Mathematical Biosciences and Engineering (MBE)

Dynamics of epidemic models with asymptomatic infection and seasonal succession

Pages: 1407 - 1424, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017073

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Yilei Tang - School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China (email)
Dongmei Xiao - School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China (email)
Weinian Zhang - Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China (email)
Di Zhu - School of Mathematical Science, Shanghai Jiao Tong University, Shanghai 200240, China (email)

Abstract: 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 $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ 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.

Keywords:  Epidemic model, asymptomatic infection, seasonal succession, basic reproduction number, threshold dynamics.
Mathematics Subject Classification:  Primary: 92D25, 34C23; Secondary: 34D23.

Received: March 2017;      Accepted: April 2017;      Available Online: May 2017.

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