key: cord-0853101-8bj94qfd
authors: Baril-Tremblay, Dominique; Marlats, Chantal; Ménager, Lucie
title: Self-isolation()
date: 2021-02-06
journal: J Math Econ
DOI: 10.1016/j.jmateco.2021.102483
sha: b6dfd5632fad4a080d0eb44e85fded2902e97769
doc_id: 853101
cord_uid: 8bj94qfd

We analyze the spread of an infectious disease in a population when individuals strategically choose how much time to interact with others. Individuals are either of the severe type or of the asymptomatic type. Only severe types have symptoms when they are infected, and the asymptomatic types can be contagious without knowing it. In the absence of any symptoms, individuals do not know their type and continuously tradeoff the costs and benefits of self-isolation on the basis of their belief of being the severe type. We show that all equilibria of the game involve social interaction, and we characterize the unique equilibrium in which individuals partially self-isolate at each date. We calibrate our model to the COVID-19 pandemic and simulate the dynamics of the epidemic to illustrate the impact of some public policies.

The herd immunity level of an infectious disease is dened as the fraction of the population that must become immune for the spread of the disease to decline and stop. Under the simplest model, it is equal to (R 0 − 1)/R 0 , where R 0 is the basic reproduction number 1 of the disease, which is often estimated to be approximately 60% for COVID-19. The gure of 60% assumes that the population is homogenous and passive, while it is well-documented 2 that the herd immunity level varies between populations consisting of people with dierent behaviors. The object of the growing epi-economic literature is to analyze the two-sided interactions between the dynamics of epidemics and individual behaviors.

One of the many features of COVID-19 is the wide variety of responses to the infection in the population, with some individuals completely asymptomatic, and others developing fatal forms within a few days. As with symptomatic individuals, asymptomatic patients are a source of the spread of infection 3 . Before being infected, there is no way of knowing whether one is of the asymptomatic type. Therefore, individuals form beliefs about their type, which they continuously update on the basis of how much they might have been exposed to the virus.

At the same time, they decide how much to expose themselves to the virus in function of their updated belief. For instance, an individual who interacts with many people without developing COVID-19 symptoms becomes more optimistic about being the asymptomatic type. As a result, she may be tempted to meet even more people and forget about social distancing.

The contribution of this paper to the epi-economic literature is to introduce learning into an epidemiological model, and to analyze the dynamics of an epidemic when individuals tradeo the costs and benets of self-isolation on the basis of their subjective beliefs.

To analyze this question, we amend the classical Susceptible-Infected-Recovered (SIR hereafter) model of Kermack and McKendrick (1927) in two ways. In its classical version, the SIR model divides an homogeneous population into three groups: susceptible, infected and recovered, with individuals transiting from one group to another one at given, exogenous rates that 1 The basic reproduction number is the average number of secondary infections caused by a single infectious individual introduced into a completely susceptible population. gradually increase the time they spend outside, maintaining the eective reproduction number below the value that accelerates the epidemic. As a result, the epidemic curve is decreasing between the time of announcement of the epidemic and the arrival of the vaccine, contrary to the well-known bell-shaped curve of the SIR model. We nd that a later announcement of the epidemic increases the number of deaths, in line with the results of Silverio et al. (2020) , who nd a positive correlation between the number of cases before lockdown and the mortality rate in Italy. We also analyze the impact of policies aiming at mitigating the transmission of the virus such as mask distributions, messaging about hygiene measures, etc. We nd that individuals compensate the decrease in the risk of infection by reducing social distances, but not to the point of accelerating the epidemic. Overall, we nd that these policies reduce the number of deaths. We also show that a more performing health system results of less self-isolation but overall decreases the number of deaths. Finally, policies subsidizing selfisolation atten the economic curve, but we nd no substantial dierence when self-isolation is encouraged at the beginning or at the end of the epidemic.

when facing a risk of infection. For instance, Philipson and Posner (1993) show that the demand for measles, mumps and rubella vaccines increases when there is a large increase in measles cases in a community, and Ahituv et al. (1996) show that the demand for condoms increases in regions where HIV is prevalent. Some papers 4 also prove that individual behaviors impact the spread of infectious diseases. In the case of COVID-19, Cowling et al. (2020) show that border restrictions and changes in individual behavior are partly responsible for reduced transmission in Hong Kong in February 2020.

