Infectious diseases present serious public health threats, necessitating effective control measures. One such approach is a voluntary vaccination policy, which respects individual decision-making. However, to optimize disease control, exploring diverse strategies and weighing their costs and benefits is vital. This study analyzes the merits and drawbacks of a mixed-strategy approach within such a policy. By combining individual and societal risk assessments, we aim to uncover potential advantages while addressing social dilemmas and efficiency deficits. This research informs decision-makers about resource allocation with regard to combating infectious diseases within voluntary vaccination frameworks.
The application of the evolutionary game [1
] theory when analyzing the dynamics of epidemic models has provided valuable insights into the behavior of infectious diseases and the effectiveness of various intervention strategies [2
]. One critical aspect of epidemic control is vaccination [5
], which aims to mitigate the spread of diseases by immunizing individuals. However, vaccination programs [8
] have associated costs and benefits that must be carefully evaluated to maximize their effectiveness. This paper uses evolutionary game theory to explore the concept of cost–benefit vaccination in the context of an epidemic model. By considering the trade-offs between the costs of vaccination and the benefits of reduced infection rates and disease burden, this study aims to shed light on the optimal strategies for vaccine distribution and allocation.
In epidemic-based evolutionary game theory models, incorporating both individual and society-based risk assessment is crucial to a comprehensive understanding of disease dynamics and the impact of intervention strategies [13
]. Individual-based risk assessment (IB-RA) involves evaluating the risks and benefits of vaccination for a single individual, considering personal factors such as susceptibility to the disease, potential side effects, and perceived benefits of vaccination. On the other hand, society-based risk assessment (SB-RA) considers the collective welfare of the community, assessing the overall benefits and costs of vaccination at a population level. By integrating these two concepts, one can develop sophisticated models that capture the complex interplay between individual decision-making and community outcomes, providing insights into optimal vaccination strategies and addressing potential dilemmas between personal and societal interests. This approach helps inform policies that promote the common good while respecting individual autonomy.
Epidemic compartmental models [15
] are instrumental when it comes to comprehending and forecasting the dissemination of contagious diseases among communities. Furthermore, they facilitate the acquisition of insights into the behavior of diseases and inform decision-making processes for effective disease control and prevention strategies. For instance, the SVIR (susceptible–vaccinated–infected–recovered) mathematical epidemiological model is a mathematical framework used to analyze and comprehend the population-wide transmission of infectious diseases [17
]. Recognizing the impact of vaccination campaigns on disease dynamics, the SVIR model incorporates an extra compartment to account for vaccinated (V) individuals. Susceptible individuals are at risk of contracting the disease, vaccinated individuals have received a vaccine and accomplished protection against infection, infected individuals are actively spreading the disease, and recovered individuals have either survived the infection and developed immunity or have been successfully treated.
Evolutionary game theory (EGT) has emerged as a valuable mathematical framework for studying cooperative behavior in various domains, including epidemics [19
]. Bauch and Bhattacharyya [21
] proposed incorporating EGT into the analysis of individual behavior within an epidemic context. Chen and Fu [22
] applied social learning theory to examine decision-making related to vaccination and self-isolation during health crises. Zho et al. [23
] studied the impact of determined individuals on voluntary vaccination behavior based on information-driven decisions and benefit–cost analysis. Lim and Zhang [24
] investigated factors influencing vaccination choices using a nonlinear public good game. Other studies explored the effects of intermediate defensive mechanisms [25
], developed analytical frameworks for vaccination games [26
], and examined the influence of individual imitation and population structure on achieving widespread immunity [27
]. Kabir et al. [28
] assessed behavioral incentives in vaccination scenarios and proposed models to guide policymaking. Various other studies employed EGT to analyze economic shutdowns during the COVID-19 pandemic [30
], examine the impact of human behavior and memory [31
], and study the effects of vaccination and treatment on epidemic transmission patterns [32
Furthermore, it is worth noting that the beneficial effects of vaccination in the EGT model have yet to be noticed by previous authors. In this study, we aim to fill this gap by incorporating the positive impact of vaccination into the EGT framework. By considering the beneficial effects of vaccination, we can elucidate how it influences human decision-making and, consequently, aid policymakers in implementing effective measures to curb the spread of infectious diseases. Moreover, integrating individual and society-based risk assessment within these sophisticated models allows for a comprehensive understanding of the intricate dynamics between individual choices and community outcomes. This approach provides valuable insights into optimal vaccination strategies while addressing potential conflicts between personal and societal interests.
