1 Introduction
Where an infectious disease is not amenable to populationlevel prevention through vaccination and risks are nontrivial, nonpharmaceutical interventions (NPIs) remain the principal tool of public health to respond to an outbreak. This is the case with novel infectious diseases that have no specific treatment and no prophylactic (vaccine) available. In the absence of pharmaceutical interventions of proven effectiveness, in particular prophylactically, the main public health response to the emerging pandemic of COVID19, a viral syndrome caused by the (+)ssRNA virus SARSCoV2 (order Nidovirales, family Coronaviridae, genus Betacoronavirus, subgenus Sarbecovirus), has rested principally on NPIs.^{14, 9, 7, 6} At their core, all NPIs share a quintessential relationship to social distancing, either through directly encouraging social distancing, limiting transmission potential by reducing public facilities for such encounters that may transmit the pathogen (’lockdowns’), reducing gatherings and social interactions that carry such risk (’largegathering bans’), suspending economic activities that inherently carry the risk of social interactions or modifying the framework of activities to reduce such interactions (e.g. transitioning to remote work).
From the perspective of game theory, social distancing can be viewed as a noncooperative game of a population
of size , where at any given time , each player adopts the strategy . For the sake of simplicity, we will assume two fundamental axioms about social distancing behaviours:
There exist two strategy choices, and (distancing and not distancing, respectively). At any given time, any agent can opt exclusively for one of these two options, i.e. the strategy set is the set .

Where a player makes a strategy choice at time , they implement it perfectly, i.e. the efficiency of every player in implementing their strategy is equal.
Then, for the entire population , the aggregate population level strategy can be described as the sum of all strategies . We can then assign a value to each strategy, whereby the strategy of distancing, , is assigned the value of 1 and the strategy of not distancing, , the value of 0.
For any given discrete time , where would be an endpoint (such as eradication, elimination, natural extinction of the pathogen, the availability of a vaccine or a combination thereof), we may then define two disjoint sets, and , where
(1)  
so that . Then, for any in discrete time , we define the overall factor of social distancing for a population at time ,
(2) 
which can also be expressed as a function of strategies as
(3) 
In other words, the overall factor of social distancing at time , is the average of the value the function assigns to each player’s strategy over the entire population . For simplicity’s sake, we will assume that the decision process takes the shape of a continuous and simultaneous game in discrete time, and agents can instantly switch strategies with no cost (other than the cost of the strategy itself). It then holds that

A player opting for strategy (social distancing) will incur , the immediate costs of distancing. These may be social (lessened social interaction), psychological (lessened access to support systems), economic (lower access to facilities to earn) or simple matters of convenience (access to amenities). While is somewhat dependent on (thus not distancing does not yield a benefit to a lone player in terms of access to amenities if all of these closed due to widespread social distancing), it can be assumed to be largely constant.

Compared to a person opting for strategy , a person opting for strategy will not incur the fixed cost , but will instead incur a relative additional cost , where denotes the constant cost of illness and is the risk of contracting illness when not distancing, a function of .
This allows us to identify the cost function for any at time for each individual as
(4)  
Then, for every population level aggregate strategy associated with the social distancing value in the way described in Equation (3), the overall social cost of can be conceptualised as
(5) 
For the entire time space , the total social cost is then
(6) 
The principal concern of this paper is not with individual action but with analysis of decision strategies on a population level. This paper will in the following conceptualise infectious disease in a population as a differential game over a differential equation form of the compartmental model first described by Kermack and McKendrick^{8}. This model has been widely adapted and adopted since its publication in 1927,^{21, 19, 3} and building on it, we will go on to identify within the statistical dynamics of that differential game the equilibria that govern ideal societal decisionmaking.
