Exactly solvable SIR models, their extensions and their application to sensitive pandemic forecasting

TL;DR

This paper provides exact solutions to the classic and generalized SIR epidemic models, applies them to COVID-19 data, and explores their dynamics, including bifurcations, to improve pandemic forecasting.

Contribution

It introduces exact analytical solutions for time-dependent SIR models and extends them to interacting regions, also analyzing bifurcations in a generalized logistic model.

Findings

Exact solutions for SIR models with time-dependent infection rates.

Application of models to Mexico's COVID-19 data with forecasting scenarios.

Identification of bifurcation to chaos in the generalized logistic model.

Abstract

The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection rates and solved explicitly when the control parameter depends on the accumulated infections at time $t$. Numerical results are presented by way of comparison. Autonomous and non-autonomous generalizations of SIR for interacting regions are also considered, including non-separability for two or more interacting regions. A reduction of simple SIR models to one variable leads us to a generalized logistic model, Richards model, which we use to fit Mexico's COVID-19 data up to day number 134. Forecasting scenarios resulting from various fittings are discussed. A critique to the applicability of these models to current pandemic outbreaks in terms of robustness is provided. Finally, we obtain the bifurcation diagram for a discretized version of Richards model, displaying period doubling bifurcation to chaos.