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

Abstract

We present a new mathematical model of disease spread reflecting some specialities 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. \ref{fig:minipage1},\ref{fig:minipage2}. 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 {\it 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 {\it 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 specialities of the covid-19 epidemic supporting our model.