Dynamics of epidemic models with asymptomatic infection and seasonal succession

  • Received: 11 March 2017 Accepted: 01 April 2017 Published: 01 October 2017
  • MSC : Primary: 92D25, 34C23; Secondary: 34D23

  • In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0 \gt 1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

    Citation: Yilei Tang, Dongmei Xiao, Weinian Zhang, Di Zhu. Dynamics of epidemic models with asymptomatic infection and seasonal succession[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1407-1424. doi: 10.3934/mbe.2017073

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  • In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and evaluated directly for this model, and uniform persistence of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0≤ 1$, and as $\mathcal{R}_0 \gt 1$ the disease-free equilibrium is unstable and there is an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infectives and symptomatic infectives are close. These theoretical results provide an intuitive basis for understanding that the asymptomatically infective individuals and the seasonal disease transmission promote the evolution of the epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.
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    © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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