Prospect Theory Based Individual Irrationality Modelling and Behavior Inducement in Pandemic Control

1  Introduction

The outbreak of COVID-19 has led to a severe public health crisis and great economic losses. Governments worldwide have taken various measures, including lockdowns and mandatory quarantines, to inhibit the spread of the disease. However, individuals may not comply with these policies, as many have their own opinions and preferences. In such public health crises, individuals tend to act irrationally, such as excessive panic in an epidemic outbreak and underestimation of the dangers of the epidemics in its later spread, which can significantly affect individuals’ decisions and ultimately affect the spread of the epidemic. Moreover, individuals’ behaviors and the epidemic affect each other. For example, individuals’ behaviors such as social distancing, wearing masks, and isolating can inhibit the spread of the epidemic, and individuals tend to choose protective behaviors when the pandemic is more severe. Therefore, it is crucial to model individuals’ behaviors during the epidemic and determine how to control the spread of the epidemic by guiding individuals’ behaviors without resorting to mandatory measures.

1.1 Literature Review

In the following, this paper reviews recent works on epidemic control over networks, individual behavior modeling during an epidemic, and irrational behavior modeling.

1.1.1 Epidemic Control

There are numerous studies on how to control the epidemic spread over networks, and many attempted to inhibit the epidemic spread on a network by removing nodes. Wang et al. pointed out that the epidemic spread on a network is highly correlated with the largest eigenvalue of the graph’s adjacency matrix [1]. Based on this, existing studies [2–5] attempted to manipulate the adjacency matrix’s eigenvalues by removing nodes to minimize the likelihood of epidemic outbreaks. Meanwhile, some studies [6–8] investigated macro-level approaches to control epidemic spread, such as restricting population movement or implementing proportional quarantine. Besides, studies [9–12] examined the impact of isolation and immunization on disease spread.

The studies on epidemic control investigated how to control the spread by isolating individuals, restricting population movement, etc. However, these studies all assumed that individuals would comply with the control policies, which is usually not the actual situation. In reality, individuals have their own ideas and may not necessarily obey policies such as isolation and movement restrictions.

1.1.2 Individual Behavior Modeling during an Epidemic

Some prior works attempted to model individual behaviors during an epidemic. The studies [13–17] showed that as the proportion of infected individuals in the environment increases, people are more likely to adopt protective behaviors. Zhang et al. assumed that individuals would be more likely to take protective behavior when there is a large proportion of infected neighbors [14], and they analyzed the effect of individual protective behavior on epidemic spread. The model proposed in [15] assumed that an individual’s adoption of protective behavior is affected by the proportion of infected neighbors, as well as regional and global infection rates. The studies [18–20] observed that information dissemination also affects individual protective behavior during the epidemic.

These studies modeled and analyzed individuals’ behavioral choices in the epidemic. However, they did not consider the common and critical issue of individual irrationality, which can significantly affect their decision-making process during an epidemic.

1.1.3 Irrational Behavior Modeling

Individuals usually resort to irrational decision-making when confronted with risks, such as the potential for infection during an epidemic. For instance, many people may be overly panicky in the early stages of an epidemic and may underestimate the risk of the epidemic in the later stages. A challenge here is how to mathematically model such irrational behaviors. Prospect theory provides theoretical models to quantify how individuals tend to overestimate small probabilities and underestimate high probabilities [21–25]. It is crucial to analyze the impact of this irrationality on individuals’ decisions. The study [26] considered risk aversion during the consensus reaching process (CRP) in group decision-making. The studies [27–29] analyzed the impact of irrationality on individuals’ decisions on whether to take vaccination during an epidemic, and they assumed that this is a one-time binary decision problem where users only make one binary decision during the entire epidemic. However, in reality, individuals can take multiple protective behaviors, such as wearing a mask, washing hands, and isolating at home, and they need to continuously decide whether to take such protective behaviors and which behavior to take during the entire epidemic. Individuals may choose to adopt the highest-level protective measures such as self-isolation when the epidemic is rapidly spreading, and choose to not take any protective behaviors when the epidemic is declining.

The existing studies modeled individuals’ irrationality and simple decisions such as whether to take a vaccination. However, they did not consider the continuous interaction and mutual influence between the epidemic spread and individual choices over time in the context of irrational behaviors.

1.2 Our Contribution

Our work differs from prior works in two aspects. First, prior works on individual behavior modeling in epidemics either ignore the impact of irrational decision-making or fail to consider the continuous interaction and mutual influence between the epidemic spread and individual behaviors. In this paper, the prospect theory is applied to model individuals’ irrational decisions during an epidemic and the co-evolution of individuals’ behaviors and the epidemic. We theoretically analyze the impact of the individuals’ irrationality on their decisions as well as the epidemic. Second, existing works on epidemic control assume individuals’ absolute compliance with the government’s policies such as mandatory isolation. In this paper, based on the individual behavioral model, we propose an effective method to guide individuals’ behaviors and control the epidemic spread. In contrast to prior approaches, our disease spread control method does not rely on mandatory measures.

The main contributions of our work are:

•   We build an M-choice epidemic-behavior co-evolution model to simulate individuals’ irrational decision-making and analyze their impact on the epidemic. We theoretically analyze the co-evolution of user behavior and epidemic and its steady state. Also, the impact of irrationality on individuals’ behaviors and disease spread is theoretically analyzed.

•   Given the above individual behavior model, we propose a behavior inducement algorithm to guide individuals’ decisions to control the epidemic.

•   We validate our individual behavior model and behavior inducement algorithm through simulations. In addition, we use real user tests to validate the conclusions about the impact of irrationality on individuals’ behavior.

The rest of the paper is organized as follows. Section 2 introduces our proposed epidemic-behavior co-evolution model. Section 3 analyzes the steady state of the epidemic-behavior dynamics and the influence of irrationality. Section 4 presents the behavior inducement method to control the disease spread. Section 5 provides the simulation results. Section 6 shows the results of real user tests, and conclusions are given in Section 8. The important notations of this paper are listed in Table 1.

2  The M-Choice Epidemic-Behavior Co-Evolution Model

During an epidemic, individual behavioral choices and the spread of the disease mutually influence each other. When the probability of infection and the potential losses are high, people tend to adopt protective behaviors, which in turn can inhibit the epidemic spread. In this section, we propose a model to capture the co-evolution of individual behavioral choices and disease spread during a pandemic. We consider irrational behavior in our model and use the model with rational behavior assumption as a baseline to analyze the impact of irrationality on the evolutionary dynamics of the epidemic and its steady states. Based on the model presented in [30], we assume that individuals have a choice among M possible behaviors based on the severity of the epidemic, and these behavioral choices, in turn, affect the epidemic spread.

