Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence

Abstract

We extend the nonstandard finite difference (NSFD) method of solutions to the study of an SIS epidemic model with standard incidence. We show that the proposed NSFD schemes preserve two essential properties of the continuous model: positivity and global asymptotic stability properties. The reproduction number of the model is calculated by the next generation matrix approach. It is worth noting that the global asymptotic stability of the disease-free equilibrium point of the proposed numerical schemes is proved theoretically by the use of an extension of the Lyapunov stability theorem. Besides, the global asymptotic stability of the endemic equilibrium point is investigated by the use of the Lyapunov indirect method and numerical simulations. Consequently, NSFD schemes which are dynamically consistent with the continuous model are obtained. Some numerical simulations are presented to validate the theoretical results and to show that the NSFD schemes are effective and appropriate for solving the continuous model. We employ the standard finite difference (SFD) method as a means of comparison to NSFD schemes. The numerical simulations indicate that the use of SFD method is not suitable as it produced solutions that do not correspond exactly to solutions of the continuous model.