Epidemic Spreading and Equilibrium Social Distancing in Heterogeneous Networks
PPandemic Risks and Equilibrium Social Distancing inHeterogeneous Networks
Hamed Amini ∗ and Andreea Minca † Abstract
We study a SIRD epidemic process among a heterogeneous population that interactsthrough a network. We give general upper bounds for the size of the epidemic startingfrom a (small) set of initially infected individuals. Moreover, we characterize the epidemicreproduction numbers in terms of the spectral properties of a relevant matrix based onthe network adjacency matrix and the infection rates. We suggest that this can be usedto identify sub-networks that have high reproduction numbers before the epidemic reachesand picks up in them. When we base social contact on a random graph with given vertexdegrees, we give limit theorems on the fraction of infected individuals. For a given socialdistancing individual strategies, we establish the epidemic reproduction number R whichcan be used to identify network vulnerability and inform vaccination policies. In the secondpart of the paper we study the equilibrium of the social distancing game and we show thatvoluntary social distancing will always be socially sub-optimal. Our numerical study usingCovid-19 data serves to quantify the absolute and relative utility gaps across age cohorts. Keywords:
Pandemic risks, SIR epidemic process, economic epidemiology, social distanc-ing, heterogeneous networks, random graphs with given vertex degrees. ∗ J. Mack Robinson College of Business, Georgia State University, Atlanta, GA 30303, USA, email: [email protected] † School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14850, USA,email: [email protected] a r X i v : . [ phy s i c s . s o c - ph ] J u l Introduction
The first wave of the coronavirus crisis has seen an unprecedented scale of lockdown mea-sures, imposed worldwide and in many cases very strictly in order to mitigate the publichealth threat. Unarguably, the economic and social impact has been devastating. The pathto reopening the economy remains uncertain, and outbreaks are expected to re-emerge assoon as measures are relaxed. While lockdown measures have been shown to have saveda tremendous number of lives, there is little disagreement that they cannot be maintainedin the long run. With a disease so contagious and widespread, the long run will indeed bemeasured in years rather than months. The path forward, at least until vaccines are proveneffective to some extent, is more likely to rely on proper guidelines from the governments,targeted quarantines, combined with the transparent information for the public rather thanstrict and un-targeted lockdowns.Despite that the public adherence to guidelines – even if those were optimal– is thedriving force in the post-lockdown world, few epidemic models incorporate individuals’decision-making. One notable exception is [Jones et al., 2020], who integrate individualdecision making in a Susceptible - Infectious - Recovered (SIR) model of contagion. Thispaper is closest in spirit to ours, and they demonstrate using U.S. micro-data that indi-viduals started to socially distance earlier than the governments mandated to do so. Theyalso investigate the optimal social distancing policy, and show that this policy should bemandated for as long as possible until an effective vaccine is available. Other works fo-cusing specifically on Susceptible - Infectious - Recovered/Dead (SIRD) modeling in thecontext of this public health crisis, e.g., [Acemoglu et al., 2020], suggest an agenda to makeepidemiological models more realistic, and in particular to address multiple sources of het-erogeneity. First, it is clear that the disease treats people very differently, and, while noage group is spared, the elderly have significantly worse outcomes. At the same time, thecontact pattern is nowhere close to the homogenous mixing of the classical SIR model.[Acemoglu et al., 2020] solve the social planner’s problem for a multi-type (multi-risk) SIRmodel and leave for further research the case where interaction occurs according to a socialgraph structure.In this paper, we set up a heterogeneous SIRD model on a random graph that underliesthe social contact structure. Individuals have different types and death risk if they contractthe disease. Moreover, types derive the connectedness profiles across the population. Forthis underlying model of contagion, we study the decision problem for each individual –parametrized by her type. Like in [Alvarez et al., 2020], we use the value of statisticallife (VSL) as the basis for the individual decision. VSL incorporates all elements that anindividual deems worth for her and is determined on the basis of how much the individualis willing to pay in order to decrease her risk of death by a small amount, e.g., a tenth ofa percent. This fits precisely into the analysis of how individuals perceive the risk of deathdue to contracting COVID-19. The average risk of death, conditional on contracting thedisease, is of the order of percentage point. The decision to socially distance is precisely in rder to mitigate the risk of death. In such an event, the loss to the individual is her VSL.In balance stands the cost to the individual from socially distancing, and we define thisas a fraction of the yearly value of statistical life (VSLY). VSLY is, roughly, VSL dividedby the remaining life expectancy. A free parameter quantifies how much of VSLY dependson the interactions over the time horizon of the decision. Clearly, the dependence of thedecision on the individual type (age) comes through a plethora of sources: VSL and VSLYthemselves, the risk of death, the connectivity profile.The risk to contract the disease depends on the fraction of potentially healthy linkagesto the total number of linkages in the system. We will call this the global network immunity.When the network immunity is high, the risk to contract the disease is low for all types.The network immunity depends on the social distancing decision of all individuals and itis obtained as a fixed point, which is our notion of voluntary social distancing ( Laissez-Faire ) equilibrium. Individuals hypothesize a level of global network immunity. Based onthe implied risk to contract the disease, their death risk if they do contract the disease,their value of statistical life and their types, they decide on the social distancing strategy.Then an actual network immunity emerges as the contagion spreads, albeit in our modelthe spread is instantaneous. The actual network immunity must match the hypothesizednetwork immunity in equilibrium. To account for the increase of the risk of death – forall types– should the hospital system be overwhelmed, the risk of death depends on thenetwork immunity in addition to the individual type, and our results are robust to variousspecifications of this dependence.Our first set of results is concerned with the size of the epidemic for a given socialdistancing strategy profile across individuals and for a given network underlying the socialinteractions (that we call interaction graph). An infection matrix is obtained from the ad-jacency matrix and the type dependent rates with which susceptible individuals seek socialcontact. Based on the spectrum of the infection matrix, we characterize the amplificationof the epidemic, namely the ratio between the fraction of infected individuals during thecontagion process and the size of the initial seed. In particular, we show that if the largestsingular value of the infection matrix is smaller than one, then the expected amplificationis of the order O ( √ n ) in the size of the network n . This represents a testable conditionwhether a given interaction graph is prone to contagion and can guide governments whereto focus an eventual vaccine or identify potential infection hotspots. In the same spirit,we extend our analysis to the case of random graphs underlying the social interaction. Weimpose mild conditions on the degree distribution of the susceptible population, whereasinfectious and recovered individuals’ degree distribution can be more arbitrary. There is amaximum degree condition, which allows for degrees that grow sub-linearly with the net-work size. Our results are in this case asymptotic, and for large networks we establishthe network immunity as the unique fixed point of an analytic function depending on thedegree distributions and the initial seed size. We then establish the asymptotic limit for thefraction of susceptible individuals. Moreover, we establish the basic reproduction numberof the epidemic R in the context of our model, defined as the expected number of links f those individuals infected by one initial seed. We show that R characterizes the spreadof epidemic in the usual way: If this is larger than one the epidemic, starting from smallinitial infected individuals, is explosive, whereas if it is below one then the epidemic willdie out.Our main results refer to the equilibrium of the social distancing game. All individualschoose their strategies and, in equilibrium, the realized network immunity – determinedusing our first set of results – must match the hypothesized network immunity. The spaceof social activity levels is discretized, with zero representing (fully) social distancing andthe maximum given by a government imposed level. Individuals are assumed myopic: theymake short term decisions which have lifelong implications or even imply death. For clar-ity, we think of the decision process as daily and of course, it suffices to scale our resultsfor weekly, monthly or other short term horizon one deems realistic for the individuals’commitment to their social distancing strategy. Individuals derive short-term utility fromsocial activity, and we assume that this scales linearly with their number of contacts. Thisis counteracted by the probability of contracting the disease (over the same time horizon),multiplied by the type-specific death (or sequela) probability given infection. The prob-ability of contracting the disease clearly depends on the individual rates of contacts andon the aggregate decisions of everyone else, through the network immunity. We show thatthere is at most one equilibrium, which can be given semi-analytically. For the case withtwo possible decisions, to socially distance or not, and when the graph is regular, the socialutility averaged over the population has a particularly simple form. For the regular graphcase, we show that the voluntary social distancing will always lead to a lower average utilitythan the social optimum, and this result holds irrespective of the functional dependence ofthe death rate on network immunity. Put simply, even when people are in full recognitionof the impact on the heath system and health outcomes of having a large outbreak, theirdecisions will have worse utility than the social optimum.We then proceed to examine numerically the gap between the Laissez-Faire equilibriumand social optimum for our model, calibrated to the Covid-19 current data. Several pointsemerge from this study: As we increase the fraction of social contact in VSLY, all agegroups will practice less and less social distancing. However, for the youngest cohorts, therate of decrease is highest. This effect could only increase if the fraction of social contactis non-constant across age groups and higher for the younger ones. Second, if individualsoverestimate network immunity (or the epidemic size is downplayed), then they will choosehigher levels of activity than if they had perfect knowledge of the state of the networkepidemic. In doing so, the epidemic becomes large.We next investigate the utility gap between the social optimum and the voluntary socialdistancing. We find that the gap is one magnitude more significant when the death ratesgiven infection depend on the global network immunity. Related literature.
Our work is part of the vast literature on SIR epidemics on randomnetworks, to name just a few [Ball and Sirl, 2016, Stegehuis et al., 2016, Janson et al., 2014, astor-Satorras et al., 2015, Ball et al., 2009, Kiss et al., 2017, Ball et al., 2010, Ball et al., 2014,Barbour et al., 2013, Britton et al., 2007, Draief and Massouli, 2010]. We continue on thesame line as [Janson et al., 2014] who studies the SIR epidemics dynamics in the configura-tion model. We partly extend his work (which on the other hand allows for time dynamics)to allow for different individual types. Our proof also is quite different and we allow forthe more general class of epidemics represented by the independent threshold model withdifferentiated types. This may be of interest in itself.The second related strand of literature is on economics of information security for homo-geneous networks, see e.g. [Gordon and Loeb, 2002, Lelarge, 2012a, Acemoglu et al., 2016]and on games on network [Jackson, 2010, Jackson and Zenou, 2015]. In [Gordon and Loeb, 2002],the authors consider a simple one-period economic model for a single individual who takesinto account a monetary loss should infection occur and a probability depending on se-curity investment to become infected. The security investment choice is analogous to thesocial distancing. [Lelarge, 2012a] gives a a sufficient condition for monotone investmentwhich guarantees that when network vulnerability is higher individuals invest more. Wemake the equivalent assumption here that there is less social distancing when network im-munity is higher. In particular, more closely related to our paper, [Acemoglu et al., 2016,Lelarge, 2012a] analyze the network security game (strong versus weak protection) for con-tact process in random networks. We generalize their results by allowing more social dis-tancing levels, moreover type dependent.Following the health emergency, several papers study the equilibrium social distanc-ing for COVID-19, see e.g., [Acemoglu et al., 2020, Farboodi et al., 2020, Jones et al., 2020,Ferguson et al., 2006, Ferguson et al., 2020, Del Valle et al., 2007, Prem et al., 2020, Miller et al., 2010,Prem et al., 2017, Mossong et al., 2008, Toxvaerd, 2020]. Our paper is to our best knowl-edge the first to allow for a network underlying social contacts and heterogeneity. Outline.
