A behavioral game-theoretic model for vaccination with fractional-order imitation and aspiration dynamics

Abstract

This research work introduces a new modeling technique that simulates the vaccination game, integrating imitation and aspiration social learning dynamics using the Caputo fractional-order derivative. We investigated a fractional-order epidemic model that incorporates imitation and aspiration dynamics in both pure and combined imitation and aspiration frameworks, considering symmetric and asymmetric aspiration levels. This approach is particularly relevant, as conventional game-theoretic models of vaccine behavior often fail to accurately represent the intricate and flexible characteristics of human decision-making processes. The Caputo fractional-order derivative enables a more precise representation of the memory and hereditary effects inherent in individuals’ decision-making processes regarding vaccination. We also examine the interaction between social influence, characterized by imitation, and individual ambition levels, offering a complete framework for analyzing the impact of these variables on vaccination behavior within a community through the evolutionary game theory (EGT) framework, and investigate how public health policy, individual risk perception, and societal dynamics affect the vaccination rate. The findings demonstrate the advantages of employing the Caputo fractional-order derivative for policymaking in capturing the complex temporal dynamics of vaccination games, offering valuable insights for developing effective vaccination strategies and public health initiatives.

Introduction

Vaccination is one of the most effective approaches for controlling infectious diseases and preventing epidemics^1,2. Although vaccination has been effective, the number of people vaccinated is frequently insufficient to achieve herd immunity. Several social, psychological, and economic variables impact this. Understanding the intricate dynamics behind vaccination practices is essential for developing efficient public health interventions. Conventional approaches to studying vaccination behavior have traditionally employed classical differential equations, assuming that changes in vaccination coverage occur smoothly and predictably. Nevertheless, the dynamics of vaccination in the real world are often characterized by irregularities and impacted by variables such as individual awareness, peer influence, and evolving desires, which are not effectively accounted for in these conventional models. In this context, for the first time, the vaccination game with imitation and aspiration dynamics, incorporating the Caputo fractional-order derivative³, presents itself as a comprehensive and cutting-edge methodology for investigating the complex dynamics arising from individual decisions regarding vaccination and the influence of social factors.

The integration of game theory into epidemiological models in recent years has yielded profound insights into the influence of individual decision-making on public health outcomes^4–8. Vaccination games^9–23, which involve individuals making decisions about vaccination based on their assessment of potential risks and benefits, have demonstrated the strategic aspect of vaccination choices. These models often include imitation dynamics^9–19, in which people imitate the immunization methods of others, and aspiration dynamics^20–23, in which personal happiness with results drives future decisions, as recent experimental research reveals that people do not necessarily imitate others, but instead make cognitive choices based on their contentment with the consequences. Occasionally, social information can lack reliability²⁴. Hence, it is reasonable to infer that specific individuals within a community can evaluate the anticipated benefits based on their level of contentment or desired achievement, rather than merely comparing themselves to others. The ambition level serves as a benchmark for comparing one’s growth in rewards²⁵. In the context of dyadic interactions in evolutionary game theory, individuals using aspiration updating compare their attained reward with a predetermined self-assessment or ambition level. If the attained payout does not meet that threshold, they choose an alternative strategy; otherwise, they stick with the existing one^20,23,26. In the context of the vaccination game, if the payment obtained by the vaccinators does not exceed their expected level, some can reconsider their decision, as can the non-vaccinators. Individuals’ ambitions regarding the vaccine issue are largely contingent upon several factors, including the cost of vaccination, its effectiveness, and the severity of the disease. While vaccine games that use the imitation mechanism provide a realistic representation of real-world situations, there are instances when they can lead to an overestimation or underestimation of the actual circumstance¹⁶.