Following the work in [19], two undirected networks are used in this paper to represent the connections among individuals. The first network is the physical contact network where the disease spreads. The second network is the information network where individuals exchange information about their current health state and behavioral choices. It is worth noting that the two networks are different. In reality, an individual may get infected by strangers in a restaurant or on a bus, while their behaviors will not be influenced by these strangers since they have not interacted with them. Similarly, individuals’ decisions may be influenced by their friends on the Internet without any physical contact with each other. To simplify the analysis, we make the assumption that both the physical contact network and the information network are regular networks consisting of N nodes, where each node represents an individual. In a regular network, each node has a fixed degree, denoted as k¯ for the physical contact network and d¯ for the information network. As individuals communicate with each other in the information network, we assume that they have knowledge of the health states of their neighbors in the information network. However, in the physical contact network, there is information exchange, and we assume that individuals do not have knowledge of the health states of their neighbors. In the following, we will use the terms “graph” and “network” interchangeably, and the terms “node”, “user”, and “individual” interchangeably.

Our model consists of two interconnected parts: the disease spread model and the behavior change model. The disease spread model quantifies how the epidemic spreads through the network given the current behaviors of all individuals, and the behavior change model describes how individuals update their behaviors based on the current number of infected individuals and the behaviors of their neighbors. The illustration of our model is in Fig. 1. In the following, the two components of our model will be introduced in detail.

Figure 1: The proposed M-choice epidemic-behavior co-evolution model

2.1 The Disease Spread Model

We use the classic Susceptible-Infected-Susceptible (SIS) model to depict disease spread, which describes the disease spread process using differential equations. In addition to depicting disease spread, this method of differential equations has a wide range of applications, such as describing the spread of viruses in the body [31,32] and the evolution of opinions [33,34]. In the SIS model, each individual can be in one of two health states: susceptible or infected. We divide time into slots of equal length. At each time slot, a susceptible individual can be infected by an infected individual at a certain infection rate, and an infected individual recovers at a certain recovery rate. We assume that susceptible individuals can adopt different protective measures to reduce their risk of infection, and they have a total of M possible behavioral options {a1,a2,...,aM}. For example, some may take the epidemic very seriously and adopt self-quarantine to avoid contact with infected individuals; some may adopt medium-level protective behaviors such as wearing masks when going out and washing hands frequently; while others may take no protective behaviors and act as if the epidemic does not exist. To simplify the analysis, we also assume that all susceptible individuals taking action aj have the same infection rate βj. For those already infected with the disease, we consider the worst-case scenario where they do not take any protective behavior such as home isolation to prevent the disease from further spreading. This assumption is grounded in the understanding that infected individuals might not possess the same level of motivation or necessity to embrace protective measures since they are already infected. We also assume that all infected people have the same recovery rate γ to simplify the analysis. Let s(t) and i(t) be the fractions of susceptible and infected individuals respectively at time t, while xj(t) denotes the fraction of individuals adopting action aj among all susceptible individuals at time t. Then the mean-field equation of the disease spread is:

dsdt=γi(t)−s(t)i(t)β¯k¯,anddidt=s(t)i(t)β¯k¯−γi(t),(1)

where s(t)+i(t)=1 and β¯=∑j=1nβjxj(t). By substituting s(t)=1−i(t) into (1), the differential equation modeling the disease spread can be obtained:

di(t)dt=i(t)(1−i(t))β¯k¯−γi(t).(2)

2.2 The Behavior Change Model

The individual behavior model quantifies the dynamics of {xj(t)},j=1...M, which represents the proportion of susceptible individuals adopting action aj at time t. The change in xj(t) can be attributed to two main factors. First, individuals may change their decisions over time in response to the severity of the epidemic and the influence of their neighbors’ behaviors. We use Dj(t) to represent this part of the change in xj(t). Meanwhile, due to nodes’ changes in their health state, the proportion of susceptible individuals adopting different behaviors may also change. For example, a susceptible individual who was taking action aj at time t−1 may become infected at time t, or an infected person recovers at time t and decides to take action aj. We use Bj(t) to represent this part of the change in xj(t). Then we have:

dxj(t)dt=Dj(t)+Bj(t).(3)

In Section 2.2.1, we focus on modeling Dj(t), where individuals’ decisions are influenced by their neighbors and the severity of the epidemic. In Section 2.2.2, we study Bj(t) and analyze how the changes in individuals’ health states may affect the change in xj(t).

2.2.1 Analysis of Dj(t)

Game theory models strategic interactions among agents, providing insights into decision-making across diverse scenarios, from social networks [35] to supply chains [36] and beyond. To model individuals’ active behavior changes in response to their neighbors’ impact and the proportion of infected individuals, we employ the evolutionary game theory, which is a useful framework to investigate the impact of neighbors on individuals’ decisions [37,38]. The basic elements of the evolutionary game theory include individual, strategy, payoff, and strategy update rules. These elements will be introduced one by one in the following.

Individual and Strategy: Each individual is represented as a node in the information network. As mentioned in Section 2.1, we assume that there are a total of M possible protective behaviors {a1,⋯,aM} for susceptible individuals, and each behavior aj corresponds to one strategy for a susceptible individual. For infected individuals, as mentioned in Section 2.1, they do not take any protective behavior such as home isolation to prevent the disease from spreading. Therefore, in this work, we focus on the analysis of susceptible individuals’ behavior and study how their decisions are affected by their neighbors and the severity of the epidemic. In each time slot, m percent of susceptible individuals are randomly chosen as focal individuals. These focal individuals observe and imitate their neighbors’ behaviors. The remaining susceptible individuals maintain their actions unchanged during this time slot.

The Payoff: In this paper, we study the protective behavior of susceptible individuals. Note that infected individuals have different health states from susceptible ones, and they adopt the same and fixed strategy of no protective behaviors. Therefore, in this work, we assume that susceptible individuals do not value these infected individuals’ decisions, and the strategies of all susceptible individuals are only affected by their susceptible neighbors. So we define the payoff for susceptible individuals only, and ignore the payoffs of infected individuals, as they do not affect susceptible individuals’ update of their strategies.

