# American Institute of Mathematical Sciences

January  2010, 13(1): 195-211. doi: 10.3934/dcdsb.2010.13.195

## Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  January 2008 Revised  August 2009 Published  October 2009

A disease transmission model of SIRS type with latent period and nonlinear incidence rate is considered. Latent period is assumed to be a constant $\tau$, and the incidence rate is assumed to be of a specific nonlinear form, namely, $\frac{kI(t-\tau)S(t)}{1+\alpha I^{h}(t-\tau)}$, where $h\ge 1$. Stability of the disease-free equilibrium, and existence, uniqueness and stability of an endemic equilibrium for the model, are investigated. It is shown that, there exists the basic reproduction number $R_0$ which is independent of the form of the nonlinear incidence rate, if $R_0\le 1$, then the disease-free equilibrium is globally asymptotically stable, whereas if $R_0>1$, then the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region for the model in which there is no latency, and periodic solutions can arise by Hopf bifurcation from the endemic equilibrium for the model at a critical latency. Some numerical simulations are provided to support our analytical conclusions.
Citation: Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195
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