In the theoretical literature, some models analyze the eect of social distance in SIR or SIS epidemiological models, either in a social optimum approach (e.g. Sethi (1978) Rampini (2020) and Bairoliya and Imrohoroglu (2020) ). The problem of individuals who tradeo the costs and benets of self-isolation in the SIR model has been studied notably by Toxvaerd (2020) , Farboodi et al. (2020) and Brotherhood et al. (2020) . In an innite horizon model, Toxvaerd (2020) characterizes the exposure level at the symmetric equilibrium and shows that self-isolation attens the epidemic curve. The remainder of this paper is organized as follows. Section 2 sets up the model. In Section 3, we solve the best-response problem of a player, analyze some properties of the equilibrium and characterize equilibria in which there is no connement at all, or always partial connement. In Section 4, we calibrate our model to t the COVID-19 pandemic, we simulate the dynamics of the epidemic in equilibrium and provide some policy analysis.

Section 5 concludes and technical proofs are gathered in the Appendix.

The population Time t ∈ [0, +∞) is continuous and discounted at a common rate r > 0.

There is a rampant disease in the population, against which a vaccine will arrive at time T > 0.

The population is a continuum of individuals who must continuously decide what fraction of their time they spend outside. An individual who stays home is protected from infection, while an individual who goes out may be infected by other individuals, with a probability that will be described later. For simplicity, we assume that an individual is contagious as long as she is infected. People know whether they have been infected only if they experience the symptoms of the disease. There are two types of individuals in the population. Individuals of type θ s , the severe type, who experience the symptoms of the disease immediately after being infected.

In contrast, individuals of type θ a , the asymptomatic type, who do not have symptoms, thus never realize that they have been infected. The proportion µ ∈ (0, 1) of asymptomatic types 5 J o u r n a l P r e -p r o o f in the population is common knowledge, but individuals do not know their own type, unless they are of type θ s and become infected.

We assume that an individual who gets symptoms self-isolates immediately until the end of the symptoms, either to protect others, or simply because she is too sick to go out. Therefore, a strategy for player i is a measurable function k i : R + → [0, 1], with the interpretation that k i (t) is the proportion of time spent outside at time t, absent symptoms by time t.

Evolution of the epidemic To model the spread of the disease, we use the SIR model from Kermack and McKendrick (1927) . At each time t, the population is divided into three groups: susceptible S(t), infected I(t) and recovered R(t), i.e., those who died from the disease or recovered and are now immune to it 5 . Accordingly, s(t) is the fraction of the population that is healthy but susceptible to be infected at time t, i(t) the fraction of the population that is infected at time t, and r(t) = 1 − s(t) − i(t) the fraction of the population that has died or recovered from the disease at t.

The disease is transmitted to a susceptible individual through contact with an infected individual at rate β ∈ (0, 1), which measures the contagiousness of the disease. Therefore, the mass of susceptible individuals who become infected between t and t + dt depends on β, but also on the size of groups S(t) and I(t) and on the behavior of the population in each group.

Given a strategy prole k := ((k j ) j ), this mass equals β × j∈S(t) k j (t)dj × j∈I(t) k j (t)dj, thus the group of susceptible evolves according to the dynamics:

wherek I (t) := 1 i(t) j∈I(t) k j (t)dj andk S (t) = 1 s(t) j∈S(t) k j (t)dj denote the average fraction of time spent outside at t by infected and susceptible individuals, respectively. Infected recover from the disease at rate γ ∈ (0, 1). We assume that asymptomatic individuals do not die from the disease, while individuals of the severe type die at rate ν, with (γ + ν) ∈ (0, 1). As the fraction of infected is also increased by −ṡ(t), the group of infected evolves according to the following dynamics:

Evolution of subjective beliefs At time t, individual i holds a subjective belief p i (t) of being type θ s , with a common prior belief p i (0) = 1 − µ for all individuals 6 . In this model, no news is good news: the subjective belief of being the severe type decreases as time passes without the arrival of symptoms, and jumps to 1 the rst time the symptoms occur. Let us now describe the law of motion of p i (t). A susceptible individual i develops symptoms in [t, t + dt)

with probability 0 when she is of type θ a ; when she is of type θ s , she develops symptoms if she meets and is infected by some individual in I(t), which occurs with instantaneous probability 7 k i (t) × βk I (s)i(s)dt. By Bayes' rule, the law of motion of the subjective belief of individual i is thus 8 :ṗ

Payos Staying home prevents one from being infected, but comes at a cost (boredom, opportunity cost of not working or working in poorer conditions, lack of physical activity, etc.).

