Nash SIR: An Economic-Epidemiological Model of Strategic Behavior During a Viral Epidemic
aa r X i v : . [ ec on . T H ] J un Nash SIR: An Economic-Epidemiological Model ofStrategic Behavior During a Viral Epidemic
David McAdams ∗ May 3, 2020
Abstract
This paper develops a Nash-equilibrium extension of the classic SIR model ofinfectious-disease epidemiology (“Nash SIR”), endogenizing people’s decisions whetherto engage in economic activity during a viral epidemic and allowing for complemen-tarity in social-economic activity. An equilibrium epidemic is one in which Nash equi-librium behavior during the epidemic generates the epidemic. There may be multipleequilibrium epidemics, in which case the epidemic trajectory can be shaped throughthe coordination of expectations, in addition to other sorts of interventions such asstay-at-home orders and accelerated vaccine development. An algorithm is providedto compute all equilibrium epidemics.
People’s choices impact how a viral epidemic unfolds. As noted in a March 2020
Lancet commentary on measures to control the current coronavirus pandemic, “How individualsrespond to advice on how best to prevent transmission will be as important as governmentactions, if not more important” (Anderson et al (2020)). Early on when pre-emptive mea-sures could be especially effective (Dalton, Corbett, and Katelaris (2020)), people are atlittle personal risk of exposure and hence may be unwilling to follow orders to “distance”themselves from others. On the other hand, as infections mount and the health-care systemis overwhelmed, people may then voluntarily take extreme measures to limit their exposureto the virus. Clearly, the way in which people’s incentives change during the course of anepidemic is essential to how the epidemic itself progresses, and how widespread are its harms. ∗ Fuqua School of Business and Economics Department, Duke University, Durham, North Carolina, USA.Email: [email protected]. I thank Carl Bergstrom, Yonatan Grad, and Marc Lipsitch for encour-agement and Sam Brown, Troy Day, Nick Papageorge, Elena Quercioli, Lones Smith, Yangbo Song, MartaWosinska, and participants at the Johns Hopkins Pandemic Seminar in April 2020 for helpful comments. equilibrium epidemics that, while highly stylized, sheds light on the interplay be-tween epidemiological dynamics, economic behavior, and the health and economic harm doneduring the course of a viral epidemic.The paper’s most important modeling innovation is to account for the economic com-plementarities of personal interaction that can be lost when agents “distance” themselvesto slow viral transmission. Such complementarities are missing from the existing literature(discussed below), but can impact the progression of an epidemic in meaningful ways. Inparticular, a positive feedback can arise in which people complying with public-health di-rectives induces others to do so as well, and vice versa. As non-essential businesses close,there is less that people are able to do outside the home, reducing their incentive to go out.Similarly, as co-workers in an office (or professors in a university) stay home, there is lessreason to go to the office yourself, especially when the work involved is collaborative and canbe managed remotely. Developed independently, this paper’s Nash-equilibrium SIR (“Nash SIR”) model gen-eralizes the Nash SIR model in Farboodi et al. (2020), by allowing for complementarities insocial-economic activity.In the traditional SIR model, the trajectory of the epidemic is completely determinedby epidemiological fundamentals. Similarly, in Farboodi et al. (2020)’s Nash SIR model, theepidemic has a unique equilibrium trajectory. By contrast, in this paper’s Nash SIR model,there may be multiple potential trajectories for the epidemic, each of which induces agents tobehave in a way that generates that epidemic trajectory. Because of this indeterminancy, theultimate harm done during an epidemic, in terms of lost lives and lost livelihoods, can hingeon what agents believe about what others believe. This paper’s model therefore highlightsthe importance of coordinating mechanisms, such as effective political leadership, in shapingexpectations during an epidemic.In addition to coordinating interventions such as a political leader’s public statements, fundamental interventions such as public policies, public-health programs, scientific effort,and new cultural practices impact the set of equilibrium-epidemic possibilities. Such impacts The opposite is true of essential work. The more that essential workers are absent, the more valuablethe work done by those who remain. More generally, there may be congestion effects associated with social-economic activity, increasing the benefit one gets as others reduce their activity. This paper abstracts fromcongestion for ease of exposition.
Relation to the literature.