The paper is structured as follows. In Section 2 we provide the modelingframework for heterogeneous SIRD epidemics and state our main results regarding the finaloutcome of the epidemic on given networks and random networks underlying the socialcontacts. In Section 3 we consider the network social distancing game. In Section 4, weillustrate how our model can be applied to the Covid-19 public health crisis and calibratethe parameters. Section 5 concludes and the Appendix A contains all the proofs.
In this section we introduce the epidemic model and state the main results, for a givenindividuals social distancing strategies profile across individuals, when contact takes placeon general networks and respectively on random networks with given vertex degrees. .1 Heterogeneous SIRD epidemics We consider a heterogeneous stochastic SIR epidemic process with a possibility of death,i.e., a SIRD (Susceptible → Infectious → Recovered/Dead) model, which is a Markovianmodel for spreading a disease or virus in a finite population. Our population is assumed tointeract via a network G ( n ) . The set of nodes [ n ] := { , , . . . , n } represents individuals orhouseholds, and the edges represent (potential) connections between individuals. Connec-tions can stem from various sources, and the network is understood to aggregate all thesesources. Individuals susceptible to the epidemic may become infected through contact withother infected neighboring individuals.The population is heterogeneous, individuals can be of different types (e.g., age, sex,blood type, etc.) in a certain type space T , large enough to classify all individuals to theavailable information. We use the notations t = ( t , t , . . . , t n ) to denote the type profilesof all individuals.Moreover, we consider a finite ensemble of social distancing strategies S = { , , , . . . , K } ,with 0 representing complete isolation. We use the notations s = ( s , s , . . . , s n ) and s − i = ( s , . . . , s i − , s i +1 , . . . , s n ) to denote the social distancing profiles of all individualsand all individuals other than i respectively.We assume that at time t = 0, all individuals have only partial information about thenetwork characteristics, the epidemic parameters and the initial conditions. An individualof type t will get utility u ( s ) t by choosing social distancing strategy s ∈ S . The socialdistancing equilibrium will be discussed in Section 3.Let us denote by n ( s ) t the number of individuals of type t ∈ T with social distancingstrategy s ∈ S so that (cid:88) t ∈T (cid:88) s ∈S n ( s ) t = n. It is understood that the network is parametrized by its size (and indeed all quantitieswe define depend on n , which we leave out from the notation for simplicity). We seek tounderstand the outcomes of a major outbreak as the size of the network becomes large.The following condition is standard: for all t ∈ T and s ∈ S , n ( s ) t n −→ µ ( s ) t as n → ∞ . (1)We let µ t := (cid:80) s ∈S µ ( s ) t be the (asymptotic) fraction of individuals with type t .The initial condition of the epidemic is given by the set of initially infected individuals I (0), the set of initially removed individuals R (0) and the set of susceptible individuals S (0). The set of initially removed individuals could be interpreted as a set of immune ornon-susceptible individuals. In the later stages of the epidemic, the set of removed nodeswill grow with the recovered or dead individuals. Note that S (0) ∪ I (0) ∪ R (0) = [ n ] andthe initial conditions may also depend on the type of each individual.Each infected individual, throughout its infection period, infects (makes infectious con- act with) any susceptible neighbor individual with type t and social distancing strategy s at the points of a Poisson process with rate β ( s ) t >
0. We assume there is no latent periodso that the contacted susceptible individuals are immediately infected and are able to infectother individuals.To simplify the analysis, we assume that the infection time ρ is constant and, withoutloss of generality, we scale the time to make the constant ρ = 1. The infected individualwith type t dies after time ρ with probability κ t and recovers with probability 1 − κ t . Notethat if ρ (cid:54) = 1, it suffices to replace the infection rates β by ρβ .We will allow in Section 3 the fatality probability to depend on the fraction (number)of infected people during the epidemic process, as the hospital system can be overwhelmed.In our baseline model an infected individual knows that he is infected and from this mo-ment he is indifferent (the social activity doesn’t depend on type for infected individuals)regarding his social activity. The assumption that the infection rate depends only on thetype and strategy of the susceptible party is implicitly assuming a conservative setting inwhich the effort to avoid infection comes from the susceptibles. One could make additionalassumptions on the infectives, on whom we could impose quarantine or we could modeladditional elements in their utility functions to entail concern for their family and friends.Here we leave these considerations aside, in order to focus on the individual’s decision whenthe utility includes only her own value of life.We also assume that the recovered individuals are no longer infectious, and moreoverimmune to further infections. Note that this remains a point of active research for Covid-19. The epidemic process continues until there are no infective individuals present in thepopulation. Each alive individual is then either still susceptible, or else they have beeninfected and have recovered.We assume that there are initially n S , n I and n R susceptible, infective and removed(recovered or dead) individuals, respectively. Moreover, for each type t ∈ T , there arerespectively n S,t , n
I,t and n R,t of these individuals with type t . Hence, we have |S (0) | = n S , |I (0) | = n I , |R (0) | = n R , n S = (cid:88) t ∈T n S,t , n I = (cid:88) t ∈T n I,t , n R = (cid:88) t ∈T n R,t and n S + n I + n R = n. We are then interested in S ( s ) ( ∞ ) and R ( s ) ( ∞ ) the final set of susceptible and removedindividuals, respectively, when the individuals follow the social distancing strategy s . Sim-ilarly, S ( s ) t,d ( ∞ ) denotes the final set of susceptible individuals with type t , degree d andsocial distancing strategy s . In this section we state some general conditions on the adjacency matrix of the interactiongraph and epidemics parameters for the size of the epidemics to be small compared to thesize of the network. et A denote the adjacency matrix of the social contact graph. The probability that aninfected individual makes infectious contact with any susceptible neighbor with type t andsocial activity s is given by 1 − e − β ( s ) t . Given the social distancing profile s , we define theinfection matrix B ( s ) as B ( s ) ij := (cid:18) − e − β ( sj ) tj (cid:19) A ij , (2)for all i, j ∈ [ n ]. Note that the infection rates are not necessarily symmetric and, in general,the matrix B ( s ) might not be symmetric even (if) the adjacency matrix A is symmetric.We first give a condition on the maximum row sum of the matrix B , which gives us an up-per bound for the expected amplification of infected individuals ( |R ( s ) ( ∞ ) |−|R (0) | ) / |I (0) | .The set of removed nodes at infinity R ( s ) ( ∞ ) \ R (0) is the set of recovered or deadduring the epidemic and is same set of all individuals who have ever been infected startingfrom the initial seed. Proposition 2.1.
Let B ( s ) i = (cid:80) nj =1 B ( s ) ij and B ( s )max = max i ( B ( s ) i ) . If B ( s )max < , then E [ |R ( s ) ( ∞ ) | ] ≤ |R (0) | + 11 − B ( s )max |I (0) | , (3) which in particular implies that for all k > , P (cid:18) |R ( s ) ( ∞ ) | − |R (0) | ≥ k − B ( s )max |I (0) | (cid:19) ≤ k . We now consider the L norm of the matrix B . Let λ max ( B ) = || B || be the largestsingular value of B , which is the square root of the largest eigenvalue of the positive-semidefinite matrix B T B . The following proposition shows that the expected amplificationis O ( √ n ) whenever the largest singular value is smaller than 1. Proposition 2.2. If λ max ( B ( s ) ) < , then E [ |R ( s ) ( ∞ ) | ] ≤ |R (0) | + 11 − λ max ( B ( s ) ) (cid:112) n |I (0) | , (4) which in particular implies that for all k > , P (cid:18) |R ( s ) ( ∞ ) | − |R (0) | ≥ k − λ max ( B ( s ) ) (cid:112) n |I (0) | (cid:19) ≤ k . For n ∈ N , let d ( n ) = ( d i ) ni =1 be a sequence of non-negative integers such that (cid:80) ni =1 d i is even. We now consider a configuration model for the underlying network. We endowthe set of individuals [ n ] := { , , . . . , n } with a sequence of degrees d ( n ) . We define arandom multigraph with given degree sequence ( d i ) n as follows. To each node i , we asso-ciate d i labeled half-edges. All half-edges need to be paired to construct the graph, this s done by randomly matching them. When a half-edge of a node i is paired with a half-edge of a node j , we interpret this as an edge between i and j . We denote the resultingrandom graph by G ( n ) and we write ( i, j ) ∈ G ( n ) for the event that there is an edge be-tween i and j . It is easy to see that conditional on the multigraph being simple graph,we obtain a uniformly distributed random graph with these given degree sequences; seee.g. [van der Hofstad, 2016, Durrett, 2007].We consider asymptotics as n → ∞ for the SIRD model on the configuration model.In the remainder of the paper we will use the notation o p and p −→ in a standard way. Let { X n } n ∈ U be a sequence of real-valued random variables on a probability space (Ω , P ). If c ∈ R is a constant, we write X n p −→ c to denote that X n converges in probability to c . Thatis, for any (cid:15) >
0, we have P ( | X n − c | > (cid:15) ) → n → ∞ . Let { a n } n ∈ N be a sequence of realnumbers that tends to infinity as n → ∞ . We write X n = o p ( a n ), if | X n | /a n converges to0 in probability. If E n is a measurable subset of Ω, for any n ∈ N , we say that the sequence {E n } n ∈ N occurs with high probability ( w.h.p. ) if P ( E n ) = 1 − o (1), as n → ∞ .We assume that there are initially n ( s ) S,t,d susceptible individuals with social distancingstrategy s ∈ S , type t ∈ T and degree d ∈ N . Further, there are n I,d and n R,d infective andrecovered individuals with degree d ∈ N , respectively. Hence, we have n S = (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 n ( s ) S,t,d , n I = ∞ (cid:88) d =0 n I,d , n R = ∞ (cid:88) d =0 n R,d , and n S + n I + n R = n .For the initially infected or removed individuals we do not need to know their distributionacross types, and only their total number of links (infected linkages) matters for the epidemicdynamics. We only need to know that their initial fraction of converge as the networkbecomes large. Similarly, we need convergence of the fraction of infected linkages. Thetype, degree and social distancing strategy distribution only matters for the susceptibleindividuals.We now describe the regularity assumptions on individual degrees under individuals typeprofile t ( n ) = ( t , t , . . . , t n ) and social distancing strategy profile s ( n ) = ( s , s , . . . , s n ) .We assume that the sequence ( s , t , d ) and the set of initially susceptible, infective andrecovered individuals satisfies the following regularity conditions:( C ) The fractions of initially susceptible, infective and recovered vertices converge to some α S , α I , α R ∈ (0 , n S /n → α S , n I /n → α I , n R /n → α R . (5)Moreover, α S > C ) The degree, type and social distancing strategy of a randomly chosen susceptibleindividual converges to n ( s ) S,t,d /n S → µ ( s ) t,d , (6) or some probability distribution (cid:16) µ ( s ) t,d (cid:17) s ∈S ,t ∈T ,d ∈ N . Moreover, this limiting distribu-tion has a finite and positive mean µ S := (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dµ ( s ) t,d ∈ (0 , ∞ ) , and the average degree of a randomly chosen susceptible individual converges to µ S as n → ∞ , i.e. (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dn ( s ) S,t,d /n S → µ S . (7)( C ) The average degree over all individuals converges to some λ ∈ (0 , ∞ ), i.e. as n → ∞ n n (cid:88) i =1 d i → λ, (8)and, in more detail, for some λ S , λ I , λ R , the average degrees over susceptible, infectiveand recovered individuals converge: (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dn ( s ) S,t,d /n → λ S , ∞ (cid:88) d =0 dn I,d /n → λ I , ∞ (cid:88) d =0 dn R,d /n → λ R . (9)( C ) The maximum degree of all individuals is not too large: d max = max { d i : i = 1 , . . . , n } = o ( n ) . (10)Our first theorem concerns the case where λ I > Theorem 2.3.
