Introduction

While COVID-19 continues to exert a significant burden across the world, SARS-CoV-2 has now established itself as an additional circulating pathogen in human populations1,2, joining many other endemic respiratory viruses, such as seasonal coronaviruses and influenza. For many of these pathogens, a hallmark of their epidemiology is that they exhibit seasonal variations in transmission rates3. The drivers of seasonality in infectious disease dynamics range from external factors, such as humidity and temperature4, to social factors, such as school terms or the timing of large social events5,6. The effect of seasonal drivers on epidemiological dynamics has been examined for a range of specific infectious diseases, ranging from airborne respiratory infections such as influenza, COVID-19, measles, and chickenpox, to vector-borne diseases, such as malaria and plague3,7,8,9,10,11. Earn et al.10 showed that an epidemic model with seasonal variations in transmission could capture key epidemiological transitions in the dynamics of measles transmission.

In parallel, a key driver of infectious disease dynamics is individual decision-making. For example, if infection levels or risks from infections are high, individuals may choose to adhere to non-pharmaceutical interventions (NPIs), such as mask-wearing or social distancing, or pharmaceutical interventions, such as vaccination or therapeutics. The decision to follow such interventions may then itself change transmission dynamics, and affect the circulating number of infections. While traditional epidemiological modeling did not explicitly take into account individual reaction to disease (e.g., refs. 12,13,14), several studies have examined these effects in more detail15,16,17,18,19,20,21,22,23,24,25,26,27.

Because of the impact of individual behavior on trajectories of the COVID-19 pandemic, there has been an increased interest in coupling epidemiological and behavioral models, and in understanding the potential feedback between these processes28,29,30. Evolutionary game theory provides a mathematical framework to model how individuals adjust their behavior in response to the state of their population. The decision to comply with a given intervention may lead to social dilemmas, which arise when the individual interests conflict with the collective interests. One example of such a situation is the prisoner’s dilemma (PD) game, where individuals can choose between cooperation and defection. In the PD game, the best strategy for a player is to defect regardless of the other player’s choice. This results in a suboptimal scenario where both players defect whereas mutual cooperation would bring a higher gain for both players. This scenario is common and challenging, as it studies how cooperation can emerge between selfish actors. Another example is the Snowdrift game (SD), which depicts competition for shared resources and escalation of conflict31,32,33. In the SD game, players can maximize their gain by choosing the opposite strategy from the other player—even if it would be beneficial for the collective if both cooperated. The dynamics of such a population are most commonly described by the replicator dynamics34,35. Recent work has shown that this approach is particularly relevant in the context of NPIs, which can lead to social dilemmas depending on the individual assessments25,27,36.

Here, we examine the influence of decision-making on epidemic dynamics with variations in seasonal transmission rates, and vice versa. To investigate this, we extend a recent framework that couples individual adherence to an NPI with epidemic dynamics25 by introducing a seasonal transmission rate. We associate adherence to an NPI to cooperation and not adhering to defection27,36,37. We then examine whether seasonality can introduce a tension between individual and population-level outcomes, i.e., a social dilemma and whether individual behavior affects the impact of seasonality on epidemiological trajectories. Social dilemmas can arise depending on the infection levels in the population and the costs for adherence. With this approach, different seasons can be characterized by different social games—even if we assume a constant number of infections, seasonality in disease transmission can imply that adherence to an NPI may be, e.g., a PD in spring and autumn, an SD game in winter and not a dilemma at all in summer. As we show below, coupling seasonal disease dynamics with behavioral dynamics can lead to complex and unexpected dynamics.