In recent years, a growing body of research has focused on the dynamics of epidemic control strategies, commonly known as the intervention game, both theoretically and numerically [33
]. Bauch et al. [21
] and others [17
] adopted an approach considering a scenario in which disease spread and individuals’ behavioral changes due to social learning evolve simultaneously within one season, resembling real-world dynamics in specific social contexts. However, Kuga et al. [13
] and Kabir et al. [14
] developed vaccination epidemic game models, respectively, which consider the spread of the disease within a single season, referred to as the “local time scale,” with strategy updates occurring at the end of each season on the “global time scale” or generation. We also investigated the concept of Social Efficiency Deficit (SED) [39
] to understand the social dilemma better. In line with this concept, we aim to construct a mathematical formulation of the vaccination game that accounts for disease transmission and strategy updates for vaccination behavior within different time scales.
3. Result and Discussion
In this discussion, we examine the impact of vaccination on reducing disease transmission and analyze the cost–benefit effects of a vaccination game. To gain a deeper understanding, we assign values to various variables for evaluation purposes. These variables include the final epidemic size
, the fraction of vaccination coverage
, the average social payoff
, the social efficiency deficit
, the deviation of
, and the deviation of vaccination coverage
at each social equilibrium. We compare the outcomes using two different strategy update rules: individual-based risk assessment (IB-RA) and society-based risk assessment (SB-RA). The comparison is presented in a 2D phase diagram illustrating the relationship between the vaccination cost
and vaccination efficiency
. Figure 2
, Figure 3
, Figure 4
, Figure 5
, Figure 6
and Figure 7
showcase the outcomes of the individual-based risk assessment (IB-RA), society-based risk assessment (SB-RA), and a combination of both strategies (intermediate) for three distinct levels of strategy selection intensity rate, (a-*)
, and (b-*)
, respectively. The figures also depict three different values of benefit rate (*-
, (*-ii) ,
, which are utilized to adjust the vaccine cost and efficiencies.
The findings from Figure 2
(*-i) with a benefit rate of
(without any benefits) indicate that during a pandemic, the majority of individuals opt for vaccination either through individual-based risk assessment (IB-RA) when
or society-based risk assessment (SB-RA) when
. This observation aligns with the findings of a previous study [13
]. However, individuals who adopt an intermediate strategy
, instead of prioritizing either IB-RA or SB-RA, follow an embedded approach to update their strategy regarding the transmission of the infection.
In panel 2(a–i), when the cost of vaccination is higher and its efficiency is negligible, a full-scale spread of the infection becomes inevitable. This is because individuals who are having doubts about its reliability and affordability tend to avoid vaccination altogether. To control the spread of the epidemic, the boundary between the monotone region and the remaining region plays a crucial role in transitioning the phase from a pandemic to a controlled (disease-free) state. In the controlled phase, a distinct blue area signifies lower infection rates due to higher efficiency and lower cost. Similarly, detailed full-phase diagrams for panels 2(b–i) and 2(c–i), representing the intermediate strategy and SB-RA approach, respectively, exhibit differences. However, the overall trend remains similar to that which is observed in Figure 2
(a–i) in certain aspects. Increasing the value of
(strategy selection intensity rate) reduces the size of the red region and expands the disease-free (blue) region. The updating rule based on global knowledge (SB-RA) rather than local knowledge (IB-RA) provides a more effective means of suppressing the spread of the disease.