In each time slot, every susceptible individual receives a payoff determined by the chosen strategy and interactions with neighbors. In this paper, we consider two scenarios where the susceptible individual is rational and irrational. Therefore, we define the payoff of different behaviors based on the expected utility theory (EUT) [39] and prospect theory (PT) [23], respectively, where EUT models the individual as a rational person and PT considers the individual’s irrationality. Here, following the prior work in [29], to simplify the analysis, we assume that either all individuals are rational or they are all irrational, and compare the two results to analyze the impact of users’ irrationality on the co-evolution of individuals’ behaviors and the epidemic.

Rational individuals’ payoff function: Expected utility theory (EUT) is an economic theory that models the decision-making of rational individuals. When an individual chooses a specific behavior, denoted as aj, it leads to L potential actual payoffs oj,1,oj,2,...,oj,L with probabilities pj,1,pj,2,...,pj,L, respectively. It is worth noting that an individual may receive more than one actual payoff for their behavior, and ∑k=1...Lpj,k≠1. For example, if an individual decides to go out for dinner during an epidemic, they will receive a positive payoff from enjoying the fine cuisine, while they may also face a negative payoff if they become infected. In addition, according to EUT, the perceived payoff may differ from the actual payoff. For example, the relationship between the perceived and actual payoffs is often not linear, and there is a phenomenon of diminishing marginal payoff [21]. In prior works in EUT, the value function uE(x) is used to model the relationship between the actual payoff x and the perceived payoff uE(x). Various forms of uE(x) have been used in the previous works, including the simplest form of uE(x)=x, as well as the power function and the exponential function form [40]. Thus, in EUT, the payoff associated with adopting behavior aj is calculated as the expected utility, which is denoted as UjEUT and is defined as:

UjEUT=∑k=1...LuE(oj,k)pj,k.(4)

In our behavior modeling and epidemic control problem, every susceptible individual adopting protective behavior aj will obtain a fixed actual payoff of cj, which is the payoff from the behavior itself, and the probability of obtaining this outcome is 1. One example is the gain from enjoying the fine cuisine of dining outside during an epidemic. If the individual is infected at the next moment, it will suffer from a loss of cn with cn<0. For an individual adopting aj, the probability of being infected at time t is approximately k¯βji(t) [41]1. Therefore, in our model, the individual who takes aj will obtain two potential actual payoffs. A payoff of cj with probability 1, and a payoff of cn with probability k¯βji(t). So the expected utility in (4) becomes

UjEUT=uE(cj)+uE(cn)k¯βji(t).(5)

Irrational individuals’ payoff function: Different from the expected utility theory (EUT), the prospect theory (PT) considers the irrational tendencies exhibited by individuals when faced with uncertainty. In PT, individuals tend to overestimate the probability of small risks and underestimate the probability of large risks [21]. Therefore, not only are the actual and the perceived payoffs different, but the actual and the perceived probabilities are also different when irrational individuals face uncertainties.

Similar to EUT, the value function uP(x) in PT can take different forms, and one commonly used form is the power function [25]:

uP(x)={xσ,if x≥0,−λ(−x)σ,if x<0,(6)

where λ reflects the individual’s sensitivity to gain and loss, and σ∈(0,1] reflects the curvature and shape of the value function. It is worth noting that the theoretical analysis in Sections 3 and 4 do not rely on the specific form of uP(x).

In addition, instead of using the actual probability pj, individuals’ perceived probability is ω(pj,α), where ω(p,α) is the probability weighting function. Following the prior work in [22], in this work, we use the following weighting function to describe the relationship between the perceived probability ω(p,α) and the actual probability p:

ω(p,α)=e(−(−lnp)α),  p∈[0,1], α∈(0,1],(7)

where α is the irrationality coefficient. A smaller α indicates that the individual is more irrational (or equivalently, less rational), and the difference between the actual and the perceived probabilities is larger. Note that when α=1, we have ω(p,1)=p, and the perceived and actual probabilities are the same. Meanwhile, from (7), we have ω(1,α)=1 define ω(0,α)=0. This ensures that ω(p,α)∈[0,1] and ω(p,α) is an increasing function of p. For simplicity, we use the mean-field method and assume all individuals have the same α.

Given the probability weighting function in (7) and the value function uP(x), when an irrational individual chooses behavior aj, which leads to L different potential payoffs {oj,k} with corresponding probabilities {pj,k}, the expected payoff is

UjPT=∑k=1...LuP(oj,k)ω(pj,k,α).(8)

In our problem, same as the analysis of UjEUT in the above, the individual who chooses behavior aj will obtain two potential actual payoffs: a payoff of cj with probability 1, and a payoff of cn with probability k¯βji(t). So (8) can be expressed as

UjPT=uP(cn)⋅ω[k¯βji(t),α]+uP(cj).(9)

Note that if the value functions of EUT and PT are identical (i.e., uE(x)=uP(x)) and the irrationality coefficient α is set to 1, then PT degenerates to EUT.

Strategy Update Rules: In each time unit, mN0(t) individuals are randomly selected as the focal individuals to update their strategies and others will keep their strategies unchanged, where m is the fraction of individuals who are chosen as the focal individuals, and N0(t) is the total number of susceptible individuals at time t in the network. The focal individuals tend to imitate their neighbors’ behavior with a high payoff. Following the work in [45], given a focal individual v with strategy aj and given v randomly chooses a neighbor z using strategy ak, the probability the individual v changes its strategy to ak is

p(aj→ak)=12+w21Umax(Uk−Uj),(10)

where w∈(0,1] measures the strength of selection, and Umax is the normalization term to ensure p(aj→ak)≤1. Uj and Uk are the payoffs of strategy aj and ak, respectively.

