By Lina Sorg
In the last 18 months, mainstream news outlets have repeatedly referred to COVID-19 as an “unprecedented pandemic.” In reality, epidemics and pandemics have long been part of human civilization. Just over a century ago, the 1918 outbreak of Spanish influenza—one of the deadliest pandemics in history—ravaged the world for several years. It affected roughly a third of the world’s population and killed an estimated 20 to 50 million individuals in three successive waves of infection.
Like the Spanish flu, COVID-19 has also occurred in several waves. One mechanism that is likely responsible—at least in part—for these oscillating surges is social distancing. During a contributed presentation at the 2021 SIAM Annual Meeting, which is taking place virtually this week, Alina Glaubitz of Dartmouth College modified a traditional epidemiological model to account for the oscillatory dynamics of social distancing. The classical susceptible-infected-removed (SIR) model served as the basis of her efforts. “Our goal was to include social distancing,” Glaubitz said. “In particular, we want to give the susceptible population the opportunity to practice social distancing or not practice social distancing.”
Figure 1. Modified susceptible-infected-removed (SIR) model that accounts for social distancing (SD) among the susceptible population.
For the sake of simplicity, Glaubitz considers a well-mixed population and assumes that social distancing works perfectly and that those who practice it will avoid infection. She broke down the “susceptible” category of the SIR model into social distancers (SD) and non-social-distancers (NSD) (see Figure 1). Here, \(\varepsilon(t)\) is the proportion of people who practice social distancing. “Keep in mind that players make individual decisions about social distancing by weighing their perceived costs and benefits,” Glaubitz said. “This decision also depends on the level of infection at each time.”
Glaubitz then applied the replicator equation from evolutionary game theory to her model. She assigned a fixed cost to social distancing and a fixed cost to infection, which yield a separate payoff for each strategy. “We essentially let players in the susceptible population encounter other players, compare their payoffs, and change their strategies if the other payoff is higher,” Galbitz said. Susceptible individuals can therefore move back and forth between the SD and NSD groups. The threshold \(I^*\) is the proportion of infected players such that the payoff of social distancing is equal to that of not social distancing. If the proportion of infected individuals is smaller than this threshold, people stop practicing social distancing techniques. But if the proportion of infected individuals is greater than the threshold, people begin to social distance because the payoff is higher than socializing. Figure 2 depicts two simulations of the threshold for different parameters.
Figure 2. Two simulations of the threshold for different parameters.
Glaubitz’s SIR-SD model reveals oscillations of the infections around \(I^*\), indicating the presence of a feedback loop between infections and social distancing. “If the infections increase, this also means that at some point we get above the threshold \(I^*\) and people start to practice social distancing,” she said. “This leads to a decrease in infections, and we fall below the threshold \(I^*\) again, which then means that people stop practicing social distancing. This then leads to another increase in infections, and so on.”
The cyclical feedback loop is therefore responsible for the model’s multiple waves of infection because players stop social distancing as soon as their individual payoff is below the threshold. Glaubitz calls this an oscillatory tragedy of the commons. “The players in our model act to maximize their own self-interests and payoffs, which is in contrast to maximizing the payoff of the population as a whole,” she said. “In this case, the payoff of the population would be maximized if all players practiced social distancing until the infection dies away.”
Figure 3. Comparison between the classical SIR model and the modified SIR-SD model that accounts for social distancing.
Glaubitz’s SIR-SD model yields a significant reduction in total infections when compared to the initial SIR model. The SIR model presents one big wave of infection, while the SIR-SD model flattens the curve with several smaller waves of infection (see Figure 3). When the proportion of susceptible players falls below the threshold, the number of infections increases more slowly than it decreases. This ultimately leads to herd immunity, after which the infection goes away. If herd immunity is reached at the beginning of a new wave of infections, a significant proportion of the population can still infect others before they become “removed.” But if herd immunity occurs at the end of a wave and the number of active infections is small, less people get sick and the total infection size is smaller.
Glaubitz concluded her presentation with a discussion of a few caveats for the model. She acknowledged that other factors besides social distancing—such as seasonality or the evolution of new variants—might be partially responsible for the oscillating waves of infection. In addition, social distancing is not a completely individual decision like the model suggests; lockdowns and other regulations are typically somewhat responsible for people’s actions. “Further, one thing that we can see in reality is that the perceived cost of social distancing or infection is not necessarily fixed within the population and the same for everyone,” Glaubitz said. Because the SIR model on which her work is based does not consider things like death, hospitalization, or testing, she is currently working on a model expansion that includes vaccination and accounts for varying levels of vulnerability to COVID-19.
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Lina Sorg is the managing editor of SIAM News.
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