A Study of Infectious Disease Models with Switching

Abstract

Infectious disease models with switching are constructed and investigated in detail. Modelling infectious diseases as switched systems, which are systems that combine continuous dynamics with discrete logic, allows for the use of methods from switched systems theory. These methods are used to analyze the stability and long-term behaviour of the proposed switched epidemiological models. Switching is first incorporated into epidemiological models by assuming the contact rate to be time-dependent and better approximated by a piecewise constant. Epidemiological models with switched incidence rates are also investigated. Threshold criteria are established that are sufficient for the eradication of the disease, and, hence, the stability of the disease-free solution. In the case of an endemic disease, some criteria are developed that establish the persistence of the disease. Lyapunov function techniques, as well as techniques for stability of impulsive or non-impulsive switched systems with both stable and unstable modes are used. These methods are first applied to switched epidemiological models which are intrinsically one-dimensional. Multi-dimensional disease models with switching are then investigated in detail. An important part of studying epidemiology is to construct control strategies in order to eradicate a disease, which would otherwise be persistent. Hence, the application of controls schemes to switched epidemiological models are investigated. Finally, epidemiological models with switched general nonlinear incidence rates are considered. Simulations are given throughout to illustrate our results, as well as to make some conjectures. Some conclusions are made and future directions are given.