Social dilemma of nonpharmaceutical interventions: Determinants of dynamic compliance and behavioral shifts

Significance

The effectiveness of nonpharmaceutical interventions (NPI) depends on widespread compliance, without which infection waves can surge, driven by pathogen evolution and changing public vigilance. Here, we reveal that bottom–up adherence to NPI measures is dynamic and varies according to context, leading to intricate temporal changes in preferences between NPI measures or inaction through an ongoing epidemic. By characterizing the dynamic adoption of NPIs against their perceived costs and effectiveness, particularly in the cases of face coverings and social distancing, our findings offer insights into overcoming barriers to NPI compliance. Most significantly, which NPI measures to recommend must align with the emergent innate preferences of the public to minimize resistance to such public health measures.

Abstract

In fighting infectious diseases posing a global health threat, ranging from influenza to Zika, nonpharmaceutical interventions (NPI), such as social distancing and face covering, remain mitigation measures public health can resort to. However, the success of NPI lies in sufficiently high levels of collective compliance, otherwise giving rise to recurrent infections that are not only driven by pathogen evolution but also changing vigilance in the population. Here, we show that compliance with each NPI measure can be highly dynamic and context-dependent during an ongoing epidemic, where individuals may prefer one to another or even do nothing, leading to intricate temporal switching behavior of NPI adoptions. By characterizing dynamic regimes through the perceived costs of NPI measures and their effectiveness in particular regarding face covering and social distancing, our work offers insights into overcoming barriers in NPI adoptions.

The emergence of new infectious diseases, like Zika (1), Ebola (2), COVID-19 (3) and Mpox (4, 5), presents a great challenge to global health and humanity. Given the time required to develop and distribute pharmaceutical solutions like vaccines, we often have to rely on nonpharmaceutical interventions (NPI) in the early stages of an epidemic (6). These include, but are not limited to, individual protective measures like personal hygiene and mask-wearing (7–10), as well as community measures such as travel restrictions (11) and quarantines (12). However, the effectiveness of these NPIs heavily depends on public compliance and adherence, making human behavior a critical factor in controlling disease spread. Therefore, it is of importance and interest to integrate behavioral response into epidemiological models to gain a deeper understanding of the initial spread of infectious diseases and their control and mitigation measures (13–15).

Mathematical models have become an increasingly important part of the understanding of and the fight against infectious diseases (16). Epidemiological models help inform our response to disease threats (17). From the influence of vaccines and vaccine compliance (18–21) to the role of disease awareness (22, 23), previous studies have made important progress in understanding the role of behavior in epidemiology (also known as behavioral epidemiology) (10, 24–29). It has been shown that these interactions can lead to rich dynamic behaviors, like the hysteresis effect (30), by using evolutionary game theory to model how behaviors evolve under the interplay between self-interest, social influence, and disease spread (10, 18, 29, 31–33).

Undoubtedly, the COVID-19 pandemic has led to a further increase in research into the influence of NPIs (6, 33, 34) as well as vaccination strategies (35) on epidemics. Unlike pharmaceutical measures like vaccination, NPIs such as social distancing and mask-wearing require individuals to continually assess the need to adhere to such measures (29, 36, 37). Thus, it is necessary to account for the socioeconomic consequences of the disease and control measures (38). Social distancing has been modeled as a population game where individuals weigh the risk of the disease against the cost of social distancing (39). Additional efforts have been made to understand the effect of social distancing on the COVID-19 epidemic (40–42), emphasizing the importance of aligning each individual’s goals with those of society as a whole.

Moreover, optimal control theory has been applied to model and inform the implementation of mandated vaccination, social distancing, and lockdown efforts (43, 44), aiming to minimize outbreak costs through centralized planning (45–49). Despite providing valuable insights into achieving population-wide optimal outcomes (50), the practical implementation of these strategies can be limited by failing to ensure widespread compliance. Beyond finding the Nash equilibrium (perfect adoption) in the consistent effort to practice social distancing, previous work emphasizes the role of dynamically changing individual compliance leading to an “oscillatory tragedy of the commons” in social distancing dynamics (51) by accounting for bounded rationality (52) and loss aversion (53).