In our work, we assume that individuals with different behaviors are uniformly distributed in the entire network. Therefore, the probability that the focal individual v chooses behavior aj is xj(t), which represents the proportion of susceptible individuals who choose behavior aj at time t in the entire network. Meanwhile, the proportion of focal individual v’s susceptible neighbors choosing behavior aj is the same as the proportion of susceptible individuals choosing behavior aj in the entire network. Then the probability that xj(t) increases by 1N0(t) due to individuals’ strategy change is

p(Δxj=1N0(t))=∑l=1Mxj(t)xl(t)p(al→aj),(11)

where N0(t)=N0s(t) is the number of susceptible individuals at time t. Similarly, the probability that xj(t) decreases by 1N0(t) due to individuals’ strategy change is

p(Δxj=−1N0(t))=∑l=1Mxj(t)xl(t)p(aj→al).(12)

Combining (10) and (12), we have

Dj(t)=mN0(t){p(Δxj=1N0(t))×1N0(t)−p(Δxj=−1N0(t))×1N0(t)}=mwUmax∑l=1Mxj(t)xl(t)(Uj−Ul).(13)

2.2.2 Analysis of Bj(t)

In reality, even if individuals do not change their behaviors, the proportion of susceptible individuals with different behaviors will change over time due to transitions in health states. Let sj(t) be the fraction of individuals who are susceptible and adopt behavior aj at time t among the entire population, that is, sj(t)=s(t)xj(t), where s(t) is the fraction of susceptible individuals among all users in the network, and xj(t) is the fraction of individuals adopting aj among all the susceptible individuals. Note that s(t)=∑j=1Msj(t), when all individuals do not change their behavior, we have

Bj(t)=ddt(Sj(t)S(t))=ddt(sj(t)∑l=1MSl(t))=∑l=1M[Sj'(t)Sl(t)−Sl'(t)Sj(t)]S2(t).(14)

Note that Sji(t), the first order derivative of sj(t), contains two parts. The first part represents the change caused by the infection of susceptible individuals, while the second part represents the change caused by the recovery of infected individuals. For the first part, as there are a total of sj(t) susceptible individuals adopting behavior aj, and each of them has probability βjk¯i(t) to be infected, we have sj1′(t)=−sj(t)βjk¯i(t). For the second part, we assume that the recovered individuals would choose their behaviors based on the ratio of different behaviors of susceptible individuals in the network, similar to the work in [46]. Therefore, γi(t) infected individuals will recover, xj(t) of whom will choose action aj, and we have Sj2'(t)=γxj(t)i(t). By combining these two parts (sj′(t)=sj1′(t)+sj2′(t)), we have

sj′(t)=−s(t)xj(t)βjk¯i(t)+γxj(t)i(t).(15)

Given (14), (15) and sj(t)=s(t)xj(t), we have

Bj(t)=∑l=1Mxj(t)xl(t)k¯i(t)(βl−βj).(16)

2.2.3 The Overall Behavior Change Dynamics

Combining (3), (13) and (16), the complete differential equations describing the dynamics of individual behavior change are

dxj(t)dt=∑l=1Mxj(t)xl(t)k¯i(t)(βl−βj)+mwUmax∑l=1Mxj(t)xl(t)(Uj−Ul), j=1...M.(17)

2.3 The Dynamics and the Steady States of the Epidemic-Behavior Co-Evolution Model

Based on the disease spread Eq. (2) and the behavior change dynamics (17), the M-choice epidemic-behavior co-evolution model is obtained as follows:

didt=i(t)(1−i(t))β¯k¯−γi(t),anddxjdt=∑l=1Mxj(t)xl(t)k¯i(t)(βl−βj)+mwUmax∑l=1Mxj(t)xl(t)(Uj−Ul), j=1...M.(18)

Here, the first differential equation represents the dynamics of individuals’ health states, and the subsequent M equations model the changes in the proportions of susceptible individuals choosing each of the M behaviors.

At the steady state, both the proportion of infected individuals i(t) and the proportions of individuals adopting different behaviors {xj(t)} reach a stable state where there are no further changes in i(t) and {xj(t)}. Even if a small group of individuals becomes infected/recovered or changes their strategies, the steady state would be restored. We denote the steady state of the M-choice model as (i∗,x1∗,⋯,xM−1∗) with xM∗=1−∑j=1M−1xj∗.

To find the steady state of our M-choice disease spread and behavior change model, we follow an approach that is similar to [46] and apply Lyapunov’s first method [47].

Definition 1. The steady-state (i∗,x1∗,⋯,xM−1∗) satisfies: for j=1,⋯,M−1,

didt|i=i∗=0, dxjdt|xj=xj∗=0, Re(λk)<0, k=1,...,M,(19)

where {λk} are the eigenvalues of the Jacobian matrix

[∂i′∂i∂i′∂x1⋯∂i′∂xM−1∂x1′∂i∂x1′∂x1⋯∂x1′∂xM−1⋮⋮⋱⋮∂xM−2′∂i∂xM−2′∂x1⋯∂xM−2′∂xM−1∂xM−1′∂i∂xM−1′∂x1⋯∂xM−1′∂xM−1]|(i*,x1*,⋯,xM−1*),(20)

and Re(x) represents the real part of x. As it is usually difficult to find the closed-form solution of (19), we often use numerical methods to find the steady states of the co-evolution of disease spread and behavioral choice.

3  Analysis of the Steady State and the Influence of Irrationality for the 2-Behavior Model

To obtain insights into the co-evolution process of epidemic spread and behavioral choice and their steady states, in this section, we consider a simple scenario where each susceptible individual can take two possible behaviors and the theoretical solution of (19) can be obtained. For example, a susceptible individual may either choose risky behavior such as continuing to go out in spite of the epidemic, or choose conservative behavior such as home isolation. Home isolation can significantly reduce the risk of being infected, but it leads to substantial economic loss and affects people’s physical and mental well-being. During the epidemic, individuals need to choose between high-cost low-risk conservative behavior and low-cost high-risk risky behavior. Their decisions are often influenced by the severity of the epidemic and the potential loss due to home isolation. People tend to choose self-isolation when the epidemic poses a greater threat to their health, and they tend to go out when the loss due to home isolation is too high (e.g., losing their jobs and income). By investigating this simple scenario, this paper aims to gain insights into the co-evolution process and its steady states, and theoretically analyze the impact of irrationality on the epidemic as well as individuals’ behaviors. For the more general scenario with more than two possible behavior choices, we use numerical solutions and simulation results to show the evolution process, and an example with three possible behavior choices is provided in Appendix A in Supplementary Material.

In this section, based on our model in the previous section, we analyze the evolution of the epidemic and the dynamics of individuals’ choices between two behaviors: the risky behavior (going out) represented by a1, and the conservative behavior (home isolation) represented by a2. As an example, we assume that the infection rate for the risky behavior is β1, while the infection rate for the conservative behavior is β2=0 as isolation ensures no infection risk. Then we analyze the steady states when the individuals are all rational or irrational, respectively, and compare their results to investigate the influence of irrationality.