Suppose that ( C ) − ( C ) hold and λ I > . Then there is a unique solution x ( s ) ∗ ∈ (0 , to the fixed point equation x = f ( s ) ( x ) , where f ( s ) ( x ) := λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dµ ( s ) t,d λ (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − . (11) Moreover, the final fraction (probability) of susceptible nodes with degree d ∈ N , type t ∈ T and social distancing strategy s ∈ S satisfies: |S ( s ) t,d ( ∞ ) | n ( s ) S,t,d p −→ (cid:16) x ( s ) ∗ + (cid:0) − x ( s ) ∗ (cid:1) e − β ( s ) t (cid:17) d . (12)We can interpret x ( s ) ∗ as the probability that a neighbor of a randomly chosen susceptibleindividual does not get infected during the epidemic. The intuition behind equation (11)is the following: either that neighbor is recovered, with probability given by the first term R λ or she is susceptible. In the latter case she has degree d and strategy s with probabilityproportional to the number of links of these nodes: dn ( s ) S,t,d λ S n which converges to α S dµ ( s ) t,d λ as n goes to infinity. To be consistent with the susceptible status of this neighbour, it mustbe that all its other d − x ( s ) ∗ ) or they wereremoved before interaction (with probability (1 − x ( s ) ∗ ) e − β ( s ) t ).The same intuition applies to (12). To be consistent with the susceptible status of anindividual of degree d , type t and strategy s , it must be that all its d neighbors are eithersusceptible (with probability x ( s ) ∗ ) or they were removed before interaction (with probability(1 − x ( s ) ∗ ) e − β ( s ) t ).Our next theorem concerns the case with initially few infective individual, i.e. |I (0) | = o ( n ) and λ I = 0. Let R ( s )0 := (cid:16) α S λ (cid:17) (cid:88) s ∈S (cid:88) t ∈T (cid:16) − e − β ( s ) t (cid:17) ∞ (cid:88) d =0 d ( d − µ ( s ) t,d . (13)This quantity represents the expected number of infective links in the second generationof the epidemic, i.e., the number of linkages of those infected by the initial seed (other thanthe link from the initial seed). It is these linkages that could propagate the epidemic. Sus-ceptible individuals are reached by the initial seed according to the size biased distribution α S dµ ( s ) t,d λ that we have seen above, and they are infected with probability 1 − e − β ( s ) t . Infact, when the initial fraction of susceptible individuals is macroscopic, R ( s )0 characterizesnot only the second generation of the epidemic, but every generation in the initial stagesof the epidemic. Initially, the contagion process behaves like a branching process, whichcould either die out or explode. The following theorem states that if R ( s )0 is below 1 thenthe epidemic will die out and otherwise a positive fraction of the population will becomeinfected. Theorem 2.4.
Suppose that ( C ) − ( C ) hold and λ I = α I = 0 . The followings hold:(i) If R ( s )0 < , then the number of susceptible individuals that ever get infected is o p ( n ) .(ii) If R ( s )0 > , , then there exists δ > such that at least δn susceptible individuals getinfected with probability bounded away zero. We end this section by the following remark. Our results in this paper are all statedfor the random multigraph G ( n ) . However, they could be transferred by conditioningon the multigraph being a simple graph (without loop and multiples edges). The re-sulting random graph, denoted by G ( n ) ∗ , will be uniformly distributed among all graphswith the same degrees sequence. In order to transfer the results, we would need (seee.g., [Janson et al., 2014]) to assume that the degree distribution has a finite second mo-ment, i.e. (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d µ ( s ) t,d ∈ (0 , ∞ ) , which from [Janson, 2009b] implies that the probability that G ( n ) is simple being bounded way from zero as n → ∞ . Moreoer, as this is also stated in [Janson et al., 2014], we suspectthat all results hold even without the second moment assumption. [Bollob´as and Riordan, 2015]have recently shown results for the size of the giant component in G ( n ) ∗ from the multigraphcase without using the second moment assumption; they prove that even with the smallprobability that the multigraph is simple, the error probabilities are even smaller. We now consider heterogeneous SIRD epidemics in percolated random graph G ( n ) wherewe first generate the random graph G ( n ) (by the configuration model) and then vaccinate(remove) individuals at random. Given a probability function ω : T × N → [0 , G ( n ) ω denote the random graph obtained by randomly and independently deleting each individualof type t ∈ T and degree d ∈ N with probability ω t,d . In particular (as an example), foredge-wise vaccination, one vaccinates the end point of each susceptible half edge with somefixed probability ν independently of all the other half-edges. Thus the probability that adegree d susceptible individual is vaccinated will be ω t,d = 1 − (1 − ν ) d . Note that in the case where the social planner does not have information on the typesand degrees, degree vaccination, where we vaccinate the nodes with highest degrees, is notpossible. Edge-based vaccination is more beneficial compared to random vaccination (seee.g. [Ball and Sirl, 2016, Janson et al., 2014]).In general, the total number of individuals vaccinated will satisfy
V /n
S p −→ (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 ω t,d µ ( s ) t,d . (14)Since vaccinating an individual would be equivalent to changing its type from susceptibleto recovered, our results apply to the number of infected individual after vaccination: Theorem 2.5.
Consider heterogeneous SIRD epidemics in percolated (vaccinated) randomgraph G ( n ) . Suppose that ( C ) − ( C ) hold. Suppose further that each initially susceptibleindividual with type t ∈ T and degree k ∈ N is vaccinated with probability ω t,d ∈ [0 , independently of the others, and let λ I = α I = 0 . Let R ( s ) ω := (cid:16) α S λ (cid:17) (cid:88) s ∈S (cid:88) t ∈T (cid:16) − e − β ( s ) t (cid:17) ∞ (cid:88) d =0 d ( d − µ ( s ) t,d (1 − ω t,d ) . (15) The following hold:(i) If R ( s ) ω < , then the number of susceptible individuals that ever get infected is o p ( n ) .(ii) If R ( s ) ω > , then there exists δ > such that at least δn susceptible individuals getinfected with probability bounded away zero. This theorem can be obtain as a corollary of theorem 2.4. Indeed, it suffices to augmentthe set of removed nodes by the set of vaccinated nodes. The assumptions ( C ) − ( C ) holdwith high probability with the new distribution for the susceptible nodes ω t,d (cid:0) − µ ( s ) t,d (cid:1) . Equilibrium social distancing
In this section we introduce the social distancing game and analyse its equilibrium.
We now consider a network social distancing game in presence of an epidemic risk. Weassume that individual i can decide on social activity level s ∈ S := { , , . . . , K } for apayoff π ( s ) i = π ( s ) t i ,d i , and faces a potential loss (cid:96) i in case it becomes infected. Clearly,deciding in a higher social activity increases the payoff, i.e., π ( s ) i is strictly increasing in s .However, 0 = β (0) t < β (1) t < · · · < β ( K ) t ≤ . (16)Further, in our baseline model, the government itself might imposes a maximum level ofactivity K g , so the activity space is S g . For example, all effective strategies will be cappedby a level K g and the strategy becomes s ∧ K g .The timeline is as follows: individuals learn their potential loss in case they become in-fected. This is private information, but the distribution of (type-dependent) losses, denotedby F t , is common knowledge. Individuals then decide on their social activity level. We as-sume that at time t = 0, all individuals have only partial information about the network.Namely, they do not observe who is connected to whom. The degree-type and epidemicparameter β ( s ) t are common knowledge. Similarly, they do not know the exact nodes thatare initially infected, but only their (asymptotic) fraction.In the network of size n , we write the utility of (susceptible) node i as u i ( (cid:96), s ) = u i ( (cid:96) , ..., (cid:96) n , s , ..., s n ) := π ( s i ) t i ,d i − (cid:96) i κ t i P n ( i ∈ I ( s ) ( ∞ )) , (17)where I ( s ) ( ∞ ) = R ( s ) ( ∞ ) / R (0) denotes the final set of all cumulative infection startingfrom initial infected seed I and P n ( i ∈ I ( s ) ( ∞ )) is over the distribution of the randomgraph G ( n ) of size n , given all nodes’ degrees, losses and social activity vector s . As in[Farboodi et al., 2020], we capture the risk of loosing life or becoming ill together by asingle function κ . We refer to κ t i as the fatality probability for an infected individual.Given the loss of life or illness, there is a random loss denoted by (cid:96) i , whose distributionmight depend on t i .We say that a social activity across susceptible individuals s ∗ = ( s ∗ , s ∗ , . . . , s ∗ n ) is a(pure-strategy) Nash equilibrium if s ∗ i ∈ arg max s ∈S u i ( (cid:96) , ..., (cid:96) n , s ∗ , ...s ∗ i − , s, s ∗ i +1 , ..., s ∗ n ) , (18)for all i ∈ [ n ].Similarly, a social activity profile s ∗ = ( s ∗ , s ∗ , . . . , s ∗ n ) is a social optimum if for all ∈ S n , n (cid:88) i =1 u i ( (cid:96), s ∗ ) ≥ n (cid:88) i =1 u i ( (cid:96), s ) . We call parameter x ∗ the global network immunity, because it represents the probabilitythat a susceptible neighbor of a randomly chosen node does not get infected. Then, for agiven random network with immunity x , a representative susceptible individual with type t ,degree d and social activity s will get infected and face losses (cid:96) with (asymptotic) probability(see Theorem 2.3) κ t ( x ) (cid:18) − (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d (cid:19) . An individual of type t and degree d will get utility u ( s ) t,d ( x ) by choosing social distancingstrategy s ∈ S . More precisely, her utility is decomposed into the utility from activity,denoted by π ( s ) t,d and a cost of life loss or path to recovery modeled by a loss random variable (cid:96) (following distribution F t ) and fatality probability κ t . We also assume that κ t = κ t ( x ∗ )depends on network immunity x ∗ . This captures the fact that recovery is impacted by theperformance of the health system, which in turn depends on the network immunity andthe size of infected population. Note that the utility from social activity depends on thechoice of the susceptible individual and also on the overall fraction of individuals choosingto interact.Given the global network immunity x ∈ [0 , t and degree d maximization problem is thus s ∗ t,d := arg max s ∈S π ( s ) t,d − (cid:96)κ t ( x ) (cid:18) − (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d (cid:19) . (19)This can be interpreted as an asymptotic Nash equilibrium when the global networkimmunity x summarizes the impact of all individuals optimal social activity levels s ∗ t,d . Thisis a fixed point problem that will be described in the following. Indeed, this is an asymptoticNash equilibrium because under partial information the limit of P n ( i ∈ I ( s ) ( ∞ )) in (17)depends on the strategies of all other players only through the global network immunity.Therefore, the strategy of each individual will be given by the strategy of the representativeindividual of her degree and type.In particular, given global network immunity x , the representative individual prefers thesocial activity level s over higher social activity level s + 1 if and only if (cid:96)κ t ( x ) (cid:18)(cid:16) x + (1 − x ) e − β ( s +1) t (cid:17) d − (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d (cid:19) > π ( s ) t,d − π ( s +1) t,d . (20)Let us define for s = 0 , , . . . , K −
1, the threshold loss functions (cid:96) ( s ) t,d ( x ) := π ( s +1) t,d − π ( s ) t,d κ t ( x ) (cid:18)(cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − (cid:16) x + (1 − x ) e − β ( s +1) t (cid:17) d (cid:19) . (21) ote that (cid:96) ( s ) t,d ( x ) > x ∈ (0 ,
1) since π ( s ) t,d and β ( s ) t are strictly increasing in s and x ∈ (0 ,
1) makes the denominator positive.Hence, the representative individual prefers the social activity level s over higher socialactivity level s + 1 if and only if (cid:96) > (cid:96) ( s ) t,d ( x ). In other words, since the loss function (cid:96) followsdistribution F t , the fraction of susceptible individuals with type t and degree d which prefersocial activity s + 1 over s is given by F t (cid:16) (cid:96) ( s ) t,d ( x ) (cid:17) .( A ) We assume in the following that (cid:96) ( s ) t,d ( x ) is a decreasing function of s , i.e., for all d ∈ N , t ∈ T , x ∈ [0 , (cid:96) (0) t,d ( x ) > (cid:96) (1) t,d ( x ) > · · · > (cid:96) ( K − t,d ( x ) . (22)Note that the above assumption is only needed if there are more than two social activitylevels, i.e. K >
2. Under this condition, the optimal individual’s social activity is threshold-type: follow the social activity level s if and only if (cid:96) ∈ ( (cid:96) ( s ) t,d ( x ) , (cid:96) ( s − t,d ( x )] (set (cid:96) ( − t,d ( x ) = ∞ ).( A ) We assume in the following that (cid:96) ( s ) t,d ( x ) is an increasing function of x , i.e. κ t ( x ) (cid:18)(cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − (cid:16) x + (1 − x ) e − β ( s +1) t (cid:17) d (cid:19) is (strictly) decreasing in x .The first assumption states that the level of loss where individuals choose to sociallyisolate is higher than the level of loss where individuals choose activity level 1, and so onfor the higher activity levels. The second assumption states that the level of loss whereindividuals choose to socially isolate is higher when the network immunity is higher. Sameholds for all levels of activity. The less the global network immunity, the more susceptibleindividuals follow social distancing. We are now ready to describe the asymptotic Nash equilibrium as a fixed point problem.In the previous section we described the representative individuals’ choice given the globalnetwork immunity.Let x e denote the expected global immunity in the random network (expected ratio ofinfected edges among all the edges). Hence, the representative individual with degree d andtype t would choose social activity level s if and only if (cid:96) ( s ) t,d ( x e ) < (cid:96) ≤ (cid:96) ( s − t,d ( x e ) . So the fraction of individuals with degree d , type t and following social activity level = 0 , , . . . , K would be ¯ γ ( s ) t,d = γ ( s ) t,d ( x e ), where¯ γ ( s ) t,d = F t (cid:16) (cid:96) ( s − t,d ( x e ) (cid:17) − F t (cid:16) (cid:96) ( s ) t,d ( x e ) (cid:17) (23)and we set F t ( (cid:96) ( − ) = F t ( ∞ ) = 1. So we have µ ( s ) t,d ( x e ) = µ t,d ¯ γ ( s ) t,d . On the other hand, given the probability distributions ¯ γ : T × N → P ( S ), followingTheorem 2.3, a node i with type t and degree d will choose social activity s ∈ S as long as (cid:96) ( s ) t,d ( x ¯ γ ∗ ) < (cid:96) i ≤ (cid:96) ( s − t,d ( x ¯ γ ∗ ) , where x ¯ γ ∗ is the smallest fixed point in (0 ,
1) of x = f ¯ γ ( x ), with f ¯ γ ( x ) := λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d ¯ γ ( s ) t,d (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − . (24)Hence, the actual fraction of individuals with type t and degree d following social activity s ∈ S is given by γ ( s ) t,d = γ ( s ) t,d ( x ¯ γ ∗ ) where γ ( s ) t,d = F t (cid:16) (cid:96) ( s − t,d ( x ¯ γ ∗ ) (cid:17) − F t (cid:16) (cid:96) ( s ) t,d ( x ¯ γ ∗ ) (cid:17) . (25)Following the above analysis, for any z ∈ [0 , , t ∈ T , d ∈ N and s ∈ S , we define f γ ( z ) ( x ) := λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d γ ( s ) t,d ( z ) (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − , (26)where we set γ ( s ) t,d ( z ) = F t (cid:16) (cid:96) ( s − t,d ( z ) (cid:17) − F t (cid:16) (cid:96) ( s ) t,d ( z ) (cid:17) . (27)In the following theorem, we show the uniqueness of (asymptotic) Nash equilibrium forthe social distancing game. Theorem 3.1.
Consider a random graph G ( n ) n satisfying ( C ) − ( C ) . Assume that ( A ) − ( A ) holds. We have at most one equilibrium, which is given by the solution of the followingequation: z = inf x ∈ [0 , { x : f γ ( z ) ( x ) = x } . (28)The proof of above theorem is provided in Section A.5. .3 The effect of isolation In this section, we assume S = { , } and consider the (extreme) case where a susceptibleindividual following social distancing s = 0 isolates and cannot get infected at all, i.e., β (0) t = 0 and β (1) t = β t for all t ∈ T . Namely, s i = 1 if node i does not quarantineand s i = 0 if node i quarantines and isolate from the network. Hence, given the network(expected) global immunity x and setting π t,d = π (1) t,d − π (0) t,d , a susceptible individual i withtype t and degree d will quarantine (isolate) from the network if and only if (cid:96) i > (cid:96) t,d ( x ) := π t,d κ t ( x ) (cid:16) − ( x + (1 − x ) e − β t ) d (cid:17) . (29)In this case, all individuals following the quarantine can be removed from the networkand the epidemic goes through all other individuals. This is also equivalent to the individualvaccination (site percolation) model. Let γ t,d denote the fraction of susceptible individuals(in equilibrium) with type t and degree d following quarantine.Hence, in this case ( A ) − ( A ) are automatically verified and the equilibrium fixed pointequations (26)-(27) can be simplified to: γ t,d = 1 − F t (cid:16) π t,d κ t ( x γ ∗ ) (cid:16) − ( x γ ∗ + (1 − x γ ∗ ) e − β t ) d (cid:17) (cid:17) , (30)where x γ ∗ is the smallest fixed point in [0 ,
1] of equation f γ ( x ) := λ R λ + α S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d (cid:104) γ t,d + (1 − γ t,d ) (cid:0) x + (1 − x ) e − β t (cid:1) d − (cid:105) . (31)In Section 4 we will investigate the solution to this equation and give the application toCovid-19 for various network parameters. Let us assume π (0) t,d = 0 (“no joy in isolation”).In this case, the social utility averaged over the population converges to1 n n (cid:88) i =1 u i ( (cid:96), s ) p −→ ¯ u social ( γ ) := (cid:88) t ∈T ∞ (cid:88) d =0 µ t,d ¯ u t,d ( γ ) , with¯ u t,d ( γ ) := π t,d (1 − γ t,d ) − κ t ( x γ ∗ ) (cid:16) − (cid:0) x γ ∗ + (1 − x γ ∗ ) e − β t (cid:1) d (cid:17) (cid:90) γ F − t (1 − u ) du (32)where, for individuals with type t and degree d , π t,d (1 − γ t,d ) is the total payoff from socialactivity and κ t ( x γ ∗ ) (cid:82) γ F − t (1 − u ) du is the average loss faced by the (1 − γ t,d )-fraction ofindividuals who are not following isolation and therefore are subject to epidemic risk. .4 Social optimum for regular homogeneous networks In this section, we consider the previous isolation setup in the case of random regular graphs,where d i = d and t i = t for all nodes i ∈ [ n ]. Hence, µ t,d = 1 , λ = d, λ R = α R d , and wesimplify the notations to κ t = κ, β t = β, π t,d = π and F t = F . We use the value at risknotation for the loss distributionVaR − γ ( L ) = F − (1 − γ ) = − inf { (cid:96) ∈ R : F ( (cid:96) ) > − γ } , which represents the minimum amount of loss in 100(1 − γ )% worst-case scenarios. Theequilibrium fixed point equations are simplified toVaR − γ ( L ) = πκ ( x γ ∗ ) (cid:16) − ( x γ ∗ + (1 − x γ ∗ ) e − β ) d (cid:17) , (33)where x γ ∗ is the smallest fixed point in [0 ,
1] of equation f γ ( x ) := α R + α S (cid:104) γ + (1 − γ ) (cid:0) x + (1 − x ) e − β (cid:1) d − (cid:105) . (34)Let us assume again π (0) = 0 and π (1) = π . The social utility averaged over the populationconverges to1 n n (cid:88) i =1 u i ( (cid:96), s ) p −→ ¯ u s ( γ ) := π (1 − γ ) − κ ( x γ ∗ ) (cid:16) − (cid:0) x γ ∗ + (1 − x γ ∗ ) e − β (cid:1) d (cid:17) (cid:90) γ VaR − u ( L ) du where π (1 − γ ) is the total payoff from social activity and κ ( x γ ∗ ) (cid:82) γ VaR − u ( L ) du is theaverage loss faced by the (1 − γ )-fraction of individuals who are not following isolation andtherefore are subject to epidemic risk.The following theorem compares the fraction of self-isolating individuals in the voluntarysocial distancing equilibrium and the optimum reached by a social planner. Theorem 3.2.
