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H AT I F I MMUN IT Y W AN E S ? M. A
LPER C¸ ENESIZ ∗ L U ´ IS G UIMAR ˜ AES † August 3, 2020
Abstract
Using a simple economic model in which social-distancing reduces contagion,we study the implications of waning immunity for the epidemiological dynamicsand social activity. If immunity wanes, we find that COVID-19 likely becomes en-demic and that social-distancing is here to stay until the discovery of a vaccine orcure. But waning immunity does not necessarily change optimal actions on the on-set of the pandemic. Decentralized equilibria are virtually independent of waningimmunity until close to peak infections. For centralized equilibria, the relevance ofwaning immunity decreases in the probability of finding a vaccine or cure, the costsof infection (e.g., infection-fatality rate), and the presence of other NPIs that lowercontagion (e.g., quarantining and mask use). In simulations calibrated to July 2020,our model suggests that waning immunity is virtually unimportant for centralizedequilibria until at least 2021. This provides vital time for individuals and policy-makers to learn about immunity against SARS-CoV-2 before it becomes critical.JEL Classification : D62; E17; I12; I18.
Keywords : COVID-19; SARS-CoV-2; Immunological Memory; Optimal Policy; Social-Distancing; Waning Immunity. ∗ Nottingham Trent University. E-mail address: [email protected]. † Queen’s University Belfast. E-mail address: [email protected]. Introduction
We do not know yet the duration of immunity against severe acute respiratory syn-drome coronavirus 2 (SARS-CoV-2) causing coronavirus infectious disease 2019 (COVID-19). But early evidence points to waning immunity against SARS-CoV-2 (Seow et al.,2020) and we know that immunity against other coronaviruses wanes within two years(Edridge et al., 2020; Huang et al., 2020; Kellam and Barclay, 2020).If immunity against COVID-19 indeed wanes, then COVID-19 likely becomes en-demic and herd immunity cannot be naturally reached. Therefore, ignoring waningimmunity may lead to costly policies with irreversible consequences. Despite theserisks, almost all the economics literature on the COVID-19 pandemic assumes perma-nent immunity. Our paper fills this gap in the literature by assessing the implicationsof waning immunity for decentralized and centralized equilibria in an economic modelof an epidemic.In the model, decision makers are constrained by disease contagion and maximizethe difference between the utility from social activity and the cost of infection. Theutility from social activity captures, in a stylized way, all the payoffs from economicand social actions that require physical proximity. Our approach is grounded in threereasons. First, the main economic impact of the pandemic has been on sectors thatrely on physical proximity (Chetty et al., 2020). Second, there are also other significantcosts of constrained social activity such as anxiety, distress, fatigue, and domestic vio-lence (Ravindran and Shah, 2020; Serafini et al., 2020). Third, contagion of virus causingrespiratory diseases is mostly unrelated with consumption and work (Ferguson et al.,2006; Eichenbaum, Rebelo and Trabandt, 2020 a ) but can be influenced by behavior.The epidemiological dynamics in the model is based on recurrence relations be-tween three (main) health states: susceptible (S), infected (I), and recovered (R) withthe flow pattern S → I → R → S (and hence the conventional labeling SIRS). An SIRSmodel nests both SIR and SIS models. The canonical SIR model (Kermack and McK- In an already large and fast-growing economics literature addressing the COVID-19 pandemic, we areonly aware of three papers allowing for waning immunity. We contrast our paper with these three papersbelow. The assumption of permanent immunity is also common outside of the economics literature: e.g.,Ferguson et al. (2020) and Wang et al. (2020). Among various approaches to study epidemics in economic models, ours follows Farboodi, Jaroschand Shimer (2020), Garibaldi, Moen and Pissarides (2020), Guimar˜aes (2020), and Toxvaerd (2020) by di-rectly modeling the choice of social activity. Another approach is to assume contacts are a function ofthe level and type of i) consumption (Eichenbaum, Rebelo and Trabandt, 2020 a , b ; Krueger, Uhlig andXie, 2020) and/or ii) labor (Eichenbaum, Rebelo and Trabandt, 2020 a , b ; Glover et al., 2020). Yet anotherapproach is treating pandemics as exogenous shifts in state variables (e.g., human capital) (Boucekkine,Diene and Azomahou, 2008). Such an approach resembles the MIT shock assessed by Guerrieri et al.(2020) in the context of the COVID-19 pandemic. See also Philipson and Posner (1993), Kremer (1996),Chakraborty, Papageorgiou and P´erez Sebasti´an (2010) and Greenwood et al. (2019) for an economic per-spective of HIV and malaria. The flow from recovered to susceptible stems from waning immunity. For an accessible review of epidemiological models, see Hethcote (2000). a ; Farboodi, Jarosch and Shimer, 2020). The canonical SIS model as-sumes that agents are never immune and, thus, is employed in studying the economicsof recurrent diseases (e.g., Goenka and Liu, 2012, 2019; Goenka, Liu and Nguyen, 2014).An SIRS model is between an SIR and an SIS model by allowing agents to be immunebut only temporarily. In light of the evidence on immunity against SARS-CoV-2 andother coronaviruses, an SIRS model is warranted to study the COVID-19 pandemic(Kellam and Barclay, 2020).In the canonical SIRS model, immunity is a binary variable: agents are either im-mune or not. And after agents lose immunity they become as susceptible as any othersusceptible agent. Waning immunity, however, does not necessarily mean that agentswho lose immunity are as unprotected as those who were never infected (Punt et al.,2018; Huang et al., 2020). Immunological memory (e.g., antibody