Global asymptotic stability of an SEIRS model and its solutions by nonstandard finite difference schemes

Abstract

. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of a recognized SEIRS model with nonlinear incidence and vertical transmission. Our main objective is to re-establish the local asymptotic stability of the disease endemic equilibrium point without the assumption proposed in [L. Qi, J. Cui, The stability of an SEIRS model with nonlinear incidence, vertical transmission and time delay, Appl. Math. Comput. 221 (2013), 360-366] and prove that this equilibrium point is not only locally asymptotically stable but also globally asymptotically stable if it exists. Therefore, the global stability of the model is established rigorously. Additionally, we design nonstandard finite difference (NSFD) schemes that preserve essential properties of the SEIRS model using the Mickens methodology. A set of numerical experiments are performed to support the validity of the theoretical results as well as to show advantages and efficiency of the NSFD schemes over the standard finite difference schemes.