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.

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 (t 0) + ∫ t 0 t d x J ˚ (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. [ABSTRACT FROM AUTHOR]