An epidemiological model with voluntary quarantine strategies governed by evolutionary game dynamics

Vaccination and Collective Action Under Social Norms

Social dynamics are an integral part of the spread of disease affecting contact rates as well as the adoption of pharmaceutical and non-pharmaceutical interventions. When vaccines provide waning immunity, efficient and timely uptake of boosters is required to maintain protection and reduce infections. How then do social dynamics affect the timely uptake of vaccines and thereby the course of an epidemic? This paper explores this scenario through a behavioural-epidemiological model. It features a tipping-point dynamic for the uptake of vaccines that combines the risk of infection, perceived morbidity risk of the vaccine, and social payoffs for deviating from the vaccination decision-making of others. The social payoffs are derived from a social norm of conformity, and they create a collective action problem. A key finding driven by this dilemma is that waves of vaccine uptake and infections can occur due to inefficient and delayed uptake of boosters. This results in a nonlinear response of the infection load to the transmission rate: an intermediate transmission rate can result in greater prevalence of disease relative to more or less transmissible diseases. Further, global information about the prevalence of the disease and vaccine uptake can increase the infection load and peak relative to information restricted to individuals' contact networks. Thus, decisions driven by local information can mitigate the collective action problem across the population. Finally, the optimal public policy program to promote boosters is shown to be one that focuses on overcoming the social inertia to vaccinate at the start of an epidemic.

Structural inequalities exacerbate infection disparities

During the COVID-19 pandemic, the world witnessed a disproportionate infection rate among marginalized and low-income groups. Despite empirical evidence suggesting that structural inequalities in society contribute to health disparities, there has been little attempt to offer a computational and theoretical explanation to establish its plausibility and quantitative impact. Here, we focus on two aspects of structural inequalities: wealth inequality and social segregation. Our computational model demonstrates that (a) due to the inequality in self-quarantine ability, the infection gap widens between the low-income and high-income groups, and the overall infected cases increase, (b) social segregation between different socioeconomic status (SES) groups intensifies the disease spreading rates, and (c) the second wave of infection can emerge due to a false sense of safety among the medium and high SES groups. By performing two data-driven analyses, one on the empirical network and economic data of 404 metropolitan areas of the United States and one on the daily Covid-19 data of the City of Chicago, we verify that higher segregation leads to an increase in the overall infection cases and higher infection inequality across different ethnic/socioeconomic groups. These findings together demonstrate that reducing structural inequalities not only helps decrease health disparities but also reduces the spread of infectious diseases overall.

Anomalous behavior of replicator dynamics for the prisoner's dilemma on diluted lattices

In diluted lattices, cooperation is often enhanced at specific densities, particularly near the percolation threshold for stochastic updating rules. However, the replicator rule, despite being probabilistic, does not follow this trend. We find that this anomalous behavior is caused by structures formed by holes and defectors, which prevent some agents from experiencing fluctuations, thereby restricting the free flow of information across the network. As a result, the system becomes trapped in a frozen state, though this can be disrupted by introducing perturbations. Finally, we provide a more quantitative analysis of the relationship between the percolation threshold and cooperation, tracking its development within clusters of varying sizes and demonstrating how the percolation threshold shapes the fundamental structures of the lattice.

Analyzing disease control through testing game approach embedded with treatment and vaccination strategies

This research introduces an expanded SEIR (Susceptible-Exposed-Infected-Recovered) model that incorporates the components of testing, treatment, and vaccination. The study utilizes an evolutionary game theory (EGT) framework to investigate the impact of human behavior on the acceptance and implementation of these interventions. The choice to undergo testing and vaccination is considered a strategic decision influenced by perceived risks and benefits. Regarding disease dynamics, adherence to vaccination and testing protocols is seen as a behavioral factor. The present study employs a finite difference method to numerically examine the impact of proactive vaccination and retroactive treatment policies on human behavior. The investigation focuses on these policies' individual and combined effects, considering various factors, including vaccination and testing costs, vaccine efficacy, awareness level, and infection rates. The findings indicate that the integration of heightened awareness and enhanced vaccination efficacy can successfully alleviate the transmission of diseases, even in situations where the expenses associated with testing and vaccination are substantial. Reducing infections in situations characterized by low or moderate awareness or vaccination effectiveness is contingent upon low testing costs. The final epidemic size (FES) negatively correlates with testing and vaccine costs, indicating that lower costs are linked to a lower FES. Optimal vaccine coverage (VC) occurs when vaccine costs are minimal and vaccine efficiency is efficient, whereas treatment coverage (TC) reaches its peak when testing costs are minimal. This research underscores the significance of considering human behavior and the intricate relationship between vaccination, testing, and treatment approaches in managing the transmission of contagious illnesses. It offers valuable perspectives for policymakers to mitigate the consequences of epidemics.