Being infected is also costly for individuals of the severe type because they suer from the symptoms, and, in the worst case, because they die from the disease. Therefore, at each time t, individuals tradeo the cost of self-isolating and the expected benet of not having the symptoms. We denote by c H the ow cost per unit of time spent at home, by c I the ow cost of having symptoms and by c D the ow cost of being dead.

Fix some strategy prole k and let us describe the expected payo to individual i at time t ≤ T when she plays the some strategy k i , denoted by v i (t; k i ). Uncertainty is solved for individual i the rst time she has symptoms. In that event, she knows that she is the severe 6 The common prior belief assumption is very strong in the case of the COVID-19 epidemic for two reasons.

First, as of today the proportion of asymptomatic has yet to be precisely estimated, not to mention commonly known. Second, even if the distribution of types in the population is still unknown, there are already evidences that the probability of being asymptomatic is conditional to individual characteristics. We discuss the implications of relaxing this assumption in section 5.3. 7 A possible interpretation of this probability is as follows. When she goes out, individual i is randomly matched with another individual who also went out, according to the uniform distribution. The probability of being matched with an infected agent at time t is thus exactly the mass of infected agents who are outside at time t, i.e.,kI (t)i(t). Finally, conditional on being matched with an infected individual, an individual has a probability β of being contaminated. 8 By Bayes' rule, player i's probability of being type θs conditionally on having no symptoms between t and t + dt is pi(t + dt) =

. Expression (3) is obtained by simplifying p i (t+dt)−p i (t) dt and taking dt to 0. 7 J o u r n a l P r e -p r o o f type, thus that she will stay at home until she recovers or passes away, thereby incurring a total cost of min{τ H ,τ D } 0 e −rt (c H + c I )dt, with τ H and τ D standing for the random times of healing and death, respectively. If she recovers (i.e., if τ H < τ D ), she becomes immune to the disease, plays k(t) = 1 forever after, thus obtains the continuation payo 0. If she dies (i.e., if τ D < τ H ), she incurs the ow cost c D forever after, thus obtains the continuation payo −c D /r. Therefore, the expected continuation payo to individual i the rst time she has symptoms is 9 :

Conditionally on having no symptoms before s ∈ [t, T ], the instantaneous payo to player i at time s is v I if she has symptoms, which occurs with subjective probability p i (s)k i (s)βk I (s)i(s), minus the cost c H scaled with the proportion of time spent in isolation, 1−k i (s). At time t, the subjective probability of not having any symptoms before i is vaccinated at time T , thus plays k i (t) = 1 for every t ≥ T and obtains the continuation

where functions s(.), i(.) and p i (.) are dened by (1), (2) and (3).

Fix a strategy prole k and an individual i. The best-response problem faced by i is the optimal control problem: 10 Using the law of motion of beliefs described by (3), we can establish two equalities.

First, i between self-isolating today and self-isolating tomorrow for player i? The best response at time t maximizes the sum of her current expected payo and of her discounted continuation payo, should no symptoms occur in the interval [t, t+dt). Therefore, her best-response payo at time t satises the Bellman equation:

Using (1 − rdt) as an approximation for e −rdt and eliminating terms to the order (dt) 2 , V i (t + dt)dt is approximated by V i (t)dt and the latter expression rewrites:

It appears that the best response of player i at time t depends on the sign of the expression c H −

To interpret this, note that two things can happen for individual i at time t: either she develops symptoms, thus obtains the payo v I today, or she does not, thus obtains tomorrow the continuation payo V i (t + dt), whose value at time t can be approximated by V i (t). More self-isolation thus aects the payo in three ways: it decreases the probability of getting v I at rate p i (t)βk I (t)i(t), increases the probability of getting V i (t) at rate p i (t)βk I (t)i(t) and increases the incurred cost at rate c H . This is why p i (t)βk

can be interpreted as the marginal expected benet of connement for player i at time t and c H as the marginal cost of connement; accordingly, player i's best response at time t depends on whether the marginal cost of connement is larger or smaller than the marginal expected benet. This is formally stated in the next proposition.

Proposition 1 (Best response). Given a strategy prole k, the best-response problem of player i admits a solution k * i , which is characterized by the pair of C 1 functions V * i : R + → R and

9 J o u r n a l P r e -p r o o f

Proof. See the Appendix.

An immediate corollary of Proposition 1 is that all equilibria feature social interaction, in the sense that, at every time, there is a mass of individuals who do not self-isolate. The reason is simple: if the rest of the population stays at home, each individual can spare the connement cost c H by going out without risking infection.

there is a non-empty set of individuals such that k * i (t) > 0 for every i in this set.

