Construction of positivity-preserving numerical method for stochastic SIVS epidemic model

In this paper we propose the balanced implicit numerical techniques for maintaining the nonnegative path of the solution in stochastic susceptible-infected-vaccinated-susceptible (SIVS) epidemic model. We can hardly acquire the explicit solution for the SIVS model, so we often use the numerical scheme to produce approximate solutions. The Euler-Maruyama (EM) method is a useful and effective means in producing numerical solutions of SIVS model. The EM method to simulate the stochastic SIVS model often results in the problem that the numerical solution is not positive. In order to eliminate the negative path of the solution in a stochastic SIVS epidemic model, we construct a numerical method preserving positivity for the SIVS model. It is proved that the balanced implicit method (BIM) can preserve positivity and we show the convergence of the BIM numerical approximate solution to the exact solution. Finally, a numerical example is offered to support the theoretical results and verify the availability of the approach.