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**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.

\[ \begin{aligned} \frac{dx}{dt}&=m-dx-\frac{y^2}{1+y^2}x,\\ \frac{dy}{dt}&=\frac{y^2}{1+y^2}x-ay. \end{aligned} \tag{1} \]

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.

Reviewer: Georgy Osipenko (St. Peterburg)

### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

92D30 | Epidemiology |

34C23 | Bifurcation theory for ordinary differential equations |

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\textit{Y. Yu} et al., Bound. Value Probl. 2009, Article ID 958016, 19 p. (2009; Zbl 1187.34062)

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