








key: cord-155015-w3k7r5z9
authors: Arazi, R.; Feigel, A.
title: Discontinuous transitions of social distancing
date: 2020-08-16
journal: nan
DOI: nan
sha: 
doc_id: 155015
cord_uid: w3k7r5z9

The 1st wave of COVID-19 changed social distancing around the globe: severe lockdowns to stop pandemics at the cost of state economies preceded a series of lockdown lifts. To understand social distancing dynamics it is important to combine basic epidemiology models for viral unfold (like SIR) with game theory tools, such as a utility function that quantifies individual or government forecast for epidemic damage and economy cost as the functions of social distancing. Here we present a model that predicts a series of discontinuous transitions in social distancing after pandemics climax. Each transition resembles a phase transition and, so, maybe a general phenomenon. Data analysis of the first wave in Austria, Israel, and Germany corroborates the soundness of the model. Besides, this work presents analytical tools to analyze pandemic waves.

Pandemics are complex medical and socioeconomic phenomena [1, 2] : near social interactions benefit both spread of disease and significant part of modern economy [3, 4] . During COVID-19 most government and individuals accepted significant limitation on interpersonal contacts (aka social distancing) to reduce pandemics at the cost of the economy. Can the dynamics of social distancing be explained as a self-contained socio-epidemiological model [5] [6] [7] or completely subjected to extrinsic effects [8] is a crucial question for sociophysics, a field that tries to describe social human behavior as a physical system.

Social distancing changes as pandemic unfolds, reflecting a change in personal and government beliefs in the future. The spread of disease increases the individual probability to become sick and may collapse the national health system. Social distancing reduces interpersonal interactions and complicates some individual production. The level of social distancing corresponds to equilibrium between future beliefs on epidemic size and economy cost [9, 10] .

To address social distancing classical epidemiological modeling is extended with economy or game theory tools [5-7, 11, 12] . SIR model [13] addresses disease spread as gas or network-like [14] interaction of susceptible, infected, and recovered individuals. An infected transmit the disease to susceptible from the moment of infection until he/she recovers. Social distancing reduces inter-person interaction and thus reduces the effectivity of transmission, but at the same time claims significant economic price. To find an equilibrium level of social distancing a utility function quantifies the negative weight of epidemics, social distancing itself, and the economic cost of social distancing. Utility function extension of the SIR model is in the focus of game theory treatment of vaccine policies [15] and recent estimates of COVID-19 economy damage due to social distancing [4, 9, 10, [16] [17] [18] . * sasha@phys.huji.ac.il

Here we show that social distancing during a pandemic may have discontinuous changes that to some extent resemble phase-transitions. In this work, epidemy starts to unfold according to the SIR model. At each moment social distancing depends on utility functions that quantify the final epidemic size and social distancing economy cost. Optimal values of social distancing correspond to the maxima of the utility function. Discontinuous transitions correspond to the abrupt changes in maxima locations. This work presents an analytical expression of the general utility function in a polynomial form. Utility function, so, reminds the free energy of a system during Ginzburg-Landau phase transition [19] [20] [21] [22] and may have some level of universality.

The first component of the utility function is economic gain or loss from a decrease or increase of social distancing. Thus social distancing parameter should address changes in the number of individuals that produce or take place in productive mingling [9, 23] .

This work associates social distancing with s th = 1/R, where R is a basic reproductive number. R is a major parameter of the SIR model -the average number of susceptible that an infected person infects. Epidemy breaks out when R > 1. Network interpretation maps SIR model on a network with edge occupancy T = 1−exp(−s th ) [14] . A utility function, as all other parameters of the model, depends on network topology and s th . This work considers only changes in s th . Association of social distancing transitions only with s th is supported by the fit of COVID-19 data.

The second component of the utility function comprises epidemy cost that favors social distancing. In this work, this cost corresponds to individual future probability to become sick. This probability is proportional to the forward amount of infected till the end of epidemic wave -Final Epidemy Size (FES). An important contribution of this work is an analytical presentation of FES derivatives due to basic reproductive number with the help of Lambert W function [24] [25] [26] .

The work proceeds with the presentation of the SIR model with induced transitions (SIRIT), almost analytical treatment of this model, the calibration of an epidemic and economy parameters of the model using time series of active cases and causalities during the 1st wave of COVID-19 in Austria, Israel, and Germany, followed by a discussion of obtained results and their implications.