count) might notbe enough to avoid a reinfection but is likely enough for the body to react faster to areinfection. For this reason, our SIRS block allows susceptible agents to differ amongthemselves based on infection history. The heterogeneity in infection history can becaptured by distinct i) probabilities of being infected, ii) recovery speed, iii) viral shed-ding, and iv) cost of infection. These possible distinctions are important as they mayprevent an endemic COVID-19.In Section 4, we analyze the simplest case in which all susceptible agents, irrespec-tive of their infection history, are alike. We reach two main conclusions. First, if im-munological memory wanes, there is no vaccine or cure, and there is no major exoge-nous change in the contagiousness of the virus, then COVID-19 becomes endemic be-cause of the continuous flow of agents into the susceptible health state. In this scenario,both a social planner and decentralized individuals choose to social-distance forever.Second, the duration of immunity may not meaningfully change optimal choices inthe initial months of the pandemic. We find that the decentralized equilibria is virtu-ally independent of waning immunity for more than six months and until close to peakinfections because agents abstract from how their actions affect the probability thatthey are reinfected later. In slight contrast, we find that the centralized equilibria mayvary with waning immunity depending on the costs of infection and the probability offinding a vaccine or cure.An endemic COVID-19, induced by waning immunity, implies a higher present valueof infection costs than a non-endemic one. In response to these higher costs, the socialplanner mandates further social-distancing. Yet, this extra social-distancing stemmingfrom waning immunity can be small in the short run. If a vaccine is expected in 18months and the costs of infection reflect an infection-fatality rate of 0.64%, we find In particular, Huang et al. (2020) report that individuals can be infected with the same human coron-aviruses one year after first infection but with lower severity. b ), Giannitsarou, Kissler and Toxvaerd(2020), and Malkov (2020). These papers, however, differ from ours in crucial aspectsincluding the object of study, approach, and modeling choices. Eichenbaum, Rebeloand Trabandt study the role of testing and quarantines in a model with health state un-certainty and check the robustness of their findings if immunity wanes; thus, they donot fully explore how the duration of immunity affects contagion in the context of thecurrent pandemic. Malkov focus on how waning immunity affects the epidemiologi-cal dynamics during the COVID-19 pandemic, but he does not allow individuals andthe social planner to endogenously react in his simulations. Giannitsarou, Kissler andToxvaerd assess the centralized problem during the pandemic in case immunity wanes,but they do not contrast the centralized and decentralized equilibria and their resultsdiffer from ours due to modeling and calibration choices. In Section 4.2, we contrastin more detail our results with those in the three papers.
We build an economic model of an epidemic in which agents face a trade-off betweensocial activity and exposure to the virus. This trade-off results from the link betweenthe epidemiological and utility-maximization blocks of the model. The link, in turn,stems from our assumption, following Farboodi, Jarosch and Shimer (2020), Garibaldi,Moen and Pissarides (2020), and Guimar˜aes (2020), that new infections depend on thenumber of susceptible and infected agents and the social activity chosen by susceptibleagents. The model is set in discrete time. The population is constant and of measureone. We focus on symmetric equilibria in which agents with the same health state be- There are two other relevant differences. As Eichenbaum, Rebelo and Trabandt and Malkov, Giannit-sarou, Kissler and Toxvaerd assume that all susceptible agents are alike irrespective of infection history.And, as Eichenbaum, Rebelo and Trabandt, Giannitsarou, Kissler and Toxvaerd place their simulations atthe start of the pandemic and assume that only one non-pharmaceutical intervention is in place (testingin the case of Eichenbaum, Rebelo and Trabandt and mandatory social-distancing in the case of Gian-nitsarou, Kissler and Toxvaerd). Our last set of simulations in which we include the effects of other NPIsbrings, thus, further insights to the current policy discussion. primary and agents that wereinfected at least once as secondary . To further ease our exposition, we use the index j ∈ { p, q } , when referring primary and secondary agents, respectively. The population in period t consists of five groups of agents: primary susceptible, s p,t ,primary infected, i p,t , recovered, r t , secondary susceptible, s q,t , and secondary infected, i q,t . The number of new infections for each type is given by β j a j,t s j,t i t , where β j is the measure of contagiousness for susceptible agents of type j with β q ≤ β p , a j,t ∈ [0 , is the social activity of susceptible agents of type j , and i t = i p,t + σi q,t (1)is the number of infected agents. We adjust i q,t with σ ≤ to allow secondary infectedindividuals shedding less virus than primary infected ones.The laws of motion governing the transitions between health states are the follow-ing: s p,t +1 = (1 − β p a p,t i t ) s p,t , (2) i p,t +1 = β p a p,t s p,t i t + (1 − γ p ) i p,t , (3) r t +1 = P j γ j i j,t + (1 − α ) r t , (4) s q,t +1 = αr t + (1 − β q a q,t i t ) s q,t , (5) i q,t +1 = β q a q,t s q,t i t + (1 − γ q ) i q,t , (6)where γ j is the probability that an infected individual of type j recovers and α is theprobability that a recovered individual loses immunity. If α = 0 and a p,t = 1 for all t , themodel reduces to the canonical SIR model. If α > , σ = 1 , β p = β q , γ p = γ q , and a j,t = 1 for all j