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

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.