P -adic (treelike) diﬀusion model for covid-19 1 epidemic 2

5 We present a new mathematical model of disease spread reﬂecting 6 specialties of covid-19 epidemic by elevating the role social clustering 7 of population. The model can be used to explain slower approaching 8 herd immunity in Sweden, than it was predicted by a variety of other mathematical models; see graphs Fig. 2. The hierarchic structure 10 of social clusters is mathematically modeled with ultrametric spaces 11 having treelike geometry. To simplify mathematics, we consider ho- 12 mogeneous trees with p -branches leaving each vertex. Such trees are endowed with algebraic structure, the p -adic number ﬁelds. We ap- ply theory of the p -adic diﬀusion equation to describe coronavirus’ 15 spread in hierarchically clustered population. This equation has ap- plications to statistical physics and microbiology for modeling dynam- ics on energy landscapes. To move from one social cluster (valley) to 18 another, the virus (its carrier) should cross a social barrier between 19 them. The magnitude of a barrier depends on the number of social 20 hierarchy’s levels composing this barrier. As the most appropriate for 21 the recent situation in Sweden, we consider linearly increasing barri- 22 ers. This structure matches with mild regulations in Sweden. The 23 virus spreads rather easily inside a social cluster (say working collec- 24 tive), but jumps to other clusters are constrained by social barriers. 25 This behavior matches with the covid-19 epidemic, with its cluster 26 spreading structure. Our model diﬀers crucially from the standard 27 mathematical models of spread of disease, such as the SIR-model. We 28 present socio-medical specialties of the covid-19 epidemic supporting 29 our purely diﬀusional model. 30

In the present paper, by using results of work [17] on the relax-126 ation dynamics for diffusion pseudo-differential equation on ultramet-127 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.
• Covid on surface. As was shown in study [58], the probability to become infected through some surface (say in a buses, metro, 147 shop) is practically zero. It was found that even in houses with 148 many infected (symptomatic) people, the viruses on surfaces (of 149 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]. public concerts, neither football matches. 5 has the following feature -the presence of superspreaders of in-     Geometrically ultrametric spaces can be represented as trees with hier-213 archic levels. Ultrametricity means that this metric satisfies so-called 214 strong triangle inequality: for any triple of points x, y, z. Here in each triangle the third side is Thus, a social state x is represented by a vector of the form: The vector representation of psychical, mental, and social states is very 261 common in psychology and sociology. i.e., not fix n and m. This gives the possibility to add new coordinates.

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The space of such vectors can be represented by rational numbers of  where, for each x, there exists n such that x −j = 0, j > n. Denote the 304 space of such sequences by the symbol Q p . On this space, a metric is 305 introduced in the following way. Consider two sequences x = (x j ) and 306 y = (y j ); let x j = y j , j < n, where n is some integer, but x n = y n .

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Then the distance between two vectors is defined as So, if n is negative, then distance is larger than 1, if n is nonnegative, 309 then distance is less or equal to 1. The ρ p is an ultrametric. We remark 310 that each ball can be identified with a ball of radius R = p n , n ∈ Z.

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Ball B 1 (0) = plays the important role and it is defined by special 312 symbol Z p . As in any ultrametric space, each ball is represented as 313 disjoint union of smaller balls,e.g., where a j ∈ Z p is constrained by condition x 0 = j and a j 0 ...j n−1 is 315 constrained by conditions x 0 = j 0 , ..., x n−1 = j n−1 , and so on. We It is natural to assume that the transition probability decreases with 362 increasing of the distance between two clusters, for example, that Here C α > 0 is a normalization constant, by mathematical reasons it 364 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 ,  We use the mathematical result from [17] (see also [18,19]) and ob-376 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 387 Fig. 1.

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Consider now a kind of "integral immunity", combination of innate 389 and adaptive components, defined as the probability of not become and its average over social cluster represented by ball B n , Here the new parameter T has the meaning of temperature. Thus 452 behavior of distance between valleys of the energy landscape is deter-453 mined by the size of the barrier for one-step jump ∆ and temperature. 454 We rewrite formula (13) for transition probability by using these pa-455 rameters: In our model, we introduce the notion of social temperature T. As in 457 physics, this parameter calibrates energy, in our case social energy.

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in the form: Thus, for large t, the average probability to become infected in social 471 cluster B n decreases quicker with increase of social temperature T.

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Decrease of the one-step jump barrier ∆ implies the same behavior.

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Immunity probability p Im (B n , t) behaves in the opposite way. It The use of purely diffusional model is supported by specialties of covid-19 epidemic, presented in section 2. Of course, this model is only approximate. But, it seems that it gives the right asymptotic of probabilities, to become infected and immune, in socially clustered society. social clusters, p Im (t) increase so slowly that there is practically no 495 hope to approach herd immunity. infected with disease; someone who infected the number of people far 499 exceeding the two to three. As was pointed out in MIT Technology