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An analytical solution for the Kermack–McKendrick model

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  • Carvalho, Alexsandro M.
  • Gonçalves, Sebastián

Abstract

We present an analytical solution for the Susceptible–Infective–Removed (SIR) model introduced initially by Kermack–McKendrick in 1927. Starting from the differential equation for the removed subjects presented by them in the original article, we rewrite it in a slightly different form to derive a formal solution, unless one integration. Then, using approximate algebraic techniques, we obtain an analytic solution for the integral. We compare the numerical solution of the differential equations of the SIR model with the analytic solution here proposed, showing an excellent agreement. Finally, the present scheme allows us to represent analytically two fundamental quantities: the time of the infection peak and the fraction of immunized to stop the epidemic.

Suggested Citation

  • Carvalho, Alexsandro M. & Gonçalves, Sebastián, 2021. "An analytical solution for the Kermack–McKendrick model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).
    • Handle: RePEc:eee:phsmap:v:566:y:2021:i:c:s0378437120309572
    DOI: 10.1016/j.physa.2020.125659
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    References listed on IDEAS

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    1. Awawdeh, Fadi & Adawi, A. & Mustafa, Z., 2009. "Solutions of the SIR models of epidemics using HAM," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3047-3052.
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