Oscillatory dynamics in the dilemma of social distancing

Social distancing as one of the main non-pharmaceutical interventions can help slow down the spread of diseases, like in the COVID-19 pandemic. Effective social distancing, unless enforced as drastic lockdowns and mandatory cordon sanitaire, requires consistent strict collective adherence. However, it remains unknown what the determinants for the resultant compliance of social distancing and their impact on disease mitigation are. Here, we incorporate into the epidemiological process with an evolutionary game theory model that governs the evolution of social distancing behaviour. In our model, we assume an individual acts in their best interest and their decisions are driven by adaptive social learning of the real-time risk of infection in comparison with the cost of social distancing. We find interesting oscillatory dynamics of social distancing accompanied with waves of infection. Moreover, the oscillatory dynamics are dampened with a non-trivial dependence on model parameters governing decision-makings and gradually cease when the cumulative infections exceed the herd immunity. Compared to the scenario without social distancing, we quantify the degree to which social distancing mitigates the epidemic and its dependence on individuals’ responsiveness and rationality in their behaviour changes. Our work offers new insights into leveraging human behaviour in support of pandemic response.

AG, 0000-0001-9609-3712; FF, 0000-0001-  Social distancing as one of the main nonpharmaceutical interventions can help slow down the spread of diseases, like in the COVID-19 pandemic. Effective social distancing, unless enforced as drastic lockdowns and mandatory cordon sanitaire, requires consistent strict collective adherence. However, it remains unknown what the determinants for the resultant compliance of social distancing and their impact on disease mitigation are. Here, we incorporate into the epidemiological process with an evolutionary game theory model that governs the evolution of social distancing behaviour. In our model, we assume an individual acts in their best interest and their decisions are driven by adaptive social learning of the real-time risk of infection in comparison with the cost of social distancing. We find interesting oscillatory dynamics of social distancing accompanied with waves of infection. Moreover, the oscillatory dynamics are dampened with a non-trivial dependence on model parameters governing decision-makings and gradually cease when the cumulative infections exceed the herd immunity. Compared to the scenario without social distancing, we quantify the degree to which social distancing mitigates the epidemic and its dependence on individuals' responsiveness and rationality in their behaviour changes. Our work offers new insights into leveraging human behaviour in support of pandemic response. by weighing the risk of infection with the cost of social distancing. The responsiveness parameter in our model modulates the relative time scale of individuals revisiting their social distancing decisions, as compared to the pace of an unfolding epidemic. We introduce bounded rationality that individuals are not necessarily using the best response but rather with some probability of changing their behaviour.
In this work, we find an interesting oscillatory tragedy of the commons in the collective dynamics of social distancing. Individuals are inclined to social distancing when the disease prevalence is above a threshold that depends on the transmissibility of the disease and the relative cost of social distancing versus contracting the disease. As the epidemic curve is being flattened, individuals consequently feel more safe not to practise social distancing, thereby causing the decline in the compliance of social distancing and further resulting in a resurgence of disease outbreaks in the population. Even though such reactive social distancing is hardly able to help reach the optimality of disease mitigation, it can avoid the overshooting of infected individuals which typically happens in an susceptible-infected-recovered (SIR) model in the absence of any interventions. We also find non-trivial dependence of the effectiveness of social distancing, measured by the fraction of susceptible individuals who would become infected without social distancing, on model parameters governing individuals' rationality and responsiveness.

Model and methods (a) Model
Our model is basically a combination of the classical SIR model with the replicator equation: In a well-mixed infinite population each individual is either susceptible, infected or recovered. Moreover, each susceptible individual can at each time choose to either practise social distancing or not to practise social distancing. If the individual practises social distancing they cannot become infected. If they do not practise social distancing they become infected in an encounter with an infected individual with probability β > 0. At each time an infected recovers with probability γ > 0 (figure 1).
We denote the proportion of susceptible individuals at time t by S(t), the proportion of infected by I(t), and the proportion of removed by R(t). Furthermore, we denote by E(t) the proportion of susceptible individuals that practise social distancing. We denote the initial conditions by I 0 = I(0), S 0 = S(0) as well as E 0 = E(0). A susceptible individual determines his strategy based on a cost-benefit analysis. Hence, by π sd we denote the payoff of social distancing, and by π nsd the payoff of no social distancing. In our model, the perceived cost of social distancing is C sd > 0 at each time t. Thus, we have π nsd depends on two factors: the perceived cost of infection that we denote by C I > 0 and the risk of infection. The risk of infection in time (t, t + 1) without social distancing is given by Therefore, the payoff of not socially distancing is given by

