Numerical analysis of linearly implicit Milstein method for stochastic SEIR models with nonlinear incidence rates

In this paper, we focus on the numerical analysis of stochastic SEIR models with nonlinear incidence rates. By reformulating the stochastic basic reproduction number $R_0^S$, it is shown that the disease extinction of deterministic models is preserved under stochastic noises. On the other hand, the total population of stochastic SEIR models is varying and even unbounded when there are some noises in the natural death rate. Therefore, as the fundamental approach, we have to present the boundedness in the 4th moment and Hölder continuity of the exact solutions for the numerical convergence analysis. Numerically, a linearly implicit Milstein method is employed to ensure the numerical positivity under the condition of $h<h_0$ and hence the numerical boundedness is obtained in the 4th moment. After the strong convergence analysis under the fundamental theory, we are much more interested in the numerical dynamic behaviors. Since the previous technique, the exponent representation of the stability function, is not available for the higher dimensional models, a logarithmic martingale estimation to the numerical disease is introduced in this paper, and hence the numerical replications of the long-time dynamic behaviors are discussed thoroughly. Finally, some numerical experiments are provided to verify the theoretical analysis and illustrate the convergence analysis of the numerical steady distribution in the future.