When we shift our attention to the benefit parameter in panels 2(*-ii) and 2(*-iii), we notice that as increases, the disease-free region, represented by the blue region, expands, meaning that a more significant portion of the population remains unaffected by the epidemic. Additionally, an increase in leads to a decrease in the final epidemic size, which refers to the total number of individuals affected by the disease. In other words, by increasing the benefit parameter , we enhance the efficacy of measures or interventions to prevent the spread of the disease. This could involve various actions such as public health campaigns, vaccination efforts, improved healthcare infrastructure, and other strategies designed to limit transmission and mitigate the epidemic’s impact.
illustrates the results obtained by assuming different information benefit rates (
) concerning a fraction of vaccinators (FOV). The depicted scenarios correspond to B values of 0.0, 0.5, and 1.0. Additionally, three types of strategy-updating aspects are shown: (a-*
) IB-RA, (b-*) intermediate, and (c-*) SB-RA. These aspects are associated with varying vaccination costs and efficiencies. In the controlled epidemic phase, we can observe a green-colored region in the diagrams, indicating a high vaccination coverage. A combination of higher efficiency and lower vaccination prices characterizes this region. It is interesting to note that even when a significant portion of the population receives the vaccination, the epidemic cannot be eliminated due to the lower reliability of the vaccine. Figure 4
likely provides additional information about the average social payoff (ASP). Although they are different from Figure 4
, the detailed full-phase diagrams follow a similar overall trend. Overall, the findings from Figure 3
and Figure 4
suggest that the information benefit rates (
), along with the selection of strategy updating aspects and associated vaccination costs and efficiencies, play crucial roles in shaping the outcomes of the epidemic control efforts. Higher
values and effective strategies can increase vaccination coverage and mitigate the epidemic’s impact. However, despite high vaccination rates, the reliability of the vaccine remains a critical factor in determining the success of epidemic elimination.
presents a 2D heat map illustrating the social efficiency deficit (SED) as it varies, with
represented. We consider every combination of
to calculate the Average Social Payoff (ASP) at the social equilibrium (NE). Additionally, we estimate the ASP at the social optimum (SO) without employing the game approach by determining the maximum ASP for each
while varying the vaccination coverage
from 0 to 1. We then calculate the difference between the ASP at SO and NE using a defined Equation (11), which gives us the SED value. In the heat map, the black region represents no SED and, therefore, no dilemma. In this case, lower efficacy does not motivate people to vaccinate, resulting in a dominant defection (D) state as the social optimum (SO), meaning everyone chooses not to commit to vaccination. Similarly, in the fraction of vaccination coverage (FOV) heat map (Figure 3
), the corresponding triangular region displays a dominant defection (D) NE, resulting in identical ASP values (Figure 4
) as observed at SO. In other words, the payoff at NE cannot be improved further, leading to the absence of any social dilemma. However, another region characterized by low cost in Figure 5
(*-i) exhibits no SED. In the FOV phase diagram (Figure 3
), the equivalent area indicates dominant cooperation (C) NE, where all individuals choose to commit to vaccination, despite the efficiency not being very high. Furthermore, the ASPs associated with this region (Figure 4
) are nearly identical at NE and SO, indicating the absence of SED and, therefore, no social dilemma occurs. On the other hand, the remaining part of Figure 5
(*-i) shows varying SED levels, indicating a social dilemma. In this region, we observe non-monotonic changes in SED when the vaccination cost is not excessively high.
An interesting observation can be made when the benefit parameter is increased, as depicted in Figure 5
(*-ii) and 5(*-
iii); the IB-RA and SB-RA strategies show contrasting tendencies. In the IB-RA strategy, as the benefit parameter
increases, a non-zero SED becomes more pronounced, indicating that a dilemma arises, implying that there is still room to reduce infection despite trade-offs. This suggests that increasing the benefit parameter in the IB-RA strategy leads to a higher likelihood of encountering a social dilemma. However, the SED is comparatively lower in the SB-RA strategy when the benefit parameter
is higher. This suggests that as
increases, the occurrence of a social dilemma diminishes in the SB-RA strategy. This observation implies that the SB-RA strategy, which involves individual group decisions based on social benefits, is more authentic and beneficial for everyone involved. Therefore, when the benefit parameter is increased, the IB-RA strategy shows a higher presence of the social dilemma, indicating trade-offs and the need for further efforts to reduce infection. However, the SB-RA strategy exhibits lower SED levels with higher values of
, indicating a reduced presence of the social dilemma. The SB-RA strategy emphasizes individual group decisions and social benefits and is considered more advantageous and authentic when it comes to managing the epidemic.