This paper follows the dominant tradition within economicsof modeling disease hosts as dynamically-optimizing agents with correct forward-looking be-liefs. A few notable examples include Geoffard and Philipson (1996), Kremer (1996), Adda(2007), Chan et al. (2016), and Greenwood et al. (2019). More recently, there has been anoutpouring of important work motivated by the SARS-CoV2 outbreak, much of it embed-ding economic models into an SIR (or closely related) framework. Some notable exam-ples include Alvarez et al. (2020), Bethune and Korinek (2020), Eichenbaum et al. (2020),Garibaldi et al. (2020), Glover et al. (2020), Jones et al. (2020), Keppo et al. (2020), Krueger et al.(2020), and Toxvaerd (2020).There is also of course an enormous literature within epidemiology that models behavioralresponse to infectious disease. However, epidemiologists have been slow to adopt economics-style modeling, usually instead making ad hoc assumptions about behavior. For instance,Bootsma and Ferguson (2007) assume that people’s intensity of social-contact avoidanceduring the 1918 flu pandemic varied depending on how many others in their communityhad recently died. An example that grapples with the dynamics of social distancing in thecurrent pandemic is Kissler et al. (2020).Thanks to the recent explosion of interest in economic epidemiology among economists,the gap between economics and infectious-disease epidemiology is closing. Farboodi et al.(2020) provide an elegant Nash-equilibrium extension of the SIR model that augments theusual system of differential equations that governs epidemiological dynamics with just twoadditional differential equations. Toxvaerd (2020) beautifully analyzes a similar equilibriumSIR model, establishing compelling features of the equilibrium trajectory. Because of theiranalytical simplicity and tight connection to existing models and methods within epidemiol-ogy, Farboodi et al. (2020), Toxvaerd (2020), and others in this fast-growing literature couldpotentially have enormous influence on infectious-disease epidemiology, marrying the fieldsand promoting further cross-fertilization of ideas.Yet there is also a danger here. This new crop of equilibrium SIR models make an im-plicit assumption that the benefit people get from social-economic activity does not dependon others’ activity. Consequently, the “activity game” that people play necessarily exhibits3egative externalities (activity increases others’ risk of infection) and strategic substitutes(increased risk of infection prompts others to be less active, Bulow et al. (1985)). As a pro-fession, we have strong insights about such games, insights that can be easily and powerfullycommunicated. These models could therefore be highly influential in terms of shaping pub-lic policy. However, the insights that we get from these models could be misguided if, infact, the activity game exhibits positive externalities and/or strategic complements. This isespecially important because, as I discuss in the concluding remarks, the qualitative natureof the game does indeed change during the course of the epidemic.The rest of the paper is organized as follows. Section 1 presents the economic-epidemiologicalmodel, along with preliminary analysis. Section 2 analyzes equilibrium epidemics in moredetail. Section 3 discusses some limitations of the model and directions for future research.
This section presents the economic-epidemiological model, divided for clarity into three parts: the epidemic , on how the epidemic process depends on agents’ behavior (Section 1.1); theeconomy, on how the epidemic impacts economic activity, both directly by making peoplesick and indirectly by changing behavior (Section 1.2); and individual and collective behavior, on how the state of the epidemic and expectations about economic activity impact Nash-equilibrium behavior at each point along the epidemic trajectory (Section 1.3).
There is a unit-mass population of hosts, referred to as “agents.” Building on the classicSusceptible-Infected-Recovered (SIR) model of viral epidemiology, each host is at each time t ≥ S ) if as-yet-unexposed to the virus;“carriage/contagious” ( C ) if asymptomatically infected; “infected/sick” ( I ) if symptomati-cally infected; “recovered from carriage” ( R C ) if immune but never sick; and “recovered fromsickness” ( R I ) if immune and previously sick. Epidemiological distance.
At each point in time t , each agent who is not sick decideshow intensively to distance themselves from others. Distancing with intensity d i ∈ [0 , αd i of “meetings” with other agents, where α ∈ (0 ,
1] isa parameter capturing the maximal effectiveness of distancing. Agents who are sick are It remains unknown whether those who recover from SARS-CoV2 infection are immune to re-infectionand, if so, for how long (Lipsitch (2020). α = 1. (The analysis extendseasily to a more general context in which sick agents also transmit the virus.)Let Ω ≡ { S, C, R C , R I } denote the set of not-sick epidemiological states. For each ω ∈ Ω,let d ω ( t ) denote the average distancing intensity of those currently in state ω at time t whochoose to distance themselves. Let d t ≡ ( d ω ( t ′ ) : ω ∈ Ω , ≤ t ′ < t ) denote the collectivedistancing behavior of the agent population up to time t , and let d ≡ ( d ω ( t ) : ω ∈ Ω , t ≥ Epidemiological dynamics.
