An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population
In this paper, we introduce and analyze a discrete–time model of an epidemic spread in a heterogeneous population. As the heterogeneous population, we define a population in which we have two groups which differ in a risk of getting infected: a low–risk group and a high–risk group. We construct our...
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AIMS Press
2024-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2024340 |
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| author | Marcin Choiński |
| author_facet | Marcin Choiński |
| author_sort | Marcin Choiński |
| collection | DOAJ |
| description | In this paper, we introduce and analyze a discrete–time model of an epidemic spread in a heterogeneous population. As the heterogeneous population, we define a population in which we have two groups which differ in a risk of getting infected: a low–risk group and a high–risk group. We construct our model without discretization of its continuous–time counterpart, which is not a common approach. We indicate functions that reflect the probability of remaining healthy, which are based on the mass action law. Additionally, we discuss the existence and local stability of the stability states that appear in the system. Moreover, we provide conditions for their global stability. Some of the results are expressed with the use of the basic reproduction number $ \mathcal{R}_0 $. The novelty of our paper lies in assuming different values of every coefficient that describe a given process in each subpopulation. Thanks to that, we obtain the pure population's heterogeneity. Our results are in a line with expectations – the disease free stationary state is locally stable for $ \mathcal{R}_0 < 1 $ and loses its stability after crossing $ \mathcal{R}_0 = 1 $. We supplement our results with a numerical simulation that concerns the real case of a tuberculosis epidemic in Poland. |
| format | Article |
| id | doaj-art-bc82387188bd43e389b950c47c2d78bd |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-bc82387188bd43e389b950c47c2d78bd2025-01-23T05:05:30ZengAIMS PressMathematical Biosciences and Engineering1551-00182024-12-0121127740775910.3934/mbe.2024340An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous populationMarcin Choiński0Institute of of Information Technology, Warsaw University of Life Sciences – SGGW, Nowoursynowska 159 Street, building 34, 02-776 Warsaw, PolandIn this paper, we introduce and analyze a discrete–time model of an epidemic spread in a heterogeneous population. As the heterogeneous population, we define a population in which we have two groups which differ in a risk of getting infected: a low–risk group and a high–risk group. We construct our model without discretization of its continuous–time counterpart, which is not a common approach. We indicate functions that reflect the probability of remaining healthy, which are based on the mass action law. Additionally, we discuss the existence and local stability of the stability states that appear in the system. Moreover, we provide conditions for their global stability. Some of the results are expressed with the use of the basic reproduction number $ \mathcal{R}_0 $. The novelty of our paper lies in assuming different values of every coefficient that describe a given process in each subpopulation. Thanks to that, we obtain the pure population's heterogeneity. Our results are in a line with expectations – the disease free stationary state is locally stable for $ \mathcal{R}_0 < 1 $ and loses its stability after crossing $ \mathcal{R}_0 = 1 $. We supplement our results with a numerical simulation that concerns the real case of a tuberculosis epidemic in Poland.https://www.aimspress.com/article/doi/10.3934/mbe.2024340discrete–time systems$ sis $ modellocal stabilityglobal stabilitypopulation heterogeneitydynamical systems |
| spellingShingle | Marcin Choiński An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population Mathematical Biosciences and Engineering discrete–time systems $ sis $ model local stability global stability population heterogeneity dynamical systems |
| title | An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population |
| title_full | An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population |
| title_fullStr | An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population |
| title_full_unstemmed | An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population |
| title_short | An inherently discrete–time $ SIS $ model based on the mass action law for a heterogeneous population |
| title_sort | inherently discrete time sis model based on the mass action law for a heterogeneous population |
| topic | discrete–time systems $ sis $ model local stability global stability population heterogeneity dynamical systems |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2024340 |
| work_keys_str_mv | AT marcinchoinski aninherentlydiscretetimesismodelbasedonthemassactionlawforaheterogeneouspopulation AT marcinchoinski inherentlydiscretetimesismodelbasedonthemassactionlawforaheterogeneouspopulation |