key: cord-0786535-njypqq2c authors: Khrennikov, A. title: Ultrametric model for covid-19 dynamics: an attempt to explain slow approaching herd immunity in Sweden date: 2020-07-08 journal: nan DOI: 10.1101/2020.07.04.20146209 sha: 78b3f2994b3f8ba5bf5ba46c2147b16bf68b12ef doc_id: 786535 cord_uid: njypqq2c We present a mathematical model of infection dynamics that might explain slower approaching the herd immunity during the covid-19 epidemy in Sweden than it was predicted by a variety of other models; see graphs Fig. 2. The new model takes into account the hierarchic structure of social clusters in the human society. We apply the well developed theory of random walk on the energy landscapes represented mathematically with ultrametric spaces. This theory was created for applications to spin glasses and protein dynamics. 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 linearly increasing (with respect to hierarchy's levels) barriers. This structure of barriers matches with a rather soft regulations imposed in Sweden in March 2020. In this model, the infection spreads rather easily inside a social cluster (say working collective), but jumps to other clusters are constrained by social barriers. This model's feature matches with the real situation during the covid-19 epidemy, with its cluster spreading structure. Clusters need not be determined solely geographically, they are based on a number of hierarchically ordered social coordinates. The model differs crucially from the standard mathematical models of spread of disease, such as the SIR-model. In particular, our model describes such a specialty of spread of covid-19 virus as the presence of "super-spreaders" who by performing a kind of random walk on a hierarchic landscape of social clusters spreads infection. In future, this model will be completed by adding the SIR-type counterpart. But, the latter is not a specialty of covid-19 spreading. In particular, the mathematical model of covid-19 epidemy dynamics of Tom Britton [7, 8] that was used by Swedish State Health Authority predicted that the herd immunity will be approached already in May. 2 As is well known and widely debated in mass-media, the Swedish government chosen its own way to handle covid-19 epidemy, namely, without any kind of lock-down. The main aim of such a policy is approaching the herd immunity. spreading. We hope that this paper may attract attention of experts munity. 93 Consequence 2. Slow decay of epidemy. 94 The problem of approaching the herd immunity is especially impor- To model cluster dynamics, we use random walk on ultrametric 101 spaces [19] . It was widely used in studied in physics and microbiology, 102 3 "Disjoint" has the meaning of separation in social space. In physical space, clusters can essentially overlap. Social separation does not complete, say of the form of walls. There are social barriers, but a person (virus-carrier) can hope over barriers, with some probability. see [19] , In our model of the covid-19 epidemy, the virus (or its carrier) 111 randomly walks in socially clustered society. In our model, it starts 112 just from a single social cluster (this assumption is used for mathe-113 matical simplicity); it spreads relatively easily inside any cluster, but 114 to approach other social clusters it should "jump over social barriers". Just before submission of this preprint, I discovered the recent 131 paper of Britton et al. [15] . One of its authors, Britton, initiated in- ). This is a step towards cou-138 pling with our model. Before to discuss similarities and differences of 139 two models in more detail, I should study article [15] more carefully. where its coordinates x m take (typically) discrete values quantifying Thus the points of the hierarchic social space can be represented by satisfies the strong triangle inequality: for any triple of points x, y, z ∈ Z p;n . Here in each triangle, the third 195 side is less or equal not only to the sum of two other sides (as usual), 196 but even to their maximum. can be selected as its center. For our modeling, it is important that the space Z p;n can be split into disjoint social clusters. (As we shall see soon, these clusters are, in fact, balls.) Each cluster is determined by fixing a few first (the most important) social coordinates, where C j = {x : x 0 = j}. This cluster representation corresponds to the first level of social hierarchy, we distinguish points by their most important coordinate. Each of clusters C j can be represented similarly as Clusters are, in fact, ultrametric balls: where a is any point of the form a 0 = i 0 , ..., a k−1 = i k−1 and arbitrary 206 coordinates a j , j = k, ..., n − 1. The probability depends on the height ∆ of the minimal social barrier, 277 but, for large m, its contribution to is not so important. By using random walk on the tree with n levels of hierarchy and 279 approaching n → ∞ one can derive the following asymptotic behavior 280 of the relaxation probability: Set κ = T ln p/∆. If κ << 1, e.g., the social temperature is low and 282 the primary social barrier ∆ is relatively large, then the probability for For linearly growing social barriers, and n → ∞, the asymptotic 295 behavior has the following form: This average distance goes to infinity. (As can be expected, higher Infectious Diseases of Humans: Dy-374 namics and Control Stochastic Epidemic Models and Their Mit-423 igation and herd immunity strategy for COVID-19 is likely 424 to fail Abstract/FREE Full Text Estimating the burden of 431 SARS-CoV-2 in France Estimates of the 434 peak-day and the number of infected individuals dur-435 ing the covid-19 outbreak in the Stockholm region Public Health Institute of Sweden: Antikroppstester avviker 441 fran prognoserna -matematikern "ser tva förklaringar Public Health Institute of Sweden: Yngre personer har 444 högre andel antikroppar. Sverige Radion A mathematical model reveals 447 the influence of population heterogeneity on herd immunity to 448 SARS-CoV-2 Using data on so-450 cial contacts to estimate age-specific transmission parameters 451 for respiratory-spread infectious agents Epidemic spreading in scale-454 free networks Network 456 frailty and the geometry of herd immunity Dynamics on Ultrametric Spaces. PHYSICAL RE Proc. Natl Human subconscious as a p-adic dynamical sys-472 tem Memory retrieval as 474 a p-adic dynamical system 476 A p-Adic Model for the Process of Thinking Disturbed by Phys-477 iological and Information Noise Information Dynamics in Cognitive, Psychologi-480 cal, Social, and Anomalous Phenomena Probabilistic pathway representation of cognitive 482 information Toward an adequate mathematical model of 485 mental space Vi får oroande rapporter om utelivet". SVT, Ny-488 heter Se hur matematikprofessorn räknar ut när Stockholm uppnar 491 flockimmunitet. SVT Nyheter Hoppfulla teorin: Sa når vi flockim-493 munitet redan i juni Det här vet vi om coronaviruset