Abstract
In this paper, by introducing environmental perturbation, we extend an epidemic model with graded cure, relapse, and nonlinear incidence rate from a deterministic framework to a stochastic differential one. The existence and uniqueness of positive solution for the stochastic system is verified. Using the Lyapunov function method, we estimate the distance between stochastic solutions and the corresponding deterministic system in the time mean sense. Under some acceptable conditions, the solution of the stochastic system oscillates in the vicinity of the disease-free equilibrium if the basic reproductive number R0≤1, while the random solution oscillates near the endemic equilibrium, and the system has a unique stationary distribution if R0>1. Moreover, numerical simulation is conducted to support our theoretical results.
Highlights
Mathematical models can improve our understanding of the dynamics of infectious diseases, predict the transmission trend, and help us formulate preventive measures. e classical SIR and SIS epidemic models established by Kermack and McKendrick are one of the most important models in epidemiology [1, 2]
Since most systems in the real world are disturbed by random and unpredictable perturbation, we introduce environmental noise of white noise type into the transmission of disease and study a stochastic version of a nonlinear SIRS epidemic model with relapse and cure. e reproduction number R0 is a threshold parameter
If the conditions of eorems 3 and 4 hold, according to R0 ≤ 1 or R0 > 1, we prove that the solution of the stochastic model oscillates in the vicinity of the disease-free equilibrium and the endemic equilibrium, respectively, and the fluctuation intensity is proportional to the white noise intensity. roughout the paper, we use numerical simulations to illustrate our theoretical results
Summary
Mathematical models can improve our understanding of the dynamics of infectious diseases, predict the transmission trend, and help us formulate preventive measures. e classical SIR and SIS epidemic models established by Kermack and McKendrick are one of the most important models in epidemiology [1, 2]. To explore infectious diseases in which infected individuals may be permanently rehabilitated or reinfected, Lahrouz et al [7] proposed a nonlinear SIR epidemic model with relapse and graded cure as follows:. The unique disease-free equilibrium E0 (μ/μ1, 0, 0) is globally asymptotically stable if R0 < 1. An increasing number of stochastic epidemic models including environmental noise have been developed [10,11,12,13].
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