Cooperation in public goods games: Leveraging other-regarding reinforcement learning on hypergraphs

Cooperation is a self-organized collective behavior. It plays a significant role in the evolution of both ecosystems and human society. Reinforcement learning is different from imitation learning, offering a new perspective for exploring cooperation mechanisms. However, most existing studies with the public goods game (PGG) employ a self-regarding setup or are on pairwise interaction networks. Players in the real world, however, optimize their policies based not only on their histories but also on the histories of their coplayers, and the game is played in a group manner. In this work, we investigate the evolution of cooperation in the PGG under the other-regarding reinforcement learning evolutionary game on hypergraph by combining the Q-learning algorithm and evolutionary game framework, where other players' action history is incorporated and the game is played on hypergraphs. Our results show that as the synergy factor r[over ̂] increases, the parameter interval divides into three distinct regions-the absence of cooperation, medium cooperation, and high cooperation-accompanied by two abrupt transitions in the cooperation level near r[over ̂]_{1}^{*} and r[over ̂]_{2}^{*}, respectively. Interestingly, we identify regular and anticoordinated chessboard structures in the spatial pattern that positively contribute to the first cooperation transition but adversely affect the second. Furthermore, we provide a theoretical treatment for the first transition with an approximated r[over ̂]_{1}^{*} and reveal that players with a long-sighted perspective and low exploration rate are more likely to reciprocate kindness with each other, thus facilitating the emergence of cooperation. Our findings contribute to understanding the evolution of human cooperation, where other-regarding information and group interactions are commonplace.

The cross-regional settlement methods in hospitals and the treatment-seeking behavior of patients with malignant tumors in China: an evolutionary game model

Background

Cross-regional settlement management is a key indicator of national health insurance system maturity. Given the significant demand for cross-regional medical treatment among Chinese patients with malignant tumors and the territorially managed health insurance system, further research is necessary to explore the relationship between hospital settlement methods and treatment-seeking behaviors among these patients. This study introduces and validates an evolutionary game model that provides a theoretical foundation for direct settlement policies in cross-regional treatment.

Methods

An evolutionary game model was constructed with patients and hospitals serving as strategic players within a dynamic system. This model integrates the patients' treatment utility, medical and nonmedical costs, and hospitals' financial and technological advancement benefits.

Results

The evolutionary stability analysis revealed seven-game outcomes between hospitals and patients with malignant tumors. The numerical simulations suggest an evolutionary convergence toward strategy (1, 0), indicating a trend where patients with malignant tumors opt for cross-regional treatment, yet hospitals choose not to implement a direct settlement policy. Parameter sensitivity analysis showed that the parameters set in this study affected player behavioral choices and game equilibria.

Conclusion

A strong demand for cross-regional medical treatment among Chinese patients with malignant tumors, and some hospitals require more incentives to implement cross-regional settlements. The key factors influencing the willingness of some patients with malignant tumors to resettle include the costs of in-area medical care, costs of cross-regional treatment without direct settlement, and the utility of cross-regional treatment. Technological advancement benefits and input costs influence some hospitals' motivation to adopt cross-regional settlements. Policy adjustments that effectively implement direct settlement policies can facilitate equilibrium, enhance the initiatives of some local health insurance management departments, improve the accessibility and efficiency of medical services, and reduce nonmedical expenses for patients.

An innovative fractional-order evolutionary game theoretical study of personal protection, quarantine, and isolation policies for combating epidemic diseases

This study uses imposed control techniques and vaccination game theory to study disease dynamics with transitory or diminishing immunity. Our model uses the ABC fractional-order derivative mechanism to show the effect of non-pharmaceutical interventions such as personal protection or awareness, quarantine, and isolation to simulate the essential control strategies against an infectious disease spread in an infinite and uniformly distributed population. A comprehensive evolutionary game theory study quantified the significant influence of people's vaccination choices, with government forces participating in vaccination programs to improve obligatory control measures to reduce epidemic spread. This model uses the intervention options described above as a control strategy to reduce disease prevalence in human societies. Again, our simulated results show that a combined control strategy works exquisitely when the disease spreads even faster. A sluggish dissemination rate slows an epidemic outbreak, but modest control techniques can reestablish a disease-free equilibrium. Preventive vaccination regulates the border between the three phases, while personal protection, quarantine, and isolation methods reduce disease transmission in existing places. Thus, successfully combining these three intervention measures reduces epidemic or pandemic size, as represented by line graphs and 3D surface diagrams. For the first time, we use a fractional-order derivate to display the phase-portrayed trajectory graph to show the model's dynamics if immunity wanes at a specific pace, considering various vaccination cost and effectiveness settings.

Heterogeneous risk attitudes and waves of infection

Many countries have experienced multiple waves of infection during the COVID-19 pandemic. We propose a novel but parsimonious extension of the SIR model, a CSIR model, that can endogenously generate waves. In the model, cautious individuals take appropriate prevention measures against the virus and are not exposed to infection risk. Incautious individuals do not take any measures and are susceptible to the risk of infection. Depending on the size of incautious and susceptible population, some cautious people lower their guard and become incautious-thus susceptible to the virus. When the virus spreads sufficiently, the population reaches "temporary" herd immunity and infection subsides thereafter. Yet, the inflow from the cautious to the susceptible eventually expands the susceptible population and leads to the next wave. We also show that the CSIR model is isomorphic to the SIR model with time-varying parameters.

Behavioral game of quarantine during the monkeypox epidemic: Analysis of deterministic and fractional order approach

This work concerns the epidemiology of infectious diseases like monkeypox (mpox) in humans and animals. Our models examine transmission scenarios, including transmission dynamics between humans, animals, and both. We approach this using evolutionary game theory, specifically the intervention game-theoretical (IGT) framework, to study how human behavior can mitigate disease transmission without perfect vaccines and treatments. To do this, we use non-pharmaceutical intervention, namely the quarantine policy, which demonstrates the delayed effect of the epidemic. Additionally, we contemplate quarantine-based behavioral intervention policies in deterministic and fractional-order models to show behavioral impact in the context of the memory effect. Firstly, we extensively analyzed the model's positivity and boundness of the solution, reproduction number, disease-free and endemic equilibrium, possible stability, existence, concavity, and Ulam-Hyers stability for the fractional order. Subsequently, we proceeded to present a numerical analysis that effectively illustrates the repercussions of varying quarantine-related factors, information probability, and protection probability. We aimed to comprehensively examine the effects of non-pharmaceutical interventions on disease control, which we conveyed through line graphs and 2D heat maps. Our findings underscored the significant influence of strict quarantine measures and the protection of both humans and animals in mitigating disease outbreaks. These measures not only significantly curtailed the spread of the disease but also delayed the occurrence of the epidemic's peak. Conversely, when quarantine maintenance policies were implemented at lower rates and protection levels diminished, we observed contrasting outcomes that exacerbated the situation. Eventually, our analysis revealed the emergence of animal reservoirs in cases involving disease transmission between humans and animals.

