Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

A stochastic epidemic model incorporating media coverage

Pages: 893 – 910

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n4.a1

Authors

Yongli Cai (School of Mathematical Science, Huaiyin Normal University, Huaian, China)

Yun Kang (Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, Az., U.S.A.)

Malay Banerjee (Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Uttar, Pradesh, India)

Weiming Wang (School of Mathematical Science, Huaiyin Normal University, Huaian, , China)

Abstract

In this paper, we investigate the effects of environment fluctuations on disease dynamics through studying the stochastic dynamics of an SIS model incorporating media coverage. The value of this study lies in two aspects: Mathematically, we show that the disease dynamics the SDE model can be governed by its related basic reproduction number $R^S_0$: if $R^S_0 \leq 1$, the disease will die out stochastically, but if $R^S_0 \gt 1$, the disease will break out with probability one. Epidemiologically, we partially provide the effects of the environment fluctuations affecting spread of the disease incorporating media coverage. First, noise can suppress the disease outbreak. Notice that $R^S_0 \lt R_0$, and it is possible that $R^S_0 \lt 1 \lt R_0$. This is the case when the deterministic model has an endemic while the SDE model has disease extinction with probability one. Second, two stationary distribution governed by $R^S_0$: If $R^S_0 \lt 1$, it has disease-free distribution which means that the disease will die out with probability one; while $R^S_0 \gt 1$, it has endemic stationary distribution, which leads to the stochastically persistence of the disease. In order to understand the role of media coverage on disease dynamics, we present some numerical simulations to validate the analytical findings. It is interesting to note that although some parameters have no role in determining $R^S_0$, however the strength of noise to the susceptible population and the parameters characterizing media affect play crucial role in determining the long term dynamics of the system.

Keywords

epidemic model, Lyapunov function, stochastic asymptotic stability, ergodic property

2010 Mathematics Subject Classification

60H10, 92D30, 93E15

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Published 6 April 2016