Stochastic dynamic effects of media coverage and incubation on a distributed delayed epidemic system with Lévy jumps

A stochastic SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic system with Lévy jumps and distributed time delay is established, which is utilized to investigate hybrid dynamic effects of media coverage and incubation on infectious disease transmission. Stochastically ultimate boundedness of the positive solution is discussed. Existence of a unique global positive solution is studied. By constructing appropriate stochastic Lyapunov functions, existence of a unique ergodic stationary distribution is investigated. Sufficient conditions are derived to discuss exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, sufficient conditions for persistence in mean and extinction of infectious disease are investigated. Finally, numerical simulations are carried out to show consistency with theoretical analysis. •A stochastic SIAM epidemic system with media coverage and incubation is proposed.•Hybrid dynamic effects of distributed time delay and Lévy jumps are investigated.•Stochastically ultimate boundedness of the positive solution is investigated.•Existence and uniqueness of an ergodic stationary distribution are discussed.•Persistence in mean and extinction of the infectious disease are investigated.