Results

Model

We consider a SIRS model, where a population is composed of susceptible (S), infected (I) and recovered (R) individuals. The model is given by the system of equations

$$\frac{{\rm{d}}S}{{\rm{d}}t}=-b\beta (t)SI+\delta R,$$
(1a)
$$\frac{{\rm{d}}I}{{\rm{d}}t}=b\beta (t)SI-\gamma I,$$
(1b)
$$\frac{{\rm{d}}R}{{\rm{d}}t}=\gamma I-\delta R,$$
(1c)

where β(t) is the transmission rate of the infection, γ is the recovery rate and δ is the rate of waning immunity. Assuming a constant population size, we can set S + I + R = 1. We define the seasonality of the transmission rate as

$$\beta (t)={\beta }_{0}-{\beta }_{1}\cos \left(\frac{2\pi }{12}t\right)$$
(2)

with a period of oscillations of 12 months (1 year). We fix time such that the minimum β0 − β1 occurs in month 0. Here, β0 is a baseline transmission rate and β1 the strength of seasonality. We illustrate the infection model in Fig. 1A.

Fig. 1: Illustration of our model.
figure 1

A Schematic of the SIRS model, describing susceptible (S), infected (I), and recovered (R) individuals and the transitions between them. The seasonal transmission rate is β(t), b ≤ 1 is the average effect of adherence to an NPI, γ is the recovery rate, and δ is the rate at which recovered individuals become susceptible again (waning immunity). B SIRS model with constant transmission rate. We show an outbreak of the infection, transient oscillations and the dynamics approaching an endemic equilibrium for the fraction of infected, susceptible, and recovered individuals as a function of time for a constant transmission rate. C SIRS model with seasonal transmission rate. We show the fraction of infected, susceptible, and recovered individuals as a function of time for a seasonal transmission rate after reaching the steady state, where β0 is the baseline transmission rate, and β1 the strength of oscillations. Increasing the oscillation strength, the number of peaks in the infection changes with a damping appearing for high enough values. Model parameters are as in Table 1. D Behavioral dynamics. We show a summary of all dilemma conditions depending on the fraction of infected individuals in the population. Non-adherers are depicted in yellow, and adherers in purple. Each game has a different level of adherence at the steady state, where no one adheres in the NA and PD games, some adhere in the SD game, and everyone adheres in the HG game. We show the thresholds for the number of infected individuals separating the games.

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In addition to the infection dynamics in the population, we consider that individuals can adhere to an NPI. The average reduction in transmission due to adherence is captured by

$$b=p{x}_{A}+(1-{x}_{A}).$$
(3)

We assume that xA is the probability of an individual to adhere to an NPI, and p is the effectiveness of adherence (0 < p < 1). We define the payoffs for adherers (A) and non-adherers (N), given by the payoff matrix

$$\begin{array}{c}A\qquad \qquad \qquad \,\,N\\ \begin{array}{c}A\\ N\end{array}\left(\begin{array}{cc}-{p}^{2}\beta (t)\xi I-c&-p\beta (t)\xi I-c\\ -p\beta (t)\xi I&-\beta (t)\xi I\end{array}\right).\end{array}$$
(4)

The individual payoffs depend on the fraction of infected individuals I, the risk perception ξ, the effectiveness of adherence p, the cost c of following the NPI, and the transmission rate β(t). If no one adheres to the NPI, the transmission rate is β(t). If an adherer interacts with a non-adherer, the transmission rate is reduced by a factor of p. If both individuals adhere, the transmission rate is reduced by a factor of p2.

Thus, the average payoff of non-adherers is given by

$${\Pi }_{N}=\left[-p\beta (t)\xi I\right]{x}_{A}+\left[-\beta (t)\xi I\right](1-{x}_{A}),$$
(5)

whereas the payoff of adherers is

$${\Pi }_{A}=p{\Pi }_{N}-c\,.$$
(6)

We assume that the change in the probability to adhere follows a replicator equation with a rate of spontaneous switching μ, leading to the equation

$$\frac{1}{{\tau }_{A}}\frac{{\rm{d}}{x}_{A}}{{\rm{d}}t}={x}_{A}(1-{x}_{A})({\Pi }_{A}-{\Pi }_{N})+\mu (1-{x}_{A})-\mu {x}_{A},$$
(7)

where τA is the time scale of the dynamics. We introduce a spontaneous switching rate μ to prevent the population from reaching absorbing states where behavior becomes fixed.

