Ultrametric diffusion equation on energy landscape to model disease spread in hierarchic socially clustered population

Andrei Khrennikov

Physica A: Statistical Mechanics and its Applications, 2021, vol. 583, issue C

Abstract: We present a new mathematical model of disease spread reflecting some specialties of the COVID-19 epidemic by elevating the role of hierarchic social clustering of population. The model can be used to explain slower approaching herd immunity, e.g., in Sweden, than it was predicted by a variety of other mathematical models and was expected by epidemiologists; see graphs Fig. 1, 2. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider trees with the constant number p>1 of branches leaving each vertex. Such trees are endowed with an algebraic structure, these are p-adic number fields. We apply theory of the p-adic diffusion equation to describe a virus spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling dynamics on energy landscapes. 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’s levels composing this barrier. We consider linearly increasing barriers. A virus spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This behavior matches with the COVID-19 epidemic, with its cluster spreading structure. Our model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model; in particular, by notion of the probability to be infected (at time t in a social cluster C). We present socio-medical specialties of the COVID-19 epidemic supporting our model.

Keywords: Energy landscapes; Ultrametric spaces; p-adic numbers; Ultrametric diffusion equation; Social barriers; Linear growing barriers; COVID-19; Epidemic; Disease spread; Herd immunity; Hierarchy of social clusters; Rigid vs. mild social restrictions (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:583:y:2021:i:c:s0378437121005574

DOI: 10.1016/j.physa.2021.126284

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