On the adoption of nonpharmaceutical interventions during the pandemic: An evolutionary game model

The adoption of behavioral nonpharmaceutical interventions (NPIs) among the public is essential for tackling the COVID-19 pandemic, yet presents challenges due to the complexity of human behaviors. A large body of literature has utilized classic game theory to investigate the population's decisions regarding the adoption of interventions, where the static solution concept such as the Nash equilibrium is studied. However, individual adoption behavior is not static, instead it is a dynamic process that involves the strategic interactions with other counterparts over time. The study of quantitatively analyzing the dynamics on precautionary behavior during an outbreak is rather scarce. This article fills the research gap by developing an evolutionary game-theoretic framework to model the dynamics of population behavior on the adoption of NPI. We construct the two-group asymmetric game, where behavioral change for each group is characterized by replicator equations. Sensitivity analyses are performed to examine the long-term stability of equilibrium points with respect to perturbation of model parameters. We found that the limiting behavior of intervention adoption in the population consists of only pure strategies in a game setting, indicating that the evolutionary outcome is that everyone either takes up the preventive measure or not. We also applied the framework to examine the mask-wearing behavior, and validated with actual data. Overall, this article provides insights into population dynamics on the adoption of intervention strategy during the outbreak, which can be beneficial for policy makers to better understand the evolutionary trajectory of population behavior.

Individual preferences, government policy, and COVID-19: A game-theoretic epidemiological analysis

Purpose

The ongoing COVID-19 pandemic imposes serious short-term and long-term health costs on populations. Restrictive government policy measures decrease the risks of infection, but produce similarly serious social, mental health, and economic problems. Citizens have varying preferences about the desirability of restrictive policies, and governments are thus forced to navigate this tension in making pandemic policy. This paper analyses the situation facing government using a game-theoretic epidemiological model.

Methodology

We classify individuals into health-centered individuals and freedom-centered individuals to capture the heterogeneous preferences of citizens. We first use the extended Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) model (adding individual preferences) and the signaling game model (adding government) to analyze the strategic situation against the backdrop of a realistic model of COVID-19 infection.

Findings

We find the following: 1. There exists two pooling equilibria. When health-centered and freedom-centered individuals send anti-epidemic signals, the government will adopt strict restrictive policies under budget surplus or balance. When health-centered and freedom-centered individuals send freedom signals, the government chooses not to implement restrictive policies. 2. When governments choose not to impose restrictions, the extinction of an epidemic depends on whether it has a high infection transmission rate; when the government chooses to implement non-pharmacological interventions (NPIs), whether an epidemic will disappear depends on how strict the government's restrictions are.

Originality/value

Based on the existing literature, we add individual preferences and put the government into the game as a player. Our research extends the current form of combining epidemiology and game theory. By using both we get a more realistic understanding of the spread of the virus and combine that with a richer understanding of the strategic social dynamics enabled by game theoretic analysis. Our findings have important implications for public management and government decision-making in the context of COVID-19 and for potential future public health emergencies.

Time delay of the appearance of a new strain can affect vaccination behavior and disease dynamics: An evolutionary explanation

The emergence of a novel strain during a pandemic, like the current COVID-19, is a major concern to the healthcare system. The most effective strategy to control this type of pandemic is vaccination. Many previous studies suggest that the existing vaccine may not be fully effective against the new strain. Additionally, the new strain's late arrival has a significant impact on the disease dynamics and vaccine coverage. Focusing on these issues, this study presents a two-strain epidemic model in which the new strain appears with a time delay. We considered two vaccination provisions, namely preinfection and postinfection vaccinations, which are governed by human behavioral dynamics. In such a framework, individuals have the option to commit vaccination before being infected with the first strain. Additionally, people who forgo vaccination and become infected with the first train have the chance to be vaccinated (after recovery) in an attempt to avoid infection from the second strain. However, a second strain can infect vaccinated and unvaccinated individuals. People may have additional opportunities to be vaccinated and to protect themselves from the second strain due to the time delay. Considering the cost of the vaccine, the severity of the new strain, and the vaccine's effectiveness, our results indicated that delaying the second strain decreases the peak size of the infected individuals. Finally, by estimating the social efficiency deficit, we discovered that the social dilemma for receiving immunization decreases with the delay in the arrival of the second strain.