The following notation is used to describe the state of theepidemic at each time t ≥
0, depending on agents’ distancing behavior: S ( t ; d t ) = mass of agents who are susceptible; C ( t ; d t ) = mass of agents who are in carriage, i.e., asymptotically infected but not sick; I ( t ; d t ) = mass of agents who are sick; R C (; d t ) = mass of agents who are immune and were not previously sick; and R I ( t ; d t ) = mass of agents who are immune and were previously sick.Because the population has unit mass, P ω ∈ Ω ω ( t ) = 1 for all t .Agents transition between epidemiological states as follows: S → C : Susceptible agents become asymptomatically infected once “exposed” tosomeone currently infected, at a rate that depends on agents’ behavior (details below); C → I : Each agent with asymptomatic infection becomes sick at rate σ >
0; and C → R C and I → R I : Each agent with infection clears their infection at rate γ > t = 0, mass ∆ > that is, S (0) = 1 − ∆, C (0) = ∆, and I (0) = R C (0) = R I (0) = 0.Epidemiological dynamics at times t > The model can be easily extended to allow for innate immunity, by allowing some mass of hosts to be instates R I and R C at time t = 0. For instance, during the “second wave” of SARS-CoV2 infections expectedto arrive in Fall 2020, some hosts may retain immunity due to exposure during the first wave in Spring 2020. S ′ ( t ; d t ) = − β (1 − αd S ( t ))(1 − αd C ( t )) S ( t ; d t ) C ( t ; d t ) (1) C ′ ( t ; d t ) = − S ′ ( t ; d t ) − ( σ + γ ) C ( t ; d t ) (2) I ′ ( t ; d t ) = σC ( t ; d t ) − γI ( t ; d t ) (3) R ′ C ( t ; d t ) = γC ( t ; d t ) (4) R ′ I ( t ; d t ) = γI ( t ; d t ) (5)Let E ( t ; d t ) ≡ ( S ( t ; d t ) , C ( t ; d t ) , I ( t ; d t ) , R C ( t ; d t ) , R I ( t ; d t )) denote the “epidemic state” attime t and E ( d ) ≡ ( E ( t ; d t ) : t ≥
0) the “epidemic process.”
Note on notation:
I use “ d t notation” in equations (1-5) to emphasize how the epidemic stateat time t depends on agents’ previous distancing behavior. However, to ease exposition, Ihenceforth suppress this notation, except where needed for clarity.Equations (2-5) are standard—reflecting agents’ progression over time into carriage andthen either to infection at rate σ or to viral clearance at rate γ , and from infection toclearance at rate γ —but equation (1) is different than in a standard SIR model.Each susceptible agent i has a potential meeting (i.e., opportunity for transmission) withanother randomly-selected agent j at “transmission rate” β >
0. Since fraction S ( t ) ofthe population is susceptible and fraction C ( t ) have unisolated infection, the flow of po-tential meetings between susceptible and infected agents across the entire population is βS ( t ) C ( t ). However, because susceptible and contagious agents distance themselves withintensity d S ( t ) and d C ( t ), respectively, each such potential meeting is avoided with proba-bility (1 − αd S ( t ))(1 − αd C ( t )). The overall flow of newly-exposed hosts is therefore β (1 − αd S ( t ))(1 − αd C ( t )) S ( t ) C ( t ), a functional form that appeared first in Quercioli and Smith(2006). End of the epidemic.
For analytical convenience, I assume that the epidemic ends at time
T > T .) I focus onthe case when T < ∞ and T is known to all agents, but the analysis can be easily extendedto a setting in which T is a random variable drawn from interval support. Information states and distancing strategies.
Agents’ distancing decisions depend onwhat they know about their own epidemiological state and the overall epidemic. This paperfocuses on the simplest non-trivial case, assuming that (i) agents know when they are sick but6therwise observe nothing about their own epidemiological state and (ii) agents have correctbeliefs about the epidemic process. The model can be extended in several natural directions,to include diagnostic testing (allowing agents to learn more about their own epidemiologicalstate) and incorrect beliefs, but such extensions are left for future work.Agent i ’s information state captures what she knows and believes, which depends onlyon (i) the time t ≥ I ), was previously sick (state R I ), orhas not yet been sick (combined state N ≡ S ∪ C ∪ R C ).An agent currently in information state ι ∈ { N, I, R I } is referred to as a “ ι -agent.” Let N ( t ) = S ( t ) + C ( t ) + R C ( t ) denote the mass of N -agents; thus, N ( t ) + I ( t ) + R I ( t ) = 1.Agent i ’s distancing strategy specifies her likelihood of distancing herself at each time t in each information state. I -agents are automatically isolated, as mentioned earlier. R I -agents know that they are immune and therefore have a dominant strategy not to distancethemselves. It remains to determine the behavior of N -agents.Let d N ( t ) denote the share of N -agents who distance themselves. Because susceptibleand contagious agents are in the same not-yet-sick information state, d N ( t ) = d S ( t ) = d C ( t )and equation (1) simplifies to: S ′ ( t ) = − β (1 − αd N ( t )) S ( t ) C ( t ) (6) Attack rate.
Each agent’s ex ante likelihood of becoming infected, referred to as the“attack rate” of the virus, is equal to lim t →∞ ( R C ( t ) + R I ( t )). The attack rate is alwaysstrictly less than one, even if a vaccine is never discovered ( T = ∞ ) and the epidemic is leftcompletely uncontrolled; see Brauer et al. (2012) and Katriel and Stone (2012) for details. Each agent’s activities fall into three broad categories: isolated activities that can be per-formed while distancing (e.g., lifting weights, collaborating online), public activities thatrequire entering public spaces but do not require interacting with others (e.g., going for awalk, getting gas), and social activities that require interacting physically with others (e.g.,meeting friends, working in an office building). An agent who distances herself with intensity d i can continue engaging in isolated activity, but forgoes fraction αd i of public and socialactivity and reduces others’ opportunities to join her in social activity. Availability for social interaction.
