Backward bifurcation in epidemic models: Problems arising with aggregated bifurcation parameters

This study addresses problems that have arisen in the literature when calculating backward bifurcations, especially in the context of epidemic modeling. Backward bifurcations are generally studied by varying a bifurcation parameter which in epidemiological models is usually the so-called basic reproduction number R₀. However, it is often overlooked that R₀ is an aggregate of parameters in the model. One cannot simply vary the aggregate R₀ while leaving all model parameters constant as has happened many times in the literature. We investigate two scenarios. For the incorrect approach we fix all parameters in the aggregate R₀ to constant values, but R₀ is nevertheless varied as a bifurcation parameter. In the correct approach, a key parameter in R₀ is allowed to vary, and hence R₀ itself varies and acts as a natural bifurcation parameter. We explore how the outcomes of these two approaches are substantially different.