By incorporating both imitation and aspiration dynamics, we investigate different types of aspiration levels, including symmetric (where both vaccinators and non-vaccinators have the same aspirations) and asymmetric (where vaccinators and non-vaccinators have different aspirations) aspirations. These variations allow us to effectively model the spread of the disease across a wide range of scenarios. Arefin et al.²⁷ examine vaccination behaviors by combining imitation and aspiration updating rules. They utilize a pre-existing model of mean-field vaccination games, as imitation, self-evaluation, or aspiration significantly influences human vaccination choices. Arefin and Tanimoto²⁸ investigate the development of cooperation in two-player games where both imitation and aspiration dynamics coexist in a population that is indefinitely big and well-mixed. The authors systematically explore several forms of aspiration dynamics, including symmetric, asymmetric, and adaptive aspirations, in the context of strong selection and combine these with imitative behavior. Unfortunately, these works mentioned above are performed through a finite difference scheme. Therefore, these models can still have limitations owing to their dependence on integer-order differential equations, which cannot ultimately capture the memory and genetic characteristics inherent in human behavior. In that case, the current work argues that the fractional calculus, more precisely, the Caputo fractional-order derivative (used for numerical simulation), significantly enhances the accuracy and flexibility of models used to understand and predict the ultimate characteristics of infectious diseases.

Nowadays, many authors apply different types of fractional-order derivatives^29–32 to explain the disease dynamics of epidemic models^33–43, owing to enhanced model accuracy, flexibility in modeling, improved parameter estimation, and robustness to uncertainty. Yunus and Olayiwola⁴⁴ formulate a fractional-order mathematical model using the Caputo-Fabrizio derivative to investigate the dynamics of Ebola virus transmission and to forecast the impact of vaccination based on empirical evidence. Olayiwola and Yunus⁴⁵ examine Nigeria’s COVID-19 outbreak based on its economic and social effects, utilizing a unique fractional-order model with the Laplace Adomian Decomposition method. This novel methodology fills the gap in the literature on the impact of COVID-19 on African nations by revealing the efficacy of control methods in limiting the spread and transmission of the virus. Using fractional-order differential equations, one can enhance the precision of modeling the long-lasting impacts of previous actions and the steady progression of vaccination tactics over time^46–49. According to the viewpoints mentioned above, the authors of^50–54 employ the Caputo^50–53 and Atangana-Baleanu-Caputo (ABC)⁵⁴ fractional-order derivative approaches with the EGT (imitation dynamics) setting in epidemic models. Ullah et al.⁵⁰ argued that an individual’s impression of the participant in a vaccination program reflects their inherent understanding of the balance between vaccine behavior, the danger of infection, and memory impact. The author of⁵³ employs enforced control methods and vaccine game theory to investigate the dynamics of diseases in a population with either temporary or declining immunity, distributed evenly and infinitely. However, to date, no work has been conducted through imitation and aspiration dynamics in the same context; nevertheless, a specific and well-defined theoretical framework is necessary in this setting, and current research is attempting to fill this gap.

This work primarily presents a vaccination game method with imitation and aspiration dynamics that leverages the adaptability and memory characteristics of fractional calculus to visualize the dynamics of the epidemic model. Our proposed model integrates standard epidemiological modeling with the complex nature of human behavior in vaccination contexts, considering how people’s choices are influenced by the imitation and aspiration protocol. Imitation, in this sense, refers to people lifting vaccination techniques based on the activities they see others doing, whereas aspiration indicates personal objectives or thresholds that drive vaccination decisions. The model integrates these components to provide a comprehensive understanding of the interplay between disease dynamics and human behavior, offering valuable insights for the advancement of public health policies. The vaccination game model is a sophisticated framework used to analyze the transmission of contagious diseases and immunization patterns in communities.

Definition

First, we deliver precise definitions of the Caputo fractional derivative, as it is associated with many applications referenced in^50–54.

We want to alter our hypothesized epidemic dynamics by using the established Caputo fractional-order derivative³ as follows,

1

where, is the well-known Gamma function, is the order of the Caputo fractional derivative.

Model and method

It is crucial to recognize and analyze the dynamic patterns of adverse outcomes resulting from any disease in the population and to forecast potential occurrences. Maintaining surveillance of the virus has been an increasingly exhausting task, and vaccination has been a common strategy for reducing the spread of the infection. The memory effect is a crucial component in several biological phenomena and is included in the models. Therefore, it is feasible to use fractional alternation since fractional derivatives primarily involve the influence of memory. A practical approach to address this problem is to use the fractional derivatives of order for the state variables. Therefore, assume that there is a uniform distribution of interactions throughout the population, meaning that every member has an equal chance of interacting with every other. Employing the Caputo fractional order compartmental modeling technique, the disease transmission patterns are described by dividing the whole population at any given time into five epidemiological states, namely, susceptible , vaccinated , susceptible infected , vaccinated infected and recovered :

2

3

4

5

6

where is a scaling constant with units , confirming that (dimensionless) produces a vaccination rate^55,56.

