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On the Integrability of the SIR Epidemic Model with Vital Dynamics

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  • Ding Chen

Abstract

In this paper, we study the SIR epidemic model with vital dynamics S.=−βSI+μN−S,I.=βSI−γ+μI,R.=γI−μR, from the point of view of integrability. In the case of the death/birth rate μ = 0, the SIR model is integrable, and we provide its general solutions by implicit functions, two Lax formulations and infinitely many Hamilton‐Poisson realizations. In the case of μ ≠ 0, we prove that the SIR model has no polynomial or proper rational first integrals by studying the invariant algebraic surfaces. Moreover, although the SIR model with μ ≠ 0 is not integrable and we cannot get its exact solution, based on the existence of an invariant algebraic surface, we give the global dynamics of the SIR model with μ ≠ 0.

Suggested Citation

  • Ding Chen, 2020. "On the Integrability of the SIR Epidemic Model with Vital Dynamics," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).
    • Handle: RePEc:wly:jnlamp:v:2020:y:2020:i:1:n:5869275
    DOI: 10.1155/2020/5869275
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    References listed on IDEAS

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    1. Freddy Dumortier & Jaume Llibre & Joan C. Artés, 2006. "Qualitative Theory of Planar Differential Systems," Springer Books, Springer, number 978-3-540-32902-2, March.
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