Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model

Abstract

We design and investigate the reliability of various nonstandard nite di erence (NSFD) schemes for the SIS epidemiological model in three di erent settings. For the classical SIS model, we construct two new NSFD schemes which faithfully replicate the property of the continuous model of having the parameter R0, the basic reproduction number, as a threshold to determine the stability properties of equilibrium points: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R0 1; it is unstable when R0 > 1 and there appears a unique GAS endemic equilibrium (EE) in this case. These schemes also preserve the positivity and boundedness properties of solutions of the classical SIS model. The schemes are further used to derive NSFD schemes for the SIS-di usion model which constitutes the second setting of the study. The designed NSFD schemes are dynamically consistent with the global asymptotic stability of the disease-free equilibrium for R0 1 and the instability of the disease-free equilibrium for R0 > 1. In the latter case, the schemes replicate the global asymptotic stability of the endemic equilibrium. Positivity and boundedness properties of solutions of the SIS-di usion model are also preserved by the NSFD schemes. In a third step, the classical SIS model is extended into a SIS-Volterra integral equation model in which the contact rate is a function of fraction of infective individuals and allows a distributed period of infectivity. The qualitative analysis is now based on two threshold parameters Rc 0 1 Rm0 . The system can undergo the backward bifurcation phenomenon as follows. The DFE is the only equilibrium and it is GAS when R0 < Rc 0; there exists only one EE, which is GAS when R0 > Rm0 with the DFE being unstable when R0 > 1; for Rc 0 < R0 < 1, the DFE is locally asymptotically stable (LAS) and coexists with at least one LAS endemic equilibrium. We design a NSFD scheme and prove theoretically and computationally that it preserves the above-stated stability properties of equilibria as well as positivity and boundedness of the solutions of the continuous model.