Proof. Fix time t and suppose that k * i (t) = 0 for all i, i.e.,k I (t) = 0. Yet, by condition (7) in Proposition 1, the best response of each individual i tok

Because the expected marginal benet of connement depends on t in expression (7), self-isolation may be worth the connement cost at some dates, but may not at some other dates. Intuitively, when c H is relatively large with respect to v I , it is a dominant strategy for individuals to go out at every date.

Proposition 3 (The no-connement equilibrium). If (1 − µ)βµv I + c H > 0, the game admits a unique equilibrium, in which all individuals play k(t) = 1 for every t ∈ [0, T ]. In this equilibrium, the players' payo at time t is

Proof. See the Appendix.

J o u r n a l P r e -p r o o f

When c H is so large that nobody self-isolates, the epidemic ends quickly but results in a large number of deaths. As a vaccine will arrive at time T , the government may want to implement policies to reduce the cost of connement, in order to create the conditions under which individuals might consider self-isolation. We thus investigate the existence of interior equilibria, in which individuals partially self-isolate at every date. We prove that there can exist only one interior equilibrium, and that it is symmetric. Proposition 4 (The interior equilibrium). Letk be the strategy dened by the system of equations for every t ∈ [0, T ]:

then the game has a unique interior equilibrium, in which all individuals playk. In this equilibrium, the players' payo at time t is

Proof. See the Appendix.

The purpose of this section is to illustrate the impact of self-isolation behaviors on the dynamics of the epidemic, and to highlight the policy lessons that can be drawn from our ndings.

To do so, we simulate the dynamics of the epidemic in the interior equilibrium described in Proposition 4, and compare it with the dynamics of the standard SIR model (referred to as the SIR model in the rest of the section), i.e., the model described by equations (1) and (2) withk S (t) =k I (t) = 1. We chose the behavioral parameters c I , c H and c D arbitrarily in such a way that the interior equilibrium exists, and we calibrate the epidemiological parameters β, γ, µ and ν to the COVID-19 pandemic.

Throughout our simulations, we assume that individuals are not aware of the epidemic until some time τ ∈ (0, T ), which can be interpreted as the moment at which the government of reported cases of COVID-19 died from the disease. As only patients with severe symptoms were tested at the beginning of the outbreak, we believe that the mortality rates measured in March 2020 are a valid estimate of the probability of death for an individual of the severe type infected by the disease 12 , i.e., ν/(ν + γ). Therefore, we set: ν/(ν + γ) = 0.034.

The dynamics of the epidemic in the interior equilibrium

We begin by analysing the impact of strategic self-isolation on the dynamics of the epidemic, which is illustrated in Figure 1 . Contrary to the now well-known bell-shaped curve of the SIR model, the epidemic curve (i.e., the graph of the percentage of infected plotted against time) continuously decreases on [τ, T ]. Therefore, the epidemic peak is reached before the population is informed about the epidemic, while in the SIR model, the fraction of infected continues to increase after time τ , reaching later a higher peak. The reason is that the population reacts to the epidemic announcement by self-isolating drastically after time τ , which results in a rapid decline in the percentage of infected. As the probability of being infected decreases, the marginal benet of 12 In our model, an infected of the severe type dies if the event Death occurs for her before the event Healing. Therefore, the probability of death (conditional on being infected and the severe type) is P (τD < τH ), with τH and τD denoting the random times of healing and death, respectively. Straightforwardly, P (τD < τH ) = 14 When an individual with no symptoms learns at time τ that the epidemic has been spreading since time 0, she updates her belief of being type θs to pi(τ ) = (1 − µ)/(1 − µ + µe β τ 0 i(t)dt ). Therefore, the larger τ , the smaller pi(τ ). 

The COVID-19 crisis has highlighted important dierences between countries in terms of health system performance, even within the OECD group. It is reasonable to assume that the state of the health system (intensive care beds capacity, possibility of inter-hospital patient transfers or of setting up eld hospitals, etc.) inuences the recovery rate γ and the mortality rate ν. We analyze the impact of the health system performance on the dynamics of the epidemic by comparing our baseline system (γ = 1/15 and ν = 0.00187367) with a more performing system where γ = 0.07334 and ν = 0.00094.

Given a xed percentage of time spent out by individuals, a better health system directly attens the epidemic curve and decreases the number of deaths. However, with strategic selfisolation a better health system has also a perverse eect via the increase in the continuation payo 17 in case of infection. Since the expected cost of having symptoms decreases with the quality of the health system, the marginal benet of connement decreases and individuals have less incentives to self-isolate. One can observe this risk compensating behavior in Figure   7 : in a more performing health system, individuals deconne earlier and the percentage of time spent outside stabilizes at a higher level.

