Asymptotic behavior and numerical simulation of a stochastic multi-group SEIR epidemic model with infinite distributed delays

In this paper, we studied the asymptotic behavior of a stochastic multi-group Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model with infinite distributed delays and Lévy jumps. First, by the methods of Lyapunov functions, Itô's formula, and the theory of stopping times, we proved the existence and uniqueness of the global positive solution to the stochastic delayed system. Furthermore, by using appropriate Lyapunov functions, graph theory and stochastic analysis, we established the asymptotic dynamical behaviors around the disease-free equilibrium $P_0$ and the endemic equilibrium $P^{\ast }$ of the deterministic system, respectively. It was shown that if the threshold $R_0<1$, the solution of the stochastic delayed system oscillates around the disease-free equilibrium $P_0$; while if $R_0>1$, the solution fluctuates around the endemic equilibrium $P^{\ast }$. Finally, numerical simulations were performed to intuitively analyze the impact of Lévy noise on the dynamical behavior of the stochastic delayed system.