and t , the model reduces to the canonical SIRS model. Under permanent immunity, α = 0 , the number of secondary susceptible agents remains zero. Underwaning immunity, α > , with σ = 1 , β p = β q , and γ p = γ q , identifying secondary agents is trivial. .2.1 Utility Maximization In this section, we detail the lifetime utility maximization problem of a primary suscep-tible agent. Agents derive utility from their social activity. The utility function, denotedby u ( a ) is single-peaked and its maximum is normalized to zero at a = 1 . The maxi-mization problem of a primary susceptible agent is given by max { a p,t ,a p,t } ∞ t =0 X ∞ t =0 X j Λ t (cid:16) s j,t u ( a j,t ) − γ j κ j i j,t (cid:17) , subject to Eqs. (2–6). In this maximization problem, the fraction of agents in eachhealth state group corresponds to the (subjective) probability of the agent being in thatstate; Λ is the discount factor; and κ j captures all the costs of recovering from the infec-tion. As primary and secondary infected agents may respond differently to the infection(e.g., differ in symptoms severity), we set κ q ≤ κ p . The decentralized optimum socialactivity is, then, governed by u ′ ( a j,t ) = β j i t ( V s j ,t − V i j ,t ) , (7) V s j ,t Λ = u ( a j,t +1 ) + V s j ,t +1 − β j a j,t +1 i t +1 ( V s j ,t +1 − V i j ,t +1 ) , (8) V i j ,t Λ = V i j ,t +1 − γ j ( κ j + V i j ,t +1 − V r,t +1 ) , (9) V r,t Λ = V r,t +1 + α ( V s q ,t +1 − V r,t +1 ) , (10)for both j ∈ { p, q } and V x,t denotes the (shadow) value of the agent in state x ∈ { s p ,s q , i p , i q , r } . Eq. (7) summarizes the trade-off. Its left-hand side is the marginal util-ity of social activity while its right-hand side is expected marginal costs resulting fromthe possibility of infection. Marginal costs depend on how likely they are exposed bymarginally increasing activity, β j i t . And it also depends on the change in the valuecaused by exposure, which is always positive, V s j ,t − V i j ,t > . Thus, susceptible agentsrestrain their social activity, a j,t ≤ , to reduce exposure risk.Eqs. (7-10), determining the behavior of primary agents, are symmetric along j .Given that these equations do not depend on the (subjective) probability of being inany health state, the same equations also determine the behavior of secondary agents.Therefore, for brevity, we do not present the utility maximization problem of secondaryagents. A decentralized equilibrium corresponds to a path of social activities, { a p,t , a q,t } , thenumber of infected agents, i t , state variables, { s p,t , s q,t , i p,t , i q,t , r t } , and shadow values, { V s p ,t , V s q ,t , V i p ,t , V i q ,t , V r,t } , that satisfy Eqs. (1–10). .3.1 Utility Maximization In this section, we present the maximization problem of the social planner. The socialplanner chooses socially optimal activity by directly influencing aggregate variables. Inparticular, the maximization problem of the social planner is given by max { a p,t ,a p,t } ∞ t =0 X ∞ t =0 X j Λ t (cid:16) s j,t u ( a j,t ) − γ j κ j i j,t (cid:17) , subject to Eqs. (1-6). Relative to the decentralized problem, Eq. (1) is the additionalconstraint because the social planner internalizes how infected individuals affect con-tagion. The socially optimum social activity is, then, governed by u ′ ( a j,t ) = β j i t ( V s j ,t − V i j ,t ) , (11) V s j ,t Λ = u ( a j,t +1 ) + V s j ,t +1 − β j a j,t +1 i t +1 ( V s j ,t +1 − V i j ,t +1 ) , (12) V i j ,t Λ = V i j ,t +1 − γ j ( κ j + V i j ,t +1 − V r,t +1 ) − σ j P j β j a j,t +1 s j,t +1 ( V s j ,t +1 − V i j ,t +1 ) (13) V r,t Λ = V r,t +1 + α ( V s q ,t +1 − V r,t +1 ) , (14)for both j ∈ { p, q } , and σ j = ( if j = p,σ if j = q. Comparing this set of equations govern-ing the optimal choice of the social planner with that governing the optimal choice ofagents in the decentralized problem (Eqs. 7-10), we can see that the only differenceis in the shadow values of the infected states. This difference reflects a key externalityemphasized in the literature: in a decentralized equilibrium, agents decide their socialactivity without considering the risk of infecting others. As a result, both V i p ,t and V i q ,t are lower in the social planner’s problem, which ( ceteris paribus ) restrains social activityrelative to the decentralized equilibrium. Part of our objective in this paper is to ana-lyze how the possibility of recovered agents losing immunity distances decentralizedand centralized choices. A centralized equilibrium corresponds to a path of social activities, { a p,t , a q,t } , the num-ber of infected agents, i t , state variables, { s p,t , s q,t , i p,t , i q,t , r t } , and shadow values, { V s p ,t , V s q ,t , V i p ,t , V i q ,t , V r,t } , that satisfy Eqs. (1-6) and Eqs. (11-14). We summarize our parameter choices in Table 1. Each period in the model correspondsto one day. The discount factor includes both a time discount rate, ρ , and the prob-ability of finding a cure-for-all, δ , that would end the problem. In particular, we set Λ = ρ δ , ρ = 0 . / , and δ = 0 . / reflecting a yearly discount rate of and8he probability of finding the cure-for-all of within a year (see, e.g., Alvarez, Argenteand Lippi, 2020; Farboodi, Jarosch and Shimer, 2020).Table 1: Benchmark Calibration Discount factor:
Λ = . /
