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

T. Harko, F. S. N. Lobo, M. K. Mak

In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify theanalysisofitsproperties. Thegeneral solutionoftheAbelequationisobtainedby usinga perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.

Applied Mathematics and Computation
Volume 236, Page 184
2014 June

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Faculdade de Ciências da Universidade de Lisboa Universidade do Porto Faculdade de Ciências e Tecnologia da Universidade de Coimbra
Fundação para a Ciência e a Tecnologia COMPETE 2020 PORTUGAL 2020 União Europeia