3.1 Steady State Analysis

3.1.1 The Steady States with All Rational Individuals

When all individuals are rational, the payoff is modeled by EUT in (5). Since x1(t)+x2(t)=1, x2(t) can replaced by 1−x1(t). With two possible behavior choices and given the EUT payoff function (5), the differential equations in (18) can be expressed as:

{didt  =β1k¯i(t)x1(t)(1−i(t))−γi(t),dx1dt=−β1k¯x1(t)(1−x1(t))i(t)+k0x1(t)(1−x1(t))⋅(uE(c1)+uE(cn)β1k¯i(t)−uE(c2)),(21)

where k0=mwUmax>0. To find the steady states of (21), we have the following Theorem 1.

Theorem 1. The steady state (iE,x1E) of (21) satisfies (22),

(iE,x1E)={(0,1),if k¯<γβ1,(1−γk¯β1,1),if k¯>γβ1,Φ1<0,(i(1),γ(1−i(1))k¯β1),if k¯>γβ1,Φ1≥0,(22)

where i(1)=k0(uE(c2)−uE(c1))(k0uE(cn)−1)k¯β1, Φ1= −k0(uE(c1)−uE(c2))+(γ−k¯β1)(k0uE(cn)−1).

When k¯=γβ1, there is no steady state.

Proof: See Appendix B in Supplementary Material.

From Theorem 1, there are three possible steady states, which correspond to three different situations in reality:

•   Case 1: The steady state (0,1) represents the extreme situation where the infection rate is too low and the epidemic will die out eventually even without any protection. So all individuals choose the risky behavior of going out. The evolution process reaches the steady state (0,1) when k¯<γβ1, which is equivalent to β1γ<1k¯. The term 1k¯ is the epidemic threshold of a homogeneous network [48]. If β¯γ<1k¯, the epidemic will die out; otherwise, it will spread out. Since β¯=β1x1(t)∈[0,β1], in this scenario, the epidemic will die out no matter which behavior individuals choose. Therefore, all individuals will choose the risky behavior in this steady state since it gives a higher payoff. Therefore, when k¯<γβ1, the stable state is i=0 and x1=1.

•   Case 2: For the steady state (1−γk¯β1,1), two constraints need to be satisfied. The first constraint is k¯>γβ1, which means that if all individuals choose the risky behavior, the epidemic will spread out. The second constraint is Φ1<0. To understand the second constraint, note that when all individuals choose the risky behavior with x1=1, the proportion of infected individuals will reach the stable state i^=1−γk¯β1, which represents the maximum extent to which the epidemic can spread (proof: see Appendix C in Supplementary Material). If for all possible values of i in the range [0,i^], we have dx1(t)dt>0 for all x1(t)∈(0,1),2 then more individuals will choose the risky behavior over time, and ultimately all individuals will choose risky behavior at the steady state with x1=1. This may happen when the payoff of risky behavior is much higher than that of the conservative behavior, with c1≫c2, or when the cost of being infected cn is very low. Note that from (21), dx1dt|0≤i≤i^,0<x1<1>0 is equivalent to Φ1<0, where Φ1 is defined in (22). Therefore, when k¯>γβ1 and Φ1<0, the steady state (1−γk¯β1,1) is reached, where all individuals choose the risk behavior, and the proportion of infected people reaches the maximum level i^.

•   Case 3: The steady state (i(1),γ(1−i(1))k¯β1) represents the scenario other than the above two extreme cases. In this scenario, the epidemic will not die out, nor will it spread to the maximum extent, and at the steady state, 0<i(1)<i^ of the individuals in the network will be infected. Meanwhile, 0<γ(1−i(1))k¯β1<1 of susceptible individuals will choose the risky behavior. This happens when k¯>γβ1 and Φ1≥0.

3.1.2 The Steady States with Irrational Individuals

Next, we consider the scenario where all individuals are “irrational”, and model their payoff function using the PT. By substituting the PT utility function (9) into (17), the dynamic of the epidemic and the behavior can be expressed as:

{didt=β1k¯i(t)x1(t)(1−i(t))−γi(t),dx1dt=−β1k¯x1(t)(1−x1(t))i(t)+k0x1(t)(1−x1(t))⋅(uP(c1)+uP(cn)⋅ω[β1k¯i(t),α]−uP(c2)).(23)

Similarly, we can get the steady state of (23) in Theorem 2.

Theorem 2. The steady state (iP,x1P) of (23) satisfies:

(iP,x1P)={(0,1),if k¯<γβ1,(1−γk¯β1,1),if k¯>γβ1,Φ2<0,(i(2),γ(1−i(2))k¯β1),if k¯>γβ1,Φ2≥0,(24)

whereΦ2= −k0(uP(c1)−uP(c2))−(γ−k¯β1)−k0uP(cn)⋅ω[k¯β1−γ,α],andi(2) satisfiesk0uP(cn)⋅ω[k¯β1i(2),α]−k¯β1i(2)+k0(uP(c1)−uP(c2))=0.(25)

When k¯=γβ1, there is no steady state.

Proof: See Appendix D in Supplementary Material.

The three stable states in PT are similar to those in EUT:

•   Case 1: The steady state (0,1) is the same as Case 1 under EUT. In this situation, the epidemic will always die out, and all individuals will choose the risky behavior.

•   Case 2: The steady state (1−γk¯β1,1) is the same as Case 2 under EUT. In situations where the payoff for the risky behavior is extremely high or when the loss of being infected is very small, all individuals will choose risky behavior, causing the epidemic to spread to its maximum extent. Note that the first constraints in (22) and (24) are the same, while the second constraints are different as Φ1≠Φ2.

•   Case 3: The steady state (i(2),γ(1−i(2))k¯β1) is similar to Case 3 under EUT, where the epidemic does not extinct, nor does it spread to the maximum range.

3.2 Analysis of Individuals’ Irrationality

In this section, we analyze the influence of individuals’ irrationality on the steady state. In the weighting function in (7), the irrationality coefficient quantifies the irrationality degree of individuals, and a smaller value of α indicates that individuals are more irrational as the difference between the actual and perceived risk becomes larger. To analyze the impact of the irrationality coefficient on individuals’ behavior, this paper compares the steady states (iP,x1P) at different α, and we have the following Theorem 3.

Theorem 3. Given the same set of system parameters (β1, γ, k0, c1, c2, cn and k¯) and the same value function uP(x), let 0<α_<α¯<1 be two irrationality coefficients, and (i¯P,x¯1P) and (i_P,x_1P) are the steady states with α¯ and α_, respectively, where the individuals with α¯ have low irrationality and individuals with α_ have high irrationality. Then we have:

3a.   When k¯<γβ1, all individuals, regardless of their irrationality degrees, will choose risky behavior with x¯1P=x_1P=1, and the epidemic will eventually die out with i¯P=i_P=0.