Based on the entries of a general payoff matrix

$$\begin{array}{c}A\qquad \,\,N\\ \begin{array}{c}A\\ N\end{array}\left(\begin{array}{cc}{a}_{AA}&{a}_{AN}\\ {a}_{NA}&{a}_{NN}\end{array}\right),\end{array}$$
(8)

different social games can be defined. Using the values in the adherence payoff matrix (Eq. (4)), we can determine the thresholds separating these games. These boundaries depend on the effectiveness of adherence p, the risk perception ξ, the cost c of adhering to the NPI, and the seasonal transmission β(t). We find the following thresholds:

  1. (i)

    If aNA > aAA, aNN > aAN, and aNN > aAA, non-adherence is always preferable. We will call this scenario the No Adherence game (NA). This case arises when the fraction of infected individuals is sufficiently small,

    $$I \,<\, \frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{(1-{p}^{2})}.$$
    (9)
  2. (ii)

    If aNA > aAA, aNN > aAN, but aNN < aAA, non-adherence is preferable, but mutual adherence is better than mutual non-adherence. This defines a PD game, resulting in the condition

    $$\frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{(1-{p}^{2})} \,<\, I \,<\, \frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{(1-p)}.$$
    (10)

    In this case, any individual favors N over A, even if the social optimum is that all individuals adhere.

  3. (iii)

    If aNA > aAA, aNN < aAN, and aNN < aAA, we have an SD game, where the population settles in a mixed configuration of N and A. The boundaries for this game are

    $$\frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{(1-p)} \,<\, I \,<\, \frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{p(1-p)}.$$
    (11)

    In this case, the social optimum is still that all individuals adhere, but individuals may still favor N over A.

  4. (iv)

    Finally, if aNA < aAA, aNN < aAN and aNN < aAA, adherence is always favored. This defines the Harmony Game (HG), which emerges for high fractions of infected individuals,

    $$I \,>\, \frac{c}{\xi }\frac{1}{\beta (t)}\frac{1}{p(1-p)}.$$
    (12)

We summarize these thresholds in Fig. 1D. For 0 < p < 1, the Stag-Hunt game cannot be reached—this requires more sophisticated assumptions on the effect of an NPI36. As the adherence dynamics follows the replicator equation, for μ → 0, the population will converge to no adherence for the NA and PD game, partial adherence for the SD, and complete adherence in the HG38.

For high transmission rates β and high costs of adherence c, the HG can only be reached for intermediate effectiveness p25. Basically, if adherence is too effective, there is no need for everyone to adhere. If adherence is too ineffective, it is not worth doing so. Table 1 summarizes the model parameters. Parameter values related to the infection dynamics were based on values from ref. 25, while values related to the behavioral dynamics were chosen to encompass all possible social games dynamics.

Table 1 Model parameters and their description
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SIRS model with constant transmission

First, we illustrate the SIRS model with a constant transmission rate and without the introduction of adherence to an NPI (i.e., b = 1). A common way to assess the growth of an infection is by defining the effective reproductive number Rt,

$${R}_{t}=\frac{\beta S}{\gamma }.$$

From Eq. (1b), we see that the fraction of infected individuals will grow if I(βS − γ) > 0. Supposing that the infection is already present in the population (I > 0), this corresponds to Rt > 1. Initially, when most of the population is still susceptible (S ≈ 1), an outbreak occurs if \({R}_{0} > \frac{\beta }{\gamma }\). In Fig. 1B, we show the infection progression over time for this scenario. An outbreak occurs, creating a wave of infection in the initial steps, and eventually establishing in the population. In this model, an endemic equilibrium is possible due to the influx of susceptibles in the population, a consequence of the waning immunity δ. A similar dynamics occurs when considering birth and death processes25.