Understanding the Impact of Vaccination and Self-Defense Measures on Epidemic Dynamics Using an Embedded Optimization and Evolutionary Game Theory Methodology

Explaining how individual choice and government policy can appear in the same context in real society is one of the most challenging scientific problems. Controlling infectious diseases requires effective prevention and control measures, including vaccination and self-defense measures. In this context, optimal control strategies incorporating vaccination and self-defense measures have been proposed using the framework of evolutionary game theory. This approach accounts for individuals' behavior and interactions in a population. It can provide insights into the effectiveness of different strategies for controlling the spread of infectious diseases. The optimal control strategy involves balancing the costs and benefits of vaccination, considering the dynamic interplay between the infected and susceptible populations. By combining evolutionary game theory with optimal control theory, we can identify the optimal allocation of resources for vaccination and self-defense measures, which can maximize the control of infectious diseases while minimizing costs. The model is utilized to analyze public health policies diseases, such as vaccination and self-defense strategies, to mitigate the spread of infectious in the context of delayed decision-making.

Attitudinal analysis of vaccination effects to lead endemic phases

To achieve endemic phases, repeated vaccinations are necessary. However, individuals may grapple with whether to get vaccinated due to potential side effects. When an individual is already immune due to previous infections or vaccinations, the perceived risk from vaccination is often less than the risk of infection. Yet, repeated rounds of vaccination can lead to avoidance, impeding the establishment of endemic phases. We explore this phenomenon using an individual-based Monte Carlo simulation, validating our findings with game theory. The Nash equilibrium encapsulates individuals' non-cooperative behavior, while the system's optimal value represents the societal benefits of altruistic cooperation. We define the difference between these as the price of anarchy. Our simulations reveal that the price of anarchy must fall below a threshold of 12.47 for endemic phases to be achieved in a steady state. This suggests that for a basic reproduction number of 10, a consistent vaccination rate greater than 89% is required. These findings offer new insights into vaccination-related decision-making and can inform effective strategies to tackle infectious diseases.

Exploring epidemic voluntary vaccinating behavior based on information-driven decisions and benefit-cost analysis

A complex dynamic interplay exists between epidemic transmission and vaccination, which is significantly influenced by human behavioral responses. We construct a research framework combining both the function modeling of the cumulative global COVID-19 information and limited individuals' information processing capacity employing the Gompertz model for growing processes. Meanwhile, we built a function representing the decision to get vaccinated following benefit-cost analysis considered the choices made by people in each scenario have an influence from altruism, free-riding and immunity escaping capacity. Through the mean-field calculation analysis and using a fourth-order Runge-Kutta method with constant step size, we obtain plots from numerical simulations. We found that only when the total number of infectious individuals proves sufficient to reach and exceed a certain level will the individuals face a better trade-off in determining whether to get vaccinated against the diseases based on that information. Besides, authoritative media have a higher decisive influence and efforts should be focused on extending the duration of vaccine protection, which is beneficial to inhibit the outbreaks of epidemics. Our work elucidates that reducing the negative payoff brought about by the free-riding behavior for individuals or improving the positive payoff from the altruistic motivation helps to control the disease in cultures that value social benefits, vaccination willingness is generally stronger. We also note that at a high risk of infection, the decision of vaccination is highly correlated with global epidemic information concerning COVID-19 infection, while at times of lower risk, it depends on the game theoretic vaccine strategy. The findings demonstrate that improving health literacy, ensuring open and transparent information on vaccine safety and efficacy as a public health priority can be an effective strategy for mitigating inequalities in health education, as well as alleviating the phenomenon that immunity escaping abilities is more likely to panic by populations with high levels of education. In addition, prosocial nudges are great ways to bridge these immunity gaps that can contribute to implementing government public health control measures, creating a positive feedback loop.

A cyclic behavioral modeling aspect to understand the effects of vaccination and treatment on epidemic transmission dynamics

Evolutionary epidemiological models have played an active part in analyzing various contagious diseases and intervention policies in the biological sciences. The design in this effort is the addition of compartments for treatment and vaccination, so the system is designated as susceptible, vaccinated, infected, treated, and recovered (SVITR) epidemic dynamic. The contact of a susceptible individual with a vaccinated or an infected individual makes the individual either immunized or infected. Inventively, the assumption that infected individuals enter the treatment and recover state at different rates after a time interval is also deliberated through the presence of behavioral aspects. The rate of change from susceptible to vaccinated and infected to treatment is studied in a comprehensive evolutionary game theory with a cyclic epidemic model. We theoretically investigate the cyclic SVITR epidemic model framework for disease-free and endemic equilibrium to show stable conditions. Then, the embedded vaccination and treatment strategies are present using extensive evolutionary game theory aspects among the individuals in society through a ridiculous phase diagram. Extensive numerical simulation suggests that effective vaccination and treatment may implicitly reduce the community risk of infection when reliable and cheap. The results exhibited the dilemma and benefitted situation, in which the interplay between vaccination and treatment evolution and coexistence are investigated by the indicators of social efficiency deficit and socially benefited individuals.

Assessing the medical resources in COVID-19 based on evolutionary game

COVID-19 has brought a great challenge to the medical system. A key scientific question is how to make a balance between home quarantine and staying in the hospital. To this end, we propose a game-based susceptible-exposed-asymptomatic -symptomatic- hospitalized-recovery-dead model to reveal such a situation. In this new framework, time-varying cure rate and mortality are employed and a parameter m is introduced to regulate the probability that individuals are willing to go to the hospital. Through extensive simulations, we find that (1) for low transmission rates (β < 0.2), the high value of m (the willingness to stay in the hospital) indicates the full use of medical resources, and thus the pandemic can be easily contained; (2) for high transmission rates (β > 0.2), large values of m lead to breakdown of the healthcare system, which will further increase the cumulative number of confirmed cases and death cases. Finally, we conduct the empirical analysis using the data from Japan and other typical countries to illustrate the proposed model and to test how our model explains reality.