Classical SIR model separates the population in three compartments: S susceptible, I infected and R recovered. The flux between compartments goes in the order S → I → R since susceptible becomes infected at frequency β −1 during an encounter with an infected one. Infected becomes recovered after time γ −1 on average. The population is well mixed and sustains the gas-like interaction of its members.

During COVID-19 pandemics, daily worldwide reports [27] include active cases and coronavirus deaths. Active cases are detected infected, which are a fraction of the total infected. Thus in this work, we redefine I as active cases, and instead of recovered R will use deceased D = M R, where M is Infected Fatality Rate (IFR). IFR is an average probability of an infected to die (for instance, reported Covid-19 IFR in Germany M ≈ 0.37% [28] ).

Here are SIR equations modified to this work:

where I is reported absolute amount of active cases [27] , A ′ is a ratio between actual and reported active cases, N is the population size, 0 < s < 1 is the normalized (s = S N ) amount of susceptible and D are reported deaths due to epidemy. When population size N and A ′ remain constant it is convenient to unite them into a single pa-

The parameter s th represents social distancing in this work. It also represents a threshold value to ratio of susceptible in population: number of infected growths when s > s th and reduces when s < s th . Besides s th = 1/R, where R = β/γ is basic reproduction number.

In this work s th changes with time according to utility function U (s th ). At each moment t the value of s current th changes if there exists s new th such that:

The utility function consists of two parts that represent pandemics cost C and economy gain G:

where C is proportion to epidemy size FES [15, 29] :

Epidemy cost changes with time as population advances in (s, I) space. Economy gain is some general function G(s th ). Let us expand economy gain in Taylor series around s current th : Epidemy cost C is expanded in Taylor series of the third order:

Coefficients a, b, c (as epidemy cost itself (4)) depend on s ht , s, I. Expansion of the third order, unlike the second order in (5), required due to non-linear behavior of FES (4) and because pandemic cost prevents significant lift of social distancing constraints (the third term in (6)). This work considers only a decrease in social distancing ∆s th < 0 which corresponds to the reopening of the economy. It is a consequence of an assumption that A close ≫ A open -even small constraint on social interactions bring significant economy damage. Transition occur if utility function (4), taking into account (5) and (6):

possesses any positive values for ∆s th < 0. To consider only relative changes in s th , we set U (s current th ) = 0. Utility function (7) together with condition (2) possesses Ginzburg-Landau like instability, see Figure 1 . First, no transition occurs if U < 0 for all s th . Second, discontinuous change in s th take place if there is single value U (s th ) > 0. Third, s th changes continuously when derivatives of (7) vanish near ∆s th = 0

Discontinuous transition occurs when there exists a single ∆s th < 0 root for U = 0 (7). This condition requires the determinant of quadratic function U/∆s th (7) to vanish: (dashed red line). Two cases (dotted lines) that make possible change of s th to many values do not exist becasue either continuous or discontinuous transition occur before. This work consideres only opening of population, that corresponds to reduction of s th . Utility function is approximately a cubic fuction of ∆s th (solid green). The transition predicted by exact calculation (solid blue) predict insignificant for this work changes in time and strength of the transition.

The corresponding ∆s th :

The new s new th is:

Coefficients a, b, c possess an analytic approximation, while condition (8) reduces to a polynome of the 4th order.

The first two equations of (1) have a solution in the form of Lambert W function [24] [25] [26] :

where (s 0 , I 0 ) are initial values for ratio of susceptible s and amount of infected I correspondingly, see Appendix A. Eq. (11) is valid for any (s, I) on the trajectory in time that initiates at (s 0 , I 0 ). The parameter s reaches it smallest value s min when there are no more infected (I = 0) at t = ∞:

Following (1), FES at any time t is:

and parameters a, b, c in (6) are the Taylor coefficients:

The parameters (14) are polynomials of log s of the order 1,2 and 3 correspondingly, see Appendix B. Thus condition (8) is a polynomial of the 4th order (quatric function) of log s, with coefficients that are the functions of (s 0 , I 0 , s, I, s th , β, A).

Consider population at state (s 0 , I 0 , s, I, s th , β, A). Figure 2 . Transitions of s th take place until (16) predicts ∆s th = 0. Following (8) and (9) it happens when the first two derivatives of (7) vanish:

. The time to these infinite number of transitions to take place remain finite because time is finite to pass between and two values of s, see (15) .

