Open Access

A Viral Infection Model with a Nonlinear Infection Rate

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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ1_HTML.gif
(1.1)

where susceptible cells ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq1_HTML.gif ) are produced at a constant rate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq2_HTML.gif , die at a density-dependent rate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq3_HTML.gif , and become infected with a rate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq4_HTML.gif ; infected cells ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq5_HTML.gif ) are produced at rate https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq7_HTML.gif ; free virus particles ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq9_HTML.gif and die at a rate https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq11_HTML.gif . Thus, the number of infected cells https://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 https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq15_HTML.gif is globally asymptotically stable if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq16_HTML.gif , and so is the infection equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq17_HTML.gif if https://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 https://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
https://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, https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ4_HTML.gif
(1.4)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq22_HTML.gif denotes the infectious dose at which https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq23_HTML.gif of the susceptible cells are infected, https://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 https://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, https://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 https://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 https://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 https://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 https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq32_HTML.gif . For simplicity, we still use variables https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq33_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq34_HTML.gif and obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq35_HTML.gif . Note that https://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 https://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, https://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 https://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 https://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
https://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 https://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 https://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
https://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 https://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 https://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
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq46_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq47_HTML.gif ;

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

Here,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ11_HTML.gif
(2.5)
Thus, the surface
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq55_HTML.gif . The Jacobian matrix at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq56_HTML.gif is
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ14_HTML.gif
(2.8)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq57_HTML.gif in this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq58_HTML.gif is valid. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq59_HTML.gif and the equilibrium https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq61_HTML.gif is
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq62_HTML.gif . Thus, the equilibrium https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq64_HTML.gif .

Theorem 2.1.

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

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

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

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq74_HTML.gif is a linear center if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq76_HTML.gif is
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ18_HTML.gif
(2.12)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ19_HTML.gif
(2.13)

which means that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq78_HTML.gif .

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

This means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq80_HTML.gif . Thus, under the condition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq81_HTML.gif and the sign of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq83_HTML.gif is always valid if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq84_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq85_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq86_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq87_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq88_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq89_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq90_HTML.gif if https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq93_HTML.gif is a saddle and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq95_HTML.gif and does not include https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq97_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq99_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq101_HTML.gif . Take a Dulac function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq104_HTML.gif , respectively. We have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ24_HTML.gif
(2.18)

which is negative if https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ25_HTML.gif
(2.19)
Note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq107_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq108_HTML.gif , using the expression of https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ26_HTML.gif
(2.20)
Indeed, since
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ28_HTML.gif
(2.22)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ29_HTML.gif
(2.23)
Thus,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ31_HTML.gif
(2.25)
which is equivalent to
https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ33_HTML.gif
(2.27)
is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq110_HTML.gif , both
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ34_HTML.gif
(2.28)

are equivalent to https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq113_HTML.gif ;

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

When https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq117_HTML.gif . The Jacobian matrix at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq118_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ35_HTML.gif
(2.29)
The determinant of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq119_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ36_HTML.gif
(2.30)
and the trace of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq120_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ37_HTML.gif
(2.31)

Thus, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq123_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq124_HTML.gif , rewrite https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq125_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq126_HTML.gif , and obtain
https://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 https://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) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq129_HTML.gif , where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq130_HTML.gif to origin, we set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq131_HTML.gif and obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq132_HTML.gif bifurcation, we assume further that

(H2) https://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 https://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 https://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)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq136_HTML.gif . Since the parameters https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ42_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq139_HTML.gif , are smooth functions in variables https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq141_HTML.gif to (3.5), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ43_HTML.gif
(3.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq143_HTML.gif , are smooth functions in variables https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ44_HTML.gif
(3.7)
Then, (3.6) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ45_HTML.gif
(3.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq146_HTML.gif , are smooth functions in https://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,
https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq149_HTML.gif by in [25, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq150_HTML.gif , Section https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq152_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ47_HTML.gif
(3.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq154_HTML.gif and https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ48_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq157_HTML.gif are in the small neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq158_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq160_HTML.gif are in the small neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq161_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ49_HTML.gif
(3.12)
Clearly, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq162_HTML.gif is the degenerate equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq163_HTML.gif of (1.6). Substituting https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ50_HTML.gif
(3.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq167_HTML.gif , are smooth functions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq168_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq170_HTML.gif . Making the change of variables https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ51_HTML.gif
(3.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ52_HTML.gif
(3.15)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq173_HTML.gif , are smooth functions in variables https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq176_HTML.gif .

Let https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ53_HTML.gif
(3.16)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq179_HTML.gif , are smooth functions of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq180_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq182_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ54_HTML.gif
(3.17)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq183_HTML.gif . Then (3.16) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ55_HTML.gif
(3.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ56_HTML.gif
(3.19)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq184_HTML.gif is smooth function in variables https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq186_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq187_HTML.gif .

By setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq188_HTML.gif to (3.18), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ57_HTML.gif
(3.20)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq189_HTML.gif is smooth function in variables https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq191_HTML.gif to (3.20), which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq192_HTML.gif , and still writing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq193_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq194_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ59_HTML.gif
(3.22)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq195_HTML.gif is smooth function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq197_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq199_HTML.gif . Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq200_HTML.gif to (3.22), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ60_HTML.gif
(3.23)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq201_HTML.gif is smooth function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq202_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq203_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq205_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ61_HTML.gif
(3.24)
If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq206_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ62_HTML.gif
(3.25)
By setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq209_HTML.gif , and rewriting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq210_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq211_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ63_HTML.gif
(3.26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_Equ64_HTML.gif
(3.27)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq212_HTML.gif is smooth function of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq214_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq217_HTML.gif of the origin (i.e., https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq222_HTML.gif ], when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq224_HTML.gif ) is shown in Figure 1.
https://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 https://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, https://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, https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq228_HTML.gif , the stability of the infection equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq229_HTML.gif depends on the value of parameter https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq232_HTML.gif . The infection equilibrium is stable if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq233_HTML.gif and becomes unstable if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq234_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq235_HTML.gif gets to the critical value, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq237_HTML.gif . When https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq239_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq241_HTML.gif as https://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.
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq243_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq244_HTML.gif is stable, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F958016/MediaObjects/13661_2009_Article_891_IEq245_HTML.gif is a saddle point, https://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 https://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 https://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

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