Backward bifurcation analysis for two continuous and discrete epidemiological models

Abstract

The bifurcation analysis of a continuous n‐dimensional nonlinear dynamical system with a nonhyperbolic equilibrium point is done by using the main theorem in the work of Castillo‐Chavez and Song. We derive an analog of this theorem for discrete dynamical systems. We design nonstandard finite difference schemes for a susceptible‐infectious‐susceptible epidemiological model with vaccination and for a malaria model. For the latter model, we sharpen the interval of the values of the disease induced death rate for which backward bifurcation may occur. Applying the discrete theorem, it is shown that each nonstandard finite difference scheme replicates the property of the continuous model of having backward bifurcation at the value one of the basic reproduction number.