SIRS model with seasonal transmission

Next, we consider a seasonal transmission rate10, according to Eq. (2). In Fig. 1C, we show the oscillations in the fraction of susceptible, infected and recovered individuals after reaching a steady state. These oscillations are driven by a seasonally oscillating transmission rate with a period of 12 months, starting at its minimum at the beginning of the year, and peaking in the middle. For sufficiently strong seasonality, several peaks of infection occur early in the year, each with decreasing amplitude. The period of these oscillations (T) is much faster than the annual cycle and arises from the dynamics of the infectious diseases. It can be calculated by analyzing perturbations around the steady state of a system with constant β (see SI). For our seasonal transmission β(t), we found numerically T ≈ 1.36 months, which is consistent with the period T ≈ 1.27 months that can be calculated analytically from a constant transmission rate β.

Seasonal SIRS model with NPI adherence

Next, we explore the consequences of having adherers in the population. In Fig. 2, we illustrate how the behavioral and epidemiological components of the model are coupled. In the top panel, we show the fraction of infected individuals over the period of one year after reaching the steady state. The bottom panel shows the effective transmission rate of the infection, which varies according to the fraction of adherence in the population. At the beginning of the year, the infection does not spread due to the minimum in the transmission rate. Consequently, there is no need for individuals to adhere and pay the associated cost to do so. Although adherence would decrease the fraction of infected individuals, both the individual and social optima are to not adhere. Therefore, there is no social dilemma in this scenario (NA region). As the transmission rate starts to increase, the fraction of infected individuals grows, and the game turns into a PD. Here, the social optimum would be to adhere, but this does not happen due to the temptation to not adhere. For even higher transmission rates, the population reaches the SD game. Now, according to the replicator equation, adherence should start to increase, but there are not enough adherers in the population to decrease the transmission rate significantly. Adherence only starts to grow substantially in the middle of the year (HG region), when the transmission rate is highest. As a consequence, this increase in adherence reduces the transmission of the infection, lowering its abundance in the population. Similar to the beginning of the year, there is no dilemma in this period. After the peak in transmissibility, the fraction of infected individuals decreases, and the population will go back to the SD, PD, and NA, respectively.

Fig. 2: Dynamic social games.
figure 2

The fraction of infected individuals oscillates after reaching the steady state. Red curves represent the SIRS model without adherence dynamics, while black curves represent the SIRS model with adherence dynamics. The background colors represent the different social games defined by the thresholds in Fig. 1D. Note that at the beginning and at the end of the year, the model with adherence dynamics shows a higher fraction of infected due to an increased fraction of susceptible individuals. The bottom panel displays the transmission rate β(t) and the effective transmission bβ(t) due to adherence. Time scale is τa = 0.25, switching rate is μ = 10−7, and other parameters are as in Table 1.

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Parameter variation

Above, we have discussed that partial adherence is expected in the SD region. Despite that, we only observe a substantial growth in adherence in the HG region. This is due to the slow time scale of adherence, which delays the individual’s responses to changes in the social game. One would expect that changing adherence faster can alter this picture compared to the SIRS dynamics. To investigate this, we explore different values of the time scale of adherence dynamics, τA. A larger value of τA indicates a faster change of adherence, whereas a smaller value of τA indicates a slower change of adherence.

Figure 3 shows the infection and adherence dynamics over time after reaching steady state oscillations, for different adherence time scales. If adherence is slow (Fig. 3A), individual’s reactions to changes in the social game are delayed. Therefore, transitioning from the SD to PD, adherence decreases only slowly, leading to a benefit in situations that would result in no adherence otherwise (NA and PD). This results in a less severe outbreak at the end of the year. When we increase the speed of adherence (Fig. 3B, C), individuals react faster when entering a social dilemma, adhering earlier in the year (SD region). On the other hand, when the population transitions out of the dilemma, individuals will stop adhering faster. This quick exposure of susceptibles to the actual transmission rate will generate a higher peak of infection at the end of the year. Theoretically, the best scenario to minimize the infection would be to have a fast growth of adherence at the beginning of the infectious season and a slow decline in adherence at the end.

