On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations
This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact t...
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MDPI AG
2019-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/3/430 |
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| author | Manuel De la Sen Asier Ibeas Santiago Alonso-Quesada Raul Nistal |
| author_facet | Manuel De la Sen Asier Ibeas Santiago Alonso-Quesada Raul Nistal |
| author_sort | Manuel De la Sen |
| collection | DOAJ |
| description | This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations. |
| first_indexed | 2024-04-13T07:41:50Z |
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| language | English |
| last_indexed | 2024-04-13T07:41:50Z |
| publishDate | 2019-03-01 |
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| series | Symmetry |
| spelling | doaj.art-d13ae73fa2f44f5bbdf1f0516cb51c362022-12-22T02:55:51ZengMDPI AGSymmetry2073-89942019-03-0111343010.3390/sym11030430sym11030430On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric ParameterizationsManuel De la Sen0Asier Ibeas1Santiago Alonso-Quesada2Raul Nistal3Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Barrio Sarriena, 48940 Leioa, Bizkaia, SpainDepartment of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, SpainInstitute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Barrio Sarriena, 48940 Leioa, Bizkaia, SpainInstitute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Barrio Sarriena, 48940 Leioa, Bizkaia, SpainThis paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations.https://www.mdpi.com/2073-8994/11/3/430epidemic modelirreducible matrixMetzler matrixdisease transition and transmission matricesdecentralized controldisease-free and endemic equilibrium pointsMoore–Penrose pseudoinversenext generation matrixpatchy environmentvaccination controls |
| spellingShingle | Manuel De la Sen Asier Ibeas Santiago Alonso-Quesada Raul Nistal On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations Symmetry epidemic model irreducible matrix Metzler matrix disease transition and transmission matrices decentralized control disease-free and endemic equilibrium points Moore–Penrose pseudoinverse next generation matrix patchy environment vaccination controls |
| title | On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations |
| title_full | On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations |
| title_fullStr | On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations |
| title_full_unstemmed | On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations |
| title_short | On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations |
| title_sort | on a sir model in a patchy environment under constant and feedback decentralized controls with asymmetric parameterizations |
| topic | epidemic model irreducible matrix Metzler matrix disease transition and transmission matrices decentralized control disease-free and endemic equilibrium points Moore–Penrose pseudoinverse next generation matrix patchy environment vaccination controls |
| url | https://www.mdpi.com/2073-8994/11/3/430 |
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