Therefore, the parameters , and have appropriate units such that the dimensions are consistent between the two sides of each equation, while the parameters and are dimensionless.

The initial conditions are as follows:

Here, the proposed model (Fig. 1) parameters are delineated as follows: denotes the disease transmission rate per contact, signifies the vaccine efficacy, reflecting the probability of evading infection post-vaccination, represents the recovery rate from infection, and indicates the vaccination adoption rate, influenced by social learning and subject to dynamic alterations through imitation or aspiration dynamics [Table 1].

Fig. 1.

Schematic diagram of the model.

Table 1.

List of parameters, variables, and their meanings.

Notation	Meaning	Values	References
	Transmission rate	0.8333	18,19,28,29,50,54
	Vaccination rate	0.01	18,19,28,29,50,54
	Vaccine efficacy rate	0.1–0.9	18,19,28,29,50,54
	Recovery rate	0.333	18,19,28,29,50,54
	Cost for vaccination	0.1–0.9	18,19,28,29,50,54
	Infection cost for individuals	1.0	18,19,28,29,50,54
	Selection intensity	0.1	27,28
	Imitation and aspiration level	0.0, 0.5, 1.0	27,28
	Vaccinated people’s aspiration level	–	Assumed
	Non-vaccinated people’s aspiration level	–	Assumed

Evolutionary dynamics

Within the framework of epidemic models, evolutionary dynamics refer to the study of how diseases change over time in response to various settings, including host immune responses, treatment strategies, and public health initiatives. This area investigates the impact of hereditary differences within a population of pathogens on their ability to cause disease, spread between individuals, and develop resistance to treatments or vaccinations. By simulating these processes, researchers can predict potential evolutionary paths of diseases, thereby aiding in the development of public health initiatives and treatments. These models often consider mutation rates, selection pressures, and the fitness landscape. They provide valuable insights into the spread of diseases and potential control or mitigation strategies.

The result of the epidemic spreading can be divided into four categories: those who are both vaccinated and healthy, individuals who are vaccinated but still infected, individuals who are unvaccinated but healthy, known as successful free riders, and individuals who are unvaccinated and infected, known as unsuccessful free riders. Table 2 displays a comprehensive representation of all fractions and their corresponding payoffs.

Table 2.

Four fractions, along with payoffs.

	Healthy	Infected
Vaccinated
Non-vaccinated

An individual’s vaccination choice is expected to be impacted by the cost of vaccination and the efficiency of the vaccine. Since vaccines are considered preventive, the number of vaccinators stays unchanged throughout an epidemic. The cost of vaccination includes the price of the vaccine and any potential side effects that may arise from its use. On the other hand, infection costs encompass the expenses associated with treating the disease, the time required for recovery, and the adverse health effects, including morbidity and mortality.

Let be a non-negative cost associated with vaccination and be a positive cost related to infection. It is reasonable to infer that the value of is greater than or equal to . If we choose , then we may assume, without any loss of generality, that the relative cost of vaccination is , where is a value between 0 and 1. Consequently, a vaccinated person incurs a cost (i.e., the payback is ), but an unvaccinated person incurs no cost. The expected outcome for a vaccinator and a non-vaccinator who gets the infection during the epidemic or pandemic is assessed as and , respectively. Individuals who benefit from the efforts of others by not being vaccinated but maintaining good health receive a reward of . It is important to note that the vaccination is believed to be imperfect, meaning that not all individuals who get the vaccine will achieve complete protection. While vaccinated individuals are protected against the disease, those who are not vaccinated or have not developed immunity are still vulnerable. If a proportion of the vulnerable individuals get the vaccination, which has an efficiency .

