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Zbl 1250.92031
Hu, Zhixing; Ma, Wanbiao; Ruan, Shigui
Analysis of SIR epidemic models with nonlinear incidence rate and treatment. (English)
[J] Math. Biosci. 238, No. 1, 12-20 (2012). ISSN 0025-5564

Summary: This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.

MSC 2000:: *92C60 Medical epidemiology
34C60 Applications of qualitative theory of ODE
37N25 Dynamical systems in biology
65C20 Models (numerical methods)

Keywords: treatment rate; backward bifurcation; stability

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Zentralblatt MATH Berlin [Germany]