Mean-Field Game Analysis of SIR Model with Social Distancing Samuel Cho Program for Quantitative and Computational Biology, Princeton University September 4, 2020 Abstract The current COVID-19 pandemic has proven that proper control and prevention of infec- tious disease require creating and enforcing the appropriate public policies. One critical policy imposed by the policymakers is encouraging the population to practice social distancing (i.e. controlling the contact rate among the population). Here we pose a mean-field game model of individuals each choosing a dynamic strategy of making contacts, given the trade-off of gaining utility but also risking infection from additional contacts. We compute and compare the mean-field equilibrium (MFE) strategy, which assumes each individual acting selfishly to maximize its own utility, to the socially optimal strategy, which maximizes the total utility of the population. We prove that the optimal decision of the infected is always to make more contacts than the level at which it would be socially optimal, which reinforces the important role of public policy to reduce contacts of the infected (e.g. quarantining, sick paid leave). Additionally, we include cost to incentivize people to change strategies, when computing the socially optimal strategies. We find that with this cost, policies reducing contacts of the in- fected should be further enforced after the peak of the epidemic has passed. Lastly, we compute the price of anarchy (PoA) of this system, to understand the conditions under which large dis- crepancies between the MFE and socially optimal strategies arise, which is when intervening public policy would be most effective. 1 Introduction The current COVID-19 pandemic has been evidence that posing a public policy to fight such a crisis is an extremely difficult and interdisciplinary task. A major component of such policies is urging people to practice social distancing. By reducing interpersonal contacts, the spread of infection can be slowed down. However, it is uncertain how to incentivize people to practice social distancing, when there are clearly numerous benefits to making contacts, such as working for income, general desire for freedom and social relationships. Given this trade-off between the additional utility from making more contacts, and the additional chance of infection from those contacts, we will compute and compare 1) the selfish strategies, arXiv:2005.06758v2 [q-bio.PE] 3 Sep 2020 in which individuals make contacts to optimize only their own utilities and 2) socially optimal strategies, in which individuals make contacts which optimize the utility of the total population. The approach of focusing on the economic causes and epidemiological consequences is referred to as economic epidemiology [1]. Recent work has focused on treating economic factors behind contact and mixing decisions as part of the disease transmission mechanism [2{6]. More specifi- cally, some previous work in this field has explicitly included contact behaviors as control variables into the classical SIR model and have shown that different dynamics can emerge with adaptive behavior [7{9]. Disease dynamics has also been studied with game theory, focusing mostly on steady-state problems or those related to vaccination [10{12]. Optimal control theory has also been used to study policy interventions on infectious disease dynamics [13, 14]. Our work here builds on these previous models to pose a mean-field game problem of social distancing, explic- itly modeling the feedback between the individuals and the population structure. The individual, which is susceptible (S), infected (I), or recovered (R), each chooses a dynamic contact strategy to 1 maximize its accumulated utility over the time period. Mean-field games is a recently developed field of mathematics, studying dynamic game-theoretic problems of infinite number of players [15, 16]. Our model is a relatively simple formulation of the mean-field games, in which we have a deterministic game with 3 discrete states (S,I,R) in continuous time. Depending on the number of contacts each individual makes, the dynamics of the infectious population follow the SIR model, and the anticipation of this population structure influences the computation of an individual's optimal strategy. We take mean-field assumptions such that 1) the number of individuals in the population is large (N ! 1), 2) the individuals are homogeneous within each compartment and the population is well-mixed, and 3) individuals engage in symmetric interactions. 