The analysis of deterministic and stochastic SIM model with general information intervention

Abstract

In this paper, we propose an epidemic model with the impact of information intervention and general incidence rate in deterministic and stochastic environment, respectively. The information intervention prompts susceptible individuals to change their behavior to protect themselves from infection. First, the asymptotic dynamics of the disease-free equilibrium and the unique endemic equilibrium are analyzed and the results indicate that the basic reproduction number is a pointy disease threshold about the stability under some conditions. Then, we formulate the corresponding stochastic model by perturbing the disease transmission rate and information response rate parameters by white noise terms and provide verifiable sufficient conditions for extinction and persistence in mean. Two sufficient conditions for extinction show that if the noise intensity of the disease transmission rate is large enough or small enough, the infected population of the stochastic system will tend to go extinct. For the persistence, we get the sufficient conditions which guarantee the infected population to be persistent in the mean. Finally, we perform some numerical simulations to compare the dynamic behaviors of the deterministic and the stochastic system.