The social planner will choose a larger fraction γ of individuals followingisolation than the market equilibrium for any fixed payoff π and fatality rate function κ . Figure 1a varies the link payoff π (this derives the gain form social participation and inthe numerical calibration will be taken to equal the value of statistical life). As the networkimmunity fixed point solution decreases, the final fraction of infected individuals decreases.This is intuitive: all else fixed, as the link payoff becomes larger, less people choose tosocially distance and this results into a large scale epidemic. Note the gap between thefraction of individuals who self-isolate in equilibrium versus the social optimum.In Figures 1b-1c individuals may (1%) overestimate the network immunity as they com-pute their optimal decision. We plot the final fraction of infected individuals and individualschoosing to self isolate when they have perfect observation of the global network immunityversus when they overestimate the network immunity. Under over-estimation, a lower frac- a)(b)(c) Figure 1:
Equilibrium solutions for regular homogeneous networks: Here d = 10 , α R =0 , α I = 0 . , α S = 0 . , β = 0 . , κ ( x ) = 0 . / (1 + x ) and L follows half-normal distribution L ∼ HN (0 , f ( (cid:96) ; σ ) = √ σ √ π exp (cid:16) − (cid:96) σ (cid:17) for (cid:96) ≥ σ = 100. ion self-isolate and the infection rates are higher. The difference between the two cases canbe seen as a value of information that allows people to optimally choose social distancing.Even when payoffs from the linkages are low, an error on the estimated network risk canlead to a large scale epidemics. Remark that around π = 0, and when the immunity isequal to 1, a small fraction of individuals can choose not to self isolate because their impacton overall immunity is small enough and with their over-estimation error, they still believethat the network is fully immune. Indeed, in the case when π = 0 and expected networkimmunity is one, because all individuals are indifferent, there are infinitely many equilibria.This marks the beginning of the epidemic.Figure 2 shows that, as expected, R is larger in the voluntary social distancing equi-librium (Laissez-Faire equilibrium). We note a strong dependence of the vaccination needs(in order to bring R below 1) on the link payoff. (a) (b) Figure 2: (a) R in voluntary social distancing equilibrium versus social optimum policyand (b) heat map for R with vaccination. Here d = 10 , α R = 0 , α I = 0 . , α S = 0 . , β =0 . , κ ( x ) = 0 . / (1 + x ) and L follows half-normal distribution L ∼ HN (0 , In this section we illustrate how our model can be applied to the Covid-19 public healthcrisis, to model for example a policy response without a strict lockdown, as for examplein Sweden or some U.S. states where lockdowns were released early. Note that our vol-untary social distancing equilibrium is one in which the population is fully informed orthe global contagion probability and decides to socially distance according to their util-ity. To assess the equilibrium choice in realistic settings we set the parameters of themodel to Covid-19 data. In reality, there is enormous uncertainty around the estimationof these parameters and in particular of the infection fatality ratios relevant to our model; ≥
80 13.4%Table 1:
Infection fatality ratio across ages [Verity et al., 2020].e.g., see [Manski and Molinari, 2020]. We follow the estimates in [Flaxman et al., 2020,Verity et al., 2020], based on case report data and aggregate case and death counts frommainland China, from Hong Kong and Macau, and international case reports. These areage-stratified and reproduced in Table 1. We focus on adults, over 20 years old, and studytheir social distancing decisions. It remains contentious whether children or very youngadults act as spreaders. For simplicity, we exclude them, but they can be included as anexogenous fraction of the population with prescribed behavior. The reason why they wouldbe exogenous is that we cannot expect the same decision making process as for adults, andwe would have to prescribe their behaviour.Following other studies on Covid-19 and its various policy analyses [Greenstone and Nigam, 2020],we use the U.S. Government value of a statistical life, age-adjusted, [Murphy and Topel, 2006,Aldy and Viscusi, 2008, Aldy and Viscusi, 2007]. The idea is to define based on the amountof U.S. dollars that one individual would pay in order to decrease her death probability by0 . π to be a fraction of value of year of statistical life. This fraction,for which we will have a free parameter, represents the part of the VSL that is due tointeractions with other people and participation in social life. The following table givesVSL across age groups, assuming $ 6.3 Million statistical life [Murphy and Topel, 2006]. Inexercises where the social utility needs to be reported, we will scale the results by 1 . ge group Value of statistical life Remaining life expectancy Value of yearly statistical life
20 - 29 6.8 54.5 0.2530 - 39 7 45 0.2840 - 49 6.7 36 0.3050 - 59 5 27 0.2760 - 69 4 19 0.2870 - 79 3 12 0.30 ≥
80 1 6 0.18Table 2:
Value of statistical life at different ages in $ Million, assuming $6.3 Millionstatistical life, based on [Murphy and Topel, 2006].In order to compute the yearly value of statistical life, we set an interest rate r = 3%and we put VSLY = r VSL1 − (1+ r ) − L , where L is the remaining life expectancy. As Table 2 shows,VSLY is not constant across age groups and VSL has a non-monotonous shape with a peakin the 30 −
39 age group.For simplicity, we have aggregated across gender and the remaining life expectancyrepresents an average. We have 7 types in the model, represented by the age groups, as thisis the primary factor driving the fatality rates reported in Table 1. As more information islearned on Covid-19, the types can be made more granular.We consider two social distancing level levels, S = { , } .The yearly value of a statistical life, which we denote VSLY t is the basis for calibrationof the payoff π ( s ) i = π ( s ) t,d . Namely, we set a free parameter ι , called value of social contact.The part 1 − ι represents the fraction of her yearly value of a statistical life independent onsocial contact. The part ι that is dependent of social contact is assumed to scale linearlywith the number of contacts. Namely, we set the daily gain as π ( s ) t,d = 1365 ∗ (1 − ι )VSLY t + ι VSLY t λ d for s = 1 , (1 − ι )VSLY t for s = 0 . (35)For the loss distribution of type t we choose the lognormal distribution, with type-dependent mean parameter m t and constant standard deviation σ , i.e. with density function f ( (cid:96) ; m t , σ ) = (cid:96)σ √ π exp (cid:16) − log( (cid:96) ) − m t σ (cid:17) , such that the mean satisfiesexp (cid:18) m t + σ (cid:19) = VSL t , (36)where we set σ = 3.Recall that in our model, real time is replaced by interaction time. The relevant quantityis the probability that the constant recovery is longer than the exponential between thearrivals of the Poisson process driving interactions. The other implicit notion of time comesin the decision of the individuals: they balance the risk of loss of life against the short term ge group Population distribution Mean (Standard deviation) for number of potential contacts
20 - 29 0.147 % 13.57 (10.60)30 - 39 0.179% 14.14 (10.15)40 - 49 0.177% 13.83 (10.86)50 - 59 0.178% 12.30 (10.23)60 - 69 0.154% 9.21 (7.96)70 - 79 0.099% 8.05 (6.895) ≥
80 0.066% 6.89 (5.83)Table 3:
Population distribution (conditional on age greater than 20) by age group (Source:United Nations, Department of Economic and Social Affairs, Population Division. WorldPopulation Prospects: The 2019 Revision); Average number of daily contacts in Europeancountries provided by [Mossong et al., 2008] .effects of social distancing. The short term horizon is fixed, assumed to be one day (andwith scaling any fixed term horizon). It would be interesting to the short term horizonrandom, driven for example by the arrival of a cure and the individuals’ beliefs over itstiming.Recall that in our analysis, we assumed that the infection time ρ is constant and equal(normalized) to 1. The infected individual with type t dies after time ρ with probability κ t and recovers with probability 1 − κ t . The fatality probabilities given infection are givenin table 1. Since the infection time ρ (cid:54) = 1, we replace the infection rates β by ρβ . Asin [Alvarez et al., 2020], we set ρ = 18 reflecting that on average the illness lasts for 18days. Consistent with estimates in the literature of the average number of people whowill contact an infected person R ≈
4, see [Ferguson et al., 2020], we set β such that theinfected individual has on average 4 contaminations, i.e., we set λ (cid:0) − e − ρβ (cid:1) = 4 = ⇒ ρ ¯ β = − log(1 − /λ )and we leave the distribution of β across types free, such that its average matches ¯ β .We will take the degree distribution as power law (Pareto) distribution, with a differentshape and scale parameters for each type. Namely, we set µ t,d /µ t ∼ C t d − ( α t +1) for d ≥ δ t (37)where α t is the shape parameter for type t and δ t is the scale parameter (minimum degree)for type t and C t is the normalization constant for type t . We calibrate this distribution tohave the mean (cid:98) m t and variance (cid:98) σ t given in Table 3, i.e., (cid:98) m t ≈ α t δ t α t − (cid:98) σ t ≈ α t δ t ( α t − α t − . (38)In figure 3 we plot the fraction of individuals who voluntary self-isolate in equilibriumfor varying levels of the parameter ι and across age groups. As the social contact as a Voluntary social distancing equilibrium fraction of individuals following isolationby varying the social contact parameter ι : Here α R = 0 , α I = 0 . , α S = 0 . κ t ( x ) = κ t as in Table 1.promotion of value of statistical life increases, and all the rest constant, less people practicesocial distancing. This is true for all age groups, but the decrease is much more significantfor the younger age groups.We next plot in Figure 4 the fraction of individuals socially distancing, as a function oftheir hypothesized network immunity. In this exercise, the mortality rates are independentof the epidemic. In green is the equilibrium network immunity, so one can easily read thefractions who socially distance for each age group, ranging from 25% among the 20 − . −
39, 40 − − a)(b) Figure 4: (a) Voluntary social distancing equilibrium vs. (b) Social optimal policy fractionof individuals following isolation as a function of expected network immunity: Here α R =0 , α I = 0 . , α S = 0 . , ι = 0 . κ t ( x ) = κ t as in Table 1. a)(b)(c) (d) Figure 5: (a) Voluntary social distancing vs. (b) Social optimal policy average utility(expressed in $1 . α R =0 , α I = 0 . , α S = 0 . , ι = 0 . κ t ( x ) = κ t given in Table 1. (c) The average utilitygain vs. (d) relative utility gain for a representative individual by following social optimalpolicy vs. voluntary social distancing over different age groups. a)(b)(c) (d) Figure 6: (a) Voluntary social distancing vs. (b) Social optimal policy average utility(expressed in $1 . α R =0 , α I = 0 . , α S = 0 . , ι = 0 . κ t ( x ) = exp(1 − x ) κ t , so that κ t (1) = κ t is givenin Table 1. (c) The average utility gain vs. (d) relative utility gain for a representativeindividual by following social optimal policy vs. voluntary social distancing over differentage groups. a)(b)(c) (d) Figure 7: (a) Voluntary social distancing vs. (b) Social optimal policy average utility(expressed in $1 . α R =0 , α I = 0 . , α S = 0 . , ι = 0 . κ t ( x ) = x ) κ t , so that κ t (1) = κ t is given in Table 1.(c) The average utility gain vs. (d) relative utility gain for a representative individual byfollowing social optimal policy vs. voluntary social distancing over different age groups. Conclusion
We have studied a heterogeneous SIRD epidemic process when a network underlies socialcontact. For given social distancing strategies, we have established results on the ampli-fication of the epidemic. Based on the network and the epidemic characteristics we havedefined a relevant infection matrix. The network structure - captured by the singular valueof the infection matrix – characterizes the amplification effects of the epidemic from theinitially infected set to a final set of infected individuals. Quantities such as the epidemicreproduction number R are established and can be used as a warning signal to identifyfor example parts of the networks that are highly vulnerable. Vaccination and targetedsocial distancing can be applied in accordance in such areas to make R smaller than one.Next, we have studied the equilibrium of the social distancing game. Our theoretical resultsestablish that the voluntary social distancing will always fall short of the social optimum.The social optimum itself is of course dependent on the type. We calibrate the model tothe characteristics of the Covid-19 epidemic, as current in the literature. We note that thegap between the utility in the social distancing equilibrium and the social optimum is duefor its most part to the fact that deaths rates given infection have a significant dependenceon what fraction of the population is infected, for instance because hospital capacity reach. References [Acemoglu et al., 2020] Acemoglu, D., Chernozhukov, V., Werning, I., and Whinston,M. D. (2020). A multi-risk sir model with optimally targeted lockdown. Working Paper27102, National Bureau of Economic Research.[Acemoglu et al., 2016] Acemoglu, D., Malekian, A., and Ozdaglar, A. (2016). Networksecurity and contagion.
