A non-standard discretized SIS model of epidemics
In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic pro...
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| Language: | English |
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AIMS Press
2022-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2022006?viewType=HTML |
| _version_ | 1818715572121108480 |
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| author | Marcin Choiński Mariusz Bodzioch Urszula Foryś |
| author_facet | Marcin Choiński Mariusz Bodzioch Urszula Foryś |
| author_sort | Marcin Choiński |
| collection | DOAJ |
| description | In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation. |
| first_indexed | 2024-12-17T19:05:26Z |
| format | Article |
| id | doaj.art-eb6a7700c9ef4850a25232d2df53798f |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-12-17T19:05:26Z |
| publishDate | 2022-01-01 |
| publisher | AIMS Press |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-eb6a7700c9ef4850a25232d2df53798f2022-12-21T21:36:00ZengAIMS PressMathematical Biosciences and Engineering1551-00182022-01-0119111513310.3934/mbe.2022006A non-standard discretized SIS model of epidemicsMarcin Choiński 0Mariusz Bodzioch1Urszula Foryś21. Institute of Information Technology, Warsaw University of Life Sciences–SGGW, Nowoursynowska 159, Warsaw 02-776, Poland2. Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, Olsztyn 10-710, Poland3. Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw 02-097, PolandIn this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.https://www.aimspress.com/article/doi/10.3934/mbe.2022006?viewType=HTMLepidemic modelingsis modelneimark-sacker bifurcationnon-standard discretization methodlocal stabilityglobal stability |
| spellingShingle | Marcin Choiński Mariusz Bodzioch Urszula Foryś A non-standard discretized SIS model of epidemics Mathematical Biosciences and Engineering epidemic modeling sis model neimark-sacker bifurcation non-standard discretization method local stability global stability |
| title | A non-standard discretized SIS model of epidemics |
| title_full | A non-standard discretized SIS model of epidemics |
| title_fullStr | A non-standard discretized SIS model of epidemics |
| title_full_unstemmed | A non-standard discretized SIS model of epidemics |
| title_short | A non-standard discretized SIS model of epidemics |
| title_sort | non standard discretized sis model of epidemics |
| topic | epidemic modeling sis model neimark-sacker bifurcation non-standard discretization method local stability global stability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2022006?viewType=HTML |
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