A comparison of node vaccination strategies to halt SIR epidemic spreading in real-world complex networks

We compared seven node vaccination strategies in twelve real-world complex networks. The node vaccination strategies are modeled as node removal on networks. We performed node vaccination strategies both removing nodes according to the initial network structure, i.e., non-adaptive approach, and performing partial node rank recalculation after node removal, i.e., semi-adaptive approach. To quantify the efficacy of each vaccination strategy, we used three epidemic spread indicators: the size of the largest connected component, the total number of infected at the end of the epidemic, and the maximum number of simultaneously infected individuals. We show that the best vaccination strategies in the non-adaptive and semi-adaptive approaches are different and that the best strategy also depends on the number of available vaccines. Furthermore, a partial recalculation of the node centrality increases the efficacy of the vaccination strategies by up to 80%.

Investigating the trade-off between self-quarantine and forced quarantine provisions to control an epidemic: An evolutionary approach

During a pandemic event like the present COVID-19, self-quarantine, mask-wearing, hygiene maintenance, isolation, forced quarantine, and social distancing are the most effective nonpharmaceutical measures to control the epidemic when the vaccination and proper treatments are absent. In this study, we proposed an epidemiological model based on the SEIR dynamics along with the two interventions defined as self-quarantine and forced quarantine by human behavior dynamics. We consider a disease spreading through a population where some people can choose the self-quarantine option of paying some costs and be safer than the remaining ones. The remaining ones act normally and send to forced quarantine by the government if they get infected and symptomatic. The government pays the forced quarantine costs for individuals, and the government has a budget limit to treat the infected ones. Each intervention derived from the so-called behavior model has a dynamical equation that accounts for a proper balance between the costs for each case, the total budget, and the risk of infection. We show that the infection peak cannot be reduced if the authority does not enforce a proactive (quantified by a higher sensitivity parameter) intervention. While comparing the impact of both self- and forced quarantine provisions, our results demonstrate that the latter is more influential to reduce the disease prevalence and the social efficiency deficit (a gap between social optimum payoff and equilibrium payoff).

Emergence of protective behaviour under different risk perceptions to disease spreading

The behaviour of individuals is a main actor in the control of the spread of a communicable disease and, in turn, the spread of an infectious disease can trigger behavioural changes in a population. Here, we study the emergence of individuals' protective behaviours in response to the spread of a disease by considering two different social attitudes within the same population: concerned and risky. Generally speaking, concerned individuals have a larger risk aversion than risky individuals. To study the emergence of protective behaviours, we couple, to the epidemic evolution of a susceptible-infected-susceptible model, a decision game based on the perceived risk of infection. Using this framework, we find the effect of the protection strategy on the epidemic threshold for each of the two subpopulations (concerned and risky), and study under which conditions risky individuals are persuaded to protect themselves or, on the contrary, can take advantage of a herd immunity by remaining healthy without protecting themselves, thanks to the shield provided by concerned individuals. This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.

Individual risk-aversion responses tune epidemics to critical transmissibility (R = 1)

Changes in human behaviour are a major determinant of epidemic dynamics. Collective activity can be modified through imposed control measures, but spontaneous changes can also arise as a result of uncoordinated individual responses to the perceived risk of contagion. Here, we introduce a stochastic epidemic model implementing population responses driven by individual time-varying risk aversion. The model reveals an emergent mechanism for the generation of multiple infection waves of decreasing amplitude that progressively tune the effective reproduction number to its critical value R = 1. In successive waves, individuals with gradually lower risk propensity are infected. The overall mechanism shapes well-defined risk-aversion profiles over the whole population as the epidemic progresses. We conclude that uncoordinated changes in human behaviour can by themselves explain major qualitative and quantitative features of the epidemic process, like the emergence of multiple waves and the tendency to remain around R = 1 observed worldwide after the first few waves of COVID-19.

Game-Theoretic Frameworks for Epidemic Spreading and Human Decision-Making: A Review

This review presents and reviews various solved and open problems in developing, analyzing, and mitigating epidemic spreading processes under human decision-making. We provide a review of a range of epidemic models and explain the pros and cons of different epidemic models. We exhibit the art of coupling between epidemic models and decision models in the existing literature. More specifically, we provide answers to fundamental questions in human decision-making amid epidemics, including what interventions to take to combat the disease, who are decision-makers, and when and how to take interventions, and how to make interventions. Among many decision models, game-theoretic models have become increasingly crucial in modeling human responses or behavior amid epidemics in the last decade. In this review, we motivate the game-theoretic approach to human decision-making amid epidemics. This review provides an overview of the existing literature by developing a multi-dimensional taxonomy, which categorizes existing literature based on multiple dimensions, including (1) types of games, such as differential games, stochastic games, evolutionary games, and static games; (2) types of interventions, such as social distancing, vaccination, quarantine, and taking antidotes; (3) the types of decision-makers, such as individuals, adversaries, and central authorities at different hierarchical levels. A fine-grained dynamic game framework is proposed to capture the essence of game-theoretic decision-making amid epidemics. We showcase three representative frameworks with unique ways of integrating game-theoretic decision-making into the epidemic models from a vast body of literature. Each of the three frameworks has their unique way of modeling and analyzing and develops results from different angles. In the end, we identify several main open problems and research gaps left to be addressed and filled.