S SD
NSD I b g R Figure 1. Schematic of our model. In contrast to the SIR model, we divide the susceptible population into those that practise social distancing (and cannot be infected) and those that do not practise social distancing (and can be infected). The dynamics of the amount of people practising social distancing is given by a replicator equation. Here as well as in later figures, we refer to those practising social distancing as SD and those that do not as NSD. (Online version in colour.) Hence, the dynamics of our model are given by the following system of ordinary differential equations (ODEs):Ṡ Here, ω is a responsiveness parameter, determining the time scale for updating the social distancing behaviour. κ is a rationality parameter. For large κ individuals change their strategy if the payoff of the other strategy is larger. For small κ only a fraction of the susceptible individuals depending on the difference in payoff change their strategy. The behaviour of this model is illustrated in figure 2 for different parameters. For this figure as well as for all other figures, we used the Matlab method ode23, which is an implementation of the Bogacki-Shampine method-an explicit Runge-Kutta (2,3) pair.

(b) Perfect adaption
In Model (2.1), the dynamics of social distancing E change to direct the amount of infected I towards the amount where π sd = π nsd , i.e. towards I * . Assuming that this adaption works perfectly, we obtain the following model given by the ODEs. This model is given by the ODEsİ , t * = −(γ − β + βI * )/βI * γ , and with initial condition An illustration of this model is given in figure 3. Then, the total amount of people that get infected R PA (∞) is given by where W denotes the Lambert W function. Thus, in the case of perfect adaption, we can achieve     In each instance, I oscillates around I * with decreasing amplitude until the peak is smaller than I * . We see that the number and amplitude of oscillations depends on the parameters ω, κ, C sd , C I as well as β and γ . (Online version in colour.) by choosing C d /C I → 0. We want to use this model of perfect adaption to understand how the total amount of infected R(∞) in Model (2.1) depends on the parameters C sd , C I , ω, κ.

Results
If I > I * , then E is increasing. If I < I * then E is decreasing. On the other hand, if E is sufficiently large, this causes a decrease in I and if E is sufficiently small this causes an increase in I. If the amount of infections is high, people are more aware of the disease and practise social distancing. As soon as the amount of infections is small again, this awareness fades and people do not feel the need to practise social distancing anymore. As a result, more people become infected again leading to a higher awareness and more people practising social distancing. We refer to this feedback loop as oscillatory tragedy of the commons. Instead of high compliance to social distancing until the disease has died of, we find a decrease in individuals practising social distancing when the amount of infected is sufficiently small. This then causes another rise of infections. We can

(b) Social distancing saves lives
When comparing Model (2.1) to the SIR model, we immediately note that the total number of infections can be significantly smaller with social distancing (figure 5). Essentially, this means that voluntary social distancing can significantly reduce the total amount of infections R(∞). However, we also note that infections after the first wave of infection only emerge due to the oscillatory tragedy of the commons. If social distancing was practised until I = 0, we would have a much smaller R(∞).
Perfect adaption and Model (2.1) significantly reduce the total amount of infections compared to the SIR model. This is especially apparent for small I * and slow adaption, i. e. small ω and κ. This is illustrated in figure 6.
Essentially, the explanation for this behaviour relates to herd immunity. Social distancing flattens the curve. Instead of one large wave of infections as in the SIR model, in Model (2.1) we can have several waves of infection with smaller peaks. An example of this is illustrated in figure 5. In the SIR model, herd immunity occurs if  Clearly, in perfect adaption and Model (2.1) the total amount of infections can be significantly smaller than in the SIR model. We see that also in Model (2.1) the amount of infections tends to grow if I * grows. This will be discussed in §c. Moreover, small adaption parameters ω and κ in Model (2.1) can significantly reduce infections compared to perfect adaption as well as larger adaption parameters. This will be discussed in §d. (Online version in colour.) Thus, I is increasing until S = γ /β and then is monotonically decreasing. Even though, we have achieved some kind of herd immunity at this point, the high number of infected I still causes a high amount of new infections after herd immunity. Thus, the total amount of infections R(∞) is significantly larger than needed to obtain herd immunity. In Model (2.1), the dynamics are much more complicated. However, what remains as in the SIR model, is that as soon as (or at latest at this point) S < γ /β the amount of infected I is monotonically decreasing, since then we havė We denote the amount of infected when herd immunity is obtained by I HI .  Together, all these factors cause a significant decrease in new infections after herd immunity is achieved. Since small I HI mostly coincides with large R HI as well as high E, we focus on I HI here. When choosing the parameters ω, κ, C sd , C I such that I HI is small, R HI and many people practice social distancing, we can even achieve R(∞) to be near the herd immunity threshold 1 − γ /β. For an illustration of this, see figure 7.
This also explains why perfect adaption causes larger total amounts of infections than Model (2.1). For perfect adaption, herd immunity is obtained for t = t * with I = I * while I HI often is significantly smaller in Model (2.1). In particular, if we have I > I * while S is close to γ /β, small increases in E cause a decrease in I. Hence, I HI < I * in most cases. For instance, in figure 7a, we have I * ≈ 0.00268 while I HI can be much smaller in Model (2.1). However, we once again remember the oscillatory tragedy of the commons, i.e. that higher compliance to social distancing when I is small could lead to much smaller R(∞).
Next, we want to analyse how the perceived cost of social distancing C sd as well as the perceived cost of infection C I influence the total amount of infections R(∞).

