key: cord-0799363-e7c40su1 authors: Khrennikov, A. title: Utrametric diffusion model for spread of covid-19 in socially clustered population: Can herd immunity be approached in Sweden? date: 2020-07-16 journal: nan DOI: 10.1101/2020.07.15.20154419 sha: b65f1068266d8fba9d8f22017e7380da896f2491 doc_id: 799363 cord_uid: e7c40su1 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. 2. 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 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 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. social barrier ∆ implies slower growth of herd immunity. 126 In the present paper, by using results of work [17] on the relax-127 ation dynamics for diffusion pseudo-differential equation on ultramet-128 ric spaces we reproduce the power low for dynamics of herd immunity 2 In particular, by models Tom Britton [32, 33] that was used by Swedish State Health Authority predicted that herd immunity will be approached already in May; Anders Tegnell also announced, starting from the end of April 2020, that Sweden would soon approach herd immunity, but it did not happen, neither in May, nor June and July. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. . https://doi.org/10.1101/2020.07.15.20154419 doi: medRxiv preprint to become infected through some surface (say in a buses, metro, 148 shop) is practically zero. It was found that even in houses with 149 many infected (symptomatic) people, the viruses on surfaces (of 150 say tables, chairs, mobile phones) were too weak to infect any- : "There is no significant risk of catching the disease when you go shopping. Severe outbreaks of the infection were always a result of people being closer together over a longer period of time, for example the apré-ski parties in Ischgl, Austria." During extended and careful study in Heidelberg (the German epicenter of the covid-19 epidemic) his team could also not find any evidence of living viruses on surfaces. "When we took samples from door handles, phones or toilets it has not been possible to cultivate the virus in the laboratory on the basis of these swabs. ... To actually 'get' the virus it would be necessary that someone coughs into their hand, immediately touches a door knob and then straight after that another person grasps the handle and goes on to touches their face." Streeck therefore believes that there is little chance of transmission through contact with so-called contaminated surfaces. 4 "We have a number of reports from countries who are doing very detailed contact tracing. They're following asymptomatic cases, they're following contacts and they're not finding secondary transmission onward. It is very rare -and much of that is not published in the literature," Van Kerkhove, WOH official said on June 6, 2020. "We are constantly looking at this data and we're trying to get more information from countries to truly answer this question. It still appears to be rare that an asymptomatic individual actually transmits onward." [60] . . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 16, 2020. . https://doi.org/10.1101/2020.07.15.20154419 doi: medRxiv preprint has the following feature -the presence of superspreaders of in- In Sweden, restaurants and night clubs were open, but such events were not of masscharacter. The presence in a night club or in a restaurant of one infection spreader has practically the same impact as say 5 spreaders, the effect of closed space. Moreover, the distance between the tables in restaurants also diminished the effect of high number of infected in the population. During the intensive phase of the covid-19 epidemic (the end of March and April 2020) restaurants terminated self-serving during lunches (so typical in Sweden). . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. Geometrically ultrametric spaces can be represented as trees with hier-214 archic levels. Ultrametricity means that this metric satisfies so-called 215 strong triangle inequality: for any triple of points x, y, z. Here in each triangle the third side is . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. Thus, a social state x is represented by a vector of the form: (2) The vector representation of psychical, mental, and social states is very For our purpose, modeling of epidemic, we can consider, for exam-266 ple, the following hierarchic system of social coordinates; for simplicity, 267 let index m = 0, 1, 2, ..., so the coordinate x 0 is the most important. It is natural to use it to denote states (e.g., Sweden, Russia, USA,...); 269 x 1 can be used for age; x 2 for chronic diseases, x 3 gender, x 4 for race, It is convenient to proceed with variable number of coordinates, 295 i.e., not fix n and m. This gives the possibility to add new coordinates. The space of such vectors can be represented by rational numbers of 297 the form (3) This is the basis of the number-theoretic representation of the space 299 of social states. We shall consider it later. Now we continue in the 