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This appendix contains the proofs of all propositions and theorems in the main text.
A.1 Proof of Proposition 2.1
Let p i = P ( i ∈ R ( ∞ ) / R (0)). Hence, p i = 1 if i ∈ I (0) and otherwise p i ≤ (cid:80) nj =1 B ( s ) ji p j ,which writes for all i = 1 , , . . . , n as p i ≤
11 ( i ∈ I (0)) + n (cid:88) j =1 B ( s ) ji p j . (39)We thus obtain n (cid:88) i =1 p i ≤ n (cid:88) i =1 i ∈ I (0)) + n (cid:88) i =1 n (cid:88) j =1 B ( s ) ji p j = |I (0) | + n (cid:88) j =1 (cid:32) n (cid:88) i =1 B ( s ) ji (cid:33) p j ≤ |I (0) | + B ( s )max n (cid:88) j =1 p j . We conclude E [ |R ( ∞ ) | ] − |R (0) | = (cid:80) ni =1 p i ≤ − B ( s )max |I (0) | . The second statement followsusing the Markov inequality. Namely, for any k > P (cid:18) |R ( s ) ( ∞ ) | − |R (0) | ≥ k − B ( s )max |I (0) | (cid:19) ≤ k . by the Markov inequality. A.2 Proof of Proposition 2.2
Recall that from (39) we have p i ≤ i ∈ I (0)) + n (cid:88) j =1 B ( s ) ji p j . Let p = [ p , p , . . . , p n ] denote the vector with components p i , be the vector with allcomponents equal to 1 and I (0) be the vector with component 1 for i ∈ I (0) and 0 for i / ∈ I (0). By Equation (39), we have p ≤ I (0) + B s ) p . Denoting by || · || the Euclidean norm, we have || p || ≤ || I (0) + B p || ≤ || I (0) || + || B p || ≤ (cid:112) |I (0) | + λ max ( B ( s ) ) || p || . e thus have for λ max ( B ( s ) ) < || p || ≤ √ |I (0) | − λ max ( B s ) ) . Furthermore by the Cauchx-Schwarz inequality, E [ |R ( ∞ ) | ] − |R (0) | = (cid:88) i ∈ [ n ] p i = || T p || ≤ || T || || p || = √ n || p || . We conclude (if λ max ( B ( s ) ) < E [ |R ( ∞ ) | ] − |R (0) | ≤ − λ max ( B ) (cid:112) n |I (0) | , and the second statement follows using the Markov inequality. A.3 Proof of Theorem 2.3
Consider the heterogeneous SIRD epidemics spreading on G ( n ) satisfying ( C ) − ( C ). Inwhat follows, instead of taking a graph at random and then analyzing the epidemics, weuse a standard coupling argument which allows us to study epidemics and the graph at thesame time, revealing its edges dynamically while the epidemic spreads.Consider a vertex i with type t i and d i (labelled) free (not yet paired) half-edges. Wecall a half-edge type ( s, t )-susceptible, infective or removed according to the type of vertexit belongs to. A key step in the proof will be to decide from the beginning on the (random)infection threshold of each susceptible individual, denoted by Θ i for individual i , defined asthe (minimum) number of infected neighbor each individual can tolerate before it becomesinfected.Since the (normalized) infection last ρ = 1 days and the meeting happens at rate β ( s ) t over all edges for a susceptible individual with type t , degree d and following social activity s , it is easy to see that P (Θ = θ ) = e − ( θ − β ( s ) t (cid:16) − e − β ( s ) t (cid:17) =: p ( s ) t,d ( θ ) , (40)for θ = 1 , , . . . , d . Hence, µ ( s ) t,d p ( s ) t,d ( θ ) will be the asymptotic fraction of susceptible individ-uals with type d , degree d , following social activity s and getting infected after exactly θ infected neighbors. We see that the model is equivalent to (type-dependent) independentthreshold model for configuration model.In the following, we first extend the results of [Amini, 2010, Amini et al., 2013, Lelarge, 2012b]on independent threshold model in configuration model, allowing for heterogeneous typesand initial nodes removal. The theorem will then imply Theorem 2.3. Theorem A.1.
Consider the type-dependent independent threshold model with thresholddistribution p ( s ) t,d ( θ ) for all susceptible individuals with type t , degree d and social distancing s , on random graph G ( n ) n satisfying ( C ) − ( C ) . Let x ( s ) ∗ be the largest fixed point solution ∈ [0 , to x = f ( s ) ( x ) where f ( s ) ( x ) := λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 dµ ( s ) t,d λ p ( s ) t,d ( θ ) P ( Bin ( d − , − x ) ≤ θ − . (41) We have for all (cid:15) > w.h.p. |R ( s ) ( ∞ ) | n ≥ α R + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 µ ( s ) t,d p ( s ) t,d ( θ ) P (cid:16) Bin ( d, − x ( s ) ∗ ) ≥ θ (cid:17) − (cid:15). Moreover, if x ( s ) ∗ is a stable fixed point of f ( s ) ( x ) , then |R ( s ) ( ∞ ) | n p −→ α R + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 µ ( s ) t,d p ( s ) t,d ( θ ) P (cid:16) Bin ( d, − x ( s ) ∗ ) ≥ θ (cid:17) , (42) and, the final fraction (probability) of susceptible nodes with degree d ∈ N , type t ∈ T andsocial distancing strategy s ∈ S satisfies: |S ( s ) t,d ( ∞ ) | n ( s ) S,t,d p −→ d (cid:88) θ =1 p ( s ) t,d ( θ ) P (cid:16) Bin ( d, − x ( s ) ∗ ) ≤ θ − (cid:17) . (43)The proof of above theorem is provided in Appendix B. We now proceed with theproof of Theorem 2.3 using the above theorem with p ( s ) t,d ( θ ) = e − ( θ − β ( s ) t (cid:16) − e − β ( s ) t (cid:17) , for θ = 1 , , . . . , d . In this case, using the binomial theorem, we have d (cid:88) θ =1 p ( s ) t,d ( θ ) P ( Bin ( d − , − x ) ≤ θ −
1) = d (cid:88) θ =1 e − ( θ − β ( s ) t (cid:16) − e − β ( s ) t (cid:17) P ( Bin ( d − , − x ) ≤ θ − (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − , which implies that f ( s ) ( x ) = λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dµ ( s ) t,d λ (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − . as in Theorem 2.3. Moreover, the final fraction (probability) of susceptible nodes withdegree d ∈ N , type t ∈ T and social distancing strategy s ∈ S satisfies: |S ( s ) t,d ( ∞ ) | n ( s ) S,t,d p −→ d (cid:88) θ =1 e − ( θ − β ( s ) t (cid:16) − e − β ( s ) t (cid:17) P (cid:16) Bin ( d, − x ( s ) ∗ ) ≤ θ − (cid:17) = (cid:16) x ( s ) ∗ + (cid:0) − x ( s ) ∗ (cid:1) e − β ( s ) t (cid:17) d . Hence, to prove Theorem 2.3, it only remains to prove there is a unique solution x ( s ) ∗ ∈ ,
1) to the fixed point equation x = f ( s ) ( x ) which is a stable solution. Note that f ( s ) (0) > α S > f ( s ) (1) = λ R /λ + λ S /λ = 1 − λ I <
1. Moreover, f ( s ) ( x ) is strictlyincreasing in x which implies there is a unique solution x ( s ) ∗ ∈ (0 ,
1) to the fixed pointequation x = f ( s ) ( x ) and this is a stable solution (since f ( s ) ( x ) is strictly increasing). A.4 Proof of Theorem 2.4
The proof of Theorem 2.4 is based on Theorem 2.3 and a theorem by Janson [Janson, 2009a]on percolation in random graphs with given vertex degrees. Suppose that ( C ) − ( C ) holdand λ I = α I = 0. We first show that if R ( s )0 <
1, then the number of susceptible individualsthat ever get infected is o p ( n ). We prove that in the subcritical case, if λ I = 0 then x < f ( s ) ( x ) for all x ∈ [0 , f ( s ) (1) = 1 (note that if λ I = 0, we have λ = λ R + λ S which implies f ( s ) (1) = 1). Further, (cid:16) f ( s ) ( x ) − x (cid:17) (cid:48) = α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 (cid:16) − e − β ( s ) t (cid:17) d ( d − µ ( s ) t,d λ (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − − ≤ (cid:16) α S λ (cid:17) (cid:88) s ∈S (cid:88) t ∈T (cid:16) − e − β ( s ) t (cid:17) ∞ (cid:88) d =0 d ( d − µ ( s ) t,d = R ( s )0 − < . Since x I ( τ ) = λx (cid:0) x − f ( s ) ( x ) (cid:1) , we infer thatlim α I → x ( s ) ∗ → , which implies that (by Theorem 2.3), the number of susceptible individuals that ever getinfected is o p ( n ).We now consider the case R ( s )0 >
1. Let us consider again the independent thresholdmodel with threshold distribution p ( s ) t,d ( θ ) := e − ( θ − β ( s ) t (cid:16) − e − β ( s ) t (cid:17) . Let only look at the structure of the subgraph obtained by removing all nodes withthreshold higher than 1. Then each susceptible individual with type t , social activity s anddegree d will remain in the percolated graph with probability p ( s ) t,d (1) = 1 − e − β ( s ) t . The result of Janson [Janson, 2009a] on site percolation in configuration model impliesthat if R ( s )0 := (cid:16) α S λ (cid:17) (cid:88) s ∈S (cid:88) t ∈T p ( s ) t,d (1) ∞ (cid:88) d =0 d ( d − µ ( s ) t,d > raph have threshold 1, the infection of any individual in the giant component will triggerthe infection to whole component which implies Theorem 2.4. A.5 Proof of Theorem 3.1
We define a function g : [0 , → [0 ,
1] via the following, g ( z ) := inf x ∈ [0 , { x : f γ ( z ) ( x ) = x } . It can be easily seen that f γ ( z ) (0) > , f γ ( z ) (1) <
1. In conjugation with the continuity of x (cid:55)→ f γ ( z ) ( x ), we conclude that for any z ∈ [0 , { x : f γ ( z ) ( x ) = x } is nonemptyand closed, and hence g ( z ) ∈ (0 ,