365 11+0 . / Cost of infection: κ p = κ q = 512 Average number of days as infected: γ − p = γ − q = 18 Infectiousness: β p = β q = 2 . / Average number of days immune: α − = 750 Relative viral shedding of secondary infected: σ = 100% As in Farboodi, Jarosch and Shimer (2020) and Guimar˜aes (2020), the utility of socialactivity is determined by: u ( a ) = log( a ) − a + 1 , (15)which guarantees that u ( a ) is single-peaked with maximum at a = 1 and u (1) = u ′ (1) =0 . We also closely follow Farboodi, Jarosch and Shimer to find the cost of infection, κ p .Assuming that the value of life is US $10 million and assessing how much agents wouldbe willing to permanently reduce their consumption to permanently lower the proba-bility of dying by . , we find that the value of life is in model units. Based onthis, we obtain κ p using the probability of dying conditional on infection. The meta-analysis of Meyerowitz-Katz and Merone (2020) suggests that about . of those in-fected with the virus, die. Thus, we set κ p = 512 .We follow Atkeson (2020) and most of the economics literature assessing the COVID-19 pandemic and assume that infected individuals remain so for 18 days, γ − p = 18 . Tocalibrate β p , we follow Acemoglu et al. (2020) and assume β p = 2 . / , implying a basicreproduction number, R , of . . This number is relatively optimistic in light of, for ex-ample, the R assumed in Alvarez, Argente and Lippi (2020) of . .At this stage, the duration of immunity against COVID-19 and how secondary agentsdiffer from primary agents is unknown. To calibrate the probability that recovered in-dividuals lose immunity, we use the evidence regarding other coronaviruses surveyedin Huang et al. (2020) and also the assumption in Eichenbaum, Rebelo and Trabandt(2020 b ) and set α − = 1 / , implying that agents have immunity for about two years.Regarding the remaining parameters, in our benchmark we simply assume that β q = β p , γ q = γ p , κ q = κ p and σ = 100% . Therefore, our benchmark calibration implies a SIRSmodel augmented with the endogenous choice of social activity.We solve the model using a shooting algorithm as detailed in Garibaldi, Moen and9issarides (2020). As a starting point, we assume that 1 in a million agents are primaryinfected, i p = 1 / , and the remaining are primary susceptible. Panels A and B of Figure 1 present how a waning immunological memory affects op-timal decentralized and centralized dynamics, respectively. The blue (solid) lines as-sume our benchmark, i.e., agents are immune for two years on average. The green(dashed) lines assume, as a lower bound and consistent with Huang et al. (2020) andKissler et al. (2020), that immunological memory lasts only 10 months. The red (dot-dashed) lines assume, as an upper bound and as in the economics literature (e.g. Al-varez, Argente and Lippi, 2020; Eichenbaum, Rebelo and Trabandt, 2020 a ; Farboodi,Jarosch and Shimer, 2020), that immunity lasts forever (implying an SIR model).Two of our findings in Figure 1 are striking. First, if immunological memory wanes( α > ), then in both centralized and decentralized equilibria, social activity is severelyand permanently curtailed until the discovery of a vaccine or cure. This results fromthe continuous flow of agents from immune to susceptible, implying a continuous flowfrom susceptible to infected and, therefore, a permanent exposure risk. Thus, if immu-nity wanes, COVID-19 reaches an endemic steady-state. In the centralized equilibrium,social activity stabilizes at about 55% lower than absent the epidemic. In the decentral-ized equilibrium, social activity reaches its minimum after about 200 days and thenrecovers slightly to its long run value, 30% lower than absent the epidemic. If agentsnever lose immunity, α = 0 , the results are very different. In this case, all agents returnto normal activity as infections asymptotically disappear. This happens faster, albeitat a higher social cost, in the case of the decentralized equilibrium, leading to muchhigher peak infections. Furthermore, in the decentralized equilibrium, approximately60% of the agents are infected at least once within three years, which differs substan-tially from about 5% in the centralized equilibrium.Second, the underlying duration of immunity barely moves the initial dynamics ofepidemiological variables and social activity for around 200 days in the decentralizedand 400 days in the centralized equilibrium. This result is partly explained by the lowaccumulation of secondary agents as few agents obtain and lose immunity in the initialmonths of the epidemic even when immunity wanes after 10 months, α = 1 / . Butother factors play important roles, especially in centralized equilibria. In decentralizedequilibria, agents do not take into account how their actions, by affecting infections,change the pace at which they might be reinfected. As a result, social activity in de-centralized equilibria is mostly affected by the dynamics of infected agents. As soonas many agents start losing immunity and become susceptible and infected again, theeffects of waning immunity become visible in optimal social activities.In centralized equilibria, however, the externalities of social activity are considered10igure 1: The Role of Immunity Duration Panel A: Decentralized Equilibrium
Days
Susceptible = 1/750 = 1/300 = 0
Days
Infected
Days
Secondary Agents
Days
Primary Activity
Days
Mean Activity
Panel B: Centralized Equilibrium
Days
Susceptible = 1/750 = 1/300 = 0
Days
Infected
Days
Secondary Agents
Days
Primary Activity
Days
Mean Activity
Note:
Susceptible agents are s p,t + s q,t ; infected agents are i p,t + i q,t ; secondary agents are s q,t + i q,t + r t ; primary activity is a p,t (which, in this case, equals secondary activity, a q,t );and mean activity is s p,t a p,t + s q,t a q,t + i p,t + i q,t + r t . in decision-making. The social planner knows that by reducing social activity, it low-ers and postpones infections and, thereby, decreases the number of secondary agentsthat lose immunity. Furthermore, the social planner is aware of the costs of the en-demic steady-state. These two factors combined motivate the social planner to con-strain social activity by more when waning immunity induces an endemic COVID-19.Yet, surprisingly, in our benchmark case, the optimal centralized social activity is al-most unmoved by the duration of immunity for 400 days.11he social planner aims to minimize the sum of the present value of the costs ofinfection