However, we nd that the performance of the health system decreases the number of 17 One can see in expression 4 that vI increases when γ increases or when ν decreases. Finally, these two policies only reduce the number of deaths at the margin (see Figure 10 ), because individuals already partially self-isolate in equilibrium. This should not be interpreted as evidence of ineciency of policies subsidizing self-isolation. On the contrary, these policies allow for a shift from equilibria in dominant strategies without self-isolation to equilibria with partial self-isolation. Most countries have implemented lockdown policies to slowdown the epidemic. As lock-21 J o u r n a l P r e -p r o o f downs cause huge collateral damage in terms of economic activity, education and access to care, some governments were tempted by strategies pursuing herd immunity at rst. The optimal timing of lockdowns is a crucial question. In a theoretical model where individuals do not chose how much time they interact with others, Kruse and Strack (2020) show that if the government has the possibility to lockdown the population during only 100 days, delaying the moment to start the lockdown might actually decrease the total number of deaths. The reason is that delaying the lockdown increases the level of herd immunity in the population, which works as a protection for those individuals who remain susceptible after the lockdown.

In contrast, we nd that the number of deaths increases with the announcement time. The reason is that, when individuals behave strategically, they atten the epidemic curve by selfisolating more when the epidemic is too fast. Therefore, delaying the announcement time postpones the moment at which the population strategically controls the epidemic course.

At present, many governments have mandated the use of face masks in public spaces, arguing that face masks are low cost and might help prevent some transmission. At the beginning of the outbreak, however, WHO ocials did not recommend mask wearing in the general population 18 , stressing that 1) masks are commonly misused, and as a result, do not oer the intended protections, and 2) wearing a mask can provide a false sense of security, leading some to become less vigilant in more important hygiene measures, such as hand washing and self-isolation. Our results conrm that making masks mandatory leads individuals to reduce social distance, which can accelerate the epidemic. However, as individuals adapt to the level of the epidemic, this reduction is moderate, and the negative eect of a higher level of social interaction is more than oset by the positive eect on virus transmission, so that mandatory masks lead to a reduction in the number of deaths.

In our model we assume an homogenous population, full immunity after healing and no incubation period. These are strong assumptions in the case of the COVID-19 pandemic, which could be relaxed for future research.

Heterogeneous population. We assume that individuals have all the same prior belief of 18 On 30 March, 2020, Dr. Mike Ryan, executive director of the WHO health emergencies program declared:

There is no specic evidence to suggest that the wearing of masks by the mass population has any potential benet. In fact, there's some evidence to suggest the opposite in the misuse of wearing a mask properly or tting it properly.

J o u r n a l P r e -p r o o f Journal Pre-proof being the severe type, thus of dying from the disease. In the case of COVID-19, it is now clear and well documented 19 that the population is divided between those who are at high risk of dying from the disease (patients with co-morbidities) and those who are at lower risk.

It is still unclear whether individuals with co-morbidities are less likely to be asymptomatic.

However, they know that, if they catch the disease and are not asymptomatic, they will have more severe symptoms and a higher chance of dying than people without co-morbidities. To capture this observable intrinsic heterogeneity in the population, we can augment the model by assuming that each individual i is of medical condition i ∈ {c,c}, with c standing for co-morbidities andc standing for no co-morbidities. An individual knows her medical condition but not whether she is the asymptomatic type. In the Appendix, we prove that any equilibrium features social interaction, and that the game admits a unique equilibrium in which no individual self-isolates when the connement cost is large enough. The main dierence is that assuming an heterogeneous population precludes the existence of interior equilibria, because the indierence condition according to which the cost of connement equals the expected benet of connement cannot be satised simultaneously for individuals with and without co-morbidities.

Waning immunity. In our model, individuals who recover from the disease become perfectly immune to the virus. This is true for many infectious diseases, but likely not for COVID-19.

Antibodies to other coronaviruses are known to wane over time (12 to 52 weeks from the onset of symptoms) and homologous re-infections have been observed (see e.g. Kellam et al (2020) ).

