Multiple-Timescale Solutions to the Susceptible-Infected-Recovered (SIR) Epidemic Model Equations in the Case of High Basic Reproduction Number

A class of multiple-timescale asymptotic solutions to the equations of the susceptible-infected-recovered (SIR) model is presented for the case of high basic reproduction number, with the inverse of the latter employed as the expansion parameter. High values of the basic reproduction number, a coefficient defined as the ratio of the infection and recovery rates built into the SIR model equations, are associated with escalating epidemics. Combinations of fast and slow timescales in the suggested multiple-timescale solutions prove adequate to reflect the acknowledged epidemic paradigm, which is characterized by the concatenation of a sharp outbreak with a subsequent protracted plateau. Explicit solutions for the numbers of the infected, susceptible, and recovered compartments of the SIR model are derived via the asymptotic treatment, and the epidemic peak timing and magnitude are assessed on this basis. The asymptotic results agree seamlessly with numerical simulations based on the SIR model.