and of social-distancing. If immunity is permanent, Panel B of Figure 1 showsthat the best option to minimize social costs is to endure high social-distancing, post-pone infections, and wait for the vaccine. If, on the other hand, immunity wanes, fu-ture infection costs increase but their present value is substantially discounted becausethe vaccine or cure is expected in 18 months. Furthermore, as social activity is heav-ily constrained even if immunity is permanent, the marginal costs of social-distancingare high and very sensitive to further increases in social-distancing due to the curva-ture of the utility function. Put differently, the social planner lacks room to maneuverto strongly react to waning immunity in the early months of the pandemic. These twofactors combined explain why waning immunity is relatively unimportant for manymonths in determining optimal social-distancing.To gain further insight, in Figure 2, we show how two key parameters change thenumber of infected agents and social activity of primary agents in centralized equilib-ria. Panel A depicts again the benchmark cases to ease comparison. Panel B depicts theresults when expected time to find a vaccine or cure is 4.5 years, implying δ is a third ofits benchmark value. Panel C depicts the results when the infection-fatality rate is ap-proximately 0.21%, implying κ j is a third of its benchmark value. This figure shows thatwaning immunity matters in these two deviations from benchmark in the centralizedequilibria. The results are particularly staggering in the case of low δ : in this scenario,peak infections occur much earlier and is more than 20 times higher when immunity ispersistent than when immunity wanes.A lengthier period to discover a vaccine or cure, captured by a lower δ , implies thatthe social planner must restrict social activity for more time to avoid infections andwait for the vaccine or cure. We find that the corresponding increase in the presentvalue of social-distancing costs greatly exceeds the increase in the present value of thecosts of infection if immunity is permanent. Therefore, the social planner allows formore infections. The opposite holds when immunity wanes. Technically, a lower δ re-duces the discount factor, increasing the present value of the infection costs causedby waning immunity and the endemic COVID-19. Therefore, the social planner reactseven stronger to the pandemic when it emerges if immunity wanes and the vaccine isexpected later in time.A reduction in the infection-fatality rate, captured by a lower, κ j , implies less costsof infection and, thus, more social activity whatever is α . But the rise in social activityincreases in the duration of immunity (i.e., decreases in α ). When the costs of infec-tion, κ j , are lower, the implied point of the reduced social activity is under the flatterrange of the curved utility. Thus, the marginal cost of additional social-distancing isalso relatively low, increasing the room to maneuver of the social planner. Therefore, Figure A1 in the Appendix shows that the way waning immunity affects decentralized equilibria reliesmuch less on κ p and δ . In an experiment (not reported), we varied the curvature of the utility function and find that thechanges in social activity brought by waning immunity decrease in the curvature.
Panel A: Benchmark
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Panel C: Low Costs of Infection ( =171)
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Note:
Infected agents are i p,t + i q,t ; primary activity is a p,t (which, in this case, equals sec-ondary activity, a q,t ). the social planner acts stronger from the onset of the pandemic to reduce the costs ofan endemic COVID-19 and gain time for the discovery of a cure or vaccine. This differ-ence in optimal choices lead to clearly different disease dynamics: the faster immunitywanes, the more the social planner postpones and reduces peak infections.In sum, waning immunity implies a persistent reduction in social activity eitherindividually chosen or mandated. But because individuals lack altruism, implying aweaker link between choice and (re)infection, the early response to the pandemic indecentralized equilibria is not dependent on waning immunity. In centralized equilib-ria, however, waning immunity may affect the early response to the pandemic depend-ing on the magnitude of the costs of infection and critically on how likely a vaccine orcure is expected to arrive. Yet, in our benchmark calibration, which we find plausible,13aning immunity barely affects early optimal choices of social activity in the central-ized equilibria. In this section, we contrast our findings with the three papers in the economics liter-ature that study waning immunity. Eichenbaum, Rebelo and Trabandt (2020 b ) studythe role of testing and quarantining in a model linking consumption and labor choicesto contagion. They also find that decentralized individuals permanently reduce theiractivity (consumption and labor supply) due to the endemic steady-state caused bywaning immunity. Furthermore, their Figure 9 suggests that, for over a year, waningimmunity is virtually irrelevant for decentralized decisions. Yet, waning immunity af-fects their centralized equilibria in a way different from ours because of the differentpolicy instruments considered. Their testing and quarantining polices rule out en-demic steady states because asymptotically all individuals are continuously tested andinfected ones are quarantined. Therefore, waning immunity neither restricts socialplanner’s actions nor permanently constrains economic activity in Eichenbaum, Re-belo and Trabandt (2020 b ).Giannitsarou, Kissler and Toxvaerd (2020) study the effects of waning immunity onsocial-distancing policy. Notable differences between our paper and theirs are as fol-lows. They assume that the pandemic ends in six years (by the