Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies
In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number <inline-fo...
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MDPI AG
2018-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/6/12/328 |
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| author | Yanli Ma Jia-Bao Liu Haixia Li |
| author_facet | Yanli Ma Jia-Bao Liu Haixia Li |
| author_sort | Yanli Ma |
| collection | DOAJ |
| description | In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number <inline-formula> <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Finally, some numerical simulations are presented to illustrate the analysis results. |
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| spelling | doaj.art-12f234b196424e1089eba67349c6815e2022-12-22T00:48:40ZengMDPI AGMathematics2227-73902018-12-0161232810.3390/math6120328math6120328Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid StrategiesYanli Ma0Jia-Bao Liu1Haixia Li2Department of General Education, Anhui Xinhua University, Hefei 230088, ChinaSchool of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, ChinaDepartment of General Education, Anhui Xinhua University, Hefei 230088, ChinaIn this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number <inline-formula> <math display="inline"> <semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics> </math> </inline-formula>, which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. Finally, some numerical simulations are presented to illustrate the analysis results.https://www.mdpi.com/2227-7390/6/12/328basic reproductive numberequilibriumstabilitySIQR epidemic modelvaccinationelimination |
| spellingShingle | Yanli Ma Jia-Bao Liu Haixia Li Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies Mathematics basic reproductive number equilibrium stability SIQR epidemic model vaccination elimination |
| title | Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies |
| title_full | Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies |
| title_fullStr | Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies |
| title_full_unstemmed | Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies |
| title_short | Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies |
| title_sort | global dynamics of an siqr model with vaccination and elimination hybrid strategies |
| topic | basic reproductive number equilibrium stability SIQR epidemic model vaccination elimination |
| url | https://www.mdpi.com/2227-7390/6/12/328 |
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