Analytical Solution of the Susceptible-Infected-Recovered/Removed Model for the Not-Too-Late Temporal Evolution of Epidemics for General Time-Dependent Recovery and Infection Rates

Abstract

The dynamical equations of the susceptible-infected-recovered/removed (SIR) epidemics model play an important role in predicting and/or analyzing the temporal evolution of epidemic outbreaks. Crucial input quantities are the time-dependent infection (a(t)) and recovery (μ(t)) rates regulating the transitions between the compartments S→I and I→R, respectively. Accurate analytical approximations for the temporal dependence of the rate of new infections J˚(t)=a(t)S(t)I(t) and the corresponding cumulative fraction of new infections J(t)=J(t0)+∫t0tdxJ˚(x) are available in the literature for either stationary infection and recovery rates or for a stationary value of the ratio k(t)=μ(t)/a(t). Here, a new and original accurate analytical approximation is derived for general, arbitrary, and different temporal dependencies of the infection and recovery rates, which is valid for not-too-late times after the start of the infection when the cumulative fraction J(t)≪1 is much less than unity. The comparison of the analytical approximation with the exact numerical solution of the SIR equations for different illustrative examples proves the accuracy of the analytical approach.