A SIR Epidemic Model Allowing Recovery

Abstract

The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, S(t) and R(t), respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of −S′(0), the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if S′(0)>0 and to the SIR model if S′(0)<0. The separating case is completely soluble and S(t) is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms.