On the Integrability of the SIR Epidemic Model with Vital Dynamics
In this paper, we study the SIR epidemic model with vital dynamics Ṡ=−βSI+μN−S,İ=βSI−γ+μI,Ṙ=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 in...
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Hindawi Limited
2020-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2020/5869275 |
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doaj-e5797f936a14469299c3e506a10e23352021-07-02T12:42:46ZengHindawi LimitedAdvances in Mathematical Physics1687-91201687-91392020-01-01202010.1155/2020/58692755869275On the Integrability of the SIR Epidemic Model with Vital DynamicsDing Chen0School of Economics and Management, Xi’an Shiyou University, Xi'an, ChinaIn this paper, we study the SIR epidemic model with vital dynamics Ṡ=−βSI+μN−S,İ=βSI−γ+μI,Ṙ=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.http://dx.doi.org/10.1155/2020/5869275 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Ding Chen |
| spellingShingle |
Ding Chen On the Integrability of the SIR Epidemic Model with Vital Dynamics Advances in Mathematical Physics |
| author_facet |
Ding Chen |
| author_sort |
Ding Chen |
| title |
On the Integrability of the SIR Epidemic Model with Vital Dynamics |
| title_short |
On the Integrability of the SIR Epidemic Model with Vital Dynamics |
| title_full |
On the Integrability of the SIR Epidemic Model with Vital Dynamics |
| title_fullStr |
On the Integrability of the SIR Epidemic Model with Vital Dynamics |
| title_full_unstemmed |
On the Integrability of the SIR Epidemic Model with Vital Dynamics |
| title_sort |
on the integrability of the sir epidemic model with vital dynamics |
| publisher |
Hindawi Limited |
| series |
Advances in Mathematical Physics |
| issn |
1687-9120 1687-9139 |
| publishDate |
2020-01-01 |
| description |
In this paper, we study the SIR epidemic model with vital dynamics Ṡ=−βSI+μN−S,İ=βSI−γ+μI,Ṙ=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. |
| url |
http://dx.doi.org/10.1155/2020/5869275 |
| work_keys_str_mv |
AT dingchen ontheintegrabilityofthesirepidemicmodelwithvitaldynamics |
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1721329852548644864 |