2 Methods
2.1 The ordinary differential equations of disease dynamics
Given a population of under the assumption that reinfection is sufficiently rare so as to be ignored at a population level, as appears to be the case for SARSCoV2,^{5, 4, 1}
), the dynamics of subpopulations with respect to infection can be modelled as a system of ordinary differential equations
(7)  
under the assumption of a closed, static population, i.e. neglecting for the time being the vital dynamics (birth, unrelated death, migration) of the population. Thus, , where represents susceptible individuals, represents infected/infectious individuals and accounts for removed individuals (mortality and recovery to immunity). In addition, due to the closed population assumption,
(8) 
Furthermore, the factors and in Equation (7) relate to each other as
(9) 
The fraction equals the basic reproduction number, . For SARSCoV2, estimates of range from 1.4 to 6.49, with studies that relied on statistical estimation of ranging from 2.20 to 3.58, with an average of 2.67^{13} , on the other hand, can be estimated as the inverse of the average number of days of illness (, sometimes also described as ). This value has been identified by studies of the initial infection dynamics of SARSCoV2 to be approximately days.^{16, 12} Even in the absence of firm evidence as to whether SARSCoV2 infection followed by recovery would engender lifelong immunity or not,^{20, 15, 11} it can be assumed in the short term – based on evidence from MERSCoV and SARSCoV – that at least in the immediate aftermath of disease and recovery, survivors remain immune,^{17} and consequently for any , i.e. the number of removed individuals () is strictly monotonously increasing over time. The inverse is true, for the same reasons, for and the number of susceptible individuals (). Numerical solutions to this system of differential equations have been calculated using odepack via SciPy 1.5.1^{22} on Python 3.6, and are described in Figure 1 describes some analytical solutions for the differential equations of Equation (7) over a range of plausible values of and , with inferred from and through the relationship described in Equation (9).
2.2 Population strategy contingent solutions to population dynamics
Given a population that then adopts an aggregate strategy at time that results in adherence (or in Reluga’s terms, investment^{18}) to social distancing, the flow from to is reduced by a corresponding factor. This allows us to rewrite Equation (7) so that for an aggregate strategy yielding , the populations can be characterised as
(10)  
Solutions to this system of differential equations have been calculated and are presented in Figure 2. Importantly, this allows us to identify the marginal utility as
(11) 
i.e. the partial derivative of over . Thus, for a populationlevel strategy associated with , there exists a marginal utility function given and that indicates the marginal utility at any given value of . This, too, can be numerically ascertained, and is shown on Figure 3.
2.3 Cost, risk and strategy
Any strategy has a cost , as stated in Section 1, and the aggregate cost of individuals each adopting, respectively, strategy , is . But since strategies are limited (one may, at any given time, either engage in social distancing or not, assuming for simplicity’s sake that those who do so are entirely successful), for any aggregate strategy resulting in a level of distancing described by ,
(12) 
where is the cost of social distancing for discrete unit time and is the cost of not distancing for the same unit time. The latter of these is not constant, as Equation (4) shows, but a function of a constant cost of infection, , and the risk of infection (), which in turn is contingent on and . Thus, Equation (12) can be reformulated (once again, in discrete time) as
(13) 
which expands to
(14) 
For a susceptible individual , the risk of infection in discrete time is the proportional likelihood of infection, or in other words,
(15) 
While quantification of costs of illness is difficult, quantification of the economic, social and emotional costs of social distancing is an even more complex task. However, we may, for values of , calculate cost fractions , where
(16) 
This cost fraction indicates the relative disequilibrium between the cost of distancing and the cost of illness when adjusted for risk – in other words, for any given value of at , the cost fraction indicates an inflection point. As long as the cost of distancing is less than , social distancing at or above is the optimum strategy. As numerical estimation of this cost fraction (Figure 4) shows, social distancing is almost always the preferred strategy at equal cost (black line). The contour lines in Figure 4 indicate what ratio the cost of distancing has to be to the cost of illness at a given value of to make distancing no longer an optimal strategy. So, for instance, a of 0.05 denotes the isorisk curve over where, at given , distancing is the preferable solution as long as its costs are less than, or equal to, 0.05 times that of illness or less.
3 Results
3.1 Strategies of social distancing
As Figure 2 indicates for empirically ascertained values of and based on the literature on SARSCoV2,^{10, 13} social distancing can have an overwhelmingly significant effect on the number of infectious individuals in a closed population, and the magnitude of this effect is dependent on the number of persons in the population already engaged in social distancing. This effect is most pronounced early in the epidemic (approx. 23 days), and the effect is most significant where less than half of the population is engaged in social distancing. Thus, unlike collective immunity in the case of vaccination, which often necessitates a fairly high level of penetration (typically estimated as ), social distancing can play a meaningful role in particular where much of the population is not yet engaged in such behaviour. This result can meaningfully inform a policy of encouraging individual social distancing early in an outbreak and dispel the misperception that marginal action is unnecessary unless a critical volume of individuals are already participating.
