## A viral infection model with a nonlinear infection rate.(English)Zbl 1187.34062

A viral infection model with a nonlinear infection rate is constructed based on empirical evidences. This leads to the study of the dynamics of the following two-dimensional autonomous system
\begin{aligned} \frac{dx}{dt}&=m-dx-\frac{y^2}{1+y^2}x,\\ \frac{dy}{dt}&=\frac{y^2}{1+y^2}x-ay. \end{aligned} \tag{1}
The main purpose of the paper is to study the effect of the nonlinear infection rate on the dynamics of (1). Qualitative analysis of system (1) 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 behavior indicates that the model can display an Allee effect and fluctuation effect, which are important for making strategies for controlling the invasion of virus. Thus, the nonlinear infection rate can induce complex dynamic behavior in the viral infection model. A brief discussion on the direct biological implications of the results is given.

### MSC:

 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34C23 Bifurcation theory for ordinary differential equations
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### References:

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