Exact Solution to a Dynamic SIR Model


We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible-infected-removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.


Mathematics and Statistics


Torres has been partially supported by FCT within CIDMA, Portugal project UID/MAT/04106/2019, and by TOCCATA FCT, Portugal project PTDC/EEI-AUT/2933/2014 . The authors are very grateful to three anonymous reviewers for several constructive comments, questions and suggestions, which helped them to improve the paper.

Keywords and Phrases

Continuous time systems; Epidemiology; Asymptotic behaviors; Closed form solutions; Deterministic epidemic models; Time-scales; Time-varying coefficients; Time domain analysis; Closed-form solution; Dynamic equations on time scales

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Document Type

Article - Journal

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© 2019 Elsevier, All rights reserved.

Publication Date

01 May 2019