3.2 Marginal utility of social distancing
Based on the key epidemiological dynamics data on the SARSCoV2 pandemic,^{10, 13} the highest marginal effect of social distancing takes place in the same early timeframe of approx. 23 days. Unsurprisingly, even without integrating the timedependent discount factor proposed by Reluga (2010),^{18} the numerical solutions indicate that the effect of social distancing is most significant where it is not yet a widely adopted strategy: for SARSCoV2, based on an initial population of 10,000 with a seed population of 0.1% infected, the greatest marginal utility is encountered where less than 20% of the population is engaged in social distancing, and the effect is significantly less noticeable once reaches 0.5 (Figure 3).
Calculations of marginal utility matter because they can guide public policy in determining what fraction of the population may feasibly be exempted from social distancing while still maintaining much of the benefit. Given the need for critical services, swuch as urgent medical care, emergency services and critical supply chain activities to continue even in the throes of a pandemic, marginal utility calculations based on empirical data on an outbreak may, along with the societal response (as expressed by , which can be empirically ascertained as well), contribute to more accurate public health measures while limiting the effect of such measures on the economy and on daytoday life.
3.3 Costs and strategy choices
The accurate direct and indirect costs both of social distancing and of failing to do so are notoriously difficult to quantify accurately. Indeed, many of these costs are by their very nature not amenable to quantification. At the same time, by quantifying the relative ratio of cost of distancing () and cost of illness (), we can identify a strategyassociated cost ratio that, given approximations or empirical estimates of those costs, can assist in societal decisionmaking with regard to social distancing. As Figure 4 shows, for most cases, the cost of social distancing would have to exceed the cost of illness by at least an order of magnitude to make it a preferable strategy. In addition, the computational solution of Equation (16) shows not only that failure to socially distance may only be a preferable choice if the costs of distancing vastly exceed the costs of illness, but that this remains the case for much of the short term (¡90 days).
Estimates of direct medical costs of COVID19 are difficult, but at least one study puts the median cost of a symptomatic infection at US$3,045 in direct costs alone,^{2} typically compounded by loss of earnings, longterm physical harm, reduction in life expectancy and quality of life and, in severe cases, the risk of mortality. The costs of social distancing are much less amenable to quantification, as these costs are primarily governed by indirect factors and intangibles, such as the cost of deferred medical treatment, secondorder effects of the psychological burden inherent in decreased social interaction and the cost of lost revenue. While quantification, thus, of both the cost of distancing and cost of illness remains an outstanding subject of research, the cost fraction calculations can assist in reasoning about the best social strategy in view of these factors once ascertained or estimated.
4 Discussion
Pandemics pose a significant challenge to public health and social decisionmaking, and the COVID19 pandemic is by no means an exception. Nonpharmaceutical interventions, such as social distancing, play a significant role in the arsenal of tools that public health authorities can bring to bear on an epidemic that is otherwise not amenable to treatment or prophylaxis. Thus, until a vaccine or a reliable therapeutic, ideally with prophylactic properties, is found, nonpharmaceutical interventions are poised to remain the mainstay of public health activity in the face of COVID19. In view of this, an increased understanding of the way NPIs that rely on social distancing affect the statistical dynamics of SARSCoV2 in a population is essential for sound decisinmaking.
This paper discussed a subject that is not devoid of controversy, both in the scientific and in the public realm. By their nature, NPIs interfere with citizens’ daytoday lives and may have complex economic, social and psychological effects. It is therefore important that strategy options are adequately explored from a quantitative perspective. It is hoped that in reinforcing the case for social distancing through an analysis of the statistical dynamics that underlie it, this paper can add to growing body of knowledge in support of social distancing as an effective and costefficient NPI where other tools are unavailable or inappropriate.
Competing interests
The author declares no competing interests.
Supplementary data
All simulations, code and data are available on Github and under the DOI 10.5281/zenodo.3959666.
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