A not-sick agent who does not distance enjoys all thebenefits of public activity, but engages in social activity only with those who are “sociallyavailable.” Let A ( t ) denote agents’ availability for social interaction at time t , averaged7cross the entire population: A ( t ) = (1 − αd N ( t )) N ( t ) + R I ( t ) = 1 − I ( t ) − αd N ( t ) N ( t ) . (7)(Recall that I -agents are completely unavailable due to sickness, while R I -agents find itoptimal not to distance themselves at all.) Economic output.
Economic activity generates benefits , a broad concept that shouldbe understood to include everything from income (work activity) and access to goods andservices (shopping) to psycho-social well-being (from interactions with friends). Sick agentsare assumed for simplicity to be incapacitated and hence unable to engage in any economicactivity; their economic benefit is normalized to zero. The benefit that well agents getdepends on their own and others’ distancing decisions, as well as how many people arecurrently sick.Let b ( d i ; A ) denote the flow benefit that agent i gets when well and choosing distance d i ∈ { , } , given population-wide average availability 0 ≤ A ≤
1. For concreteness, I assumethat b ( d i ; A ) = a + a (1 − αd i ) + a (1 − αd i ) A. (8) Discussion: meaning of the economic parameters.
The parameters a , a , a > a captures the baseline level of benefits that a wellagent gets while quarantined in the home; a captures the extra benefits associated withbeing able to leave the home, e.g., the extra pleasure and health benefit of walking outside,the extra convenience of shopping in person rather than online; and a captures the extrabenefits associated with sharing the same physical space with others, e.g., eating out at arestaurant rather than at home, hugging a friend rather than just talking on the phone. (Putdifferently: a is the cost associated with everyone else being quarantined; a is the cost ofquarantining yourself, in a world where everyone else is quarantined; and a is the cost ofbeing sick, in a world where everyone is quarantined.) These parameters can be changedin many ways. For instance, a restaurant service that delivers safely-prepared fresh-cookedmeals would increase a and reduce a , as would improved virtual-meeting technology thatenhances remote collaboration. Discussion: functional form of economic benefits.
The assumption that the benefits of publicand social activity are linear in own and others’ availability simplifies the presentation but isnot essential for the analysis or qualitative findings. For instance, suppose that agents were8o prioritize their activities, e.g., by leaving the home only to get urgently-needed supplies,or to visit only with their dearest friends. In that case, each agent’s benefit from public andsocial activity would naturally be a concave function of her own personal distance and ofothers’ availability. The analysis can be easily adapted to allow for such concavity, but atthe cost of complicating the presentation.
Economic losses due to the virus.
If the virus did not exist, then no one would becomesick and everyone would choose not to distance. All agents would then get constant floweconomic benefit b (0; 1) = a + a + a and, since the population has unit mass, overalleconomic activity would also be b (0; 1). The virus reduces economic activity directly, bymaking people sick, and indirectly, by inducing not-yet-sick agents to distance themselves.Distancing in turn creates two sorts of economic harm: “private harm” that distancingoneself reduces one’s own public and social activity, and “social harm” that distancing oneselfreduces others’ social activity.Let b t ( d i ) be shorthand for each well agent’s flow economic benefit at time t . b t ( d i )depends on (i) how many people are recovered from sickness, R I ( t ), and how many arecurrently sick, I ( t ) = 1 − N ( t ) − R I ( t ), (ii) what fraction d N ( t ) of not-yet-sick agents aredistancing, and (iii) her own distancing choice d i ∈ { , } : b t ( d i ) = a + a (1 − αd i ) + a (1 − αd i ) A ( t )= a + a (1 − αd i ) + a (1 − αd i )((1 − αd N ( t )) N ( t ) + R I ( t ))All agents suffer economically throughout the epidemic, relative to the no-virus benchmarkcase in which everyone gets flow benefit a + a + a : Sick: I -agents are incapacitated and get zero economic benefit. These agents lose a + a + a . Previously sick: R I -agents do not distance, but have less opportunity for social in-teraction due to others’ distancing behavior. These agents lose social-activity benefit a (1 − A ( t )). Not-yet-sick: N -agents choose distancing intensity d N ( t ), reducing their public andsocial activities by a factor of (1 − αd N ). These agents lose public-activity benefit a αd N and lose social-activity benefit a (1 − (1 − αd N ) A ( t )) = a (1 − A ( t ) + αd N A ( t )).Let Γ E ( t ) denote the lost economic activity at time t , across the entire population. Overalleconomic loss across the entire epidemic is Γ E = R ∞ Γ E ( t ) dt . (If future losses are discounted9y discount factor 0 < δ ≤
1, then the overall economic loss has present value Γ E = R ∞ δ t Γ E ( t ) dt at time 0. I focus on the case without discounting for ease of exposition.) Lemma 1. Γ E ( t ) = a I ( t ) + a (1 − A ( t )) + a (1 − A ( t ) ) .Proof. See the Appendix.
