Backward bifurcations and multistationarity

TL;DR

This paper introduces a two-parameter unfolding criterion (Theorem 1) for systems $\dot x=f(x,\alpha_1,\alpha_2)$ arising in epidemiology, providing conditions under which a fold bifurcation moves through the positive orthant to yield two positive steady states (one stable, one unstable). The sign of $ce$ relative to zero determines backward ($ce<0$) versus forward ($ce>0$) bifurcations, and additional conditions on $c$ and $e$ imply the existence of two positive equilibria with ${\cal R}_0<1$ or ${\cal R}_0>1$. The framework is then applied to three models—Brauer’s vaccination model, Martcheva’s cholera model, and an in-host hepatitis C model—demonstrating new rigorous results on multistationarity, stability, and the possibility of a continuum of equilibria in limiting cases. The work connects center-manifold theory with epidemiological bifurcation analysis and outlines practical avenues for establishing bistability in diverse disease models, while also identifying open problems and potential extensions to more complex unfoldings. The results have implications for understanding disease persistence, control strategies, and the qualitative behavior of infection dynamics under parameter variation.

Abstract

The theory of backward bifurcations provides a criterion for the existence of positive steady states in epidemiological models with parameters where the basic reproductive ratio is less than one. It is often seen in simulations that this phenomenon is accompanied by multistationarity, i.e. the existence of more than one positive steady state, but the latter circumstance is not implied by the general theory. The central result of this paper is a theorem which gives a criterion for the existence of one stable and one unstable positive steady state for parameters where the basic reproductive ratio is less than one. It also gives a criterion for the existence of one stable and one unstable positive steady state in the case that the basic reproductive ratio is greater than one. These steady states arise in a bifurcation. It is shown that in one case, a model for the in-host dynamics of hepatitis C, this result can be used as a basis for showing the existence of parameters for which there are two positive steady states, one of which is stable. Thus, in particular, multistationarity is proved in that case. It is also shown to what extent this theorem can be applied to some other models which have been studied in the literature and what new results can be obtained. In that context the new approach is compared with those previously known.