1) is well-defined.Now we show that z (cid:55)→ g ( z ) is decreasing in z , which implies that (28) has at most onesolution. Suppose we have 0 < z < z < F t (cid:16) (cid:96) ( − t,d (cid:17) = 1 and ) f γ ( z ) ( x ) . = λ R λ + α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d γ ( s ) t,d ( z ) (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − = λ R λ + α S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d K (cid:88) s =0 (cid:16) F t (cid:16) (cid:96) ( s − t,d ( z ) (cid:17) − F t (cid:16) (cid:96) ( s ) t,d ( z ) (cid:17)(cid:17) (cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − = λ R λ + α S (cid:88) t ∈T ∞ (cid:88) d =0 dλ µ t,d K (cid:88) s =0 F t (cid:16) (cid:96) ( s − t,d ( z ) (cid:17)(cid:18)(cid:16) x + (1 − x ) e − β ( s ) t (cid:17) d − − (cid:16) x + (1 − x ) e − β ( s − t (cid:17) d − (cid:19) . Hence, by using ( A ) − ( A ), f γ ( z ) ( x ) is strictly increasing in x and strictly decreasingfunction of z (since F is strictly increasing cdf function). So that we have f γ ( z ) ( x ) >f γ ( z ) ( x ) for any x ∈ [0 , f γ ( z ) ( g ( z )) − g ( z ) < f γ ( z ) ( g ( z )) − g ( z ) = 0 . Combing with the fact that f γ ( z ) (0) ≥ x (cid:55)→ f γ ( z ) ( x ), there existsan x < g ( z ) such that f γ ( z ) ( x ) = x , which implies that g ( z ) < g ( z ). A.6 Proof of Theorem 3.2
Recall that γ e (the equilibrium without social planner) is such that κ ( x ) (cid:16) − (cid:0) x γ ∗ + (1 − x γ ∗ ) e − β (cid:1) d (cid:17) VaR − γ e ( L ) = π hile the social planner chooses γ s which maximisez ¯ u social ( γ ), i.e. γ s = arg max γ ∈ [0 , (cid:26) ¯ u social ( γ ) := π (1 − γ ) − κ ( x γ ∗ ) (cid:90) γ VaR − u ( L ) du (cid:27) . Since x γ ∗ is increasing in γ and κ ( . ) is a decreasing function, κ ( x γ ∗ ) is a decreasing functionof γ and we have¯ u (cid:48) social ( γ e ) = − dκ ( x γ ∗ ) dγ (cid:90) γ VaR − u ( L ) du + κ ( x γ e ∗ )VaR − γ e ( L ) − π ≥ κ ( x γ e ∗ )VaR − γ e ( L ) − π = π (cid:32) − (cid:0) x γ ∗ + (1 − x γ ∗ ) e − β (cid:1) d − (cid:33) ≥ , and the theorem follows. B General independent threshold epidemics on G ( n )In this section we present the proof of Theorem A.1. B.1 Markov chain transitions
We first describe the dynamics of the (independent threshold) epidemic on G ( n ) as a Markovchain, which is perfectly tailored for asymptotic study. At time 0 the threshold of each sus-ceptible individual is distributed randomly, according to (type dependent) distribution 40.For θ ∈ N , let n ( s ) t,d,θ denotes the number of susceptible individuals with type t ∈ T ,degree d and social activity s ∈ S which are given threshold θ = 1 , , . . . , d . Hence, n ( s ) t,d,θ /n S p −→ µ ( s ) t,d p ( s ) t,d ( θ )as n → ∞ . At a given time step T , individuals are partitioned into infected I ( T ), susceptible S ( T ) and removed R ( T ). We further partition the class of susceptible nodes according totheir type, social activity and threshold S ( T ) = (cid:91) t,d,s,θ S ( s ) t,d,θ ( T ) . At time zero, I (0) and R (0) contains respectively the initial set of infected and recoveredindividuals. Hence, by ( C ), we know |I (0) | /n → α I and |R (0) | /n → α R as n → ∞ .At each step we have one interaction only between two individuals, yielding at least oneinfected. Our processes at each step is as follows : • Choose a half-edge of an infected individual i ; • Identify its partner j (i.e. by construction of the random graph in the configurationmodel, the partner is given by choosing a half-edge randomly among all available alf-edges); • Delete both half-edges. If j is currently uninfected with threshold θ and it is the θ -thdeleted half-edge from j , then j becomes infected.Let us define S ( s ) t,d,θ,(cid:96) ( T ), 0 ≤ (cid:96) < θ , the number of susceptible individuals with type t , degree d , social activity s , threshold θ and (cid:96) removed half-edges (infected neighbors) attime T . We introduce the additional variables of interest: • X S ( T ): the number of (alive) susceptible half-edges belonging to susceptible individ-uals at time T ; • X I ( T ): the number of (alive) half-edges belonging to infected individuals at time T ; • X R ( T ): the number of (alive) half-edges belonging to initially recovered individualsat time T ; • X ( T ) = X S ( T ) + X I ( T ) + X R ( T ): the total number of (alive) half-edges at time T .Hence, by Condition ( C ), we have (as n → ∞ ) X I (0) /n −→ λ I , X I (0) /n −→ λ I , X R (0) /n −→ λ R and X (0) /n −→ λ. Hence, X (0) = (cid:80) ni =1 d i denote the total number of half-edges in the network and, sinceat each step we delete two half-edges, the number of existing (alive) half-edges at time T will be X ( T ) = X (0) − T. (44)It is easy to see that the following identities hold: X S ( T ) = (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 θ − (cid:88) (cid:96) =0 ( d − (cid:96) ) S ( s ) t,d,θ,(cid:96) ( T ) , (45) X I ( T ) = X (0) − T − X R ( T ) − X S ( T ) . (46)The contagion process will finish at the stopping time T ∗ which is the first time T ∈ N where X I ( T ) = 0. The final number of susceptible individual with type t , social distancing s , degree d will be S ( s ) t,d ( T ∗ ) = ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 S ( s ) t,d,θ,(cid:96) ( T ∗ ) . By definition of our process S ( T ) = (cid:110) S ( s ) t,d,θ,(cid:96) ( T ) (cid:111) t,d,s,θ,(cid:96) and X R ( T ) represent a Markovchain. We write the transition probabilities of the Markov chain. There are four possibilitiesfor the B , the partner of a half-edge of an infected individual A :1. B is infected, the next state is S ( T + 1) = S ( T ) and X R ( T + 1) = X R ( T );2. B is initially recovered. The probability of this event is X R ( T ) X (0) − T . The changes for thenext state will be X R ( T + 1) = X R ( T ) − . B is uninfected of type t , degree d , social distancing strategy s , threshold θ and thisis the ( (cid:96) + 1)-th deleted half-edge with (cid:96) + 1 < θ . The probability of this event is ( d − (cid:96) ) S ( s ) t,d,θ,(cid:96) ( T ) X (0) − T . The changes for the next state will be S ( s ) t,d,θ,(cid:96) ( T + 1) = S ( s ) t,d,θ,(cid:96) ( T ) − ,S ( s ) t,d,θ,(cid:96) +1 ( T + 1) = S ( s ) t,d,θ,(cid:96) ( T ) + 1 . B is uninfected of type t , degree d , social distancing strategy s , threshold θ and thisis the θ -th deleted incoming edge. The probability of this event is ( d − θ +1) S ( s ) t,d,θ,(cid:96) ( T ) X (0) − T .The changes for the next state will be S ( s ) t,d,θ,θ − ( T + 1) = S ( s ) t,d,θ,θ − ( T ) − . Let ∆ T be the difference operator: ∆ T X := X ( T + 1) − X ( T ). We obtain the followingequations for the expectation states variables, conditional on F T (the pairing generated bytime T ), by averaging over the possible transitions: E [∆ T X R |F T ] = − X R ( T ) X (0) − T , (47) E (cid:104) ∆ T S ( s ) t,d,θ, |F T (cid:105) = − dS ( s ) t,d,θ, ( T ) X (0) − t , E (cid:104) ∆ T S ( s ) t,d,θ,(cid:96) |F T (cid:105) = ( d − (cid:96) + 1) S ( s ) t,d,θ,(cid:96) − X (0) − t − ( d − (cid:96) ) S ( s ) t,d,θ,(cid:96) X (0) − t . (48)The initial condition satisfies X R (0) /n −→ λ R , S ( s ) t,d,θ,(cid:96) (0) /n p −→ α S µ ( s ) t,d p ( s ) t,d ( θ )11( (cid:96) = 0) , as n → ∞ . Remark that we are interested in the value of S ( s ) t,d,θ,(cid:96) ( T ∗ ), where T ∗ is the firsttime that X I ( T ∗ ) = 0. In case T ∗ < X (0), the Markov chain can still be well defined for t ∈ [ T ∗ , X (0)) by the same transition probabilities. However, after T ∗ it will no longer berelated to the epidemic process and the value X I ( T ), representing for t ≤ T ∗ the numberof alive half-edges belonging to infected individuals, becomes negative. We consider fromnow on that the above transition probabilities hold for T < X (0).We will show next that the trajectory of these variables throughout the algorithm isa.a.s. (asymptotically almost surely, as n → ∞ ) close to the solution of the deterministicdifferential equations suggested by these equations. B.2 Fluid limit of the epidemic process
Consider the following system of differential equations (denoted by (DE)): (cid:48) R ( τ ) = − x R ( τ ) λ − τ , ( s ( s ) t,d,θ, ) (cid:48) ( τ ) = − ds ( s ) t,d,θ, ( τ ) λ − τ , ( s ( s ) t,d,θ,(cid:96) ) (cid:48) ( τ ) = ( d − (cid:96) + 1) s ( s ) t,d,θ,(cid:96) − ( τ ) λ − τ − ( d − (cid:96) ) s ( s ) t,d,θ,(cid:96) λ − τ , (DE) , with initial conditions x R (0) = λ R , s ( s ) t,d,θ,(cid:96) (0) = α S µ ( s ) t,d p ( s ) t,d ( θ )11( (cid:96) = 0) . Lemma B.1.