Table 2 enables us to calculate the mean payoff for individuals who provide vaccines , those who do not administer vaccines , and the overall average societal payoff (ASP) in the following manner:

7

8

9

Evolutionary vaccination dynamics

Evolutionary vaccination dynamics in epidemic models include the interaction between vaccination techniques and the adaptive changes of diseases. Vaccines cause the evolution of vaccine-resistant variants by exerting selection pressure on microorganisms. Mathematical frameworks often describe these dynamics, including vaccination effectiveness, coverage, and population behavior. The models assist in forecasting the dissemination of the primary and resistant variants, providing valuable insights to enhance vaccination programs and minimize the likelihood of resistance. Additionally, it emphasizes the need to maintain a high level of vaccination coverage and develop more advanced vaccines to stay ahead of constantly evolving infections.

In contrast to other studies, our approach incorporates imitative and aspiration procedures into the vaccine dynamics. We consider the average impact of both dynamics in an infinite and well-mixed population. The fundamental Eqs^57–60. is used to represent the progression of vaccination decision dynamics in the mean-field method, including both processes in the following manner:

10

The parameter , which belongs to the interval , represents the relative significance of the imitation mechanism compared to the aspiration mechanism. The equation on the right side of Eq. (10) is divided into two parts. The first part represents the imitative protocol, while the second represents the aspiration protocol. It is evident that when is equal to 0 ( is equal to 1), the pure aspiration (imitative) dynamics are restored. In addition, represents the probability of non-vaccinated individuals switching to vaccinated and represents the probability of vaccinated individuals switching to non-vaccinated individuals. Under the aspiration protocol, and represent the switching probabilities of non-vaccinated and vaccinated individuals. We use Fermi functions to calculate these transition probabilities. The probability is as follows:

Here, the transition propensities are dimensionless probabilities, but they appear multiplied by rates when plugged into dynamics (since they generate flows per fractional time). The parameters and quantify the ambition of vaccinators and non-vaccinators, respectively, whereas represents the selection intensity. Typically, a greater ambition level indicates that players are driven by greed, whereas a lower aspiration reflects altruistic individuals who have lesser expectations for their contributions. However, while considering the vaccine issue, an individual’s degree of ambition can be influenced by several real-world scenarios, including cost, effectiveness, and the current state of the pandemic. A larger indicates that vaccinators have more substantial expectations about the benefits of vaccination. In other words, they anticipate a bigger average payoff from being vaccinated. Individuals who receive a vaccination can expect greater benefits if the vaccine is believed to have a high level of effectiveness. Again, if the vaccination does not match their desired effectiveness, players may alter their strategy; otherwise, they will maintain their existing approach. Conversely, if the vaccine’s effectiveness appears modest, vaccinated and unvaccinated individuals can adjust their expectations. In addition, the pandemic state might also influence people’s degree of expectancy; for example, in the case of a significant disease outbreak, such as the COVID-19 pandemic, people are more likely to seek vaccination if the vaccine is readily available. Suppose the disease’s severity is relatively low. In that case, the outbreak is under control, or herd immunity seems to have been achieved; those who choose not to vaccinate can anticipate more benefits (i.e., an enormous payback) by skipping vaccination (engaging in free riding). Also, the anticipation levels can be progressively modified in response to events by carefully assessing the total benefits derived from vaccination. This situation can be described as adaptive ambitions, where both those who have received vaccinations and those who have not can modify their levels of ambition. They do this by considering their goals and the outcomes they accomplished. More specifically, imitation refers to people modifying their vaccination choices based on the accomplishments of others. At the same time, aspiration denotes personal satisfaction levels that dictate whether one persists with or changes their approach. In essence, imitation captures social learning, whereas aspiration embodies self-evaluation of outcomes, explaining the evolution of collective behavior via social influence and individual expectations.

