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In this paper, the exact analytical solution of theSusceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametricform. By using the exact solution we investigate some explicit modelscorresponding to fixed values of the parameters, and show that the numericalsolution reproduces exactly the analytical solution. We also show that thegeneralization of the SIR model, including births and deaths, described by anonlinear system of differential equations, can be reduced to an Abel typeequation. The reduction of the complex SIR model with vital dynamics to an Abeltype equation can greatly simplify the analysis of its properties. The generalsolution of the Abel equation is obtained by using a perturbative approach, ina power series form, and it is shown that the general solution of the SIR modelwith vital dynamics can be represented in an exact parametric form.

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates