of codimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq135_HTML.gif if (H1) and (H2) hold; that is, it is a Bogdanov-Takens singularity.

          Proof.

          Under assumptions (H1) and (H2), it is clear that the linearized matrix of (3.3)
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ41_HTML.gif
          (3.4)
          has two zero eigenvalues. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq136_HTML.gif . Since the parameters http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq137_HTML.gif satisfy the assumptions (H1) and (H2), after some algebraic calculations, (3.3) is transformed into
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ42_HTML.gif
          (3.5)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq139_HTML.gif , are smooth functions in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq140_HTML.gif at least of the third order. Using an affine translation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq141_HTML.gif to (3.5), we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ43_HTML.gif
          (3.6)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq143_HTML.gif , are smooth functions in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq144_HTML.gif at least of order three. To obtain the canonical normal forms, we perform the transformation of variables by
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ44_HTML.gif
          (3.7)
          Then, (3.6) becomes
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ45_HTML.gif
          (3.8)

          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq146_HTML.gif , are smooth functions in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq147_HTML.gif at least of the third order.

          Obviously,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ46_HTML.gif
          (3.9)

          This implies that the origin of (3.3), that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq148_HTML.gif of (1.6), is a cusp of codimension http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq149_HTML.gif by in [25, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq150_HTML.gif , Section http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq151_HTML.gif ].

          In the following we will investigate the approximating BT bifurcation curves. The parameters http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq152_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq153_HTML.gif are chosen as bifurcation parameters. Consider the following perturbed system:
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ47_HTML.gif
          (3.10)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq155_HTML.gif are positive constants while (H1) and (H2) are satisfied. That is to say,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ48_HTML.gif
          (3.11)
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq157_HTML.gif are in the small neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq158_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq160_HTML.gif are in the small neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq161_HTML.gif , where
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ49_HTML.gif
          (3.12)
          Clearly, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq162_HTML.gif is the degenerate equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq163_HTML.gif of (1.6). Substituting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq164_HTML.gif into (3.10) and using Taylor expansion, we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ50_HTML.gif
          (3.13)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq165_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq167_HTML.gif , are smooth functions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq169_HTML.gif at least of order three in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq170_HTML.gif . Making the change of variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq171_HTML.gif to (3.13) and noting the conditions in (3.11) and expressions in (3.12), we have
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ51_HTML.gif
          (3.14)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ52_HTML.gif
          (3.15)

          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq173_HTML.gif , are smooth functions in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq174_HTML.gif at least of the third order, and the coefficients depend smoothly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq176_HTML.gif .

          Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq177_HTML.gif . Using (3.11) and (3.12), after some algebraic calculations, we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ53_HTML.gif
          (3.16)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq179_HTML.gif , are smooth functions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq181_HTML.gif at least of the third order in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq182_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ54_HTML.gif
          (3.17)
          Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq183_HTML.gif . Then (3.16) becomes
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ55_HTML.gif
          (3.18)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ56_HTML.gif
          (3.19)

          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq184_HTML.gif is smooth function in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq185_HTML.gif at least of order three, and all the coefficients depend smoothly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq187_HTML.gif .

          By setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq188_HTML.gif to (3.18), we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ57_HTML.gif
          (3.20)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq189_HTML.gif is smooth function in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq190_HTML.gif at least of the third order and
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ58_HTML.gif
          (3.21)
          Now, introducing a new time variable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq191_HTML.gif to (3.20), which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq192_HTML.gif , and still writing http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq193_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq194_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ59_HTML.gif
          (3.22)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq195_HTML.gif is smooth function of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq196_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq197_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq198_HTML.gif at least of three order in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq199_HTML.gif . Setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq200_HTML.gif to (3.22), we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ60_HTML.gif
          (3.23)
          where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq201_HTML.gif is smooth function of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq202_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq203_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq204_HTML.gif at least of order three in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq205_HTML.gif and
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ61_HTML.gif
          (3.24)
          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq206_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq207_HTML.gif , it is easy to obtain the following results:
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ62_HTML.gif
          (3.25)
          By setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq209_HTML.gif , and rewriting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq210_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq211_HTML.gif , we obtain
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ63_HTML.gif
          (3.26)
          where
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ64_HTML.gif
          (3.27)

          and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq212_HTML.gif is smooth function of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq215_HTML.gif at least of order three in variables http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq216_HTML.gif .

          By the theorem of Bogdanov in [26, 27] and the result of Perko in [25], we obtain the following local representations of bifurcation curves in a small neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq217_HTML.gif of the origin (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq218_HTML.gif of (1.6).

          Theorem 3.2.