discovery of a vaccine),ruling out any endemic steady state. As a result, social activity returns to normal in theirsimulations. Moreover, the costs of infection and social-distancing are much lowerin their model. They assume that the costs of infection are 10% lower output by in-fected and zero output by deceased only for the time span of the pandemic. The costsof social-distancing are quadratic and finite in a mandated full-lockdown, which pro-vide a vast room to maneuver for the social planner to act. Therefore, when immunitywanes, they obtain deferment of peak infections and a negative relation between im-munity duration and mandated social distancing (similar to our results in the low costof infection case, Figure 2, Panel C).Malkov (2020) studies how waning immunity affects the dynamics of an epidemio-logical model under different calibrations of the basic reproduction number. He con-cludes that until close to peak infections, waning immunity barely changes the dis-ease dynamics. Although Malkov does not include endogenous decision making in hismodel, his findings are relatively close to our findings in the decentralized equilibriaas waning immunity also only matters close to peak infections. But his findings differsubstantially from our results in the centralized equilibria. In this case, the social plan-ner takes into account the future costs of waning immunity in his early response to thepandemic, which in turn, leads to different disease dynamics.14 Heterogeneous Susceptible Agents
So far, we have analyzed an SIRS model augmented with endogenous social activity. Us-ing our benchmark calibration, in Figure 3, we illustrate how our results change whensecondary susceptible and infected agents differ from their primary counterparts inthree aspects. Figure 4 complements our illustration in Figure 3 by showing how ourresults differ if δ and κ p are low. Green (dashed) lines show the case in which secondarysusceptible agents are less likely to be infected than primary susceptible agents; red (dot-dashed) lines show the case in which secondary infected individuals shed 75%less virus than primary infected; yellow (dotted) lines show the case in which the costsof infection are 75% lower for secondary agents; and blue (solid) lines show the bench-mark. In the first two cases, even though all agents eventually lose immunity, asymp-totic R is below 1 and, thus, the epidemic will asymptotically disappear as secondaryagents gradually replace primary agents. In the case of κ q = 0 . κ p , the cost of a rein-fection is much lower but the flows between states do not asymptotically converge tozero. That is, asymptotically, individuals are continuously infected but suffering muchless than in the beginning of the epidemic. In this case, COVID-19 converges to an en-demic steady-state, which is similar to that of other coronaviruses giving rise to flu-likesymptoms (Edridge et al., 2020; Huang et al., 2020; Kellam and Barclay, 2020).Figure 3 and Panel A in Figure 4 show that if secondary and primary agents dif-fer, there are little changes to the optimal social activity of primary susceptible agentsfor approximately a year and a half in both centralized and decentralized equilibria.This contributes to a similar path for the number of susceptible (both primary and sec-ondary) agents for many months. Thus, as in the previous section, our benchmarkcalibration implies that any uncertainty caused by waning immunity is not much rele-vant for several months after the start of the epidemic.Our results in the centralized equilibria depend, again, on δ and κ p . When it is un-likely to discover a vaccine or cure (low δ ), the early response to the pandemic criticallydepends on whether COVID-19 becomes endemic. If it becomes endemic (benchmarkand κ q = 0 . κ p ), the social planner restricts social activity further as the present valueof the costs of the endemic steady-state are larger. But if COVID-19 does not becomeendemic ( β q = 0 . β p or σ = 0 . ), the social planner is more lenient. A low cost ofinfection of primary agents, κ p , grants room for maneuver for the social planner to actearly to endemic steady-states due to the curvature of the utility function. Therefore,mandated social-distancing visibly increases with the overall costs of the pandemic inPanel C of Figure 4.The optimal behavior of secondary susceptible agents is much different from thatof primary susceptible agents irrespective of δ and κ p . If it is unlikely that secondaryagents are reinfected ( β q is low), they restrain social activity by much less than primaryones, which is problematic from a social perspective because they expose other agents This implies a reduction of 75% in R . Different combinations of changes in β j and γ j leading to thesame fall in R imply similar results. Panel A: Decentralized Equilibrium
Days
Susceptible
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Infected
Days
Secondary Agents
Days
Primary Activity
Days
Secondary Activity
Days
Mean Activity
Panel B: Centralized Equilibrium
Days
Susceptible
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Infected
Days
Secondary Agents
Days
Primary Activity
Days
Secondary Activity
Days
Mean Activity
Note:
Homogeneous refers to the case in which secondary and primary agents are alike.Susceptible agents are s p,t + s q,t ; infected agents are i p,t + i q,t ; secondary agents are s q,t + i q,t + r t ; primary activity is a p,t ; secondary activity is a q,t ; and mean activity is s p,t a p,t + s q,t a q,t + i p,t + i q,t + r t . (especially primary) significantly. Thus, even if susceptible agents are unlikely to bereinfected, policymakers should be aware that these agents are likely to be excessivelyactive.This problem of excessive social activity in the decentralized equilibrium is evenworse if κ q = 0 . κ p . As agents are not altruistic, they only care about their own risks. Alower cost of reinfection then significantly lowers their incentives to social-distance. Incontrast, the social planner would like secondary agents to substantially constrain their16igure 4: Heterogeneous Susceptible Agents - Centralized Equilibria Panel A: Benchmark
