Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations
In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold <inline-formula><math xmlns="...
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/7/1712 |
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| author | Hui Chen Xuewen Tan Jun Wang Wenjie Qin Wenhui Luo |
| author_facet | Hui Chen Xuewen Tan Jun Wang Wenjie Qin Wenhui Luo |
| author_sort | Hui Chen |
| collection | DOAJ |
| description | In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></semantics></math></inline-formula> can be used to control the stochastic dynamics of the model. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the disease will be extinct with probability 1; whereas if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model. |
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| language | English |
| last_indexed | 2024-03-11T05:31:18Z |
| publishDate | 2023-04-01 |
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| series | Mathematics |
| spelling | doaj.art-f90460870a954480841db51d8591deb12023-11-17T17:09:36ZengMDPI AGMathematics2227-73902023-04-01117171210.3390/math11071712Stochastic Dynamics of a Virus Variant Epidemic Model with Double InoculationsHui Chen0Xuewen Tan1Jun Wang2Wenjie Qin3Wenhui Luo4Department of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, ChinaDepartment of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, ChinaDepartment of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, ChinaDepartment of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, ChinaDepartment of Mathematics, Yunnan Minzu University, 2929, Yuehua Street, Chenggong District, Kunming 650500, ChinaIn this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup></semantics></math></inline-formula> can be used to control the stochastic dynamics of the model. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, the disease will be extinct with probability 1; whereas if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>R</mi><mn>0</mn><mo>*</mo></msubsup><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model.https://www.mdpi.com/2227-7390/11/7/1712epidemic modelvaccine inoculationextinctionstationary distribution |
| spellingShingle | Hui Chen Xuewen Tan Jun Wang Wenjie Qin Wenhui Luo Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations Mathematics epidemic model vaccine inoculation extinction stationary distribution |
| title | Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations |
| title_full | Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations |
| title_fullStr | Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations |
| title_full_unstemmed | Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations |
| title_short | Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations |
| title_sort | stochastic dynamics of a virus variant epidemic model with double inoculations |
| topic | epidemic model vaccine inoculation extinction stationary distribution |
| url | https://www.mdpi.com/2227-7390/11/7/1712 |
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