Ergodic Stationary Distribution and Threshold Dynamics of a Stochastic Nonautonomous SIAM Epidemic Model with Media Coverage and Markov Chain

A stochastic nonautonomous SIAM (Susceptible individual-Infected individual-Aware individual-Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Levy jumps on infectious disease transmission. The uniform upper bound and lower bound of the positive solution are studied. Based on defining suitable random Lyapunov functions, we researched the existence of a nontrival positive T-periodic solution. Sufficient conditions are derived to discuss the exponential ergodicity based on verifying a Foster-Lyapunov condition. Furthermore, the persistence in the average sense and extinction of infectious disease are investigated using stochastic analysis techniques. Finally, numerical simulations are utilized to provide evidence for the dynamical properties of the stochastic nonautonomous SIAM.