3b.   When k¯>γβ1, Φ2≥0 is not simultaneously satisfied for α¯ and α_, we have the following conclusions.

If 1−γk¯β1≤1k¯β1e, there are two possibilities.

–    When Φ2<0 for both α¯ and α_, all individuals, regardless of their irrationality degrees, will choose risky behavior with x¯1P=x_1P=1.

–   When Φ2<0 for α¯ and Φ2≥0 for α_, all individuals with low irrationality will choose the risky behavior with x¯1P=1. Meanwhile, only a subset of individuals with high irrationality will choose risky behavior with x_1P<1.

On the contrary, if 1−γk¯β1≥1k¯β1e, there are two possibilities.

–   When Φ2<0 for both α¯ and α_, all individuals, regardless of their irrationality degree, will choose risky behavior with x¯1P=x_1P=1.

–   When Φ2<0 for α_ and Φ2≥0 for α¯, all individuals with high irrationality will choose risky behavior with x_1P=1. Meanwhile, only a subset of individuals with low irrationality will choose risky behavior with x¯1P<1.

3c.   When k¯>γβ1, Φ2≥0 for both α¯ and α_, the epidemic will neither die out nor spread to the maximum extent, and a fraction of individuals will choose the risky behavior at the steady state.

–   In addition, if i¯P≤1k¯β1e, we have i¯P≥i_P,x¯1P≥x_1P, i.e., compared to individuals with high irrationality, fewer individuals with low irrationality would choose the conservative behavior.

–   On the contrary, if i¯P≥1k¯β1e, we have i¯P≤i_P,x¯1P≤x_1P, i.e., compared to individuals with high irrationality, more individuals with low irrationality would choose the conservative behavior.

Proof: See Appendix E in Supplementary Material.

To better understand Theorem 3, note that in (24), Case 1 represents the situation where the infection rate is too low and the epidemic will eventually die out no matter how individuals choose their behaviors. Therefore, individuals’ irrationality will not affect the outcome when k¯<γβ1, as stated in Theorem 3a.

For Theorem 3b, when k¯>γβ1 and Φ2≥0 are not simultaneously satisfied for α¯ and α_, if 1−γk¯β1≤1k¯β1e, the percentage of individuals with high irrationality choosing the risky behavior will be less than or equal to the percentage of individuals with low irrationality, i.e., x_1P≤x¯1P. This is because, in this scenario, the risk of being infected is low (i.e., i¯P=i_P=1−γk¯β1≤1k¯β1e), and higher irrationality makes individuals overestimate this small probability of risk, causing them to be more conservative. On the contrary, if 1−γk¯β1≥1k¯β1e, the percentage of individuals with high irrationality choosing the risky behavior will be larger than or equal to the percentage of individuals with low irrationality, i.e., x_1P≥x¯1P. This is because, in this scenario, the risk of being infected is high (i.e., i¯P=i_P=1−γk¯β1≥1k¯β1e), and higher irrationality makes individuals underestimate this large probability of risk, causing them to be more adventurous.

For Theorem 3c, when k¯>γβ1 and Φ2≥0 for both α¯ and α_, both (i¯P,x¯1P) and (i_P,x_1P) belong to Case 3 in Theorem 2, where some individuals get infected while the rest do not. In this case, if the risk of getting infected is low when stable (i.e., i¯P≤1k¯β1e), higher irrationality can motivate individuals to adopt conservative behaviors as they tend to overestimate this small risk, resulting in a decrease in the probability of being infected. On the contrary, if the risk of getting infected is high when stable (i.e., i¯P≥1k¯β1e), higher irrationality can reduce individuals’ cautiousness as they tend to underestimate this large risk, causing more people to be infected.

Note that from Section 2.2, if the value functions of EUT and PT are identical (i.e., uE(x)=uP(x)), then EUT can be considered as a special case of PT with α=1. Therefore, we can also apply Theorem 3 to make comparisons between rational individuals (following EUT) and irrational individuals (following PT).

In summary, irrationality tends to make individuals become more extreme, that is, risk-averse when the risk is small and risk-seeking when the risk is high.

4  Behavior Inducement to Control the Disease Spread

In this section, based on our previous analysis in Sections 2 and 3, we study how to guide individuals’ behaviors and control epidemic spread through policy design and develop effective behavior inducement algorithms.

4.1 The Optimal Behavior Inducement Algorithms

We first discuss measures that governments can take to guide individuals’ behaviors during an epidemic. For example, they can incentivize or penalize certain behaviors, such as subsidizing risky behaviors (e.g., going out) to boost the economy, penalizing risky behaviors, or encouraging conservative behaviors (such as staying at home and wearing masks) to control the disease spread. In our model, this means the parameters c1 and c2 can be changed. In addition, during an epidemic, individuals usually have different perceptions of the loss of being infected, which are largely due to the various propaganda efforts. So we assume that the parameter cn can also be adjusted. Moreover, note that propaganda via social networks and media often affects individuals’ irrationality [49], and thus, we assume that the irrationality coefficient α can be changed as well. In this work, to simplify the analysis, we consider the simple scenario where these parameters c1, c2, cn, and α can be changed to the desired values. Our future work will consider a more practical scenario where the optimization parameters are the actions the government can take (such as rewarding or punishing specific behaviors through policies) instead of the exact values of these parameters.

Next, we discuss the goals of behavior guidance. The first goal is to control the spread of the disease at the steady state. For example, the government may wish to keep the number of infected people as low as possible. Here, the loss caused by the epidemic is represented as l1(iP), where (iP,x1P) is the steady state of PT. Also, if a large percentage of individuals choose conservative behavior such as self-isolation, it will significantly affect the economy and individuals’ mental health. Therefore, the second loss term considered in our paper is the loss due to such conservative behavior l2(x1P). Furthermore, note that changing the values of c1, c2, cn, and α through behavior inducements such as propaganda, subsidies, and penalties will incur costs. Therefore, the third goal is to minimize the cost associated with behavior guidance l3(δδ), where δδ=[Δα,Δcn,Δc1,Δc2] is the intervention vector quantifying the extent to which these variables are changed. Given c1, c2, cn, and α before behavior guidance, the adjusted parameters are

α′=α+Δα, cn′=cn+Δcn, c1′=c1+Δc1, and c2′=c2+Δc2.(26)