After the limit of transition, utility function preserves continuous transition state. Otherwise if s th remain constant the utility function would make possible many values of s th with U (s th ) > 0, see Figure 1 . At the region of continuous transitions, at each moment equation:

defines s th , and (1) is solved numerically.

Transitions of s th result in time derivative discontinuities of s and I. These derivative discontinuities can be detected, see Figure 2 .

SIRIT model provided a successful fit of COVID-19 data in Austria, Germany and Israel. Austria and Israel are countries with similar population sizes and with similar policies during the initial stages of the 1st wave. Germany is a country with about ×10 population size that still demonstrates SIR like behavior during the first wave.

The main purpose of the fit is to show that selfcontained model SIRIT is capable of description dynamics of the 1st COVID-19 wave. Full optimization of the COVID-19 data fit and its validation is out of the scope of this work.

The fit proceeds in the following steps: First, a small region around the greatest number of infected, see (16) and (17) .

Complete dynamics of the first wave active cases and susceptible, see Figure 3 , follows eqs. (1) and the fitted s 0 , I 0 , s th , β, A, N together with the transitions locations t i , s i tr and strengths s i th . See Figure 3 for changes in s th . To fit casualties, an effective population size N is chosen to fit reported coronavirus death at the 100th day of the first wave. It predicted quite small N less than 1/10 Austrian population. This result to be addressed during the discussion. Besides, there exists some time shift between calculated and reported coronavirus death.

Two alternative fits of Austria COVID-19 data are brought in Table I. Table I summarizes the results for all three countries. All countries demonstrated a low size of the effective population. One of the assumptions that s 0 ≈ 1. In the case of Israel, it required to be constrained. Neither of deviations from the fit refutes the main results of this work.

The work introduces SIRIT, a standard Susceptible-Infected-Recovered (SIR) model extended with a utility function that predicts induced transitions (IT) of social distancing. The model provides almost analytical treatment and reasonable but an ambiguous fit of COVID-19 1st wave active cases and casualties. Let's summarize and discuss the main assumptions, results, and implications of this work together with some alternative approaches.

The validity of the predicted discontinuous dynamics of social distancing and results depend on the specific choice of the social distancing parameter and the choice of a utility function.

The choice of s th = 1/R as a social distancing parameter is not unique. In the framework of SIR, for instance, another valid candidate is β -the probability of disease transmission per unit time [6, 7] . This parameter depends both on clinics of infection and the mechanism of interpersonal interactions. The fit of the first social distancing transitions (deviations from SIR model) after COVID-19 climax in Austria, Germany, and Israel, but, demonstrates that β remains constant during the transitions. The purpose of the fit is to show that there is a possibility to fit the real data using many transitions. A) Active cases -reported (dashed green) and calculated with many transitions (solid blue). Before transitions (red bars) classical SIR fits well the reported active cases. There is a significant deviation of SIR from reported cases after the first transition (dashed blue). SIRIT model with many transitions fir well entire range of the first wave, though the fit was obtained using a small range around the greatest of active cases and characteristics of the first transition (horizontal error bars). B) Susceptibles and Socal distance parameter s th . At each transition s th changes its value. Series of discontinuous transitions is followed by a continuous change region. C) Coronavirus deaths. There exists a time delay between reported and calculated deaths. This delay can be explained by the long course of COVID-19.

An interesting conclusion of this work is that change in social distancing corresponds to γ -a rate of becoming immune. Transition corresponds to changes in s th = γ/β while β remains constant. It seems a fallacy because γ seems to depend only on epidemy clinics. Even so, social distancing affects γ -society on alert removes contagious individuals by distancing from confirmed sick or even from asymptomatic cases that had contact with a sick person.

In an alternative way, 1/s th = R can serve as a social distancing parameter. This choice does not change major predictions or analytical developments of this work. The parameter 0 < s th < 1 that can be compared with the fraction of susceptible in a population serves better the purpose of this work.

The price of pandemics can go beyond the final epidemy size (FES). For instance, one may include derivatives of infected in time as a psychological factor that affects individual decision making. There is a lot of room to make utility function more complicated. Relative weights of different pandemic characteristics on individual or government decision making is an important question for future investigations and out of the scope of this work.