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Panel C: Low Costs of Infection ( =171)
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Note:
Homogeneous refers to the case in which secondary and primary agents are alike.Infected agents are i p,t + i q,t ; primary activity is a p,t ; secondary activity is a q,t . activity because their viral shedding and probability of infection are unchanged andmany susceptible agents are still primary susceptible. The scenario of κ q = 0 . κ p also shows that agents asymptotically constrain social activity, even in the decentral-ized equilibrium, because COVID-19 becomes endemic and the costs of infection re-main high (these costs imply a probability of dying of 0.16% in the benchmark). Ifthese costs were lower, closer to those of endemic human coronaviruses, agents in adecentralized equilibrium would behave almost as if there was no virus which is whatwe observed until the COVID-19 pandemic. In this regard, secondary agents are similar to young agents in models that breakdown agents basedon age (Acemoglu et al., 2020; Gollier, 2020). In those models, because young agents know that they areless likely to suffer if infected, they are too active from a social perspective as they increase exposure ofolder individuals. σ = 0 . . Recall that σ measures how likely sec-ondary infected shed virus onto susceptible. Since σ pertains only to the externalitycaused by secondary agents’ actions, it does not affect decisions in the decentralizedequilibrium: secondary susceptible agents act as primary susceptible agents. A socialplanner, in contrast, would allow secondary agents to enjoy relatively more social ac-tivity. Both primary and secondary agents, however, benefit indirectly from the lowerviral-shedding of secondary infected agents, which allows them to enjoy more socialactivity, converging asymptotically to full social activity in both equilibria. Following the SARS-CoV2 outbreak, governments around the world have combinedseveral NPIs to change the natural course of the pandemic. To account for this change,in this section, we base our simulations on initial conditions matching the current (epi-demiological) state of the COVID-19 pandemic.In the (new) initial conditions, we accommodate a compromise between the epi-demiological state in the US and four European countries, France, Italy, Spain, and theUK, as of 1 July 2020. On this day, the fraction of (currently) infected population wasapproximately 0.46% in the US; 0.09% in France and 0.02% in Italy. These numbersare likely to be understated as authorities fail to test and identify many of infected andespecially asymptomatic people (see references in Stock, 2020 for evidence on the pro-portion of asymptomatic). Bearing in mind the understatement and cross-country dif-ferences in the numbers, we find a compromise at i = 0 . . To set the initial numberof recovered agents, we look at the evidence from antibody surveys. In France, Spain,and the UK, antibody surveys suggest that slightly more than 5% of the population hasantibodies against SARS-CoV-2. Given that the fraction of infected population ratio istwo to three times higher in the US than in France, Spain, and the UK, we find a com-promise at r = 6% .In all countries that we examined for this section, identified infected individuals arequarantined. This NPI naturally reduces contagion and we model it as an exogenousreduction in the social activity of some infected agents. In particular, we assume that50% of infected agents, which is within the current estimated range of asymptomaticcases, are identified and cannot enjoy maximum social activity. In case infected in-dividuals are identified, they enjoy 40% of normal social activity, which increases theexpected costs of infection. Thus, average social activity of infected individuals fallsby 30%. Other NPIs, like mandatory mask use, differ across countries. In France andthe UK, mask use is only mandatory in public transport, whereas in Spain, it is manda- Statistics consulted in https://coronavirus.jhu.edu/map.html on 2 July 2020. See the ONS COVID-19 Infection Survey for the UK; for France and Spain, see Salje et al. (2020) andPoll´an et al. (2020). In ourmodel, we treat mask use (mandatory or not) as an exogenous reduction in contagious-ness, β p and β q , by 30%. In sum, these NPIs reduce β p and β q by slightly over 50%. We depict the results in Figures 5 and 6. Blue (solid) lines assume the benchmarkvalues for the the rest of the parameters. Green (dashed) lines assume that agentsare permanently immune. Red (dot-dashed) lines assume that in their contagious-ness and cost of infection, secondary agents differ substantially from primary agents: β q = 0 . β p , σ = 0 . , and κ q = 0 . κ p . Compared to our previous simulations,the other NPIs significantly elevate social activity because of the fall in contagiousness.Furthermore, the simulations suggest that individuals and policymakers do not needto know the duration of immunity and how secondary agents differ from primary onesuntil at least 2021 even if δ and κ p are low. Thus, the combination of lower conta-giousness and relatively high initial infections reduce the relevance of waning immu-nity even in centralized equilibria, making the the social planner less responsive to fu-ture infection costs. This suggests that if current NPIs remain in place, there is stillsubstantial time to learn about the duration of immunity. Yet, given the implications ofthe mortality rate for social activity, and consistent with Hall, Jones and Klenow (2020),learning about the actual infection-fatality rate seems highly important.
