Exact closed-form solution of a modified SIR model

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.

arXiv comment17 pages. New section added

References in corpus (6)

• Effective containment explains sub-exponential growth in confirmed cases of recent COVID-19 outbreak in Mainland China
• Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates
• Accurate closed-form solution of the SIR epidemic model
• Size and timescale of epidemics in the SIR framework
• How to reduce epidemic peaks keeping under control the time-span of the epidemic
• Hamiltonian structure of compartmental epidemiological models