Cooperation dynamics under pandemic risks and heterogeneous economic interdependence

The spread of COVID-19 and ensuing containment measures have accentuated the profound interdependence among nations or regions. This has been particularly evident in tourism, one of the sectors most affected by uncoordinated mobility restrictions. The impact of this interdependence on the tendency to adopt less or more restrictive measures is hard to evaluate, more so if diversity in economic exposures to citizens' mobility are considered. Here, we address this problem by developing an analytical and computational game-theoretical model encompassing the conflicts arising from the need to control the economic effects of global risks, such as in the COVID-19 pandemic. The model includes the individual costs derived from severe restrictions imposed by governments, including the resulting economic interdependence among all the parties involved in the game. By using tourism-based data, the model is enriched with actual heterogeneous income losses, such that every player has a different economic cost when applying restrictions. We show that economic interdependence enhances cooperation because of the decline in the expected payoffs by free-riding parties (i.e., those neglecting the application of mobility restrictions). Furthermore, we show (analytically and through numerical simulations) that these cross-exposures can transform the nature of the cooperation dilemma each region or country faces, modifying the position of the fixed points and the size of the basins of attraction that characterize this class of games. Finally, our results suggest that heterogeneity among regions may be used to leverage the impact of intervention policies by ensuring an agreement among the most relevant initial set of cooperators.

Criticality and Griffiths phases in random games with quenched disorder

The perceived risk and reward for a given situation can vary depending on resource availability, accumulated wealth, and other extrinsic factors such as individual backgrounds. Based on this general aspect of everyday life, here we use evolutionary game theory to model a scenario with randomly perturbed payoffs in a prisoner's dilemma game. The perception diversity is modeled by adding a zero-average random noise in the payoff entries and a Monte Carlo simulation is used to obtain the population dynamics. This payoff heterogeneity can promote and maintain cooperation in a competitive scenario where only defectors would survive otherwise. In this work, we give a step further, understanding the role of heterogeneity by investigating the effects of quenched disorder in the critical properties of random games. We observe that payoff fluctuations induce a very slow dynamic, making the cooperation decay behave as power laws with varying exponents, instead of the usual exponential decay after the critical point, showing the emergence of a Griffiths phase. We also find a symmetric Griffiths phase near the defector's extinction point when fluctuations are present, indicating that Griffiths phases may be frequent in evolutionary game dynamics and play a role in the coexistence of different strategies.

Identification and Control of Game-Based Epidemic Models

The effectiveness of control measures against the diffusion of the COVID-19 pandemic is grounded on the assumption that people are prepared and disposed to cooperate. From a strategic decision point of view, cooperation is the unreachable strategy of the Prisoner’s Dilemma game, where the temptation to exploit the others and the fear of being betrayed by them drives the people’s behavior, which eventually results in a fully defective outcome. In this work, we integrate a standard epidemic model with the replicator equation of evolutionary games in order to study the interplay between the infection spreading and the propensity of people to be cooperative under the pressure of the epidemic. The developed model shows high performance in fitting real measurements of infected, recovered and dead people during the whole period of COVID-19 epidemic spread, from March 2020 to September 2021 in Italy. The estimated parameters related to cooperation result to be significantly correlated with vaccination and screening data, thus validating the model. The stability analysis of the multiple steady states present in the proposed model highlights the possibility to tune fundamental control parameters to dramatically reduce the number of potential dead people with respect to the non-controlled case.

Behavioural response to heterogeneous severity of COVID-19 explains temporal variation of cases among different age groups

Together with seasonal effects inducing outdoor or indoor activities, the gradual easing of prophylaxis caused second and third waves of SARS-CoV-2 to emerge in various countries. Interestingly, data indicate that the proportion of infections belonging to the elderly is particularly small during periods of low prevalence and continuously increases as case numbers increase. This effect leads to additional stress on the health care system during periods of high prevalence. Furthermore, infections peak with a slight delay of about a week among the elderly compared to the younger age groups. Here, we provide a mechanistic explanation for this phenomenology attributable to a heterogeneous prophylaxis induced by the age-specific severity of the disease. We model the dynamical adoption of prophylaxis through a two-strategy game and couple it with an SIR spreading model. Our results also indicate that the mixing of contacts among the age groups strongly determines the delay between their peaks in prevalence and the temporal variation in the distribution of cases. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

Modeling COVID-19 Transmission Dynamics With Self-Learning Population Behavioral Change

Many regions observed recurrent outbreaks of COVID-19 cases after relaxing social distancing measures. It suggests that maintaining sufficient social distancing is important for limiting the spread of COVID-19. The change of population behavior responding to the social distancing measures becomes an important factor for the pandemic prediction. In this paper, we develop a SEAIR model for studying the dynamics of COVID-19 transmission with population behavioral change. In our model, the population is divided into several groups with their own social behavior in response to the delayed information about the number of the infected population. The transmission rate depends on the behavioral changes of all the population groups, forming a feedback loop to affect the COVID-19 dynamics. Based on the data of Hong Kong, our simulations demonstrate how the perceived cost after infection and the information delay affect the level and the time period of the COVID-19 waves.

Effect of delayed awareness and fatigue on the efficacy of self-isolation in epidemic control

The isolation of infectious individuals is a key measure of public health for the control of communicable diseases. However, involving a strong perturbation of daily life, it often causes psychosocial distress, and severe financial and social costs. These may act as mechanisms limiting the adoption of the measure in the first place or the adherence throughout its full duration. In addition, difficulty of recognizing mild symptoms or lack of symptoms may impact awareness of the infection and further limit adoption. Here we study an epidemic model on a network of contacts accounting for limited adherence and delayed awareness to self-isolation, along with fatigue causing overhasty termination. The model allows us to estimate the role of each ingredient and analyze the tradeoff between adherence and duration of self-isolation. We find that the epidemic threshold is very sensitive to an effective compliance that combines the effects of imperfect adherence, delayed awareness and fatigue. If adherence improves for shorter quarantine periods, there exists an optimal duration of isolation, shorter than the infectious period. However, heterogeneities in the connectivity pattern, coupled to a reduced compliance for highly active individuals, may almost completely offset the effectiveness of self-isolation measures on the control of the epidemic.