The basic reproduction number, , is often used to measure the risk of disease spread. Adding Eq. (4) and Eq. (5), we get

At if , then the total infection will start increasing. If that quantity is < 0, then the total infection will begin decreasing. Consequently, we can define the equation for as follows:

11

Result and discussion

This paper introduces the Caputo fractional-order modeling technique to simulate vaccination games, including imitation and aspiration dynamics, in the same context for the first time. The fractional-order characteristics of the model offer a more adaptable framework for capturing the memory effects and long-range interdependencies often observed in social and epidemiological phenomena. The findings suggest that imitation dynamics, in which individuals replicate the actions of their peers, usually increase vaccination coverage in groups with a predominantly pro-vaccination sentiment. Influenced by people’s contentment with their results, aspiration dynamics create variations in vaccination rates, particularly in communities exhibiting vaccine reluctance. This variability can result in oscillating patterns in vaccination rates over time. The model demonstrates that fractional-order dynamics significantly enhance the modeling of complex real-world vaccination scenarios by more effectively capturing the interaction between human behavior and disease transmission. The results suggest that the proposed method can be a valuable resource for public health authorities in developing more effective vaccination programs. To illustrate these scenarios, we present our model outcomes through line graphs, 3D surfaces, and 2D phase diagrams. For simulation purposes, we set the scaling factor to 1.

Figures 2, 3 and 4 depicts the time-based disease dynamics of the proposed epidemic model under Caputo fractional-order imitation and aspiration-driven vaccination behavior, utilizing pure aspiration dynamics , balanced social imitation, and individual aspiration () and a pure imitation learning framework , with both vaccinators and non-vaccinators exhibiting identical or symmetric ambition levels . The fractional-order parameter signifies memory effects in decision-making, where a lower denotes enhanced memory (non-local impact) and increased inertia in behavioral modification. The columns represent several vaccination scenarios characterized by vaccination cost and effectiveness : high-cost and low-efficacy (left), moderate-cost and moderate-efficacy (middle), and low-cost and high-efficacy (right). The infected, vaccinated, and recovered individuals are represented in the top, middle, and bottom rows, respectively.

Fig. 2.

Dynamics of infected, vaccinated, and recovered individuals for fractional-order and vaccination cost and efficacy (), under pure aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 3.

Dynamics of infected, vaccinated, and recovered individuals for fractional-order and vaccination cost and efficacy (), under symetric imitation and aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 4.

Dynamics of infected, vaccinated, and recovered individuals for fractional-order and vaccination cost and efficacy (), under pure imitation () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

In the first column of Fig. 2 (subplots a(i), b(i), c(i)), when vaccination is costly and exhibits poor efficacy , aspiration-based people exhibit a sluggish adoption of vaccination, particularly at reduced values of . This results in a prolonged and elevated infection peak (a(i)) and restricted vaccine adoption (b(i)). The recovery (c(i)) is sluggish and partial, with diminished values (indicating greater memory) exacerbating delays and diminishing the efficacy of interventions owing to historical reluctance and inertia in altering methods. In the second column (subplots a(ii), b(ii), c(ii)), when both cost and effectiveness are modest , the infection curve (a(ii)) exhibits a diminished and earlier peak relative to the first example. The acceptance of vaccination (b(ii)) rises consistently, with higher values (e.g., ) facilitating faster uptake owing to reduced memory effects. The recovery curves (c(ii)) exhibit a similar pattern, enhancing overall population health. This regime illustrates the advantages of aspiration-driven decision-making when the cost of vaccination is permissible. In the third column (subplots a(iii), b(iii), c(iii)), when vaccination is inexpensive and highly efficacious , all trajectories converge toward ideal outcomes. Infection rates are very low (a(iii)), vaccine coverage nears saturation (b(iii)), and recovery is almost total (c(iii)). Nonetheless, fractional memory effects are apparent: reduced values lead to diminished reaction times, indicating that a robust memory impedes prompt behavioral adjustment even in optimal circumstances. Lastly, under the pure aspiration framework, the Caputo fractional-order parameter plays a crucial role in regulating the pace and efficacy of vaccination behavior. Reduced values result in heightened memory dependency, leading to slower and more cautious modifications to strategy. The results are significantly influenced by vaccine cost and efficacy: aspiration dynamics thrive in optimal circumstances, but inertia from partial memory constrains responsiveness, especially when vaccination trade-offs are suboptimal.