Fig. 3: Coupled model for behavior and seasonal epidemics.
figure 3

Fraction of infected and adherers over time after reaching the steady state for different time scales of adherence. Red curves represent the SIRS without adherence, and black curves represent the SIRS with adherence. The background colors represent the different social games defined by the thresholds in Fig. 1D. A–C Three time scales of adherence: slow, τA = 0.25, intermediate, τA = 1, and fast, τA = 4. If the dynamics of adherence is slow, the peak in the fraction of infected can be avoided when the transmission rate goes down towards the end of the year (A). Similarly, by making adherence faster, the peak of infection increases (C). Switching rate is μ = 10−7, and other parameters are as in Table 1.

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Finally, we explore how the switching rate μ affects the time dynamics of the infection. Without switching, a population can get trapped in either complete adherence or no adherence. By introducing switching, there will always be individuals adhering when they should not and vice versa, which can have a substantial effect on behavioral dynamics39. Figure 4 shows the fraction of infected individuals over time after reaching the steady state, for different values of μ. When the switching rate is low (Fig. 4A), there are not many individuals reacting opposite to the social game. Thus, there are not many adherers at the beginning of the year, just as in the intermediate adherence case (Fig. 3B). As the switching rate increases (Fig. 4B, C), adherence starts to grow earlier in the year, a similar situation to the fast adherence case (Fig. 3C). At the same time, the peak of infection at the end of the year is avoided, resembling the slow adherence case (Fig. 3A). By considering a spontaneous switching rate, partial adherence will be reached independently of the social game. Therefore, there are fewer infected individuals even in the NA and PD regions. Also, since adherence never completely vanishes, the outbreak at the end of the year can be avoided. On the other hand, complete adherence cannot be reached anymore, which is the most beneficial scenario for decreasing transmission in the population.

Fig. 4: Effect of spontaneous switching rates on seasonal epidemics.
figure 4

Fraction of infected individuals over time after reaching the steady state for different switching rates μ. Red curves represent the SIRS without adherence, and black curves represent the SIRS with adherence and random switching between behaviors at rate μ. The background colors represent the different social games defined by the thresholds in Fig. 1D. A–C represent three different switching rates μ = 10−7, 10−3, 10−1, respectively. As the switching rate increases, it is possible to avoid the peak of infection at the end of the season, similar to the slow adherence case. Furthermore, a higher switching rate also decreases the fraction of infected individuals earlier in the season compared to the SIRS model without adherence, similar to the fast adherence case. Time scale is τa = 1, and other parameters are as in Table 1.

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To summarize the effect of different time scales of adherence and spontaneous switching rates on the infection dynamics and their impact on mitigation outcomes, we also compute the average fraction of infected individuals per year. Figure 5 shows the average fraction of infected individuals per year normalized by the fraction of infected individuals without adherence for different values of the time scale τA and switching rate μ. When τA is zero, we recover the case without adherence. For the range of parameters explored, we see a decrease in the average fraction of infected individuals compared to the case without adherence. Despite that, we observe a peak in the average fraction of infected for intermediate values of τA. For smaller time scales (slow adherence), there is a benefit due to the delayed response, as discussed before. For higher values (fast adherence), we actually see a benefit from increasing the time scale compared to the intermediate case. Even with a second peak at the end of the year (e.g., as shown in Fig. 3C), the fast response to the infection decreases the infected levels over the year. For sufficiently high switching rates, the effect of the time scale of adherence is lost, since the sensitivity to the different social games decreases, as discussed before.

