Threshold dynamics of an uncertain SIRS epidemic model with a bilinear incidence

Abstract

This paper mainly discusses the extinction and persistent dynamic behavior of infectious diseases with temporary immunity. Considering that the transmission process of infectious diseases is affected by environmental fluctuations, stochastic SIRS models have been proposed, while the outbreak of diseases is sudden and the interference terms that affect disease transmission cannot be qualified as random variables. Liu process is introduced based on uncertainty theory, which is a new branch of mathematics for describing uncertainty phenomena, to describe uncertain disturbances in epidemic transmission. This paper first extends the classic SIRS model from a deterministic framework to an uncertain framework and constructs an uncertain SIRS infectious disease model with constant input and bilinear incidence. Then, by means of Yao-Chen formula, α-path of uncertain SIRS model and the corresponding ordinary differential equations are obtained to introduce the uncertainty threshold function R 0 * as the basic reproduction number. Moreover, two equilibrium states are derived. A series of numerical examples show that the larger the value of R 0 * , the more difficult it is to control the disease. If R 0 * ≤ 1 , the infectious disease will gradually disappear, while if R 0 * > 1 , the infectious disease will develop into a local epidemic.