This work extends upon previous studies on NPIs in infectious disease control by incorporating a dual behavioral response (social distancing and face covering) compared to models typically with only one possible behavioral response for mitigating the spread of an infectious disease (7–9, 33, 51). To our knowledge, prior work has examined parallel interventions of the same type, such as choices between two kinds of face coverings (10). However, there is a need to investigate multiple concurrent NPI measures of different types and to understand how behavioral choices of individuals shift during an ongoing epidemic. Here, we provide insights into dynamic compliance levels and behavioral shifts that are driven by the costs and effectiveness associated with each NPI option. To this end, we investigate the nontrivial effects of a dual behavioral response on disease mitigation.

In particular, we introduce face covering (FC) as a less strict form of NPI compared to social distancing (SD). Our findings reveal that FC as a third strategy next to SD and no social distancing (NSD) exhibits a significantly different dynamic behavior than SD alone. Our model shows how individuals generally opt for either SD or FC but do not revert to more stringent measures (SD) once FC is preferred during an ongoing epidemic. While SD is generally preferred over NSD for high infection levels above an epidemic threshold, FC is preferred over NSD only within a range between two disease prevalence thresholds. Once the infection level surpasses the smaller threshold, FC is preferred, but once the second threshold is exceeded, NSD becomes preferred over FC again. Moreover, FC is only adopted if its effectiveness is sufficiently high compared to SD. Once the relative effectiveness of FC drops below a certain threshold, it does not get adopted at all. The existence of such bifurcation behavior is one of the important insights from the current study.

Model

As illustrated in Fig. 1, we incorporate human behavioral choices regarding NPI measures into an epidemiological model, using social imitation dynamics based on pairwise comparisons to account for the decision of whether to practice SD, wear a FC, or do nothing at all (NSD). This significantly extends our prior work focusing on a single response with social distancing (51), and results in a model of protection with dual behavioral responses.

Fig. 1.

Model schematic. (A) shows an overview of the model diagram: The susceptible population is divided into three groups according to their NPI adoption choices: those practicing social distancing (SD), those practicing no social distancing but wearing a FC, and those neither practicing social distancing nor wearing a face covering (“doing nothing,” NSD). individuals not practicing social distancing can be infected, though face covering reduces the risk of infection from $\beta$ to ${\beta }_{\text{FC}}$. (B) shows the dynamics of the adoption strategies within the susceptible population that are governed by replicator-like equations, with their proportions evolving over time in a simplex triangle. (C) plots the concurrent epidemic spreading dynamics in the infected population shown as the disease prevalence $I(t)$, the fraction of infected individuals. The correspondence between the oscillations in behavioral adoptions in (B) and the infections in (C) indicates dynamic compliance and behavior shifts as a consequence of behavior–disease interactions.

In response to the threat of infection, each susceptible individual can opt for an NPI measure, either SD or FC, or they can choose to take no measure at all by doing nothing (NSD). The spread of the disease depends on individuals’ NPI choices: SD provides full protection, whereas FC offers only partial protection. Specifically, individuals doing nothing (NSD) become infected at rate $\beta$, whereas individuals wearing an FC become infected at a lower rate ${\beta }_{\text{FC}}<\beta$ (Fig. 1A). We use the ratio $0<{\beta }_{\text{FC}}/\beta <1$ to represent the relative risk of infection; the smaller the ratio, the greater the effectiveness of face coverings in reducing disease transmission. Infected individuals recover at a rate $\gamma$.

The behavioral adoption dynamics of NPIs are influenced by the disease prevalence $I(t)$, the fraction of infected individuals, as the infection level determines the risk of infection and the costs associated with each NPI choice. Individuals weigh the costs of complying with each NPI choice for future times against the cost of infection $C_{\text{I}}$: Practicing SD incurs a cost at a rate of $C_{\text{SD}}$, while wearing an FC incurs a cost at a rate of $C_{\text{FC}}$. We have $C_{\text{FC}}<C_{\text{SD}}<C_{\text{I}}$ in our model. The risk of infection per time unit $(t,t+1)$ when wearing an FC is $1-exp(-{\beta }_{\text{FC}}I(t))$, whereas that for NSD is $1-exp(-\beta I(t))$. These considerations lead to expected payoffs for NPI choices:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\pi }_{\text{SD}}& =-C_{\text{SD}},\hfill \\ \hfill {\pi }_{\text{NSD}}& =-C_{\text{I}}[1-exp(-\beta I(t))],\hfill \\ \hfill {\pi }_{\text{FC}}& =-C_{\text{FC}}-C_{\text{I}}[1-exp(-{\beta }_{\text{FC}}I(t))].\hfill \end{array}\end{array}$$ [1]

