Analysis of an epidemiology model with a removal-rate parameter

Abstract

Abstract An epidemiology model is presented that interpolates between the classical SIS and SIR models. In contrast to those models, in the SI $$-\varepsilon \,\hbox {R}-(1-\varepsilon $$ - ε R - ( 1 - ε )S model individuals may become infected a finite number of times that is greater than one. Extensions of the model in which one or both of the S and I compartments is divided into sub-compartments allow the computation of the proportion of susceptibles who never get infected or who become infected exactly once. In the singular limit $$\varepsilon \rightarrow 0$$ ε → 0 in which the SI $$-\varepsilon \, \hbox {R}-(1-\varepsilon $$ - ε R - ( 1 - ε )S model approaches the SIS model, the number of susceptibles converges uniformly in time but the number of infectives does not converge to the number of infectives in the SIS model. However, appropriately renormalized solutions of the SI $$-\varepsilon $$ - ε R $$-(1-\varepsilon $$ - ( 1 - ε )S model differ for all time by at most $$O(\varepsilon )$$ O ( ε ) from the corresponding solutions to a modified SIS model in which the basic reproduction number decays over time.