(c) Larger cost of infection and smaller cost of social distancing reduce infections
As one might expect, if the cost of infection C I increases or the cost of social distancing C sd decreases, this induces an increase in the amount of people practising social distancing and thereby a decrease in infections. This behaviour becomes quite apparent in figure 8. We can observe a similar tendency for C I . Though, here we have larger oscillations in the total size of infections. These oscillations decrease in their amplitude and level off at R PA (∞). An example for this behaviour can be seen in figure 9.
An explanation for this behaviour is connected to the observation made in §b that is illustrated in figure 7. To reduce R(∞), (among other factors) I HI has to be small. One way to achieve this is to reduce I * , the threshold that I oscillates around. Therefore, smaller C sd as well as larger C I tend to cause a decrease in R(∞). However, this does not yet explain how the oscillations occur. For this purpose, we have a look at figure 10. Here, we see that reducing I * has two opposing effects on R(∞):  Figure 11. Development of an oscillation in the total number of infections. In Model (2.1) with β = 0.9, γ = 0.6, κ = 0.5, C I = 100, C sd = 1 and initial condition I 0 = 0.01, E 0 = 0.2, we observe the amount of infected I. As can be seen in figure 12, R(∞) has a local minimum in this case approximately at ω = 5. In (a), we have ω = 5, herd immunity is achieved for t ≈ 50 and at this time, we have I HI ≈ 0.0083. The total amount of infections is given by R(∞) ≈ 0.4316. For (b) ω = 6 herd immunity is achieved at approximately the same time, with II HI ≈ 0.0094 and total amount of infections R(∞) ≈ 0.4385. For (c) ω = 7 herd immunity is achieved at approximately the same time, with I HI ≈ 0.0106 and total amount of infections R(∞) ≈ 0.4432. Finally, for (c) ω = 8 herd immunity is achieved at approximately the same time, with I HI ≈ 0.0101 and total amount of infections R(∞) ≈ 0.4395. For increasing ω, we observe that the first three waves of infection decrease in duration and intensity. However, a fourth wave of infection develops. This leads to an increase in the amount of infections when herd immunity is achieved and therefore an increase in the total amount of infections for ω = 6, respectively, ω = 7 compared to ω = 5. For ω = 8, however, herd immunity is only obtained after the fourth wave of infection is already decreasing again. Therefore, we have a decrease in the total amount of infections for ω = 8 compared to ω = 7. Note also that the amount of infected when herd immunity is achieved, is in all cases smaller than I * . If we have I > I * while S is close to γ /β, small increases in E cause a decrease in I. Hence, I is mostly smaller than I * when herd immunity occurs. This also induces that R(∞) is mostly smaller than R PA (∞) as can also be seen in figure 12.