In Fig. 3, for lower values, the infection peaks are exceptionally high and sustained in the high-cost, low-efficacy scenario (left column), as the strong memory slows the vaccination response. The accumulation of recovered individuals is directly restricted by the low vaccination rates, which have failed to increase substantially over time. We observe a significant decrease in infection peaks and an increase in vaccination and recovery rates as increases in the moderate scenario (middle column), indicating that moderate memory systems can respond adaptively when vaccine parameters are reasonable, as the balance between imitation and aspiration is maintained. More rapid behavioral alignment and epidemic control result from higher values. The impact of the fractional order is still evident. However, the infection remains limited across all values in the low-cost, high-efficacy scenario (right column): A lower results in a delayed adoption of vaccines, whereas a higher (closer to 1.0) leads to quicker coverage and herd immunity. The recovery and vaccination curves exhibit a rapid increase and eventual saturation for , whereas memory-dominated responses exhibit delayed but eventual convergence to similar states. Finally, it demonstrates that memory strength and vaccine attributes are essential to a hybrid imitation-aspiration strategy with a memory-based approach (Caputo-fractional order). Low values impede behavioral change, particularly in inadequate vaccination conditions. Nevertheless, the resilience of aspiration-imitation learning in long-term epidemic control is underscored by the fact that even memory-driven aspiration can result in near-optimal outcomes when vaccines are incredibly effective and affordable, albeit at a much slower pace.

Figure 4 demonstrates that the memory effect of lower values is seen in every situation. The infection peaks get higher and more delayed as falls, indicating stronger memory and slower response, while vaccination and recovery happen more slowly, demonstrating how, even in imitation-dominant dynamics, non-local memory effects impede the population’s capacity to adapt swiftly. Infection peaks are firm and noticeable in the left column when vaccination conditions are bad (high cost, low efficiency). Lower values cause longer outbreaks with slower recovery. Due to quicker imitation, vaccination coverage is somewhat improved for larger but stays low throughout all curves. The imitation mechanism promotes quicker behavioral convergence in the middle column, which has a modest cost and effectiveness. Compared to the left panel, the infection peak is much smaller, and the number of vaccinated people rises gradually over time. A consistent hierarchy is evident in the recovered population: larger and quicker recovery is achieved with higher values, indicating the importance of short memory in imitation-based epidemic control. With positive vaccine qualities (low cost, high effectiveness), the pandemic is successfully controlled across all values, as seen in the right column. The curves for vaccination and recovery climb rapidly, particularly for , and the infection peaks are modest. However, the flatter curves for vaccination and recovery at indicate that lower alpha still reduces the behavioral response, even in this optimal environment. Overall, Fig. 4 shows that although fractional memory significantly impacts the speed and efficiency of response, epidemic control increases with increased vaccination accessibility and efficacy under pure imitation learning. Strong memory hinders convergence to optimal outcomes, even when people use payoff-maximizing tactics, highlighting the crucial role that fractional order plays in determining the dynamics of public health in real-time.

Figure 5 illustrates the proposed epidemic model’s infection, vaccination, and recovery dynamics, which are regulated by Caputo fractional-order behavioral adaptation. Individuals update their vaccination strategies using a hybrid mechanism integrating imitation and aspiration dynamics. The imitation weight is set to , suggesting that social learning and individual aspiration have an equal impact. The aspiration thresholds vary slightly: vaccinators have a greater satisfaction level than non-vaccinators , making vaccinators more readily satisfied with their selections. This asymmetry adds a bias that favors the retention of immunization practices when the social or personal payoffs are relatively acceptable. In addition, Fig. 5 is split into three columns depending on the various vaccination cost and effectiveness conditions: Left column: (high cost, poor effectiveness). Middle column: (mid-cost and efficacy). Right column: (low cost and high effectiveness). Each row represents a distinct result over time: infected people are in the top row, vaccinated persons are in the middle, and recovered individuals are at the bottom. The fractional order controls the power of memory in decision-making, with each line representing a distinct value. Lower values indicate high non-local memory and delayed behavioral reactions. In the high-cost, low-efficacy scenario (left column), infection prevalence peaks are bigger and longer-lasting for lower , which suggests that memory-induced inertia hampers timely vaccine uptake. The vaccinated population is low for all , but especially suppressed for , where imitation prevails, and people perceive no motivation to vaccinate under dire circumstances. The slight rise in ambition among vaccinators enables moderate uptake for , demonstrating the relevance of aspiration in sustaining vaccination even when imitation fails. Infection peaks are significantly lowered in the moderate scenario (middle column). Vaccination rates steadily rise over time, with higher uptake found for lower owing to longer memory-supporting awareness and aspiration-based switching. The hybrid method (imitation + aspiration) demonstrates that aspiration enhances vaccine adoption, particularly around and , where the dynamics balance responsiveness with cautious behavioral inertia. The outbreak is successfully controlled in the favorable scenario (right column), with all curves indicating effective containment. Vaccination coverage increases progressively, with smaller values indicating slower initial acceptance, eventually leading to high uptake. Higher leads to a continuous rise in the recovered population. The advantageous vaccination settings enable both imitation and aspiration processes to function effectively, with aspiration prevailing during early adaptation and ultimately enhancing long-term success. In conclusion, Fig. 5 demonstrates that in an epidemic model driven by Caputo fractional-order hybrid learning , a moderately ambitious vaccinator population () continue positive vaccination behavior even when imitation fails owing to inadequate vaccine conditions. The interaction between fractional memory and behavioral thresholds has a significant influence on the speed and stability of epidemic control, especially under marginal circumstances of vaccination cost-effectiveness.