Each agent seeks to minimize her own total losses during the course of the entire epidemic.Let l i ( t ) denote agent i ’s flow loss at time t . As discussed earlier: l i ( t ) = a + a + a if i is sick; l i ( t ) = a (1 − A ( t )) if i is well and not distancing, where A ( t ) is others’ availabilityfor social interaction; and l i ( t ) = a α + a (1 − A ( t ) + αA ( t )) if i is well and distancing.Let L ω ( t ) denote agent i ’s expected future total losses starting from time t if in epi-demiological state ω ∈ { S, C, I, R C , R I } , referred to as “continuation losses from state ω .”(Continuation losses depend on future agent behavior and the future trajectory of the epi-demic, but I suppress such notation as much as possible for ease of exposition.) A susceptibleagent who becomes infected at time t will not notice this transition but, at that moment,her continuation losses change from L S ( t ) to L C ( t ). Let H ( t ) ≡ L C ( t ) − L S ( t ) denote the“harm of susceptible exposure” at time t .Let p i ( t ) denote agent i ’s subjective belief about her own likelihood of being susceptible attime t , conditional on being not-yet-sick. At time t , mass N ( t ) of agents are not-yet-sick, ofwhom mass S ( t ) remain susceptible. Thus, N -agents’ average likelihood of being susceptibleis S ( t ) N ( t ) . For simplicity, I will focus on epidemics with symmetric behavior by all those in thesame information state at each point in time, in which case p i ( t ) = S ( t ) N ( t ) . Gain from distancing: reduced exposure.
Suppose that, at time t , agent i distanceswith intensity d i ∈ [0 ,
1] and other N -agents distance themselves with intensity d N ∈ [0 , i is then exposed to the virus at rate β (1 − αd i )(1 − αd N ) C ( t ), compared to beingexposed at rate β (1 − αd N ) C ( t ) if not distancing at all. The “gain from distancing” at time t , denoted GAIN t ( d N ), is therefore GAIN t ( d i , d N ) = αd i β (1 − αd N ) C ( t ) × H ( t ) × S ( t ) N ( t ) . (9) The analysis can be trivially extended to allow for discounting of future losses. t ( d N ) = d GAIN t ( d i ,d N )d d i is then M G t ( d N ) = α (1 − αd N ) βS ( t ) C ( t ) H ( t ) N ( t ) . (10)Note that the marginal gain from distancing is decreasing in d N . This is because, as othersdistance themselves more, agents face less risk of exposure. Economic cost of distancing: reduced activity.
If other N -agents choose distancingintensity d N , agent i gets flow economic benefit a + a (1 − αd i )+ a (1 − αd i )((1 − αd N ) N ( t )+ R I ( t )) when choosing distancing intensity d i , compared to a + a + a ((1 − αd N ) N ( t )+ R I ( t ))when not distancing at all. The “cost of distancing” at time t , denoted COST t ( d i , d N ), istherefore COST t ( d i , d N ) = a αd i + a αd i ((1 − αd N ) N ( t ) + R I ( t )) . (11)The marginal cost of distancing M C t ( d N ) = d COST t ( d i ,d N )d d i is then M C t ( d N ) = a α + a α ((1 − αd N ) N ( t ) + R I ( t )) . (12)Note that the marginal cost of distancing is decreasing in d N . This is because, as othersdistance themselves more, there are fewer opportunities for social activity.Because the marginal gain and the marginal cost of distancing are each decreasing in d N ,the game that N -agents play may exhibit “strategic substitutes” or “strategic complements”(Bulow et al. (1985)), depending on the epidemic state. By contrast, in Quercioli and Smith(2006), there are no sources of strategic complementarity. “Distancing game” among agents. At each time t , the not-yet-sick N -agents play a game when deciding whether or not to distance. (Sick I -agents are incapacitated, whilepreviously sick R I -agents obviously prefer not to distance. Thus, only N -agents have a non-trivial decision.) I assume that N -agents play a Nash equilibrium (NE) of this game, andfocus on symmetric NE in which all N -agents choose the same distancing intensity. Proposition 1.
Given epidemic state E ( t ) = ( S ( t ) , C ( t ) , I ( t ) , R C ( t ) , R I ( t )) and harm fromsusceptible exposure H ( t ) = L C ( t ) − L S ( t ) , the “time- t distancing game” played by not-yet-sick agents has a unique symmetric NE, in which agents choose distancing intensity d ∗ N ( t ) .In particular: (i) if M G t (0) ≤ M C t (0) , then d ∗ N ( t ) = 0 ; (ii) if M G t (1) ≥ M C t (1) , then ∗ N ( t ) = 1 ; and (iii) otherwise, if M G t (0) > M C t (0) and M G t (1) < M C t (1) then d ∗ N ( t ) = βS ( t ) C ( t ) H ( t ) N ( t ) − a − a ( N ( t ) + R I ( t )) α (cid:16) βS ( t ) C ( t ) H ( t ) N ( t ) − a N ( t ) (cid:17) ∈ (0 , . (13) Proof.
