An Ultrametric Random Walk Model for Disease Spread Taking into Account Social Clustering of the Population

Entropy

Klaudia Oleschko¹  Andrei Khrennikov² 

[1] Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, 76230 Queretaro, Mexico;International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, SE-351 95 Växjö, Sweden;

关键词: disease spread;    herd immunity;    hierarchy of social clusters;    ultrametric spaces;    trees;    social barriers;   

DOI  :  10.3390/e22090931

来源: DOAJ

【 摘 要 】

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, $1-t^{-a},$ for approaching herd immunity, where the parameter a is proportional to inverse of one-step barrier $\Delta .$ We consider linearly increasing barriers (with respect to hierarchy), i.e., the m-step barrier ${\Delta }_m=m\Delta .$ We also introduce a quantity characterizing the process of infection distribution from one level of social hierarchy to the nearest lower levels, spreading entropy $\mathcal{E}.$ The parameter a is proportional to $\mathcal{E}.$

【 授权许可】

Unknown