It is likely that immunity against COVID-19 eventually wanes and those that are im-mune today will face the risk of a reinfection (Edridge et al., 2020; Huang et al., 2020;Kellam and Barclay, 2020; Seow et al., 2020). This scenario is especially problematic ifCOVID-19 becomes endemic as other human endemic coronaviruses. We show thatif COVID-19 reaches an endemic steady-state and a vaccine or cure is not discovered,social-distancing is here to stay. But, on the bright side, we also show that optimaldecentralized and centralized choices do not necessarily depend on waning immuno-logical memory for many months following the initial outbreak/contagion. This is es-pecially the case if a vaccine is expected soon, the costs of infection are already largein the short run, and other NPIs that lower contagiousness remain in place. Beforemaking irreversible decisions, individuals and policymakers seem to have time to learnmore about immunological memory against SARS-CoV-2 and answer the call for sero-logical studies from Kellam and Barclay (2020), Kissler et al. (2020), and Lerner et al. At the time that we write this paper, France and the UK have announced mandatory mask use in shops. Crucially, R is still above one as the pandemic would asymptotically disappear if R < . But R permanently below one seems unlikely as pointed by the second wave of infections in Australia and SouthKorea. In these simulations, we assume that the initial fraction of secondary susceptible and infected indi-viduals is zero. Although there are slightly visible differences in terms of optimal primary activity if δ and κ p are low,the implied dynamics of infected individuals is almost unchanged. Optimal secondary activity dependsmuch more on the scenario for waning immunity, but there are very few agents that are secondary sus-ceptible. Days
Susceptible = 1/750 = 0Optimistic
Days -3 Infected
Days
Secondary Agents
Days
Primary Activity
Days
Secondary Activity
Days
Mean Activity
Panel A: Decentralized Equilibrium
Days
Susceptible = 1/750 = 0Optimistic
Days
012 10 -3 Infected
Days
Secondary Agents
Days
Primary Activity
Days
Secondary Activity
Days
Mean Activity
Panel B: Centralized Equilibrium
Note:
Optimistic refers to the case in which β q = 0 . β p , σ = 0 . , and κ q = 0 . κ p .Susceptible agents are s p,t + s q,t ; infected agents are i p,t + i q,t ; secondary agents are s q,t + i q,t + r t ; primary activity is a p,t ; secondary activity is a q,t ; and mean activity is s p,t a p,t + s q,t a q,t + 0 . i p,t + i q,t ) + r t . (2020).Yet, in 6-12 months, we do need to know more about how antibodies and T-cellsdefend the human body against SARS-CoV-2. In particular, we must know how longimmunity lasts and whether individuals that were infected (secondary agents) differsubstantially from those that were never infected (primary agents). The longer immu-nity lasts, the less demanding should social-distancing be. And, in the limit, if immu-nity lasts a lifetime, then COVID-19 does not reach an endemic steady-state and social-20igure 6: What if It Was Today? - Centralized Equilibria Panel A: Benchmark
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Panel C: Low Costs of Infection ( =171)
Note:
Optimistic refers to the case in which β q = 0 . β p , σ = 0 . , and κ q = 0 . κ p . Infectedagents are i p,t + i q,t ; primary activity is a p,t ; secondary activity is a q,t . distancing will sooner or later be unwarranted. Furthermore, if secondary agents maybe reinfected but are somewhat protected against the virus, then COVID-19 may notbecome endemic. Yet, the way in which secondary agents differ from primary agentsis crucial to design policy. For example, if most of the gains from the additional pro-tection are private – because secondary agents are less likely to die or less likely to bereinfected – then secondary agents are excessively active from a social viewpoint. If, onthe other hand, most of the gains from the additional protection are social – becausesecondary agents shed less virus – then the decentralized and centralized equilibria arecloser and less social-planning is required.Even though most of the economics literature assumes permanent immunity, thissimplification may not have dire consequences in the short run. If a vaccine or cure is21xpected soon, the costs of infection are not small, and other NPIs are in place, thenour model suggests that the optimal response in the initial months of the pandemicis virtually independent of waning immunity. The same is true if secondary agents,despite no longer immune, develop a strong protection against SARS-CoV-2 or shedmuch less virus. But, if these conditions do not hold, many of the policy prescriptionsneed to be revised as they rely on the possibility of herd immunity.22 eferences Acemoglu, Daron, Victor Chernozhukov, Iv´an Werning, and Michael D Whinston.
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Robustness Checks of Decentralized Equilibria
Figure A1: The Role of Immunity Duration - Decentralized Equilibria
Panel A: Benchmark
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Panel C: Low Costs of Infection ( =171)
Days
Infected = 1/750 = 1/300 = 0
Days
Primary Activity
Note:
Infected agents are i p,t + i q,t ; primary activity is a p,t (which, in this case, equals sec-ondary activity, a q,t ). Panel A: Benchmark
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Panel C: Low Costs of Infection ( =171)
Days
Infected
Homogeneous q = 0.25 p =0.25 q = 0.25 p Days
Primary Activity
Days
Secondary Activity
Note:
Homogeneous refers to the case in which secondary and primary agents are alike.Infected agents are i p,t + i q,t ; primary activity is a p,t ; secondary activity is a q,t . Panel A: Benchmark
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Panel B: Low Probability of Discovering a Vaccine or Cure ( =0.22/365)
Days -3 Infected = 1/750 = 0Optimistic
Days
Primary Activity
Days
Secondary Activity
Panel C: Low Costs of Infection ( =171)
Note:
Optimistic refers to the case in which β q = 0 . β p , σ = 0 . , and κ q = 0 . κ p . Infectedagents are i p,t + i q,t ; primary activity is a p,t ; secondary activity is a q,t ..