Overall, the heat map in Figure 5
helps visualize the distribution of SED levels across different values of
. The absence of SEDs in certain regions reflects scenarios where there is no social dilemma. At the same time, SED in other areas indicates situations in which trade-offs and non-monotonic changes occur in the decision-making process regarding vaccination strategies.
From an in-depth perspective, Figure 6
provides a detailed analysis of the deficiencies of FES and FOV in the 2D heatmap, illustrating the relationship between vaccine efficiency and vaccination cost. Panel A represents the deficiency of FES, while Panel B shows the deficiency of FOV denoted by
, respectively. To calculate the FES and FOV at NE, we consider every combination of
and determine their values. Next, we estimate the FES and FOV at the social optimum (SO) to maximize the overall societal welfare, varying the vaccination coverage
from 0 to 1. This allows us to determine the maximality of ASP for different
values at SO. Finally, we obtain the values of
by taking the difference between the FES/FOV values at SO and NE. These values indicate the extent of deficiency or improvement in FES and FOV when moving from NE to SO.
The Social Efficiency Deficit (SED) solely informs us about the presence or absence of a dilemma situation. In contrast, FES (
) and FOV (
) deficiencies offer actionable insights to address this dilemma. When aiming to reduce the spread of disease, two main approaches can be considered: reducing FES, indicated by the red portion in the figure, to improve the overall SED; or implementing vaccination strategies to mitigate the SED. However, it is essential to acknowledge that FES is, to some extent, influenced by FOV. Increasing vaccination coverage (FOV) generally leads to a decrease in FES. Nevertheless, factors such as free riding and the effects of herd immunity can introduce unexpected dynamics that complicate the relationship between FOV and FES. Comparing Panel A (
) and Panel B (
), it becomes evident that the
region is more extensive and prominent than the corresponding the
region. This observation implies that relying solely on implementing vaccination measures may not be sufficient to eradicate infections or resolve the dilemma entirely. It indicates that additional strategies or interventions beyond vaccination are required to effectively mitigate the spread of the disease and alleviate the dilemma. By studying Figure 6
, we gain valuable insights that complement the findings from Figure 5
(SED). These insights suggest possible strategies and courses of action that may reduce infection rates while effectively addressing the underlying dilemma. Therefore, Figure 6
serves as a crucial tool for guiding decision-making processes and identifying appropriate measures to combat the spread of the disease and tackle the challenges posed by the social equilibrium dilemma.
, Figure 8
and Figure 9
depict 2D graphs illustrating the relationship between
for two scenarios: (Panel A) deviation from Individual-Based Risk Assessment (IB-RA), denoted as
, and (Panel B) deviation from Society-Based Risk Assessment (SB-RA), denoted as
. These figures provide insights into decision-making outcomes and the effects of changing strategies. To calculate the FES, FOV, and ASP for
, we systematically consider every combination of
and determine IB-RA (
) and SB-RA (
) values. In doing so, we evaluate the baseline outcomes for these two decision-making approaches. Subsequently, we estimate the FES, FOV, and ASP for the dynamic equation of
(as defined in Equation (8)). This allows us to examine decision-making outcomes for different values of theta and assess the influence of dynamic decision-making on these measures. Finally, we calculate the values of
by comparing the FES, FOV, and ASP values obtained for the constant values of
with those obtained for the dynamic of
. These calculated values highlight how decisions change from myopic or individual-based associations
to more community-based associations
. These calculations demonstrate the impact of different decision-making approaches on FES, FOV, and ASP, shedding light on how myopic or community-based decision-making strategies influence individual choices and outcomes.