A susceptible individual updates their NPI choices (i.e., behavioral strategies consisting of NSD, SD, and FC) based on the pairwise comparison process (54), using the expected payoffs as given in (Eq. 1). For instance, an individual practicing SD switches to wearing an FC with the probability ${\varphi }_{\text{SD}\to \text{FC}}$ given by the Fermi function:

$$\begin{array}{c}\hfill {\varphi }_{\text{SD}\to \text{FC}}=\frac{1}{1+exp[-\kappa ({\pi }_{\text{FC}}-{\pi }_{\text{SD}})]},\end{array}$$ [2]

where $\kappa$ is a rationality parameter.

We use the replicator-like equations (55) to account for the updating dynamics of individual NPI adoption strategies, leading to the evolutionary dynamics of the three strategies, as visualized in Fig. 1B, within a simplex triangle. In parallel, the progression of the epidemic, shown as $I(t)$ in Fig. 1C, is determined by the collective levels of NPIs. Nontrivial, rich dynamical behaviors arise as a result of such closed-loop feedback between NPI adoption dynamics and epidemic spreading, with their concurrency being controlled by the relative time-scale parameter $\omega$ (see Materials and Methods and SI Appendix for details).

Results

When contemplating which NPI option (SD vs. FC) to comply with or doing nothing at all (NSD), individuals are more likely to imitate the choices of others with higher expected payoffs. Such payoff comparisons lead to pairwise dominance relations—an ordinal ranking of the three behavioral response strategies: SD, FC, and NSD (Fig. 2A). Which strategy is better off depends on the current disease prevalence $I(t)$, as it determines infection risks and, consequently, expected payoffs.

Fig. 2.

Determinants of dynamic NPI compliance and behavioral shifts. (A) shows the critical epidemic thresholds, given by the four possible critical disease prevalence levels $I_i^{\ast }$, $i=1,2,3,4$, that determine which NPI behavioral choice (SD, FC, and NSD) is more favorable in terms of expected payoffs. This leads to a social dilemma of NPIs, where the success of an individual’s NPI choice depends on the collective outcome of their behavioral choices, as these impact epidemic spreading dynamics, disease prevalence, and consequently, individuals’ risks of infection associated with each NPI option. The arrows indicate dominance: For example, if the disease prevalence $I(t)>I_1^{\ast }$, SD yields a better-expected payoff than NSD, and thus the arrow points from NSD to SD, meaning NSD individuals are more likely to adopt the choice of SD individuals. Notably, there can exist two epidemic thresholds, $I_3^{\ast }$ and $I_4^{\ast }$, for the comparison of NSD and FC. When this is the case, only for $I_3^{\ast }<I(t)<I_4^{\ast }$ is FC a better option than NSD. (B) presents a region plot across the parameter space $({\beta }_{\text{FC}}/\beta ,C_{\text{SD}}/C_{\text{I}})$ that is divided into four general cases for determining the ranking order of the four epidemic thresholds $I_i^{\ast }$. Two more critical parameter values are worth noting: the critical cost of social distancing, $C_{\text{SD}}$ compared to the cost of infection $C_{\text{I}}$ and the critical effectiveness of face covering in preventing transmission ${\beta }_{\text{FC}}$ compared to doing nothing (NSD) $\beta$. The former determines which three cases out of the four are admissible when varying ${\beta }_{\text{FC}}$ and the latter determines whether two epidemic thresholds $I_3^{\ast }$ and $I_4^{\ast }$ exist at all. In other words, a minimal effectiveness of FC is required for FC to possibly be an advantageous option. In (B), the values of the model parameters per month are $\beta =10$, $C_{\text{I}}=10$, $C_{\text{FC}}=1$.