Fig. 5: Impact of behavioral dynamics on yearly infection dynamics and mitigation outcomes.
figure 5

Average fraction of infected individuals over a year normalized by the average fraction of infected individuals in the model without adherence after reaching the steady state, as a function of the time scale of adherence τa. On average, adherence changes reduce the number of infections in all cases. Different curve colors indicate different values of the switching rate μ. Independent of μ, all curves show a minimum at low τa and a maximum in the relative average fraction of infected individuals throughout the year. As seen in Fig. 3A, slow adherence (low values of τa) benefits the population due to a slow response to the infection. This actually results in more individuals adhering to the intervention. Similarly, for fast adherence (high values of τa), we also see a benefit from having a fast response to the infection. Even with a second outbreak at the end of the year (e.g., Fig. 3C), overall, the infection is less severe on average. Other parameters are as in Table 1.

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Discussion

Epidemiological models allow us to understand and predict the spread of infectious diseases. At the same time, decision-making plays a crucial role in human interactions. By coupling both subjects, a more complex situation arises, where the infection dynamics is affected by individual’s choices to adhere to interventions that reduce transmission, and these choices are, in turn, based on the infection levels in the population.

Previous studies have combined epidemiological and behavioral models by considering a population composed of susceptible, infected and recovered individuals, where individuals may pay a cost adhere to an NPI with a certain probability, which decreases the transmission rate of the infection25,27,36. The social optimum, which takes this cost into account, does not necessarily translate to the best epidemiological outcome. Depending on the state of the infection in the population, adherence to an intervention can be a social dilemma. Our work builds upon these prior contributions by introducing a seasonal transmission rate. More generally, periodically driven social games are still not well explored in the field of evolutionary game theory, with few works on the topic40.

We have shown that different periods of the year are characterized by different social games and dilemmas depending on the levels of the infection. If the infection is sufficiently severe, individuals will behave accordingly and will adhere to the intervention, thus decreasing infection levels. However, if the infection is too mild, it is not worth paying the cost of adherence. In general, we observe that adherence to an NPI can decrease the infection levels in the population. On the other hand, increased adherence can also lead to more susceptible individuals in a population, leading to additional surges of the epidemic when adherence reduces.

We focus on two different aspects of adherence, the time scale and the switching rate. The former is related to how fast people react to the seasonality in transmission rates and the latter to a change in behavior independently of the current game. Fast reactions can decrease infection levels due to a rapid reduction of the high transmission rate early in the year. Despite that, at the end of the year, a second outbreak emerges due to a sudden exposure of susceptibles when people stop adhering too fast. Overall, throughout the year, we see a benefit in this case compared to not having adherence. Slow reactions can also bring a benefit. In our model, if people have a delayed response, adherence can exist in situations where it would not otherwise, decreasing infection levels. For the switching rate, we observe similar results as the different time scales. A high value corresponds to always having adherence independent of the game. This results in adherence earlier in the year, while at the same time decreasing the second outbreak peak.

Our approach makes certain assumptions that should be discussed: Our model assumes individuals have homogeneous costs of adherence to an NPI, while in reality individuals have heterogeneous costs depending on their life situation. We also assume that the reduction in the transmission rate when an adherer interacts with a non-adherer is the same when a non-adherer interacts with an adherer. However, as explored in previous works36, NPIs may not be symmetric in their protection and different individuals may have different types of NPIs available to them. Finally, our approach to model the seasonality of the transmission rate ignores other complex factors, such as temperature, humidity, and human behavior, which might alter the functional form of the seasonal transmission5. Additionally, geographical variability could also lead to different seasonal patterns, especially when comparing tropical and temperate regions.

In conclusion, our work explores how such seasonal social dilemmas can arise from coupling decision-making and seasonal forcing of an infectious disease3,10,25, illustrating that such situations could naturally occur. Introducing mechanisms for the maintenance of cooperation in situations where the dilemma can be temporally reduced or even vanish will be an interesting area to be explored by evolutionary game theorists. Furthermore, more refined epidemiological models will benefit from taking into account how individuals react to the spread of an infectious disease28. Ultimately, understanding the interplay between human behavior and infection dynamics in the context of changing risk will be essential for developing public health strategies and interventions.