Dynamic Games of Social Distancing During an Epidemic: Analysis of Asymmetric Solutions

Individual behaviors play an essential role in the dynamics of transmission of infectious diseases, including COVID-19. This paper studies a dynamic game model that describes the social distancing behaviors during an epidemic, assuming a continuum of players and individual infection dynamics. The evolution of the players' infection states follows a variant of the well-known SIR dynamics. We assume that the players are not sure about their infection state, and thus, they choose their actions based on their individually perceived probabilities of being susceptible, infected, or removed. The cost of each player depends both on her infection state and on the contact with others. We prove the existence of a Nash equilibrium and characterize Nash equilibria using nonlinear complementarity problems. We then exploit some monotonicity properties of the optimal policies to obtain a reduced-order characterization for Nash equilibrium and reduce its computation to the solution of a low-dimensional optimization problem. It turns out that, even in the symmetric case, where all the players have the same parameters, players may have very different behaviors. We finally present some numerical studies that illustrate this interesting phenomenon and investigate the effects of several parameters, including the players' vulnerability, the time horizon, and the maximum allowed actions, on the optimal policies and the players' costs.

Linking the disease transmission to information dissemination dynamics: An insight from a multi-scale model study

During the outbreak of emerging infectious diseases, information dissemination dynamics significantly affects the individuals' psychological and behavioral changes, and consequently influences on the disease transmission. To investigate the interaction of disease transmission and information dissemination dynamics, we proposed a multi-scale model which explicitly models both the disease transmission with saturated recovery rate and information transmission to evaluate the effect of information transmission on dynamic behaviors. Considering time variation between information dissemination, epidemiological and demographic processes, we obtained a slow-fast system by reasonably introducing a sufficiently small quantity. We carefully examined the dynamics of proposed system, including existence and stability of possible equilibria and existence of backward bifurcation, by using the fast-slow theory and directly investigating the full system. We then compared the dynamics of the proposed system and the essential thresholds based on two methods, and obtained the similarity between the basic dynamical behaviors of the slow system and that of the full system. Finally, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 3.25. Numerical analysis suggested that information transmission about COVID-19 pandemic caused by media coverage can reduce the peak size, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.

Prioritising COVID-19 vaccination in changing social and epidemiological landscapes: a mathematical modelling study

BACKGROUND

During the COVID-19 pandemic, authorities must decide which groups to prioritise for vaccination in a shifting social-epidemiological landscape in which the success of large-scale non-pharmaceutical interventions requires broad social acceptance. We aimed to compare projected COVID-19 mortality under four different strategies for the prioritisation of SARS-CoV-2 vaccines.

METHODS

We developed a coupled social-epidemiological model of SARS-CoV-2 transmission in which social and epidemiological dynamics interact with one another. We modelled how population adherence to non-pharmaceutical interventions responds to case incidence. In the model, schools and workplaces are also closed and reopened on the basis of reported cases. The model was parameterised with data on COVID-19 cases and mortality, SARS-CoV-2 seroprevalence, population mobility, and demography from Ontario, Canada (population 14·5 million). Disease progression parameters came from the SARS-CoV-2 epidemiological literature. We assumed a vaccine with 75% efficacy against disease and transmissibility. We compared vaccinating those aged 60 years and older first (oldest-first strategy), vaccinating those younger than 20 years first (youngest-first strategy), vaccinating uniformly by age (uniform strategy), and a novel contact-based strategy. The latter three strategies interrupt transmission, whereas the first targets a vulnerable group to reduce disease. Vaccination rates ranged from 0·5% to 5% of the population per week, beginning on either Jan 1 or Sept 1, 2021.

FINDINGS

Case notifications, non-pharmaceutical intervention adherence, and lockdown undergo successive waves that interact with the timing of the vaccine programme to determine the relative effectiveness of the four strategies. Transmission-interrupting strategies become relatively more effective with time as herd immunity builds. The model predicts that, in the absence of vaccination, 72 000 deaths (95% credible interval 40 000-122 000) would occur in Ontario from Jan 1, 2021, to March 14, 2025, and at a vaccination rate of 1·5% of the population per week, the oldest-first strategy would reduce COVID-19 mortality by 90·8% on average (followed by 89·5% in the uniform, 88·9% in the contact-based, and 88·2% in the youngest-first strategies). 60 000 deaths (31 000-108 000) would occur from Sept 1, 2021, to March 14, 2025, in the absence of vaccination, and the contact-based strategy would reduce COVID-19 mortality by 92·6% on average (followed by 92·1% in the uniform, 91·0% in the oldest-first, and 88·3% in the youngest-first strategies) at a vaccination rate of 1·5% of the population per week.

INTERPRETATION

The most effective vaccination strategy for reducing mortality due to COVID-19 depends on the time course of the pandemic in the population. For later vaccination start dates, use of SARS-CoV-2 vaccines to interrupt transmission might prevent more deaths than prioritising vulnerable age groups.

FUNDING

Ontario Ministry of Colleges and Universities.