The first critical epidemic threshold $I_1^{\ast }$ arises when comparing ${\pi }_{\text{NSD}}$ and ${\pi }_{\text{SD}}$: We have ${\pi }_{\text{SD}}>{\pi }_{\text{NSD}}$ for $I(t)>I_1^{\ast }$. The second critical epidemic threshold $I_2^{\ast }$ arises when comparing ${\pi }_{\text{SD}}$ and ${\pi }_{\text{FC}}$: We have ${\pi }_{\text{SD}}>{\pi }_{\text{FC}}$ for $I(t)>I_2^{\ast }$. Regardless of the order of $I_1^{\ast }$ and $I_2^{\ast }$, SD becomes more favorable to individuals under sufficiently high infection levels, as it no longer provides a better prospect for individuals to either practicing NSD or choose the lighter measure, FC, due to the elevated risk of infection.

Interestingly, multiple thresholds, denoted by $I_3^{\ast }$ and $I_4^{\ast }$, can occur when comparing ${\pi }_{\text{NSD}}$ and ${\pi }_{\text{FC}}$ (SI Appendix, Fig. S1). As presented in SI Appendix, the bifurcation condition for the existence of $I_3^{\ast }$ and $I_4^{\ast }$ requires that the critical effectiveness of FC $\epsilon$, expressed as the complement of the ratio of transmission rates $\epsilon =1-{\beta }_{\text{FC}}/\beta$, exceeds the threshold value ${\epsilon }^{\ast }$, which satisfies the equation:

$$\begin{array}{c}\hfill {\left(1-\epsilon \right)}^{\frac{1-\epsilon }{\epsilon }}-{\left(1-\epsilon \right)}^{\frac{1}{\epsilon }}=\frac{C_{\text{FC}}}{C_{\text{I}}}.\end{array}$$ [3]

When FC provides sufficient protection by reducing transmission, such that ${\beta }_{\text{FC}}/\beta <1-{\epsilon }^{\ast }$, the third and fourth thresholds $I_3^{\ast }$ and $I_4^{\ast }$ exist: We have ${\pi }_{\text{FC}}>{\pi }_{\text{NSD}}$ for $I_3^{\ast }<I(t)<I_4^{\ast }$. Otherwise, for lower disease prevalence $I(t)<I_3^{\ast }$ or higher infection levels $I(t)>I_4^{\ast }$, inaction (NSD) is more favorable than wearing an FC (${\pi }_{\text{FC}}<{\pi }_{\text{NSD}}$; see Fig. 2A).

The order of these epidemic thresholds provides insights into subtle, context-dependent preferences for NPIs, leading to scenarios where FC and/or SD may be preferable. Dynamic compliance levels emerge from the bottom up and, in turn, change the infection level $I(t)$ and cause behavioral shifts during an ongoing epidemic. Overall, our analysis, as detailed in SI Appendix, shows that there are four possible ranking orders of these epidemic thresholds $I_i^{\ast }$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\text{case} \text{(i)}\hfill & :I_2^{\ast }<I_1^{\ast }<I_3^{\ast }<I_4^{\ast },\hfill \\ \text{case} \text{(ii)}\hfill & :I_3^{\ast }<I_1^{\ast }<I_2^{\ast }<I_4^{\ast },\hfill \\ \text{case} \text{(iii)}\hfill & :I_2^{\ast }<I_1^{\ast },\text{neither} I_3^{\ast }\text{nor} I_4^{\ast }\text{exists},\hfill \\ \text{case} \text{(iv)}\hfill & :I_3^{\ast }<I_4^{\ast }<I_2^{\ast }<I_1^{\ast }.\hfill \end{array}\right.\end{array}$$ [4]

Fig. 2B plots the regions corresponding to these four cases across the parameter space $({\beta }_{\text{FC}}/\beta ,C_{\text{SD}}/C_{\text{I}})$ for fixed $C_{\text{FC}}$. Besides the critical effectiveness of FC aforementioned, another critical threshold—the relative cost of SD to infection, $C_{\text{SD}}/C_{\text{I}})$—for determining whether SD is preferable is worth noting. This critical ${\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$ satisfies the equation as follows:

$$\begin{array}{c}\hfill \frac{{\beta }_{\text{FC}}}{\beta }=\frac{log(1-\frac{C_{\text{SD}}-C_{\text{FC}}}{C_{\text{I}}})}{log(1-\frac{C_{\text{SD}}}{C_{\text{I}}})}.\end{array}$$ [5]

Letting ${\beta }_{\text{FC}}/\beta$ be the critical value found in (Eq. 3), i.e., $1-{\epsilon }^{\ast }$, we obtain ${\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$.

