Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# P-adic (treelike) Diffusion Model for COVID-19 Epidemic

Version 1 : Received: 15 July 2020 / Approved: 16 July 2020 / Online: 16 July 2020 (04:14:42 CEST)

How to cite: Khrennikov, A. P-adic (treelike) Diffusion Model for COVID-19 Epidemic. Preprints 2020, 2020070340. https://doi.org/10.20944/preprints202007.0340.v1 Khrennikov, A. P-adic (treelike) Diffusion Model for COVID-19 Epidemic. Preprints 2020, 2020070340. https://doi.org/10.20944/preprints202007.0340.v1

## Abstract

We present a new mathematical model of disease spread reflecting specialties of covid-19 epidemic by elevating the role social clustering of population. The model can be used to explain slower approaching herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. \ref{GROWTH2}. The hierarchic structure of social clusters is mathematically modeled with ultrametric spaces having treelike geometry. To simplify mathematics, we consider homogeneous trees with $p$-branches leaving each vertex. Such trees are endowed with algebraic structure, the $p$-adic number fields. We apply theory of the $p$-adic diffusion equation to describe coronavirus' spread in hierarchically clustered population. This equation has applications to statistical physics and microbiology for modeling {\it dynamics on energy landscapes.} To move from one social cluster (valley) to another, the 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. As the most appropriate for the recent situation in Sweden, we consider {\it linearly increasing barriers.} This structure matches with mild regulations in Sweden. The 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. We present socio-medical specialties of the covid-19 epidemic supporting our purely diffusional model.

## Keywords

energy landscapes; ultarmetric spaces; p-adic numbers; ultarmetric diffusion equation; social barriers; linear growing barriers; covid-19; epidemic; disease spread; herd immunity; hierarchy of social clusters

## Subject

Computer Science and Mathematics, Applied Mathematics