An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population
We present a mathematical model of disease (say a virus) spread that takes into account the hierarchic structure of social clusters in a population. It describes the dependence of epidemic’s dynamics on the strength of barriers between clusters. These barriers are established by authorities as preve...
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MDPI AG
2020-08-01
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| Online Access: | https://www.mdpi.com/1099-4300/22/9/931 |
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| author | Andrei Khrennikov Klaudia Oleschko |
| author_facet | Andrei Khrennikov Klaudia Oleschko |
| author_sort | Andrei Khrennikov |
| collection | DOAJ |
| description | We present a mathematical model of disease (say a virus) spread that takes into account the hierarchic structure of social clusters in a population. It describes the dependence of epidemic’s dynamics on the strength of barriers between clusters. These barriers are established by authorities as preventative measures; partially they are based on existing socio-economic conditions. We applied the theory of random walk on the energy landscapes represented by ultrametric spaces (having tree-like geometry). This is a part of statistical physics with applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy levels composing this barrier. Infection spreads rather easily inside a social cluster (say a working collective), but jumps to other clusters are constrained by social barriers. The model implies the power law, <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>a</mi></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> for approaching herd immunity, where the parameter <i>a</i> is proportional to inverse of one-step barrier <inline-formula><math display="inline"><semantics><mrow><mo>Δ</mo><mo>.</mo></mrow></semantics></math></inline-formula> We consider linearly increasing barriers (with respect to hierarchy), i.e., the <i>m</i>-step barrier <inline-formula><math display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>m</mi></msub><mo>=</mo><mi>m</mi><mo>Δ</mo><mo>.</mo></mrow></semantics></math></inline-formula> We also introduce a quantity characterizing the process of infection distribution from one level of social hierarchy to the nearest lower levels, spreading entropy <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">E</mi><mo>.</mo></mrow></semantics></math></inline-formula> The parameter <i>a</i> is proportional to <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">E</mi><mo>.</mo></mrow></semantics></math></inline-formula> |
| first_indexed | 2024-03-10T16:51:14Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T16:51:14Z |
| publishDate | 2020-08-01 |
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| spelling | doaj.art-d37aa356fba242eba3feccd30f9435792023-11-20T11:19:36ZengMDPI AGEntropy1099-43002020-08-0122993110.3390/e22090931An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the PopulationAndrei Khrennikov0Klaudia Oleschko1International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, SE-351 95 Växjö, SwedenCentro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, 76230 Queretaro, MexicoWe present a mathematical model of disease (say a virus) spread that takes into account the hierarchic structure of social clusters in a population. It describes the dependence of epidemic’s dynamics on the strength of barriers between clusters. These barriers are established by authorities as preventative measures; partially they are based on existing socio-economic conditions. We applied the theory of random walk on the energy landscapes represented by ultrametric spaces (having tree-like geometry). This is a part of statistical physics with applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy levels composing this barrier. Infection spreads rather easily inside a social cluster (say a working collective), but jumps to other clusters are constrained by social barriers. The model implies the power law, <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>a</mi></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> for approaching herd immunity, where the parameter <i>a</i> is proportional to inverse of one-step barrier <inline-formula><math display="inline"><semantics><mrow><mo>Δ</mo><mo>.</mo></mrow></semantics></math></inline-formula> We consider linearly increasing barriers (with respect to hierarchy), i.e., the <i>m</i>-step barrier <inline-formula><math display="inline"><semantics><mrow><msub><mo>Δ</mo><mi>m</mi></msub><mo>=</mo><mi>m</mi><mo>Δ</mo><mo>.</mo></mrow></semantics></math></inline-formula> We also introduce a quantity characterizing the process of infection distribution from one level of social hierarchy to the nearest lower levels, spreading entropy <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">E</mi><mo>.</mo></mrow></semantics></math></inline-formula> The parameter <i>a</i> is proportional to <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">E</mi><mo>.</mo></mrow></semantics></math></inline-formula>https://www.mdpi.com/1099-4300/22/9/931disease spreadherd immunityhierarchy of social clustersultrametric spacestreessocial barriers |
| spellingShingle | Andrei Khrennikov Klaudia Oleschko An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population Entropy disease spread herd immunity hierarchy of social clusters ultrametric spaces trees social barriers |
| title | An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population |
| title_full | An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population |
| title_fullStr | An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population |
| title_full_unstemmed | An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population |
| title_short | An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population |
| title_sort | ultrametric random walk model for disease spread taking into account social clustering of the population |
| topic | disease spread herd immunity hierarchy of social clusters ultrametric spaces trees social barriers |
| url | https://www.mdpi.com/1099-4300/22/9/931 |
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