Intuitively, this ${\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$ separates the two branches of the bifurcation for ${\beta }_{\text{FC}}/\beta <{({\beta }_{\text{FC}}/\beta )}^{\ast }$ (SI Appendix), corresponding to the boundary of case (ii) between case (i) (lower branch) and case (iv) (upper branch). At this boundary, we have $I_1^{\ast }=I_2^{\ast }$ and the three NPI options yield equal payoffs: ${\pi }_{\text{NSD}}={\pi }_{\text{FC}}={\pi }_{\text{SD}}$. When this holds, we naturally have $I_1^{\ast }=I_2^{\ast }=I_3^{\ast }$ (lower branch) or $I_1^{\ast }=I_2^{\ast }=I_4^{\ast }$ (upper branch). Which branch is admissible is determined by the critical ${\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$ (SI Appendix, Fig. S2). Summarizing, for small $C_{\text{SD}}/C_{\text{I}}<{\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$, we have three general cases beyond the extreme boundary case $I_1^{\ast }=I_2^{\ast }=I_3^{\ast }$, consisting of case (i), case (ii) and case (iii) when varying the relative ratio ${\beta }_{\text{FC}}/\beta$. And for large $C_{\text{SD}}/C_{\text{I}}>{\left(C_{\text{SD}}/C_{\text{I}}\right)}^{\ast }$, we have three general cases beyond the extreme boundary case $I_1^{\ast }=I_2^{\ast }=I_4^{\ast }$, consisting of case (ii), case (iii) and case (iv) when varying the relative ratio ${\beta }_{\text{FC}}/\beta$ (Fig. 2B).

The analysis of epidemic thresholds and the four cases of their specific ranking orders provides clearer insights into NPI behavioral adoption dynamics for any given choice of model parameters, as the corresponding case in which the system lies in can be determined accordingly (Materials and Methods and SI Appendix). Fig. 3 presents a microscopic view of how the switching dynamics among SD, FC, and NSD—as well the dominant strategy offering the best payoff—depend on the infection level $I(t)$ within each interval specified by the epidemic thresholds for each case.

Fig. 3.

Disease-prevalence-dependent behavior adoption dynamics. Individuals’ adoption preferences for NPIs change as a result of the prevalence of infection. We show how the pairwise dominance relations between the three different options (strategies) depend on disease prevalence $I(t)$, which determines the infection risks associated with each option and their expected payoffs. In particular, which NPI measure is preferred or inaction at all depends on whether $I(t)$ is smaller or larger than the four possible epidemic thresholds. For very low disease prevalence of infected individuals, $I(t)$, the dynamics always tend toward taking no measures. For very large prevalence of infected individuals, $I(t)$, the dynamics tend toward social distancing. The behavior in between depends on the order of the critical epidemic thresholds, which are generally categorized into four cases: (A) (i) $I_2^{\ast }<I_1^{\ast }<I_3^{\ast }<I_4^{\ast }$, (B) (ii) $I_3^{\ast }<I_1^{\ast }<I_2^{\ast }<I_4^{\ast }$, (C) (iii) $I_2^{\ast }<I_1^{\ast }$, with neither $I_3^{\ast }$ nor $I_4^{\ast }$ existing, and (D) (iv) $I_3^{\ast }<I_4^{\ast }<I_2^{\ast }<I_1^{\ast }$, corresponding to Fig. 2B. The dominant strategies (i.e., those with the highest expected payoff) are highlighted with bold red borders for each interval of $I(t)$.

In cases (i) and (iii) representing scenarios with low effectiveness of FC and low cost of SD, only SD can be favored by the population when the disease prevalence exceeds the smaller threshold between $I_1^{\ast }$ and $I_3^{\ast }$, which is $I_1^{\ast }$; that is, when $I(t)>I_1^{\ast }$ (Fig. 3A and C). In contrast, cases (ii) and (iv) represent a trade-off between FC effectiveness and SD cost, with $I_3^{\ast }$ being the smaller threshold. In these cases, both FC and SD can be preferred: The population opts for FC when $I(t)>I_3^{\ast }$ but shifts to SD at sufficiently high infection levels (Fig. 3B and D).