300 vector framework. To use fruitfully ultrametric models, we have to construct a com- where, for each x, there exists n such that x −j = 0, j > n. Denote the 305 space of such sequences by the symbol Q p . On this space, a metric is 306 introduced in the following way. Consider two sequences x = (x j ) and 307 y = (y j ); let x j = y j , j < n, where n is some integer, but x n = y n . Then the distance between two vectors is defined as So, if n is negative, then distance is larger than 1, if n is nonnegative, 310 then distance is less or equal to 1. The ρ p is an ultrametric. We remark 311 that each ball can be identified with a ball of radius R = p n , n ∈ Z. Ball B 1 (0) = plays the important role and it is defined by special 313 symbol Z p . As in any ultrametric space, each ball is represented as 314 disjoint union of smaller balls,e.g., 6 Income did not play any role in Sweden during the covid-19 epidemic. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. . https://doi.org/10.1101/2020.07.15.20154419 doi: medRxiv preprint where a j ∈ Z p is constrained by condition x 0 = j and a j 0 ...j n−1 is 316 constrained by conditions x 0 = j 0 , ..., x n−1 = j n−1 , and so on. We is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. where p(x|y; t) is the transition probability: the probability that the 358 virus being present in (elementary) cluster y would jump to cluster x. 359 We suppose that this probability does not depend on time t and it is It is natural to assume that the transition probability decreases with 363 increasing of the distance between two clusters, for example, that Here C α > 0 is a normalization constant, by mathematical reasons it 365 is useful to select distance's power larger than one. Hence, The integral operator in the right-hand side is the operator of frac- To formulate the Cauchy problem, we have to add some initial probability distribution. We select the uniform probability distribution concentrated on a single ball, initially infected social cluster B n , CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. We use the mathematical result from [17] (see also [18, 19] ) and ob-377 tain that the average probability has the power behavior: Thus the average probability to become infected in a social cluster We present some graphs corresponding to different values of α at 388 Fig. 1 . Consider now a kind of "integral immunity", combination of innate 390 and adaptive components, defined as the probability of not become 391 12 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. . https://doi.org/10.1101/2020.07.15.20154419 doi: medRxiv preprint and its average over social cluster represented by ball B n , CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. Here the new parameter T has the meaning of temperature. Thus 453 behavior of distance between valleys of the energy landscape is deter-454 mined by the size of the barrier for one-step jump ∆ and temperature. 455 We rewrite formula (13) for transition probability by using these pa-456 rameters: In our model, we introduce the notion of social temperature T. As in 458 physics, this parameter calibrates energy, in our case social energy. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 16, 2020. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 16, 2020. . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 16, 2020. CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 16, 2020. . https://doi.org/10.1101/2020.07.15.20154419 doi: medRxiv preprint An SIR epidemic model with 592 free boundary Estimating the number of pharmaceutical interventions (NPIs) to reduce COVID-19 mor-618 tality and healthcare demand Basic estimation-prediction techniques for Covid-621 19, and a prediction for Stockholm The disease-627 induced herd immunity level for Covid-19 is substan-628 tially lower than the classical herd immunity level Mit-634 igation and herd immunity strategy for COVID-19 is likely 635 to fail Abstract/FREE Full Text Estimating the bur-642 den of SARS-CoV-2 in France Using data on so-645 cial contacts to estimate age-specific transmission parameters 646 for respiratory-spread infectious agents Net-SARS-CoV-2 Proc p-Adic numbers and replica symmetry 668 breaking Estimates of the 672 peak-day and the number of infected individuals dur-673 ing the covid-19 outbreak in the Stockholm region Public Health Institute of Sweden: Antikroppstester avviker 679 fran prognoserna -matematikern "ser tva förklaringar Public Health Institute of Sweden: Yngre personer har 682 högre andel antikroppar. Sverige Radion Vi får oroande rapporter om utelivet". SVT, Ny-685 heter Se hur matematikprofessorn räknar ut när Stockholm uppnar 688 flockimmunitet Public Health Institute of Sweden Directionality Theory and 701 the Entropic Principle of Natural Selection So far, no transmission of the virus Coronavirus-Update" mit Christian Drosten Coronavirus spread by asymptomatic peo-711 ple appears to be rare What's a coronavirus superspreader Social Laser