          Let the assumptions (H1) and (H2) hold. Then (1.6) admits the following bifurcation behaviors:

          1. (i)

            there is a saddle-node bifurcation curve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq219_HTML.gif ;

             
          2. (ii)

            there is a Hopf bifurcation curve http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq220_HTML.gif ;

             
          3. (iii)

            there is a homoclinic-loop bifurcation curve HL http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq221_HTML.gif .

             
          Concretely, as the statement in [28, Chapter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq222_HTML.gif ], when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq223_HTML.gif , the orbital topical structure of the system (3.26) at origin (corresponding system (1.6) at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq224_HTML.gif ) is shown in Figure 1.
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Fig1_HTML.jpg
          Figure 1

          The bifurcation set and the corresponding phase portraits of system (3. 26) at origin.

          4. Discussion

          Note that most infection experiments suggest that the infection rate of microparasitic infections is an increasing function of the parasite dose, usually sigmoidal in shape. In this paper, we study a viral infection model with a type of nonlinear infection rate, which was introduced by Regoes et al. [11].

          Qualitative analysis (Theorem 2.1) implies that infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq225_HTML.gif is always stable if the virus is noncytopathic, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq226_HTML.gif , or cytopathic in vivo but its cytopathic effect is less than or equal to an appropriate value, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq227_HTML.gif . When the cytopathic effect of virus is greater than the threshold value, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq228_HTML.gif , the stability of the infection equilibrium http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq229_HTML.gif depends on the value of parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq230_HTML.gif , which is proportional to the birth rate of susceptible cells http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq231_HTML.gif and is in inverse proportion to the infectious dose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq232_HTML.gif . The infection equilibrium is stable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq233_HTML.gif and becomes unstable if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq234_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq235_HTML.gif gets to the critical value, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq236_HTML.gif , the infection equilibrium is a linear center, so the oscillation behaviors may occur.

          If our model (1.6) does not have a limit cycle (see Theorem 2.2 and Corollary 2.3), its asymptotic behavior is determined by the stability of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq237_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq238_HTML.gif is stable, there is a region outside which positive semiorbits tend to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq239_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq240_HTML.gif tends to infinity and inside which positive semi-orbits tend to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq241_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq242_HTML.gif tends to infinity; that is, the virus will persist if the initial position lies in the region and disappear if the initial position lies outside this region. Thus, besides the value of parameters, the initial concentration of the virus can also affect the result of invasion. An invasion threshold may exist in these cases, which is typical for the so-called Allee effect that occurs when the abundance or frequency of a species is positively correlated with its growth rate (see [11]). Consequently, the unrescaled model (1.5) can display an Allee effect (see Figure 2), which is an infrequent phenomenon in current viral infection models though it is reasonable and important in viral infection process.
          http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Fig2_HTML.jpg
          Figure 2

          Illustrations of the Allee effect for (1. 5). Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq243_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq244_HTML.gif is stable, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq245_HTML.gif is a saddle point, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq246_HTML.gif is stable. Note that SM is the stable manifolds of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq247_HTML.gif (solid line), UM is the unstable manifolds of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq248_HTML.gif (dash line), and the phase portrait of (1.6) is divided into two domains of extinction and persistence of the virus by SM.

          Furthermore, when infection equilibrium becomes a degenerate singular point, we have shown that the dynamics of this model are very rich inside this region (see Theorems 3.1 and 3.2 and Figure 1). Static and dynamical bifurcations, including saddle-node bifurcation, Hopf bifurcation, homoclinic bifurcation, and bifurcation of cusp-type with codimension two (i.e., Bogdanov-Takens bifurcation), have been exhibited. Thus, besides the Allee effect, our model (1.6) shows that the viral oscillation behaviors can occur in the host based on the appropriate conditions, which was observed in chronic HBV or HCV carriers (see [2931]). These results inform that the viral infection is very complex in the development of a better understanding of diseases. According to the analysis, we find that the cytopathic effect of virus and the birth rate of susceptible cells are both significant to induce the complex and interesting phenomena, which is helpful in the development of various drug therapy strategies against viral infection.

          Declarations

          Acknowledgments

          This work is supported by the National Natural Science Fund of China (nos. 30770555 and 10571143), the Natural Science Foundation Project of CQ CSTC (2007BB5012), and the Science Fund of Third Military Medical University (06XG001).

          Authors’ Affiliations

          (1)
          School of Science, Dalian Jiaotong University
          (2)
          Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela
          (3)
          Departamento de Psiquiatría, Radiología y Salud Pública, Facultad de Medicina, Universidad de Santiago de Compostela
          (4)
          Department of Computers Science, Third Military Medical University

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          © Yumei Yu et al. 2009

          This article is published under license to BioMed Central Ltd. 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.