The proof is in the Appendix.Uniqueness of symmetric NE is not obvious, since the time- t distancing game may haveeither strategic substitutes or strategic complements, depending on the epidemic state andthe harm of susceptible exposure. However, uniqueness arises because N -agents have adominant strategy whenever the game has strategic complements. Equilibrium epidemics.
Let E ( d N ) denote the epidemic process that results when N -agents choose distancing intensity d N ( t ) at each time t , as determined by the system (2-6). E ∗ is referred to as an equilibrium epidemic process (or “behaviorally-constrained epidemic”)if (i) E ∗ = E ( d ∗ N ) and (ii) given the epidemic process E ∗ , the time- t distancing game has asymmetric NE in which N -agents choose distancing intensity d ∗ N ( t ), for all t ≥ This section characterizes all equilibrium epidemics with symmetric agent behavior (or,more simply, “equilibrium epidemics”) and provides an algorithm for computing them. Theanalysis is organized as follows. First, the augmented system of differential equations thatgoverns economic-epidemiological dynamics in the Nash SIR model is presented. This sys-tem builds on the well-known system that governs epidemiological dynamics in the SIRmodel. Second, for any given “final prevalences” ( S ( T ) , C ( T ) , I ( T ) , R I ( T )) at time T whendistancing ends (due to a perfect vaccine being introduced), there is at most one equilibriumepidemic having these final prevalences. At any given time t , the epidemic is characterized by: (i) the mass of agents in each epidemi-ological state ( S ( t ), C ( t ), I ( t ), R C ( t ), R I ( t )); (ii) the welfare of agents in each epidemiologi-cal state (as captured by state-contingent “continuation losses” L S ( t ), L C ( t ), L I ( t ), L R C ( t ), L R I ( t )); and (iii) the distancing behavior of agents who are not yet sick ( d ∗ N ( t )). I do not know if an equilibrium epidemic can exist with asymmetric behavior by symmetric agents. pidemiological dynamics. The dynamics of the epidemic state E ( t ) = ( S ( t ), C ( t ), I ( t ), R C ( t ), R I ( t )) up until time T are determined by equations (2-6) and depend on N -agents’distancing behavior. After the vaccine is introduced at time T , equations (2-5) remainunchanged but, because there is no further transmission of the virus, S ′ ( t ) = 0. Distancing behavior.
Lemma 1 characterizes N -agents distancing behavior d N ( t ) at eachtime t , depending on the epidemic state E ( t ) and the harm of susceptible exposure H ( t ) = L C ( t ) − L S ( t ). Welfare dynamics.
It remains to characterize how the continuation losses associated witheach epidemiological state change over time.
State R I . Agents who have recovered from sickness remain well and choose not to distance.Such an agent still suffers from the fact that others are distancing, losing social-activitybenefit a ( αd ∗ N ( t ) N ( t ) + I ( t )) relative to the no-virus benchmark in which everyone earnsflow benefit a + a + a . Because these losses are “sunk” once experienced, and because R I -agents do not transition to any other state, L ′ R I ( t ) = − a ( αd ∗ N ( t ) N ( t ) + I ( t )) . (14)After time T when new transmission stops, all social distancing stops, i.e., d ∗ N ( t ) = 0 for all t > T . However, well agents still suffer from not being able to engage socially with thosewho are sick. In particular, L R I ( t ) = R t ′ ≥ t a I ( t ′ )d t ′ for all t ≥ T . State I . Sick agents incur flow loss a + a + a and transition to the recovered state R I atrate γ . Thus, L ′ I ( t ) = − ( a + a + a ) + γ ( L I ( t ) − L R I ( t )) . (15) State R C . Agents who have recovered from carriage never learn that they are immune, and socontinue to distance themselves throughout the entire epidemic. In particular, these agentslose public-activity benefit a αd ∗ N ( t ), lose social-activity benefit a (1 − (1 − αd ∗ N ( t ))((1 − αd ∗ N ( t )) N ( t ) + R I ( t )), and never transition to another state. Thus, L ′ R C ( t ) = − a α − a (1 − (1 − αd ∗ N ( t ))((1 − αd ∗ N ( t )) N ( t ) + R I ( t )) . (16)After time T , because all social distancing stops and R C -agents do not become sick, their The analysis can be extended in a straightforward way to allow for the possibility of re-infection, forinstance, by having recovered agents transition back at some rate to the susceptible state. R I -agents. So, L R C ( t ) = L R I ( t ) for all t ≥ T . State C . Agents with asymptomatic infection incur the same flow losses due to social dis-tancing as all not-yet-sick agents (including those in state R C ), but transition to sickness atrate σ and to asymptomatic recovery at rate σ . Thus, L ′ C ( t ) = L ′ R C ( t ) + γ ( L I ( t ) − L C ( t )) + σ ( L R C ( t ) − L C ( t )) . (17) State S . Susceptible agents incur the same flow losses as all other not-yet-sick agents, butbecome asymptomatically infected at rate β (1 − αd ∗ N ( t )) S ( t ) C ( t ). Thus, L ′ S ( t ) = L ′ R C ( t ) + β (1 − αd ∗ N ( t )) S ( t ) C ( t )( L C ( t ) − L S ( t )) . (18)After time T , S -agents remain susceptible and only suffer from not being able to interactwith others who are sick, the same as R I -agents. So, L S ( t ) = L R I ( t ) for all t ≥ T . Suppose for a moment that an