Comments on some analytical and numerical aspects of the SIR model

Abstract

The SIR (Susceptible-Infected-Removed) epidemiological model was introduced in 1927 by Kermack and McKendrick in 1927 to study the plague and cholera epidemics in London and Bombay [1] . For decades the model evaded the effort s of the community to derive explicit solution. The analytical solution of the SIR model was only obtained in 2014 [2] as an equation of state between the model variables in the phase space. Recently, Barlow and Weinstein have introduced numerical solutions based on asymptotic approximants [3] . The SIR model has been typically used to study the spread of various infectious diseases (see the monograph of Martcheva [4] ). However, outside epidemiology, the SIR model is also extensively used in modelling of online social networks, viral marketing, diffusion of ideas, spread of computer viruses, financial network contagion, etc. (see recent survey by Rodrigues and the references therein [5] ). In their article "Analytical features of the SIR model and their applications to COVID-19", Kudryashov et al. [6] aimed to establish the exact relationships among S, I and R population variables, as well as to suggest a form for the exact solution of the SIR model. While the article itself is well-written, one of the equations given there is wrong ( Eq. (14 )), which invalidates further the putative solution ( Eq. (15 )). Notably, the wrong equation reads 1 alpha I - C \ alpha I-C JJ Y = exp gamma - 1 gamma W - e gamma - gamma (1) gamma

In their article, N. A. Kudryashov, M. A. Chmykhov, M. Vigdorowitsch, "Analytical features of the SIR model and their applications to COVID-19", Applied Mathematical Modelling 90 (2021) 466-473, aimed to establish the relationships among S, I and R populations, as well as to suggest a form for the exact solution of the SIR model. One of the equations given there is not correct, which leads to an error in the solution. The objective of the present letter is to present the correct parametric solution and to discuss its numerical approximation by direct quadrature.

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