Since the irrationality coefficient should fall within (0,1] and the payoff of being infected should be negative, we have 0<α+Δα≤1 and cn+Δcn<0. The goal is to find the optimal δδ to minimize the total loss. The optimization problem is:

minδδ l1(iP(δδ))+l2(x1P(δδ))+l3(δδ)s.t.0<α+Δα≤1,cn+Δcn<0.(27)

In this work, we do not specify the specific forms of l1(iP), l2(x1P), and l3(δδ), and they are assumed to be differentiable, i.e., ∂l1∂i, ∂l2∂x1, and ∂l3∂δδ exist. Moreover, we assume that l3(δδ) satisfies

{∂l3∂Δα>0if  Δα>0,∂l3∂Δα<0if  Δα<0,(28)

and the same constraint also holds for ∂l3∂Δcn, ∂l3∂Δc1 and ∂l3∂Δc2. This implies that compared to the case where no behavior guidance is taken with δδ=0, increasing or decreasing any of the variables (c1, c2, cn, and α) will result in an increase in l3(δδ), and l3(δδ) has the minimum value at 00 with no behavior guidance.

From (24), there are three possible steady states. To solve the optimization problem (27), we need to consider all three possible steady states and analyze the optimal solution for each, which is very complicated. Then we introduce Theorem 4 to simplify this problem.

Given the system parameters and δδ, let (i0,x10) and (iP,x1P) be the steady states without and with behavior inducement, respectively. Let δδ(3) be the optimal adjustment parameter when the steady state after adjustment (iP,x1P) belongs to Case 3 in Theorem 2. That means

δδ(3)=argminδδ l1(iP(δδ))+l2(x1P(δδ))+l3(δδ)s.t.0<α+Δα≤1,cn+Δcn<0,k¯>γβ1,Φ2≥0,(29)

where Φ2 is defined in (24).

From our analysis in Appendix F in Supplementary Material, we have the following Theorem 4.

Theorem 4. For the optimization problem (27), if l3(δδ) satisfies (28), given δδ(3) defined in (29), let (i0,x10) and (iP,x1P) be the steady states without and with behavior guidance, respectively. Then the optimal solution of (27) is either 00 or δδ(3). Specifically,

•   if i0=0 and x10=1, i.e., the original steady state without behavior guidance belongs to Case 1 in (24), the optimal solution is 00. That means no behavior guidance is needed, and the objective function in (27) is minimized at 00.

•   If i0=1−γk¯β1 and x10=1, i.e., the original steady state belongs to Case 2 in (24), the optimal solution is either 00 or δδ(3), and the optimal solution can be obtained by comparing l1(i0)+l2(x10) with l1(iP)+l2(x1P)+l3(δδ(3)).

•   If 0<i0<1−γk¯β1 and 0<x10<1, i.e., the original steady state belongs to Case 3, the optimal solution is δδ(3).

From Theorem 4, we only need to model and solve the problem for Case 3 in (24), which greatly reduces the complexity of our problem.

4.2 Solving the Optimization Problems

In this section, we consider the following scenario and use it as an example to demonstrate how to model and solve the optimization problem in (27). Consider the scenario where the government wants to control epidemic spread and reduce its impact on the economy and people’s mental health. It means that, at the steady state after behavior inducement (iP,x1P), the percentage of infected people iP is no more than im, and the ratio of people choosing the risky behavior x1P is at least xm. That is, there are two constraints 0≤iP≤im and 1≥x1P≥xm. Here, we assume that 0≤im≤1 and 0≤xm≤1.

Since the infection rate β1, the recovery rate γ, and the average degree of the networks k¯ are assumed to be fixed and cannot be changed, it is possible that there is no δδ that can make the steady-state (iP,x1P) satisfy both constraints simultaneously. For example, from the analysis in Section 3, when the infection rate is very high, it is unlikely to control the epidemic to a very small range while everyone goes out without any protective measures. Therefore, given β1, γ and k¯, the first step is to determine if the two constraints 0≤iP≤im and 1≥x1P≥xm are feasible, i.e., whether it is possible to find δδ that make the steady state satisfy both constraints. Section 4.2.1 investigates how to determine the feasibility of the two constraints. If they are feasible, Section 4.2.2 explains the details of the optimization problem and proposes a fast algorithm to solve it. If they are not feasible, Section 4.2.3 investigates how to reformulate the problem and let the steady state (iP,x1P) be as close as possible to the constraints.

4.2.1 The Feasibility Test

From Theorem 4, we only need to get the optimal solution in Case 3 and then we can get the optimal solution of the whole space by comparing it with 00. From (24), if the steady state (iP,x1P) belongs to Case 3, it should satisfy

x1P=γ(1−iP)k¯β1.(30)

Note that in (30), x1P is an increasing function of iP. Therefore, given 0≤iP≤im, we have x1P=γ(1−iP)k¯β1∈[γk¯β1,γ(1−im)k¯β1]. If xm≤γ(1−im)k¯β1, then it is possible to find δδ whose corresponding steady-state (iP,x1P) satisfies 0≤iP≤im and 1≥x1P≥xm simultaneously, and thus the two constraints are feasible. Otherwise, the two constraints are infeasible and cannot be satisfied at the same time.

Fig. 2 presents two examples of the feasible region of (im, xm). The green area encompasses all (im, xm) pairs where the constraints are feasible, while the red area comprises all infeasible constraints. If our behavior inducement algorithm attempts to reduce the infection proportion below im while increasing the proportion of individuals choosing risky behaviors above xm, the point (im, xm) should fall within the green area, i.e., satisfying xm≤γ(1−im)k¯β1 for this goal to be feasible.