SARS-CoV-2 IgM and IgG antibody levels may remain over the course of seven weeks or at least in 80% of the cases until day 49 (see Xiao et al (2020) and Zeng et al (2020) ). Therefore, it is reasonable to assume that individuals are immune immediately after recovery, but may lose their immunity after a random period of time, probably before the vaccine arrives. One possible way to introduce waning immunity into our model would be to assume that healed individuals become susceptible again at some rate η ∈ (0.1). This would change the dynamics ∆) . The best-response problem of individuals would be an optimal control problem with time lag in the control variable, which is very dicult to solve. However, we believe that the equilibria of the game would be similar, as the uncertainty that people might be infected and contagious but asymptomatic is already present in our model with the uncertainty about the type. Proofs for Section 2 and Section 3 Lemma 1. Let τ H and τ D be independent random variables distributed according to f (t) =

γe −γt and f (t) = νe −νt , respectively. The following equality holds:

The random variable min{τ H , τ D } is distributed according to f (t) = (γ + ν)e −(γ+ν)t . Therefore,

Moreover, 

Formally, it is the solution of the optimal control problem:

where K denotes the set of piecewise continuous functions from R + into [0, 1]. Making the change of variable x(t) := e − t 0 p i (s)k i (s)βk I (s)i(s)ds , player i's problem can be rewritten as:

As F (t, x(t), k(t)) is negative and bounded below by v I , the objective is well dened. Furthermore, by standard results, the problem admits at least one solution. Applying Pontryagin's maximum principle, the optimal control k * and the associated trajectory x * must satisfy the following conditions:

Lemma 2 (Necessary conditions). If (x * , k * ) is a solution of P(k), then there exists a continuous, piecewise continuously dierentiable function V : R + → R such that:

(ii) For any admissible control k, H(t, x * (t), k(t), V (t)) ≤ H(t, x * (t), k * (t), V (t)), 30 J o u r n a l P r e -p r o o f where H(t, x(t), k(t), V (t)) := F (t, x(t), k(t))+V (t)f (t, x(t), k(t)) is the (discounted 21 ) Hamiltonian of the problem.

Observing that

the necessary conditions are rewritten as

The latter condition can be more conveniently rewritten as:

be larger than every admissible control, which is true only if k * (t) = 1. If, on the contrary,

Let us now prove that necessary conditions are also sucient.

Lemma 3 (Sucient conditions). Consider a continuous, piecewise continuously dierentiable function V : R + → R and a pair (x * , k * ) satisfying conditions (i) and (ii). For any

Proof. Using the change of variable y(t) = ln µ x(t)−µ , the maximization problem P(k) is rewritten as:

y (t) = k(t)βk I (t)i(t) and y(0) = ln µ 1−µ . (9), we obtain player i's belief at time t in equilibrium k * :

It follows that all individuals playing an interior strategy have the same belief at each date. By the belief dynamics in Proposition 1, p * i = p * j implies that k * i = k * j , thus that k * is symmetric.

• Let us now characterize the strategyk of the interior (symmetric) equilibrium whenever it exists. As a preliminary, let us describe the dynamics of the population in a symmetric equilibrium in which all individuals playk. Since individuals completely self-isolate when they have symptoms, every susceptible individual as well as every infected individual of the asymptomatic type playsk(t) at time t. As a result,k S (t) =k(t) andk I (t) = µk(t), which, once plugged into equations (1) and (2), yields:

Moreover, individuals have all the same law of motion of beliefs:

The expression ofk(t) in Proposition 4 is obtained by plugging V *

andk I (t) = µk(t) into (9).

• Finally,k is well dened as an interior strategy if and only ifk(t) ∈ (0, 1) for every t ∈ [0, T ].

In this section we augment the model by assuming that each individual i is of health condition i ∈ {c,c}, with c standing for presence of co-morbidities andc for absence of co-morbidities. An individual knows her health condition but not her type (asymptomatic or severe). For simplicity, we assume that the probability of being the asymptomatic type is independent of the health condition. Assuming that individuals with co-morbidities are more likely to be the severe type, though more realistic, would not change the form of the best-responses and the nature of our results. Proof. The proof is a direct adaptation of the proof of Proposition 3, in which we establish that players cannot play dierent interior strategies in equilibrium, and that an interior strategy

Fix a health condition ∈ {c,c}, and suppose that all individuals with condition play an interior strategy. By Proposition 3, they all play the same strategy and hold the same belief p(t), which satises:

The latter condition cannot be satised for v I = v I , thus no player with condition = can play an interior strategy.

The next proposition states that the outcome in which individuals with co-morbidities self-isolate while those without comorbidities partially self-isolate exist in equilibrium. Ifk(t) < 1 for all t ≤ T , then k * c (t) = 0 and k * c (t) =k(t) for every t. They estimated the value of R 0 for the Diamond Princess's crew members and passengers to be R 0 = 2.28, using the maximum likelihood method and assuming γ = 7.5.

