An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

TL;DR

This paper introduces an explicit, unconditionally boundary-preserving Milstein-type scheme for stochastic SIS epidemic models, ensuring accurate long-term behavior reproduction and strong convergence without parameter restrictions.

Contribution

It develops a novel logarithm transformation-based scheme that preserves boundary and dynamics properties unconditionally, improving numerical simulation of stochastic epidemic models.

Findings

Scheme is explicit and unconditionally preserves boundary and dynamics.

Achieves strong convergence rate of order one.

Numerical experiments confirm theoretical properties.

Abstract

In the present article, we construct a logarithm transformation based Milstein-type method for the stochastic susceptible-infected-susceptible (SIS) epidemic model evolving in the domain (0,N). The new scheme is explicit and unconditionally boundary and dynamics preserving, when used to solve the stochastic SIS epidemic model. Also, it is proved that the scheme has a strong convergence rate of order one. Different from existing time discretization schemes, the newly proposed scheme for any time step size h > 0, not only produces numerical approximations living in the entire domain (0,N), but also unconditionally reproduces the extinction and persistence behavior of the original model, with no additional requirements imposed on the model parameters. Numerical experiments are presented to verify our theoretical findings.