(Online version in colour.)
This leads to the oscillations, that we observed in figure 9. When reducing C sd we first see a decrease in R(∞) (caused by smaller waves of infection and a smaller I HI ) followed by an increase (induced by a new wave of infections that leads to an increase in I HI ). Overall, we thus have oscillations in R(∞) depending on ω. An example of this is illustrated in figure 11. With increasing ω the deviations of I from I * are decreasing due to faster adaption. This leads to a decrease in the amplitude of the oscillations in R(∞) and to R(∞) levelling off approximately at R PA (∞). As explained before, we mostly have I HI < I * . Therefore, larger deviations from perfect adaption where I HI = I * cause a decrease in R(∞). An example of this behaviour is illustrated in figure 12.

(d) Faster responses and higher rationality increase infections
The rationality parameter κ has a nearly similar effect as ω on the dynamics of our model. In Model (2.1), a large rationality parameter κ means that individuals change their strategy as soon as the payoff of infection becomes larger than the payoff of social distancing and vice versa. Therefore, large κ induce faster adaption of E and therefore smaller oscillations of I around I * . An increase in κ thus causes a decrease in the duration in the waves of infection as well as smaller

Discussion and conclusion
Social distancing is often used in combination with other control measures such as mask wearing, and testing and isolation. It is worthy of further investigation to account for individual preferences in their adoption choices when multiple interventions for disease mitigation are available. Generally speaking, individuals become less vigilant and feel less need to follow disease intervention measures suggested by public health officials, if the epidemic curve is being bent down, but as a result, the uptick of cases in turn causes individuals to increase their compliance levels. The feedback loop of this sort gives rise to an oscillatory dynamics of behavioural compliance and disease prevalence, as reported in the present work. Similar phenomena have previously been studied in the context of eco-evolutionary dynamics where the payoff structure of individual interactions can be regulated by the environmental feedback [43][44][45]. Social distancing can be regarded as an altruistic behaviour that incurs a cost to oneself but collectively benefits other community members especially these vulnerable in the population. Thus monetary or non-monetary means can be used to incentivize non-compulsory social distancing. For example, during the COVID-19 health crisis, governments have subsidized the cost of staying at home through tax reduction or other stimulus packages for both workers and their employers [28]. Besides, an individual who opts for social distancing can create a positive psychological reward, which in fact reduces the perceived overall cost of social distancing. As shown in several experimental works [46][47][48], encouraging altruistic social distancing, especially if people can afford to do so, through promoting a strong sense of community, empathy and compassion [49], can lead to desired compliance of social distancing. In this sense, promoting human cooperation in the social dilemma of disease control is a new promising direction for future work.
While our proof-of-principle model offers enlightening insights into understanding compliance issues in the dilemma of social distancing, targeted social distancing can be investigated by further accounting for individual heterogeneity as the attack rate and mortality rate of infectious diseases, such as the influenza [50,51] and the COVID-19 pandemic [52], are age-dependent. Thus, extending our model with an age structure will be useful to quantify the heterogeneity in both the risk of infections and the cost of social distancing for each age group. This consideration parameterized using realistic contact mixing matrices in a social network [53] as well as with an age structure (more generally, multilayer networks [54]) can be used to optimize social network-based distancing protocol (targeted social distancing) [34]. Further work along this direction is promising and will help provide practical guidance. Moreover, it appears that instead of the actual likelihood to get infected, one's perceived likelihood to get infected influences the decision whether to engage in social distancing and face covering [46]. Variation in individual risk assessment might therefore influence the results in our model and be an interesting extension to the model in future work.
In sum, we analyse and characterize oscillatory dynamics in the dilemma of social distancing, which arises from the non-trivial feedback between disease prevalence and behavioural intervention. Our results suggest an oscillatory tragedy of the commons in disease control when individuals act in their own right without coordination or in the absence of centralized institutions to enforce their compliance, a phenomenon that has been observed in past pandemics like the Spanish flu [55] and seems to repeat in the current COVID-19 pandemic [56]. Our work provides new insight into the dual role of human behaviour that can fuel, or fight against, the pandemic [57]. To resolve the dilemma of disease control from global pandemics to resurgence of common diseases (like measles which has become endemic in some regions [58]), a deep understanding of pertinent behavioural aspect in disease control and prevention, and large-scale human cooperation in particular, is urgently needed and will help to better inform pandemic support in the future [49].
Ethics. Ethical assessment is not required prior to conducting the research reported in this paper, as the present study does not have experiments on human subjects and animals, and does not contain any sensitive and private information.