equilibrium exists with final epidemic state E ( T ). HereI discuss how to determine numerically whether an equilibrium epidemic exists with this“final condition” and, if so, to compute the entire epidemic trajectory.Observe first that the final epidemic state uniquely pins down the trajectory of the epi-demic after time T . Because there is no new transmission, no one distances and subsequentepidemiological dynamics are trivial: contagious agents leave state C at rate γ + σ , fraction σγ + σ becoming sick; sick agents recover at rate γ ; and others remain in their current state.Moreover, because C ( T ) and I ( T ) together determine ( I ( t ) : t ≥ T ), they also determine L R I ( t ) = L R C ( t ) = L S ( t ) = R t ′ ≥ t a I ( t ′ )d t ′ for all t ≥ T , which in turn determine L C ( t ) and L I ( t ) after T .Having determined L S ( T ) and L C ( T ), we now know H ( T ) = L C ( T ) − L S ( T ), the harm ofsusceptible exposure just before the vaccine is introduced. Together with the final epidemicstate, this uniquely determines N -agents’ equilibrium distancing intensity just before thevaccine is introduced, as characterized in Proposition 1.Having determined N -agent behavior d ∗ N ( t ), we now can determine: S ′ ( T ) (equation (6))and all other epidemiological dynamics, which remain unchanged (equations (2-5)); L ′ R I ( T ),which in turn determines L ′ I ( T ) (equations (14,15)); and L ′ R C ( T ), which together with L ′ I ( T )determines L ′ C ( T ), which in turn determines L ′ S ( t ) (equations (16,17,18)). In this way, any14 andidate epidemic can be uniquely traced backward over time, from the given final epidemicstate (“final condition”), until one of three things happens: (i) the trajectory hits an invalidboundary , in which case no equilibrium epidemic exists with the given final condition; (ii)the backwards trajectory “ends” at the desired initial epidemic state E (0) = (1 − ∆ , ∆ , , , E (0) = (1 − ∆ , ∆ , , , This paper introduces Nash SIR, an economic-epidemiological model of a viral epidemicthat builds on the classic Susceptible-Infected-Recovered (SIR) model of infectious-diseaseepidemiology. The model departs from the previous literature by focusing on the complemen-tarities associated with the social-economic activity that can be lost when agents distancethemselves to prevent the spread of infection.
A changing game.
An important complicating feature of this paper’s model is that, asthe epidemic progresses through its course, the basic strategic structure of the “distancinggame” that agents play changes over time. For instance, very early in the epidemic wheninfection remains rare, the distancing game exhibits negative externalities, since agents getlittle health benefit but suffer substantial economic harm when others distance themselves.However, that changes once infection grows more common, as others’ distancing generatesgreater health benefit. Moreover, the game can shift between having strategic substitutesand strategic complements.
Complementarity and multi-dimensionality of agent actions.
This paper focuses ona simple context in which the only way to protect oneself from infection is to avoid public andin-person social activity. However, people can also prevent transmission in other ways, suchas wearing a mask. Bearing that in mind, it would be interesting to generalize the analysisto allow agents to decide both (i) how much to curtail their public and social activities(“avoidance,” as in this paper), and (ii) how much to change their behavior during suchactivities (“vigilance,” as in Quercioli and Smith (2006)). The game that agents play in thisricher context has an interesting strategic structure, with agents’ vigilance decisions alwaysbeing strategic substitutes, agents’ avoidance decisions potentially being either strategic An “invalid boundary” is reached if S ( t ), C ( t ), I ( t ), R C ( t ), or R I ( t ) equals zero at any time t > Asymmetry and social inequality.
This paper assumes that agents are symmetric forease of exposition, but this assumption appears to entail meaningful loss of generality. Inparticular, assuming that all agents are the same at the start of the epidemic obscuresimportant issues related to inequality and social justice. To see why, suppose that agentsbelong to one of two social classes: “elites” who are able to earn income and care forthemselves from home (higher a ) and “non-elites” whose income and well-being hinge moreon being in public social spaces (higher a ). With less to lose by staying at home, elites willdistance themselves relatively early during the epidemic. Having distanced less in the past,non-elites will then be more likely than elites to already have been exposed to the virus—further reducing their relative incentive to distance. In the end, the equilibrium trajectoryof the epidemic could exacerbate pre-existing inequality, with non-elites bearing the bruntof the burden of the epidemic, being more likely to become sick and suffering more from theeconomic contraction associated with elite-driven distancing.16 eferences Adda, J´erˆome , “Behavior towards health risks: An empirical study using the Mad Cowcrisis as an experiment,”
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Mathematical proofs
Proof of Lemma 1.
