Stochastic Periodic Solution of a Susceptible-Infective Epidemic Model in a Polluted Environment under Environmental Fluctuation.

Yu Zhao,Xu Ma,Jiangping Li

doi:10.1155/2018/7360685

Abstract

It is well known that the pollution and environmental fluctuations may seriously affect the outbreak of infectious diseases (e.g., measles). Therefore, understanding the association between the periodic outbreak of an infectious disease and noise and pollution still needs further development. Here we consider a stochastic susceptible-infective (SI) epidemic model in a polluted environment, which incorporates both environmental fluctuations as well as pollution. First, the existence of the global positive solution is discussed. Thereafter, the sufficient conditions for the nontrivial stochastic periodic solution and the boundary periodic solution of disease extinction are derived, respectively. Numerical simulation is also conducted in order to support the theoretical results. Our study shows that (i) large intensity noise may help the control of periodic outbreak of infectious disease; (ii) pollution may significantly affect the peak level of infective population and cause adverse health effects on the exposed population. These results can help increase the understanding of periodic outbreak patterns of infectious diseases.

Highlights

In Northern China, coal fire-power industries and heating systems, as well as vehicle emissions, all conduce to air pollution, which has threatened the survival of exposed human population and affected the transmission of infectious diseases [1, 2]
Humans are exposed to some kinds of infectious diseases because the diseases propagate through a polluted environment
Examples include measles spreading through air pollution, snail fever spreading through water pollution, and diarrhea spreading through food pollution

Summary

Introduction

In Northern China, coal fire-power industries and heating systems, as well as vehicle emissions, all conduce to air pollution (airborne fine particulate matter PM2.5, PM10, and SO2, etc.), which has threatened the survival of exposed human population and affected the transmission of infectious diseases [1, 2]. Considering the periodic variation and pollution exposure of epidemic models and exploring the existence of stochastic periodic solutions are meaningful to predict and control the outbreaks of infectious diseases. Such analysis has benefited from the theoretical contributions about the nonautonomous stochastic system [24, 25]. Jiang et al [21] considered a stochastic nonautonomous competitive Lotka-Volterra model in a polluted environment and derived sufficient criteria for the existence and global attractivity of a nontrivial positive periodic solution.

Mathematical Analysis

Numerical Simulation

Conclusion