Exact closed-form solution of a modified SIR model

TL;DR

This paper derives an exact closed-form solution for a modified SIR epidemiological model where recovered individuals are removed, providing analytical expressions for populations and analyzing differences from the original model.

Contribution

It presents the first exact analytical solutions for a modified SIR model with recovered individuals removed, including time-dependent transmission rates.

Findings

Populations are generalized logistic functions with the same characteristic time.

Explicit solutions enable computation of key epidemiological quantities.

Differences from the original SIR model are explained via conserved quantities.

Abstract

The exact analytical solution in closed form of a modified SIR system where recovered individuals are removed from the population is presented. In this dynamical system the populations $S(t)$ and $R(t)$ of susceptible and recovered individuals are found to be generalized logistic functions, while infective ones $I(t)$ are given by a generalized logistic function times an exponential, all of them with the same characteristic time. The dynamics of this modified SIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed. The main differences between this modified SIR model and original SIR one are presented and explained in terms of the zeroes of their respective conserved quantities. Moreover, it is shown that the modified SIR model with time-dependent transmission rate can be also solved in closed form for certain realistic transmission rate functions.