Proof.
Recall that A ( t ) = (1 − αd N ( t )) N ( t ) + R I ( t ) and hence 1 − A ( t ) = I ( t ) + αd N ( t ) N ( t ). Isolated activity:
Sick agents get no benefit, while well agents get full benefit a . The overalleconomic loss due to reduced isolated activity at time t is therefore a I ( t ). Public activity:
Sick agents get no benefit, well agents who do not distance get full benefit a , and well agents who distance get benefit a (1 − α ). Since fraction d N ( t ) of N -agentsdistance and no R I -agents distance, the overall economic loss due to reduced public activityat time t is therefore a ( I ( t ) + αd N ( t ) N ( t )) = a (1 − A ( t )). Social activity:
Sick agents get no benefit, well agents who do not distance get benefit a A ( t ),and well agents who distance get benefit a (1 − α ) A ( t ) (and hence lose a (1 − A ( t ) + αA ( t ))).The overall economic loss due to reduced social activity at time t is therefore a times I ( t ) + (1 − A ( t ))( R I ( t ) + (1 − d N ( t )) N ( t )) + (1 − A ( t ) + αA ( t )) d N ( t ) N ( t )= I ( t ) + (1 − A ( t ))( R I ( t ) + (1 − d N ( t )) N ( t )) + ((1 − A ( t ))(1 − α ) + α ) d N ( t ) N ( t )= 1 − A ( t ) + (1 − A ( t )) ( R I ( t ) + (1 − d N ( t )) N ( t ) + (1 − α ) d N ( t ) N ( t ))= (1 − A ( t )) × (1 + R I ( t ) + (1 − αd N ( t )) N ( t ))= (1 − A ( t )) × (1 + A ( t )) = 1 − A ( t ) as desired. Proof of Proposition 1.
Proof. (i)
No distancing: If M G t (0) ≤ M C t (0), then the time- t distancing game has asymmetric NE in which all agents choose not to distance, i.e., d ∗ N ( t ) = 0. To establishuniqueness, note by equations (10-12) that M G t (0) ≤ M C t (0) implies βS ( t ) C ( t ) H ( t ) N ( t ) ≤ a + a ( N ( t ) + R I ( t )). But then M G t (1) = α (1 − α ) βS ( t ) C ( t ) H ( t ) N ( t ) ≤ α (1 − α )( a + a ( N ( t ) + R I ( t ))) < α ( a + a ((1 − α ) N ( t ) + R I ( t )))= M C t (1) 19ince M G t ( d N ) and M C t ( d N ) are each linear in d N , the fact that M G t (0) ≤ M C t (0) and M G t (1) < M C t (1) implies that M G t ( d N ) < M C t ( d N ) for all d N ∈ (0 , N -agents have a dominant strategy not to distance.(ii) Maximal distancing: If M G t (1) ≥ M C t (1), then a symmetric NE exists in whichall agents choose to distance as much as possible, i.e., d ∗ N ( t ) = 1. To establish uniqueness,note by equations (10-12) that M G t (1) ≥ M C t (1) implies (1 − α ) βS ( t ) C ( t ) H ( t ) N ( t ) ≥ a + a ((1 − α ) N ( t ) + R I ( t )). But then M G t (0) = α βS ( t ) C ( t ) H ( t ) N ( t ) ≥ α − α ( a + a ((1 − α ) N ( t ) + R I ( t ))) > α ( a + a ( N ( t ) + R I ( t )))= M C t (0)Since M G t ( d N ) and M C t ( d N ) are each linear in d N , the fact that M G t (1) ≥ M C t (1) and M G t (0) > M C t (0) implies that M G t ( d N ) > M C t ( d N ) for all d N ∈ [0 , N -agents have a dominant strategy to distance.(iii) Intermediate distancing: If M G t (0) > M C t (0) and M G t (1) < M C t (1), then itmust be that M G ′ t ( d N ) = − α βS ( t ) C ( t ) H ( t ) N ( t ) < − α a N ( t ) = M C ′ t ( d N ) and hence that thereexists a unique d ∗ N ( t ) ∈ (0 ,
1) such that
M G t ( d ∗ N ( t )) = M C t ( d ∗ N ( t )), M G t ( d N ) > M C t ( d N )for all d N < d ∗ N ( t ), and M G t ( d N ) < M C t ( d N ) for all d N > d ∗ N ( t ). In particular, solving M G t ( d ∗ N ( t )) = M C t ( d ∗ N ( t )) yields d ∗ N ( t ) = βS ( t ) C ( t ) H ( t ) N ( t ) − a − a ( N ( t ) + R I ( t )) α (cid:16) βS ( t ) C ( t ) H ( t ) N ( t ) − a N ( t ) (cid:17) ..