Figure 2: The range of im and xm when (a) β1γ=23 (b) β1γ=12 with k¯ = 10. The green area represents the feasible constraints, and the red area represents the infeasible constraints

4.2.2 Solving the Optimization Problem with Feasible Constraints

If 0≤iP≤im and 1≥x1P≥xm are feasible, we discuss how to solve the optimization problem. Given the two constraints, we adopt the exterior-point method and transform the two constraints into a penalty function:

μ[P(iP−im)+P(−iP)+P(xm−x1P)+P(x1P−1)],(31)

where μ is a parameter that determines the intensity of the penalty, and P(x)=△(max{0,x})2. We let l1(iP)=μ[P(iP−im)+P(−iP)] and l2(x1P)=μ[P(xm−x1P)+P(x1P−1)], then the optimization problem (27) becomes:

minδδ μ[P(iP−im)+P(−iP)+P(xm−x1P)+P(x1P−1)]+l3(δδ),s.t.0<α+Δα<1,①cn+Δcn<0.②(32)

According to Theorem 4, the optimization problem can be transformed into the problem in Case 3 and then we can get the optimal solution of the whole space by comparing it with 00. Here we only consider the situation where k¯>γβ1.3 Then the optimization problem in Case 3 becomes:

minδδ μ[P(iP−im)+P(−iP)+P(xm−x1P)+P(x1P−1)]+l3(δδ),s.t.0<α+Δα<1,① cn+Δcn<0,② (1−iP)k¯β1x1P−γ=0,③ k0(uP(c1+Δc1)−uP(c2+Δc2))+(γ−k¯β1)+k0uP(cn+Δcn)⋅ω[k¯β1−γ,α+Δα]≤0,④ k0uP(cn+Δcn)⋅ω[k¯β1iP,α+Δα]−k¯β1iP+k0(uP(c1+Δc1)−uP(c2+Δc2))=0.⑤(33)

Here, ω[p,α]=e(−(−lnp)α). Constraints ③, ④, and ⑤ guarantee that the steady state belongs to Case 3 in Theorem 2. The method for solving the problem is provided in Appendix G in Supplementary Material.

4.2.3 Reformulation of the Optimization Problem When the Constraints Are Infeasible

If the constraints 0≤iP≤im and 1≥x1P≥xm are not feasible, we reformulate the optimization problem and make (iP, x1P) as close to (im, xm) as possible. Specifically, we transform the two constraints 0≤iP≤im and 1≥x1P≥xm into (iP−im)2+(x1P−xm)2 in the objective function, so that the steady-state (iP,x1P) is close to (im,xm) in the i−x1 plane. Then the optimization problem (33) becomes

minδδ (iP−im)2+(x1P−xm)2+l3(δδ),s.t.  0<α+Δα<1,cn+Δcn<0,(1−iP)k¯β1x1P−γ=0,   k0(uP(c1+Δc1)−uP(c2+Δc2))+(γ−k¯β1)+k0uP(cn+Δcn)⋅ω[k¯β1−γ,α+Δα]≤0,k0uP(cn+Δcn)⋅ω[k¯β1iP,α+Δα]−k¯β1iP+k0(uP(c1+Δc1)−uP(c2+Δc2))=0.(34)

We can use the same method in Section 4.2.2 to solve (34), and the details are provided in Appendix H in Supplementary Material.

5  Simulation Results

In this section, we first run simulations to validate our steady state analysis of the co-evolution process and the effect of irrationality on individuals’ behaviors and epidemic spread in Section 3. Then we validate the effectiveness of the behavior guidance algorithms proposed in Section 4. As there are few previous works have analyzed how to model irrational individuals in a pandemic, we do not compare our method with other works in this section.

5.1 Simulations of the Steady States of EUT and PT

Theorem 1 and Theorem 2 give theoretical analyses of the steady states when individuals are rational and irrational, respectively. To validate the two theorems, as an example, we conducted simulations on regular networks with 500 nodes. The physical contact network has a fixed degree of 10, while the information network has a degree of 20. We observe similar trends on other types of networks and with other parameters. We set the recovery rate to γ=0.03 as an example and let the infection rate β1 vary. We first run the simulation to validate the three cases of the steady state of EUT and PT. Since risky behaviors such as going out and not wearing a mask are the default behaviors most people take in their daily lives, we set c1=0; conservative behaviors such as isolation and wearing masks can be regarded as behaviors with losses, so we set c2<0. In this work, we let c1=0, c2=−1, and cn=−20. In order to facilitate the comparison between EUT and PT, we set uE(x)=uP(x) and use the power function in (6) with σ=0.65 and λ=1 as an example. For other value functions, we observe the same trend and omit the results here. For each simulation setup, we repeat the experiment 50 times and show the average result below.

Fig. 3 shows the simulation and theoretical results of (iE,x1E) and (iP,x1P) with different infection rates β1, where the theoretical results are calculated using Theorem 1 and Theorem 2. It can be seen that the simulation results match the theoretical results very well. The red area represents Case 1 in Theorem 1 and Theorem 2, wherein the infection rate is too low, resulting in no disease spread, then all individuals choose the risky behavior. Moving to the green area denoting Case 2 in Theorem 1 and Theorem 2, we observe a gradual rise in the proportion of infected individuals corresponding to an increasing infection rate. Despite this, as the infection rate is not high enough and the payoff for risky behavior remains high, individuals tend to choose risky behavior. Moving to the blue area denoting Case 3 in Theorem 1 and Theorem 2, the infection rate continues to increase, causing the payoff of the risky behavior to gradually decrease. Consequently, more individuals tend to choose conservative behavior, leading to a decline in the percentage of infected individuals.

Figure 3: Simulation results of the steady states with rational and irrational individuals

Then we validate Theorem 3 and investigate the impact of irrationality on the steady states using the same simulation setup as before. Fig. 4 shows the average results of 50 simulation runs. We observe the same trend for other types of networks and other parameter settings. Note that in Fig. 4, α=1 (EUT) corresponds to the scenario with rational individuals, as explained in Section 2.2.1. In Fig. 4, as β1γ increases, Point A is the boundary point separating Case 1 and Case 2, and Points B, C, and D are the boundary points separating Cases 2 and 3 with α=0.6, α=0.8, and α=1 (EUT), respectively. Then we analyze the results of α=0.6 and α=0.8 as an example. We use (i_P,x_1P) and (i¯P,x¯1P) to represent the steady states for α=0.6 and α=0.8. From Figs. 4a and 4b, we have:

Figure 4: Simulation results of the steady states with different β1γ. (a) iE and iP (b) x1E and x1P

•   Before Point A, i.e., when β1γ<1k¯, irrationality does not influence individuals’ behavior and the steady states in Case 1, as both curves with α=0.6 and α=0.8 have x¯1P=x_1P=1 and i¯P=i_P=0. Also, the boundary points separating Case 1 and Case 2 (Point A in Figs. 4a and 4b) are the same with different values of α. The results validate Theorem 3a.

•   Between Points A and B, β1γ>1k¯, Φ2<0 for both α=0.6,0.8. Therefore, from Theorem 3b, both individuals with low irrationality (α=0.8) and individuals with high irrationality (α=0.6) choose the risky behavior with x_1P=x¯1P=1 while the epidemic spreads to the maximum extent with i_P=i¯P=1−γk¯β1, and this corresponds to Case 2 for both (i_P,x_1P) and (i¯P,x¯1P).