Panels A of Figure 7
and Figure 8
show negative values for
(red) and positive values for
(green) along the boundary between the disease-free equilibrium and the endemic equilibrium. The negative
indicates that relying on dynamic decision-making (using dynamic
) rather than Individual-Based Rationality (IB-RA) reduces the disease. This observation is also evident in the positive values of
, where disease reduction is observed when entirely positive values occur. Furthermore, as the values of theta increase, the red region (
) in Figure 7
and the green region (
) in Figure 8
expand, implying that as cooperation intensity increases, individuals are more inclined to adopt a community-based decision strategy to reduce the risk of infection. Similarly, increased benefits also lead to a similar tendency towards a community-based decision strategy. Suppose we shift our focus to the Average Social Payoff (ASP). In that case, we can observe a similar trend in FES and FOV, referring to the dynamics of decision making and the resulting outcomes regarding disease reduction align with the patterns observed in FES and FOV. Thus, adopting a dynamic decision-making approach (using dynamic
) instead of relying solely on IB-RA can reduce disease, increasing cooperation intensity, and these benefits further promote adopting community-based decision strategies.
Now, Panel B in Figure 7
and Figure 8
reveals both negative (red) and positive (blue) values for
(deviation from SB-RA) along the boundary region between the disease-free and endemic equilibrium. Notably, in regions characterized by low vaccine cost and low
) is observed, indicating a preference for the SB-RA strategy. Conversely, in regions characterized by higher cost and higher
) is present, indicating a stronger inclination towards relying on the dynamic decision parameter,
. This observed tendency can be attributed to the interplay between higher cost and higher reliability of vaccination. Individuals are more inclined towards vaccination in regions with higher costs and excellent vaccination reliability. However, the higher cost presents a dilemma for individuals, prompting them to seek the benefits of free-riding through the attainment of herd immunity. Consequently, individuals are more prone to relying on the dynamic decision parameter,
, in these circumstances. Interestingly, as the benefit (
) increases (sub-panels (*-ii) and (*-iii)), the red region that represents negative
diminishes. This signifies that increased benefits can help alleviate the dilemma and reduce infection rates by encouraging individuals to participate in vaccination programs. In other words, higher benefits create more substantial incentives for individuals to overcome the dilemma and choose vaccination as a preventive measure.
As a final step, Figure 10
provides a comprehensive 2D phase-plane analysis, exploring the impact of varying
(vaccine efficiency) and benefit
across relative vaccination costs
. Sub-panels (a-*), (b-*), and (c-*) correspond to different values of
, respectively). Across all cases, it is evident that vaccine efficiency (
) influences both FES and FOV, with higher
values leading to a reduction in FES. This outcome aligns with expectations, as higher vaccination reliability makes individuals more inclined to take vaccines. Additionally, as both
increase, FES decreases significantly. This result is intuitive, as individuals are more likely to participate in vaccination programs when both the reliability of the vaccine and the associated benefits are high. Consequently, a substantial reduction in FES is observed. Furthermore, the relative vaccination cost (
) is crucial in reducing infection rates, but only when the vaccine cost is relatively low. In such cases, the impact of vaccination costs on infection reduction is significant. Interestingly, as
(the updating process parameter) increases, a paraboloid-shaped region emerges in the vaccination (and FES) region. This is accompanied by an increase in FOV and a reduction in FES. The presence of the paraboloid-shaped region suggests the occurrence of the non-free-riding effect, mainly when vaccine efficiency is relatively intermediate. These findings underscore the complementary relationship between vaccine benefits, individual strategy selection processes, and their impact on disease control. Policymakers can consider the costs and benefits associated with participation in vaccination programs. However, the selection of individual strategies is determined by the nature of the updating process. This indicates the importance of understanding the decision-making dynamics and designing effective strategies to encourage vaccine uptake and mitigate disease spread. The results also highlight the role of these factors in shaping FES and FOV while revealing the effects of the updating process and the presence of free-riding dynamics. These insights can inform policymakers in devising effective strategies to promote vaccine participation and combat the spread of infectious diseases.