On a backward bifurcation of an epidemic model with capacities of treatment and vaccination

Abstract

This paper presents an epidemic model with capacities of treatment and vaccination to discuss their effect on the disease spread. It is numerically shown that a backward bifurcation occurs in the basic reproduction number $\mathscr{R}_0$, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when $\mathscr{R}_0 < 1$, if the capacities are relatively small. This epidemiological implication is that, when there is not enough capacity for treatment or vaccination, the requirement $\mathscr{R}_0 < 1$ is not sufficient for effective disease control and disease outbreak may happen to a high endemic level even though $\mathscr{R}_0 < 1$.