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Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

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  • Lei Wang
  • Zhidong Teng
  • Long Zhang

Abstract

We study a class of discrete SIRS epidemic models with nonlinear incidence rate F(S)G(I) and disease‐induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease‐free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number ℛ0 1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when ℛ0 > 1, the disease in the model is permanent. Some special cases of F(S)G(I) are discussed. Particularly, when F(S)G(I) = βSI/(1 + λI), it is obtained that the endemic equilibrium is globally asymptotically stable if and only if ℛ0 > 1. Furthermore, the numerical simulations show that for general incidence rate F(S)G(I) the endemic equilibrium may be globally asymptotically stable only as ℛ0 > 1.

Suggested Citation

  • Lei Wang & Zhidong Teng & Long Zhang, 2014. "Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:249623
    DOI: 10.1155/2014/249623
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    References listed on IDEAS

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    1. Vargas-De-León, Cruz, 2011. "On the global stability of SIS, SIR and SIRS epidemic models with standard incidence," Chaos, Solitons & Fractals, Elsevier, vol. 44(12), pages 1106-1110.
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