Fig. 5.

Dynamics of infected, vaccinated, and recovered individuals for fractional-order and vaccination cost and efficacy (), under symetric imitation and aspiration () social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

The nine-panel figure (Fig. 6) illustrates the modulation of disease and behavior co-evolution by Caputo-type fractional memory (fractional order ) under conditions of moderate imitation and asymmetrical aspiration forces (vaccinators’ ambition ; non-vaccinators’ ambition ). Columns represent ever more advantageous vaccine cost and efficacy: Column (i), elevated expense and inadequate effectiveness (), column (ii), intermediate cost and efficacy (), and column (iii), low cost/high effectiveness (). Colors (redblack) indicate a decreasing fractional order ; a lower signifies an enhanced memory, meaning agents and infections exhibit a more gradual response to current circumstances and a greater response to historical influences. Epidemic (first row, a(·)): With costly, ineffective vaccinations, the epidemic peaks at around 20% of the population for ; however, every decrease in mitigates and postpones the peak: memory impedes transmission, and behavioral adaptation occurs. In the balanced situation (ii), the peak is already minimal ; however, the same monotonic attenuation with memory is seen. With inexpensive, efficient vaccinations, infections seldom exceed 0.35% even when , and the curves rapidly converge to zero as vaccination prevails. Immunization (middle row, b(·)): Aspiration oscillations are apparent: a rapid initial spike, a fatigue-induced decline, and gradual convergence. A reduced attenuates fluctuations and increases steady-state coverage as the extended memory amplifies prior pro-vaccination benefits. Scenario (ii) demonstrates a gradual increase; memory continues to enhance the ultimate coverage. All individuals are vaccinated promptly with almost flawless vaccinations, rendering variations in values insignificant. Recovery (bottom row, c(·)): The accumulation of recoveries correlates with infections, resulting in the ordering of curves in (i) and (ii) reflecting the infection plots: a higher (indicating less memory) leads to an increase in cases and a bigger recovered population, while more memory (represented by the black curve) has the least epidemic impact. In (iii), recoveries stabilize below 1% for all due to the suppression of infections at the source. In recapitulation, the panels together demonstrate that implementing Caputo-fractional memory consistently diminishes short-term dynamics while enhancing long-term epidemic control, with its advantages being magnified when vaccination is already appealing (low , high ). The interaction of imitation (a) and aspiration operates via two mechanisms: economic and technological incentives, as well as collective memory, to transform behavioral adoption and disease prevalence.

Fig. 6.