As such, in case (i), the infection is mitigated predominantly through SD, similar to previous models that only consider SD considered (51). In case (ii), FC is the predominant strategy for mitigation, but individuals might also practice SD if the infection becomes more prevalent. Specifically, we may observe SD in the initial stages of the epidemic due to the surge in infection levels, with FC becoming more common in later stages. In case (iii), FC is never preferred by the population, and SD is adopted when $C_{\text{SD}}$ is below the threshold $C_{\text{SD}}^{\ast }$, but gains little traction otherwise. In case (iv), only FC will ever be adopted, as the population stops FC before SD ever is adopted.

In Fig. 4, we demonstrate how dynamic compliance levels and behavioral shifts emerge from the bottom up, together with the infection spread, for a range of six scenarios based on the four cases mentioned above.

Fig. 4.

Dynamic adoption preferences and compliance with each NPI, including social distancing and face covering. For very small disease prevalence, the dynamics tend toward no measures (NSD), while for very large disease prevalence, they shift toward social distancing (SD). In between, FC becomes the preferred option for some cases. For each case, the time window during which there exists a predominant NPI measure (either FC or SD) is highlighted by the corresponding shaded color. The values of the model parameters per month are $\beta =10$, $\gamma =4$, $\omega =5$, $\kappa =5$, $C_{\text{I}}=10$, $C_{\text{FC}}=1$, and (A) $C_{\text{SD}}=2$, ${\beta }_{\text{FC}}=1$, (B) $C_{\text{SD}}=2$, ${\beta }_{\text{FC}}=4$, (C) $C_{\text{SD}}=2$, ${\beta }_{\text{FC}}=5$, (D) $C_{\text{SD}}=2$, ${\beta }_{\text{FC}}=8.5$, (E) $C_{\text{SD}}=8.5$, ${\beta }_{\text{FC}}=7$, and (F) $C_{\text{SD}}=8.5$, ${\beta }_{\text{FC}}=8.5$.

When FC is highly effective—specifically, when ${\beta }_{\text{FC}}/\beta$ is sufficiently small—FC becomes the preferred strategy for mitigating the spread of the infection. This situation corresponds to case (ii), as shown in the magenta area of Fig. 2B and also in Fig. 3B. Despite oscillations, the population effectively dampens the infection dynamics, as illustrated in Fig. 4A.

As the effectiveness of FC decreases but the cost of social distancing ($C_{\text{SD}}$) remains low, there is a transition from SD to FC as the primary measure to slow down the infection. This scenario occurs close to the boundary between cases (i) and (ii), represented by the magenta and blue areas in Fig. 2B. Here, individuals initially adopt SD and later switch to FC as the infection progresses (Fig. 4B).

With a further decrease in FC effectiveness (blue area in Fig. 2B) as in case (i) (Fig. 3A), SD becomes the dominant strategy controlling the disease dynamics, although some usage of FC may still be present. This case highlights the predominance of SD and is depicted in Fig. 4C.

For case (iii) with low $C_{\text{SD}}$, FC is no longer effective enough to provide any advantage (the teal area below the horizontal line in Figs. 2B and 3C). Consequently, only SD is adopted as a measure to control the infection spread (Fig. 4D).

When FC effectiveness is not too low and SD is very costly—characterized by case (iv) as also illustrated in Fig. 3D—the population practices only a limited level of FC. This situation is represented by the yellow area in Fig. 2B and is illustrated in Fig. 4E.

Finally, for case (iii) with high $C_{\text{SD}}$, FC offers no advantage (teal area above the horizontal line in Fig. 2B), and individuals have no incentive to adopt any preventive measures against the infection. This absence of mitigation strategies is shown in Fig. 4F.

These scenarios highlight a critical insight: Once the population transitions to using FC due to their effectiveness or as a compromise between no measures and the high cost of SD, there is no return to more stringent measures. This behavioral shift indicates a one-way trajectory in public health behavior during an epidemic, where once less stringent measures like FC are adopted due to their perceived cost-effectiveness, the population does not revert to more restrictive interventions like SD, even as the dynamics of the infection and the relative costs and effectiveness of NPIs vary (cf. Fig. 3B and D).

