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 R-0. However, it is often overlooked that R-0 is an aggregate of parameters in the model. One cannot simply vary the aggregate R-0 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 R-0 to constant values, but R-0 is nevertheless varied as a bifurcation parameter. In the correct approach, a key parameter in R-0 is allowed to vary, and hence R-0 itself varies and acts as a natural bifurcation parameter. We explore how the outcomes of these two approaches are substantially different.