As a preliminary, we have checked that the interior strategy prole is an equilibrium with the set of values A and DP . We rst simulate the dynamics of the epidemic with each set of values. We observe that the fraction of infected, the fraction of time spent outside, the eective reproduction number and the fraction of deaths of the baseline simulation is framed between the two alternative set of values, and that the curves have the same shape (see Figure   11 , Figure 12 , Figure 13 and Figure 14 ). As R 0 is greater with the set A than with the set 38 J o u r n a l P r e -p r o o f Journal Pre-proof DP , individuals self-isolate more with the former set than in the later. However, in all cases, individuals drastically reduce their contacts after the announcement; when the spread of the epidemic is under control, they gradually increase the time spent outside to a plateau, which maintains the eective reproduction number close to 1, thus containing the spread of infection.

Next, we simulate our policy analysis with the two alternative set of values and obtain the same qualitative results:

• Delaying the epidemic announcement accelerates the epidemic (see Figure 15 ) and increases the number of deaths (see Figure 17 ). This is because individuals drastically self-isolate after the announcement, then gradually reduce their level of self-isolation to a level maintaining the eective reproduction number approximately equal to 1 (see Figure 16 ). 

We nd also nd that decreasing βµ induces less self-isolation and reduces the number of deaths (see Figure 19 and Figure 20 ).

• We multiply the value of γ by 1.1 for each set of values and the value of ν by 0.5 in each set of values. While the dynamics of the infection follow a very similar trajectory in the two scenarios, the infection is stronger in set A than in set DP (see Figure 21 ). We also nd that a better health system induces less self-isolation ( Figure 22 ) but reduces the number of deaths (see Figure 23 ). 

Optimal targeted lockdowns in a multi-group SIR model

The responsiveness of the demand for condoms to the local prevalence of AIDS

−βkS(t)s(t)kI (t)i(t) + ηr(t), i(t) = βkS(t)s(t)kI (t)i(t) − (γ + (1 − µ)ν)i(t) andṙ(t) = (γ + ν)

Mask use, hand hygiene, and seasonal inuenza-like illness among young adults: a randomized intervention trial

A simple planning problem for covid-19 lockdown (No. w26981)

Macroeconomic Consequences of Stay-At-Home Policies During the COVID-19 Pandemic

A mathematical model reveals the inuence of population heterogeneity on herd immunity to SARS-CoV-2

An economic model of the Covid-19 epidemic: The importance of testing and age-specic policies

Public avoidance and epidemics: insights from an economic model

A mathematical analysis of public avoidance behavior during epidemics using game theory

Facemasks and hand hygiene to prevent inuenza transmission in households: a cluster randomized trial

Impact assessment of non-pharmaceutical interventions against coronavirus disease 2019 and inuenza in Hong Kong: an observational study

Internal and external eects of social distancing in a pandemic (No. w27059)

Restarting the economy while saving lives under Covid-19

Adaptive human behavior in epidemiological models

Economic considerations for social distancing and behavioral based policies during an epidemic

Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand

Health versus wealth: On the distributional eects of controlling a pandemic (No. w27046)

COVID-19: Insight into the asymptomatic SARS-COV-2 infection and transmission

The dynamics of humoral immune responses following SARS-CoV-2 infection and the potential for reinfection

A contribution to the mathematical theory of epidemics

Optimal control of an epidemic through social distancing

Clinical features of COVID-19 in elderly patients: A comparison with young and middle-aged patients

Optimal mitigation policies in a pandemic: Social distancing and working from home (No. w26984)

Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship

Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19)

Private choices and public health: The AIDS epidemic in an economic perspective

Prevalence of SARS-CoV-2 in Spain (ENE-COVID): a nationwide, population-based seroepidemiological study

Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus Infected Pneumonia

Sequential lifting of covid-19 interventions with population heterogeneity (No. w27063)

Game theory of social distancing in response to an epidemic

COVID-19 and Italy: what next

Presenting characteristics, comorbidities, and outcomes among 5700 patients hospitalized with COVID-19 in the

The Optimal Control of Infectious Diseases via Prevention and Treatment, mimeo

Comorbidity and its Impact on Patients with COVID-19

Optimal quarantine programmes for controlling an epidemic spread

Timing of national lockdown and mortality in COVID-19: The Italian experience

Diabetes in COVID-19: Prevalence, pathophysiology, prognosis and practical considerations