Discussion

The complexity of human behavior in response to the infection risks and intervention costs associated with infectious diseases has been investigated in related works. It has been shown that age differences in the cost of infection can introduce chaotic behavior into epidemiological models (56). Previous studies have approached mask-wearing and social distancing from network (7, 40) and game-theoretical perspectives (10, 56). Prior work has also incorporated vaccination (29) and threshold-dependent tipping dynamics (37) into models of social distancing behavior, highlighting the complex interplay between various measures and the spread of infection.

Our work builds upon these prior contributions by introducing a model that considers two sorts of NPI measures—SD and FC—and their interaction. This approach allows for a more detailed exploration of the strategic decisions individuals make in the face of an epidemic and provides insights into how multiple concurrent measures can influence the trajectory of disease spread. The model reveals that once less stringent measures like FC are adopted due to their perceived benefits, populations do not revert to more restrictive interventions such as SD, suggesting a one-way transition trajectory in NPI adoption behavior.

However, the model also has certain limitations that should be discussed. First, the model does not account for factors such as seasonality (57), the emergence of new strains through mutation (58), age-related differences in infection risk (56), or heterogeneity in the costs associated with social distancing or face covering (8). Moreover, the influence of asymptomatic individuals (9) and vaccination (59) on the dynamics of disease spread and control measures is not considered. These factors could significantly impact the effectiveness and societal acceptance of NPIs.

Another limitation lies in the assumption that the costs of social distancing and mask-wearing are constant over time. In reality, as the pandemic evolves, public perception and tolerance of these measures may shift, leading to changes in compliance levels (60). Additionally, the model does not incorporate top–down governmental interventions such as mask mandates or lockdown orders, which can play a crucial role in shaping population behavior.

The use of replicator dynamics has been growing in the exploration of social learning processes and the diffusion of behaviors across various societal challenges, from enforcing norms through peer punishment (61) to promoting responsible antibiotic usage (62). In particular, prior work has demonstrated that integrating evolutionary game theory with epidemiological processes can be fruitful in shedding light on disease interventions (10, 18, 29, 30). Nevertheless, this approach has limitations, such as neglecting intrinsic individual heterogeneity, as recently discussed in ref. 63, and thus needs to be revisited in order to empirically validate modeling predictions.

The “Swiss cheese” model of protection (64), popularized during the COVID-19 pandemic, emphasizes a multilayered approach to risk management. Each layer of protection has its flaws (“holes”), but together they provide a stronger defense. While our current model does not account for individuals using multiple NPIs simultaneously, it does capture dynamic behavioral shifts in the community where different individuals adopt different measures: Some practice SD, others wear FC, while some do neither (NSD). This reflects a population-level multilayered protection. Extending the model to include simultaneous use of multiple measures at the individual level—thereby forming a “Swiss cheese” model from that perspective—presents a promising direction for future research.

In conclusion, our model offers insights into the dynamics of NPI adoption and compliance, particularly emphasizing that their context-dependent adoption preferences should be taken into account by top–down public health interventions. By extending the current work, future research aiming to overcome barriers in NPI adoptions can incorporate a broader range of factors that influence human behavior and disease spread. Understanding the interaction between different sorts of preventive measures and their impact on epidemiological outcomes will be crucial for designing effective public health strategies in response to current and future epidemics.

Materials and Methods

Epidemiological Model with Dual Behavior Response.

Combining the SIR epidemic model with the replicator-like equations governing the NPI adoption dynamics based on the pairwise comparison process, we obtain the following system of ordinary differential equations (ODE):

