The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (mu/alpha) this model shows backward bifurcations, where mu is the death rate and alpha is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant beta. This eradication effort is explicitly expressed in terms of alpha,r, and mu As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R0 is smaller than the eradication effort for r > 1 + (mu/alpha) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.