Exact closed-form solution of a modified SIR model

Ballesteros, Angel; Blasco, Alfonso; Gutierrez-Sagredo, Ivan

The exact analytical solution in closed form of a modified SIR system is presented. This is, to the best of our knowledge, the first closed-form solution for a three-dimensional deterministic compartmental model of epidemics. 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 nonlinear 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. We recall that both models have been recently used in order to describe the essentials of the dynamics of the COVID-19 pandemic.