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

2014/03/10 by Tiberiu Harko, Francisco S. N. Lobo, M. K. Mak · 8 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · #COVID-19 epidemiological studies #Fractional Differential Equations Solutions #Mathematical and Theoretical Epidemiology and Ecology Models #nlin.CD #q-bio.PE

paper · pdf · doi:10.1016/j.amc.2014.03.030

published as Applied Mathematics and Computation, 236, 2014, 184-194 · 13 pages, 4 figures, accepted for publication in Applied Mathematics and Computation

arxiv created 2014/03/10 · openalex publication_date 2014/04/04 · arxiv updated 2014/04/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

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 the analysis of its properties. The general solution of the Abel equation is obtained by using a 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.

Citations

• A note on Exact solution of SIR and SIS epidemic models
• A contribution to the mathematical theory of epidemics

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