The system of ordinary differential equations ( DE ) admits the unique solution x R ( τ ) , s ( τ ) := (cid:110) s ( s ) t,d,θ,(cid:96) ( τ ) (cid:111) s,t,d,θ,(cid:96) in the interval ≤ τ < λ/ , with x R ( τ ) = λ R x, s ( s ) t,d,θ,(cid:96) ( τ ) := µ ( s ) t,d p ( s ) t,d ( θ ) (cid:18) d(cid:96) (cid:19) x d − (cid:96) (1 − x ) (cid:96) , (49) where x = (cid:112) − τ /λ and ≤ (cid:96) < θ .Proof. Let u = u ( τ ) = − ln( λ − τ ). Note that u (0) = − ln( λ ), u is strictly monotoneand so is the inverse function τ = τ ( u ). We write the system of differential equations withrespect to u : x (cid:48) R ( u ) = − x R ( u ) , ( s ( s ) t,d,θ, ) (cid:48) ( u ) = − ds ( s ) t,d,θ, ( u ) , ( s ( s ) t,d,θ,(cid:96) ) (cid:48) ( u ) = ( d − (cid:96) + 1) s ( s ) t,d,θ,(cid:96) − ( u ) − ( d − (cid:96) ) s ( s ) t,d,θ,(cid:96) ( u ) . Then we have x R ( u ) = λ R e − ( u − u (0)) = λ R λ √ λ − τ √ λ = λ R x,ddu ( s ( s ) t,d,θ,(cid:96) +1 e ( d − (cid:96) − u − u (0)) ) = ( d − (cid:96) ) s ( s ) t,d,θ,(cid:96) ( u ) e ( j − (cid:96) − γ − γ (0)) , and by induction, we find s ( s ) t,d,θ,(cid:96) ( u ) = e − ( d − (cid:96) )( u − u (0)) (cid:96) (cid:88) r =0 (cid:18) d − r(cid:96) − r (cid:19) (cid:16) − e − ( u − u (0)) (cid:17) (cid:96) − r s ( s ) t,d,θ,r ( u (0)) . y going back to τ , we have s ( s ) t,d,θ,(cid:96) = x d − (cid:96) (cid:96) (cid:88) r =0 s ( s ) t,d,θ,r (0) (cid:18) d − r(cid:96) − r (cid:19) (1 − x ) (cid:96) − r . Then, by using the initial conditions, we find (for 0 ≤ (cid:96) < θ ) s ( s ) t,d,θ,(cid:96) ( τ ) := α S µ ( s ) t,d p ( s ) t,d ( θ ) (cid:18) d(cid:96) (cid:19) x d − (cid:96) (1 − x ) (cid:96) . A key idea to prove Theorem 2.3 is to approximate, following [Wormald, 1995], theMarkov chain by the solution of a system of differential equations in the large networklimit. We summarize here the main result of [Wormald, 1995].For a set of variables x , ..., x b and for D ⊆ R b +1 , define the stopping time T D = T D ( x , ..., x b ) = inf { t ≥ , ( t/n ; x ( t ) /n, ..., x b ( t ) /n ) / ∈ D} . Lemma B.2 ([Warnke, 2019, Wormald, 1995]) . Given integers b, n ≥ , a bounded domain D ⊆ R b +1 , functions ( f (cid:96) ) ≤ (cid:96) ≤ b with f (cid:96) : D → R , and σ -fields F n, ⊆ F n, ⊆ . . . , supposethat the random variables (cid:0) Y (cid:96)n ( t ) (cid:1) ≤ (cid:96) ≤ b are F n,t -measurable for t ≥ . Furthermore, assumethat, for all ≤ t < T D and ≤ (cid:96) ≤ b , the following conditions hold(i) (Boundedness). max ≤ (cid:96) ≤ b | Y (cid:96)n ( t + 1) − Y (cid:96)n ( t ) | ≤ β, (ii) (Trend-Lipschitz). | E [ Y (cid:96)n ( t +1) − Y (cid:96)n ( t ) |F n,t ] − f (cid:96) ( t/n, Y n ( t ) /n, ..., Y (cid:96)n ( t ) /n ) | ≤ δ , wherethe function ( f (cid:96) ) is L -Lipschitz-continuous on D ,and that the following condition holds initially:(iii) (Initial condition). max ≤ (cid:96) ≤ b | Y (cid:96)n (0) − ˆ y (cid:96) n | ≤ αn, for some (cid:0) , ˆ y , . . . , ˆ y b (cid:1) ∈ D .Then there are R = R ( D , L ) ∈ [1 , ∞ ) and C = C ( D ) ∈ (0 , ∞ ) such that, whenever α ≥ δ min { C, L − } + R/n , with probability at least − be − nα / (8 Cβ ) we have max ≤ t ≤ σn max ≤ (cid:96) ≤ b | Y (cid:96)n ( t ) − x (cid:96) ( t/n ) n | < e CL αn, where (cid:0) x (cid:96) ( t ) (cid:1) ≤ (cid:96) ≤ b is the unique solution to the system of differential equations dx (cid:96) ( t ) dt = f (cid:96) ( t, x , ..., x b ) with x (cid:96) (0) = ˆ y (cid:96) , for (cid:96) = 1 , ..., b, and σ = σ (ˆ y , . . . , ˆ y b ) ∈ [0 , C ] is any choice of σ ≥ with the property that ( t, x ( t ) , ..., x b ( t )) has (cid:96) ∞ -distance at least e LC α from the boundary of D for all t ∈ [0 , σ ) . e apply Lemma B.2 to the epidemic process described in Section B.1. Let us define,for 0 ≤ τ ≤ λ/ x S ( τ ) = (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 θ − (cid:88) (cid:96) =0 ( d − (cid:96) ) s ( s ) t,d,θ,(cid:96) ( τ ) , (50) x I ( τ ) = λ − τ − x R ( τ ) − x S ( τ ) . (51)with s ( s ) t,d,θ,(cid:96) and x R given in Lemma B.1. With Bin ( d, x ) denoting a binomial variable withparameters d and x , we have x S ( τ ) = α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =0 µ ( s ) t,d p ( s ) t,d ( θ )( dy ) P ( Bin ( d − , − x ) ≤ θ − , (52)and, using x = (cid:112) − τ /λ and Equation 51, x I ( τ ) = λ − τ − λ R x − α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 µ ( s ) t,d p ( s ) t,d ( θ )( dy ) P ( Bin ( d − , − x ) ≤ θ − λx (cid:32) x − λ R λ − α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 dµ ( s ) t,d λ p ( s ) t,d ( θ ) P ( Bin ( d − , − x ) ≤ θ − (cid:33) = ( λx ) (cid:16) x − f ( s ) ( x ) (cid:17) . Since x ∗ is the largest solution in (0 ,
1) to the fixed point equation x = f ( s ) ( x ), we have x ∗ = (cid:112) − τ ∗ /λ where τ ∗ is the smallest τ ∈ (0 , λ/
2) such that x I ( τ ) = 0. B.3 Proof of Theorem A.1
We now proceed to the proof of Theorem A.1. We base the proof on Lemma B.2.We first need to bound the contribution of higher order terms in the infinite sums (52).Fix (cid:15) >
0. By Condition ( C ), λ S = (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dµ ( s ) t,d < ∞ Then, there exists an integer ∆ (cid:15) , such that (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =∆ (cid:15) dµ ( s ) t,d < (cid:15), which implies that for all 0 ≤ τ ≤ λ/ (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =∆ (cid:15) d (cid:88) θ =1 dµ ( s ) t,d p ( s ) t,d ( θ ) P ( Bin ( d − , − x ) ≤ θ − < (cid:15). (53) ecall that the number of susceptible vertices with type t ∈ T , social distancing s ∈ S and degree d is n ( s ) S,t,d . Again by condition ( C ), (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 dn ( s ) S,t,d /n → λ S < ∞ . Therefore, for n large enough, (cid:80) s ∈S (cid:80) t ∈T (cid:80) ∞ d =∆ (cid:15) dn ( s ) S,t,d /n < (cid:15). and for all 0 ≤ T ≤ X (0)2 , (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =∆ (cid:15) ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 dS ( s ) t,d,θ,(cid:96) ( T ) /n < (cid:15). (54)For ∆ ≥
1, we denote y ∆ := (cid:16) x R ( τ ) , s ( s ) t,d,θ,(cid:96) ( τ ) (cid:17) d< ∆ , s ∈S , ≤ (cid:96)<θ ≤ d and Y ∆ n := (cid:16) X R ( T ) , S ( s ) t,d ( T ) (cid:17) d< ∆ , s ∈S , ≤ (cid:96)<θ ≤ d , both of dimension b (∆), and x R ( τ ) , s ( s ) t,d,θ,(cid:96) ( τ ) are solutions to a system (DE) of ordinarydifferential equations. Let x ( s ) ∗ = max { x ∈ [0 ,
1] : f ( s ) ( x ) = x } . For the arbitrary constant (cid:15) > D (cid:15) as D (cid:15) = { (cid:0) τ, y K (cid:15) (cid:1) ∈ R b ( K (cid:15) )+1 : − (cid:15) < τ < λ/ − (cid:15) , − (cid:15) < x R ( τ ) < λ, − (cid:15) < s ( s ) t,d,θ,(cid:96) ( τ ) < } . (55)The domain D (cid:15) is a bounded open set which contains the support of all initial values of thevariables. Each variable is bounded by a constant times n ( C = 1). By the definition ofour process, the Boundedness condition is satisfied with β = 1. The second condition of thetheorem is satisfied by some δ n = O (1 /n ). Finally the Lipschitz property is also satisfiedsince λ − τ is bounded away from zero. Then by Lemma B.2 and by convergence of initialconditions, we have : Corollary B.3.
For a sufficiently large constant C , we have P ( ∀ t ≤ nσ H ( n ) , Y K (cid:15) n ( t ) = n y K (cid:15) ( t/n ) + O ( n / )) = 1 − O ( b ( K (cid:15) ) n − / exp( − n − / )) (56) uniformly for all t ≤ nσ H ( n ) where σ H ( n ) = sup { τ ≥ , d ( y K (cid:15) ( τ ) , ∂D (cid:15) ) ≥ Cn − / } . When the solution reaches the boundary of D (cid:15) , it violates the first constraint, determinedby ˆ τ = λ/ − (cid:15) . By convergence of X (0) n to λ , there is a value n such that ∀ n ≥ n , (0) n > λ − (cid:15) , which ensures that ˆ τ n ≤ X (0) / ≤ T = nτ ≤ n ˆ τ and n ≥ n : | X I ( T ) /n − x I ( τ ) | ≤ | X (0) /n − λ | + | X R ( T ) /n − x R ( τ ) | (57)+ (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 d | S ( s ) t,d,θ,(cid:96) ( T ) /n − s ( s ) t,d,θ,(cid:96) ( τ ) |≤ (cid:88) s ∈S (cid:88) t ∈T ∆ (cid:15) (cid:88) d =0 ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 d | S ( s ) t,d,θ,(cid:96) ( T ) /n − s ( s ) t,d,θ,(cid:96) ( τ ) | + 3 (cid:15). (58)and similarly, the total number of susceptible individuals at time T satisfies | S ( T ) /n − s ( τ ) | ≤ (cid:88) s ∈S (cid:88) t ∈T ∆ (cid:15) (cid:88) d =0 ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 | S ( s ) t,d,θ,(cid:96) ( T ) /n − s ( s ) t,d,θ,(cid:96) ( τ ) | + 3 (cid:15)., (59)where, by Lemma B.1, s ( τ ) = (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 ∞ (cid:88) θ =1 θ − (cid:88) (cid:96) =0 s ( s ) t,d,θ,(cid:96) ( τ ) (60)= α S (cid:88) s ∈S (cid:88) t ∈T ∞ (cid:88) d =0 d (cid:88) θ =1 µ ( s ) t,d p ( s ) t,d ( θ ) P ( Bin ( d, − x ) ≤ θ − . (61)We obtain by Corollary B.3 thatsup T ≤ ˆ τn | X I ( T ) /n − x I ( τ ) | ≤ (cid:15) + o L (1) , and (62)sup T ≤ ˆ τn | S ( T ) /n − s ( τ ) | ≤ (cid:15) + o L (1) . (63)We now study the stopping time T n and the size of the epidemic |R s ) ( ∞ ) / R (0) | .Consider x ∗ = (cid:112) − τ ∗ /λ is a stable fixed point of f ( s ) ( x ). Then by definition of x ∗ and by using the fact that f ( s ) (1) ≤
1, we have f ( s ) ( x ) > x for some interval ( x ∗ − ˜ x, x ∗ ).Then x I ( τ ) = ( λx ) (cid:16) x − f ( s ) ( x ) (cid:17) is negative in an interval ( τ ∗ , τ ∗ + ˜ τ ). Let (cid:15) such that 2 (cid:15) < − inf τ ∈ ( τ ∗ ,τ ∗ +˜ τ ) x I ( τ ) and denoteˆ σ the first iteration at which it reaches the minimum. Since x I (ˆ σ ) < − (cid:15) it follows thatwith high probability X I (ˆ σn ) /n <
0, so T n /n = τ ∗ + O ( (cid:15) ) + o L (1). The conclusion followsby taking the limit (cid:15) →0.