The first wave of COVID-19 in Austria, Germany, and Israel was fitted using SIRIT in two different ways: The first, series of discontinuous transitions with con-stant economy parameters. The parameters A open , B open are fitted by the first transition. The second, economic weights are fitted for every candidate transition (devia- Figure 5 . Fit of Israel COVID-19 1st wave. The results are similar to the case of Austria. The fit is valid until the beginning of the 2nd wave about at the 70th day of the first one A) Active cases. A significant deviation exists between reported and calculated active cases even before the start if the second wave. B) Susceptible and s th . The social distancing parameter s th remains a bit higher in Israel rather than in Austria or Germany. C) Coronavirus deaths. The time delay between reported and calculated cases is smaller than in the case of Austria. It can be explained either by late or early reports of coronavirus tests or reported deaths in Israel or Austria. D) Alternative fit with two transitions. tion from SIR model). Both these scenarios include discontinuous changes in social influence and have the same first transition. All fit attempts of this work need parameters of the SIR model to be constant from pandemic climax (greatest number of infected) till the first transition. Deviations from the fit may reflect a change of regulation and test policies during the first wave.

Analysis of the first wave predicts small, less than 1/10, effective population size in all three tested countries, see Table I . An estimate of effective population size depends on the choice of M -Infected Fatality Rate (IFR). Increase/reduction in M causes proportional reduction/increase in effective population size N and predicted ration A ′ between reported and real numbers of infected. These values in Table I correspond to M = 0.1%. The reported value of M for Germany is 0.37% [28] . Thus N and A ′ maybe about ×4 lower than in Table I . Nevertheless M as low as 0.17% were reported [31] . All other prediction or results of this work, including the graphs, are independent of M . Mortality rate M > 0.3% causes non-physical A ′ < 1 in case of Israel.

An explanation of low N may be that the ini-tial lockdown separated the population in disconnected domains [32, 33] and the wave of epidemy occurred in a limited number of domains. The other possible explanation is that a significant part of the population is immune to COVID-19 [34] . Finally, SIR approach with quasi-constant parameters may be an oversimplified presentation of reality. Small effective population size during the first wave may indicate a danger of abrupt transition to a bigger population size when s th reduces below some critical value. It may result in a significant second wave of the epidemy. The critical value of the basic reproduction number was reported for some interaction networks [33, 35] , while the majority of the networks lack it [32, 36] .

To conclude, this work predicts observable transitions of social distancing and provides tools for quantitative analysis of pandemic waves. Observable phenomena are essential to test the validity of human behavior modeling. The tools, as SIR model itself, contribute to social epidemiology [37] and, to spread of non-contagious but "going-viral" phenomena [38] .

Appendix A: Analytic solution of (1)

The first two equations of (1) may be rewritten as:

using transformation ∂ log I ∂t = z, ∂z ∂t = ∂z ∂ log I ∂ log I ∂t = ∂z ∂ log I z, x = log I. Integration of (A1) results in: Let us notice that s th , f (s th ) and W (f (s th )) are constant along trajectory (s t , I t ) until value of s th changes by a transition. The values f (s th ) and W (f (s th )): Derivatives of log W (f ) due to f are invariant along (s t , I t ) trajectory, see Appendix B. Derivatives of f due to s th are polynomials of log [s t ]. Consider the first derivative of (A5) taking into account (A8):

where:

Expression (A2) can be rewritten in the form: AI 0 − s min + s th log s s 0 + s 0 −3s min s 2 th (3AI 0 − 5s min + 3s 0 ) + s min s th (AI 0 − s min + s 0 ) (AI 0 + 8s min + s 0 ) + s min s th log s s 0 × (C3) s th (2AI 0 + 7s min + 2s 0 ) + 4s min (AI 0 − s min + s 0 ) + s th log s s 0 (2s min + s th ) − 9s 2 th + 2s 2 min (AI 0 − s min + s 0 ) 2 − 12s min s 3 th + 6s 4 th

Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases

The Lancet infectious diseases

Appendix B: Derivatives of Lambert W function All derivatives of W due to f are constant along any trajectory in (s, i) space:The final expressions are invariant until s th changes because they depend on f and W only, see (A7).Appendix C: Final expressions for a,b,cThe first:The second: 