Prosocial behavior of wearing a mask during an epidemic: an evolutionary explanation

In the midst of the COVID-19 pandemic, with limited or no supplies of vaccines and treatments, people and policymakers seek easy to implement and cost-effective alternatives to combat the spread of infection during the pandemic. The practice of wearing a mask, which requires change in people's usual behavior, may reduce disease transmission by preventing the virus spread from infectious to susceptible individuals. Wearing a mask may result in a public good game structure, where an individual does not want to wear a mask but desires that others wear it. This study develops and analyzes a new intervention game model that combines the mathematical models of epidemiology with evolutionary game theory. This approach quantifies how people use mask-wearing and related protecting behaviors that directly benefit the wearer and bring some advantage to other people during an epidemic. At each time-step, a suspected susceptible individual decides whether to wear a facemask, or not, due to a social learning process that accounts for the risk of infection and mask cost. Numerical results reveal a diverse and rich social dilemma structure that is hidden behind this mask-wearing dilemma. Our results highlight the sociological dimension of mask-wearing policy.

Optimal governance and implementation of vaccination programmes to contain the COVID-19 pandemic

Since the recent introduction of several viable vaccines for SARS-CoV-2, vaccination uptake has become the key factor that will determine our success in containing the COVID-19 pandemic. We argue that game theory and social network models should be used to guide decisions pertaining to vaccination programmes for the best possible results. In the months following the introduction of vaccines, their availability and the human resources needed to run the vaccination programmes have been scarce in many countries. Vaccine hesitancy is also being encountered from some sections of the general public. We emphasize that decision-making under uncertainty and imperfect information, and with only conditionally optimal outcomes, is a unique forte of established game-theoretic modelling. Therefore, we can use this approach to obtain the best framework for modelling and simulating vaccination prioritization and uptake that will be readily available to inform important policy decisions for the optimal control of the COVID-19 pandemic.

Prioritising COVID-19 vaccination in changing social and epidemiological landscapes: a mathematical modelling study

Background

During the COVID-19 pandemic, authorities must decide which groups to prioritise for vaccination in a shifting social–epidemiological landscape in which the success of large-scale non-pharmaceutical interventions requires broad social acceptance. We aimed to compare projected COVID-19 mortality under four different strategies for the prioritisation of SARS-CoV-2 vaccines.

Methods

We developed a coupled social–epidemiological model of SARS-CoV-2 transmission in which social and epidemiological dynamics interact with one another. We modelled how population adherence to non-pharmaceutical interventions responds to case incidence. In the model, schools and workplaces are also closed and reopened on the basis of reported cases. The model was parameterised with data on COVID-19 cases and mortality, SARS-CoV-2 seroprevalence, population mobility, and demography from Ontario, Canada (population 14·5 million). Disease progression parameters came from the SARS-CoV-2 epidemiological literature. We assumed a vaccine with 75% efficacy against disease and transmissibility. We compared vaccinating those aged 60 years and older first (oldest-first strategy), vaccinating those younger than 20 years first (youngest-first strategy), vaccinating uniformly by age (uniform strategy), and a novel contact-based strategy. The latter three strategies interrupt transmission, whereas the first targets a vulnerable group to reduce disease. Vaccination rates ranged from 0·5% to 5% of the population per week, beginning on either Jan 1 or Sept 1, 2021.

Findings

Case notifications, non-pharmaceutical intervention adherence, and lockdown undergo successive waves that interact with the timing of the vaccine programme to determine the relative effectiveness of the four strategies. Transmission-interrupting strategies become relatively more effective with time as herd immunity builds. The model predicts that, in the absence of vaccination, 72 000 deaths (95% credible interval 40 000–122 000) would occur in Ontario from Jan 1, 2021, to March 14, 2025, and at a vaccination rate of 1·5% of the population per week, the oldest-first strategy would reduce COVID-19 mortality by 90·8% on average (followed by 89·5% in the uniform, 88·9% in the contact-based, and 88·2% in the youngest-first strategies). 60 000 deaths (31 000–108 000) would occur from Sept 1, 2021, to March 14, 2025, in the absence of vaccination, and the contact-based strategy would reduce COVID-19 mortality by 92·6% on average (followed by 92·1% in the uniform, 91·0% in the oldest-first, and 88·3% in the youngest-first strategies) at a vaccination rate of 1·5% of the population per week.

Interpretation

The most effective vaccination strategy for reducing mortality due to COVID-19 depends on the time course of the pandemic in the population. For later vaccination start dates, use of SARS-CoV-2 vaccines to interrupt transmission might prevent more deaths than prioritising vulnerable age groups.

Funding

Ontario Ministry of Colleges and Universities.

Public perceptions of multiple risks during the COVID-19 pandemic in Italy and Sweden

Knowing how people perceive multiple risks is essential to the management and promotion of public health and safety. Here we present a dataset based on a survey (N = 4,154) of public risk perception in Italy and Sweden during the COVID-19 pandemic. Both countries were heavily affected by the first wave of infections in Spring 2020, but their governmental responses were very different. As such, the dataset offers unique opportunities to investigate the role of governmental responses in shaping public risk perception. In addition to epidemics, the survey considered indirect effects of COVID-19 (domestic violence, economic crises), as well as global (climate change) and local (wildfires, floods, droughts, earthquakes, terror attacks) threats. The survey examines perceived likelihoods and impacts, individual and authorities' preparedness and knowledge, and socio-demographic indicators. Hence, the resulting dataset has the potential to enable a plethora of analyses on social, cultural and institutional factors influencing the way in which people perceive risk.