2 Model 2.1 SIR dynamics The focus of this work is on understanding the effect of social distancing behavior and the cor- responding utility trade-offs on the epidemiological dynamics, so we will keep the epidemiology as simple as possible as a baseline model. We will devote a section to including an exposed class (SEIR), but leave further expansion on the epidemiology as future work. We consider an SIR model without births or deaths. The recovered (R) is an absorbing state, where individuals gain permanent immunity from the disease. For now, we do not consider ex- posed period or asymptomatic or presymptomatic transmissions. Without birth or death, the population is fixed, and we let xz be the fraction of the population in compartment z 2 fS; I; Rg. Then the dynamics of the SIR system is 8x_ = −C(·)βx x <> S S I x_ I = C(·)βxSxI − µxI > :x_ R = µxI (1) β is the likelihood that infection happens given a contact between S and I, and µ is the constant rate of recovery from infection. C(·) is the rate that an S individual and an I individual make contact, which is dependent on whether individuals choose to socially distance or not. Let us write the number of contacts made by a z 2 fS; I; Rg individual at time t as cz(t). It is evident that C(·) should increase as cS(t) and cI (t) increase, but there are two intuitive ways of writing an expression for C(·), analogous to frequency-dependent or density-dependent disease transmission. i) Frequency-dependent Looking only at the transmission term (suppressing notation for dependence on t for brevity), cSxScI xI x_ S = −β (2) cSxS + cI xI + cRxR where a contact by S has probability cI xI of being with I. Therefore, the transmission cS xS +cI xI +cRxR term depends on the relative frequency between the contact strategies. ii) Density-dependent Similarly the transmission term is 0 x_ S = −β cSxScI xI (3) Here, the transmission term depends on not only the relative frequency, but also the absolute density of the contact strategies. If the whole population is doubling or halving contacts, the transmission should increase or decrease accordingly, and this density-dependent formulation is consistent with such intuitions. For the rest of this paper, we will take the density-dependent formulation. (Note that β0 term here is technically different from the β term above, but for simplicity, any β referred to for the rest of this paper is β0 of the density-dependent formulation.) 2 2.2 Utility of contacts cz(t) is the contact strategy which is explicitly chosen by an individual. Large cz(t) means going out to work and socializing, while smaller cz(t) means refraining from those activities (i.e. social distancing). uz(cz) denotes the utility received by a z individual from making cz contacts, which is a combination of economic gains and personal well-being. The realistic conditions that we im- pose are the following: Starting from zero contacts, increasing the number of contacts results in increased utility, which is the baseline universal need for social interactions, as well as monetary gain from economic activities. We assume that the marginal increase in utility decreases with more contacts, until eventually, more contacts become detrimental. Therefore, the appropriate functional form of uz(cz) must be a concave function with an inte- rior maximum. One appropriate function form with interpretations of the parameters is uz(cz) = 2 γ (bzcz − cz) − az where cz 2 [0; bz][8]. bz is a parameter of how the disease impacts the marginal economic productivity of the individual. Large bz means that the individual has the choice to make more contacts, as well as receive higher marginal utility from making contacts. az is a parameter of the baseline cost of being infected, such as an individual's general propensity to be healthy. These two parameters decompose the effect of the disease into economic cost and health cost. Some assumptions on these parameters are that bS = bR > bI and aS = aR < aI , since the infected become impaired in economic productivity, as well as suffer the health cost compared to the S and R. γ changes the concave shape of the function, and γ 2 (0; 1] ensures that uz(cz) is concave everywhere in the domain. From this utility function, we see that each individual has some optimal level of social contacts. bz The utility of a z individual is maximized at 2 , which is each individual's optimal contact strategy in the absence of adaptive behavior in response to the risk of infectious disease. Figure 1: uz(cz) is shown where z 2 fS; Rg are healthy (green) and z 2 fIg is infected (red) and the parameters are bS = bR = 10, bI = 6, aS = aR = 0, aI = 1, and γ = 0:25. A healthy individual gains utility from making more contacts, but eventually does not want to make more, when cS = 5. The infected suffers the baseline cost, but also gains some utility from making contacts, although at a lower rate compared to the healthy. 2.3 Value function Over time period, t 2 [0;T ], the total utility of an individual is the sum of utility gained at each time point. For example, a recovered individual with continuous contact decision c(t) receives total R T utility 0 uR(c(t))dt.