Rational disinhibition and externalities in prevention

Antiphospholipid Antibodies in Critically Ill Patients With COVID?19. Arthritis & Rheumatology

Does comorbidity increase the risk of patients with COVID-19: evidence from meta-analysis

Do Face Masks Create a False Sense of Security? A COVID-19 Dilemma

Measuring voluntary social distancing behavior during the COVID-19 pandemic

Antibodies in Infants Born to Mothers With COVID-19 Pneumonia

Estimation of the reproductive number of novel coronavirus (COVID-19) and the probable outbreak size

Therefore, the Hamiltonian of the problem can be rewritten as:H(t, y, k, V ) = µe −y(t) k(t)βk I (t)i(t)v I − µ(1 + e −y(t) )c H (1 − k(t)) + V (t)k(t)βk I (t)i(t) Let us dene the functionĤ (y) = max k H(t, y, k, V ). Straightforwardly,Since v I < 0,Ĥ(y) is concave in y. Therefore, necessary conditions are also sucient by the Arrow-Kurz theorem (see e.g. Arrow and Kurz (1970) ).

Fix a player i, a date t and a valuek I (t). As infected individuals of the severe typeFinally, i(t) < 1 and v I ≤ V * i (t) ≤ 0. Therefore, the following chain of inequalities holds:implies that k * i (t) = 1 is the unique best response for player i to everyk I (t) by condition (7) in Proposition 1. Therefore, the condition (1 − µ)βµv I + c H > 0 guarantees that the game has a unique equilibrium in dominant strategies, in which k * i (t) = 1 for every i, t.Let us now determine the players' payo in this equilibrium. As a rst step, let us compute the players' belief at time t. Plugging k i (t) = 1 andk I (t) = µ into the belief dynamics (3), we obtain that player's belief is the solution of the ODE:with initial condition p(0) = 1 − µ. Integrating between 0 and t, we obtainwhich, after straightforward simplications, yields:with z(t) := e βµ t 0 i(u)du . Now, plugging the latter value of p(t), k i (t) = 1 andk I (t) = µ into the Euler condition in Proposition 1, we obtain that the players' payo is the solution of thewith initial condition V (T ) = 0. Multiplying both hands of the latter ODE by a(t) :=, we obtain:Integrating (8) between t and T and using that V (T ) = 0, we obtain:which, after simplications, reduces to the expression in Proposition 3.

Consider an interior strategy prole k * = (k * i ) i , i.e., such that k * i (t) ∈ (0, 1) ∀ i, t. • Let us rst prove that, if k * is an equilibrium, then it is symmetric, i.e., k * i = k * j ∀ i, j. Fix player i. By condition (7) in Proposition 1, an interior strategy k * i is a best response tok I (.)if and only if, for every t ∈ [0, T ],Plugging (9) into the Euler condition, player i's equilibrium payo satises in this case:Integrating (10) between t and T and using the terminal condition V * i (T ) = 0, we obtain player i's payo at time t in equilibrium k * :All individuals of the asymptomatic type recover at rate γ when they are infected. Individuals of the severe type with health condition ∈ {c,c} recover at rate γ and die at rate ν , with γ + ν ∈ (0, 1). To capture the eect of co-morbidities on the course of the disease, we assume that individuals of the severe type without co-morbidities recover faster: γc ≥ γ c , are less likely to die: νc ≤ ν c , and suer less from the disease: cc I ≤ c c I .At each time t the population is divided into ve groups: susceptible with and without co-morbidities: S c (t) and Sc(t), infected with and without co-morbidities: I c (t) and Ic(t), and recovered: R(t). Using the same notation as in the main model, the dynamics of the epidemic are governed by the following equations for each ∈ {c,c}: As v I increases with γ and decreases with c I or ν , it is smaller for an individual with comorbidities, thus v c I < vc I . Like in the case of an homogenous population, individual i decides whether to go out after comparing the connement cost with her expected marginal benet of connement, thus the best-response of individual i at time t is: As p * c (t) = 1 − µ and k * c (t) =k(t),As p * c (t) ≤ 1 − µ, v c I < vc I and V * c (t) = V * c (t), the above expression is negative. Therefore, the best response of players with condition c tok is to play k * c (t) = 0 for every t. The only equilibrium condition isk(t) < 1 for every t.

Because epidemiological parameter measurement (β, γ, R 0 and µ) is sensitive to the context, it is natural to ask whether the main insights of the baseline simulation are sensitive to