Centre Manifold Theory for some continuous and Discrete Epidemiological models

Abstract

In mathematical epidemiology, the threshold theory introduced by W.O. Kermack and A.G. McKendrick (1927) can be expressed in terms of the basic reproduction number R0. This is defined as the average number of secondary infections that occur when one infective is introduced into a susceptible host population. In this setting and for many diseases, the prediction of the likelihood of persistence or dying out of the disease within the population reads as follows: the disease-free equilibrium is locally asymptotically stable (LAS) when R0 < 1, it is unstable when R0 > 1 and at least one endemic equilibrium (EE) which is LAS is born in this case. In other words, at R0 = 1, a forward bifurcation occurs. However, some diseases undergo the backward bifurcation phenomenon whereby, for R0 < 1, the LAS disease-free equilibrium coexists with a small positive unstable EE and a large positive LAS EE. In this thesis, we study theoretically, numerically, and computationally the existence of the backward bifurcation phenomenon for dynamical systems, with emphasis on a “simple” SIS model with vaccination and a “complex” malaria model. We re-centre the reduction theorem in C. Castillo-Chavez and B. Song (2004) and highlight its advantage over the legendary power series approximations in the use of the Centre Manifold Theory (CMT). We propose and prove a Centre Manifold-based theorem for the existence of a backward bifurcation for discrete dynamical systems. We construct nonstandard finite difference (NSFD) schemes and prove that they preserve the backward bifurcation property of the continuous models. We make the results more specific for the SIS and malaria models for which we also provide numerical simulations that support the theory. In particular we prove for the malaria model a conjecture by Chitnis et al. (2006) for the existence of the backward bifurcation.