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Bifurcation analysis of an SIRS epidemic model with standard incidence rate and saturated treatment function. (English) Zbl 1474.34304

Summary: An epidemic model with standard incidence rate and saturated treatment function of infectious individuals is proposed to understand the effect of the capacity for treatment of infective individuals on the disease spread. The treatment function in this paper is a continuous and differential function which exhibits the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. It is proved that the existence and stability of the disease-free and endemic equilibria for the model are not only related to the basic reproduction number but also to the capacity for treatment of infective individuals. And a backward bifurcation is found when the capacity is not enough. By computing the first Lyapunov coefficient, we can determine the type of Hopf bifurcation, i.e., subcritical Hopf bifurcation or supercritical Hopf bifurcation. We also show that under some conditions the model undergoes Bogdanov-Takens bifurcation. Finally, numerical simulations are given to support some of the theoretical results.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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