Epidemics: towards understanding undulation and decay

Abstract

Undulation (usually called waves) of infection levels in epidemics, is not well understood. In this paper we propose a mathematical model that exhibits undulation (oscillation) and decay towards a stable state. The model is a re-interpretation of the original SIR-model obtained by postulating different constitutive relations whereby classical logistic growth with recovery is obtained. The recovery relation is based on the premise that it is only achieved after some time. This leads to a differential–difference (delay) equation which intrinsically exhibits periodicity in its solutions but not necessarily decay to asymptotic equilibrium. Limit cycles can indeed occur. An appropriate linearization of the governing equation provides a firm basis for heuristic reasoning as well as confidence in numerical calculations.