$$\begin{array}{cc}\hfill \dot{S}(t)& =-\beta \mathcal{G}(t)S(t)I(t)-{\beta }_{\text{FC}}\mathcal{F}(t)S(t)I(t),\hfill \\ \hfill \dot{I}(t)& =\beta \mathcal{G}(t)S(t)I(t)+{\beta }_{\text{FC}}\mathcal{F}(t)S(t)I(t)-\gamma I(t),\hfill \\ \hfill \dot{R}(t)& =\gamma I(t),\hfill \\ \hfill \dot{\mathcal{E}}(t)& =\omega \mathcal{E}(t)\mathcal{F}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{SD}}-{\pi }_{\text{FC}})\right]\hfill \\ \hfill & +\omega \mathcal{E}(t)\mathcal{G}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{SD}}-{\pi }_{\text{NSD}})\right]\hfill \\ \hfill \dot{\mathcal{F}}(t)& =\omega \mathcal{F}(t)\mathcal{E}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{FC}}-{\pi }_{\text{SD}})\right]\hfill \\ \hfill & +\omega \mathcal{F}(t)\mathcal{G}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{FC}}-{\pi }_{\text{NSD}})\right]\hfill \\ \hfill \dot{\mathcal{G}}(t)& =\omega \mathcal{G}(t)\mathcal{E}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{NSD}}-{\pi }_{\text{SD}})\right]\hfill \\ \hfill & +\omega \mathcal{G}(t)\mathcal{F}(t)tanh\left[\frac{\kappa }{2}({\pi }_{\text{NSD}}-{\pi }_{\text{FC}})\right]\hfill \end{array}$$ [6]

By further substituting the expected payoffs for each NPI choice, (Eq. 1), into (Eq. 6), we obtain the coupled system of ODEs, as shown in SI Appendix.

Numerical Simulations.

We use the initial conditions $S(0)=0.99$, $I(0)=0.01,R(0)=0$ as well as $\mathcal{E}(0)=0.1,\mathcal{F}(0)=0.1,\mathcal{G}(0)=0.8$ for Fig. 4. Model parameters used to produce Fig. 1B and C are the same as in Fig. 4B. The values of the model parameters per month used in numerical simulations are provided in the figure captions. The numeric solutions of the system of ODEs are obtained using the standard ODE solver ode45() in Matlab. The code to reproduce our results and animations is publicly available at https://github.com/fufeng/NPIdilemma.

Determinants of Dynamic Compliance Levels and Behavioral Shifts.

To analyze how adoption dynamics evolves and the resulting dynamic preferences given the cost and effectiveness of each NPI, we focus on critical conditions for the epidemic thresholds, defined by the critical infection levels, $I_i^{\ast }$ for $i=1,\cdots ,4$, through the pairwise comparison of payoffs between the three options at any given time point $t$ (Fig. 2A and SI Appendix, Figs. S1 and S2).

Using these epidemic thresholds, given the disease prevalence $I(t)$, individuals tend to switch to the dominant NPI option, which is thus more likely to prevail in the population. From a social planner’s perspective, the social optimum minimizes the total expected costs of infections and NPIs. The smaller of the two critical thresholds, $I_s^{\ast }=min\{I_1^{\ast },I_3^{\ast }\}$, above which adopting FC or SD becomes more favorable than doing nothing (NSD) corresponds to the most ideal compliance levels needed to keep the epidemic at this low prevalence. However, individuals act from their own best interest and they will switch to the more favorable option once the disease prevalence exceeds $I_s^{\ast }$, which leads to overcommitment at the population level. On the other hand, as the epidemic curve bends downward and drops below $I_s^{\ast }$, individuals are more likely to opt out of NPIs, leading to a further uptick in disease prevalence. Similarly to oscillatory tragedy of the commons, we observe this oscillatory dynamics in our simulations, which are influenced by the responsiveness and rationality level parameters (Movies S1–S6). This also gives rise to nonmonotonic behavior in the total fraction of infections, measured by the final epidemic size $R(\infty )$ (see SI appendix, Figs. S3 and S4 for details).

As detailed in SI Appendix, we also extend the model to take into account the impact of waning immunity acquired from infection (SI Appendix, Fig. S5 and Movie S7).

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4A.

Movie S2.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4B.

Movie S3.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4C.

Movie S4.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4D.

Movie S5.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4E.

Movie S6.

Animation for the NPI adoption dynamics within the simplex triangle S₃ over time, corresponding to the case shown in the main Figure 4F.

Movie S7.

Animation for the NPI adoption dynamics within the simplex triangle S3 over time with the same parameter choices as in the main Figure 4B except ω = 30 and κ = 30.

Data, Materials, and Software Availability

Matlab code data have been deposited in GitHub (https://github.com/fufeng/NPIdilemma) (65). All other data are included in the manuscript and/or supporting information.