Approximate Solutions of the SIR-Model for Describing the Coronavirus
https://doi.org/10.1134/S2304487X20050089
Abstract
A popular mathematical susceptible–infectious–recovered (SIR) model is used to describe the spread of the coronavirus. It is shown that using the first two integrals of the nonlinear system of three differential equations transforms it to one autonomous first-order differential equation with separable variables and two algebraic equations for calculating infected and recovered people with coronavirus. A feature of the reduced differential equation is that it is one-parameter model and its behavior is determined by a dimensionless parameter δ = β/(αN) depending on the transmission coefficient α, on the coefficient for the characterization of recovery β, and on the amount of contacting community N. It is demonstrated that the general solution of the SIR model can be represented in the form of quadrature. The influence of the dimensionless parameter δ and the influence of the infected patients on the characteristics describing the spread of coronavirus are investigated. Asymptotic dependences are given for the number of people that can become ill after contacts with infected people S(t), the number of ill peoples I(t), and the number of recovered an dead peoples R(t) depending on the initial number of infected people and the dimensionless parameter of the mathematical model. The results can be useful for considering the spread of COVID-19.
Keywords
coronavirus,
SIR-model,
modified mathematical model,
nonlinear differential equation,
first integral,
solution
Supporting agencies: This work was supported by the Russian Foundation for Basic Research (project no. 18-29-10025) and State Assignment of the Ministry of Higher Education and Science of the Russian Federation No. 0723-2020-0036
About the Authors
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
M. A. Chmykhov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Kudryashov N.A., Chmykhov M.A. Approximate Solutions of the SIR-Model for Describing the Coronavirus. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2020;9(5):404-411. (In Russ.) https://doi.org/10.1134/S2304487X20050089
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