Methods for deriving necessary and sufficient conditions for backward bifurcation

Backward bifurcation has significant implications for disease control. Deriving necessary and sufficient conditions for backward bifurcation is of paramount importance to understand the reasons for its occurrence and devise effective control strategies. In this paper, we review the methods that lead to necessary and sufficient conditions for backward bifurcation in infectious disease models. We review separately the methods that apply to ODEs and methods that apply to PDEs. We further propose a new method, applicable to both ODEs and PDEs. We illustrate the methods on three examples: a novel ODE model of cholera with vaccination, a PDE version of the cholera model with vaccination, and on an eight equation model of dengue, taken from the literature.