Dynamics of infected, vaccinated, and recovered individuals for fractional-order and vaccination cost and efficacy (), under symetric imitation and aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Figure 7 depicts disease transmission, vaccination patterns, recovery, and average social payoffs in a fractional-order epidemiological model characterized by pure aspiration behavior. The fractional order () affects the memory effect in population dynamics; a higher fractional order (approaching 1) signifies reduced memory and more immediate responses. In contrast, lower fractional order values enhance memory effects, leading past states to influence future behavior more. The upper row (a*) illustrates infected individuals: when fractional order increases, infection first escalates rapidly, suggesting that expedited decision-making (reduced memory) initially intensifies infection. Nevertheless, infections diminish beyond a crucial fractional order, indicating that enhanced reactivity at elevated orders reduces disease transmission. The vaccination behavior (row 2, b*) exhibits a distinct rise with fractional order, peaks at , indicating that populations with shorter memory and greater flexibility are more inclined to accept vaccination, resulting in a larger total vaccinated percentage at elevated fractional orders. The number of recovered people (row 3, c*) progressively rises with fractional order, illustrating a clear correlation: expedited behavioral adaptations and elevated vaccination rates inherently result in greater recovery. The average social payoffs (row 4, d*) exhibit intricate dynamics: at lower fractional orders, payoffs remain relatively stable, suggesting that extended decision-making and memory effects result in negligible short-term variations in payoffs. At intermediate fractional orders, payoffs significantly diminish, illustrating transitional dynamics whereby populations incur substantial short-term costs before achieving stabilization at higher orders, using more effective methods and resulting in improved overall social outcomes. Finally, Fig. 7 illustrates the complex interaction between fractional order (memory effects), vaccination behavior, and epidemiological outcomes, underscoring that rapid adaptability (higher fractional orders) typically results in enhanced epidemic control, greater vaccination rates, increased recovery, and improved social outcomes, notwithstanding temporary difficulties during intermediate fractional-order states.

Fig. 7.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Figure 8 illustrates the effects of fractional orders (α) on an epidemic model that includes vaccination behavior motivated only by pure aspiration (imitation parameter, ). In this scenario, vaccinator aspiration () and non-vaccinator aspiration () are both negative (− 0.5), indicating diminished expectations, but the vaccination cost ( ) and vaccine effectiveness () are modest (). The first row (a*) illustrates that infection dynamics respond to fractional order, specifically in the infected row. As the fractional order () climbs from zero to 1, infections first escalate, reaching a significant peak at before progressively diminishing. The last time snapshot () clearly illustrates a pronounced surge in infections at intermediate fractional orders, highlighting the key domain where memory effects, represented by fractional calculus, substantially enhance disease transmission. The vaccination profile (b*) has a comparable, although opposite, response to infection levels. Vaccination uptake incrementally rises with fractional order, notably accentuated in mid to upper fractional orders (). The final-time figure () indicates a significant increase in vaccination rates at these orders, suggesting that immunization becomes more appealing as the danger of disease becomes more apparent. The peak vaccination rate, however, diminishes significantly around , indicating saturation or equilibrium effects between perceived risk and preventive measures. The recovered graph (c*) exhibits a gradual, unidirectional rise with elevated fractional orders. Reduced fractional orders initially hinder recovery owing to constrained disease transmission. At intermediate fractional orders, recovery escalates rapidly as diseases reach their peak and immunization becomes widespread. As fractional orders increase, the final simulation time (300th day) figure () demonstrates stability at elevated recovery levels, indicating a balanced interplay of disease, vaccination, and resultant immunity. Average social payoffs (subfigure, d*) decrease with increasing fractional order, reaching substantially negative values at moderate levels (approximately ), indicating that the aggregate societal costs (infection, vaccine expenses, and disease-related hardship) are most significant under this crucial regime. At elevated fractional orders, albeit still negative, social payoffs stabilize, indicating an equilibrium where vaccination behavior sufficiently mitigates diseases to reduce societal burden, but still entails considerable costs. In conclusion, Fig. 8 elucidates how fractional dynamics (memory effects) significantly influence epidemic outcomes and behavioral responses. Intermediate fractional orders indicate a high-risk, high-cost scenario, necessitating tailored interventions, such as efficient vaccination programs, to mitigate peak infection loads and associated societal expenses.

Fig. 8.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Figure 9 depicts the influence of altering the fractional order () on a coupled behavioral-epidemic system governed by pure aspiration-driven imitation dynamics (i.e., imitation parameter ) with identical negative aspiration values for both vaccinators and non-vaccinators (). The scenario presumes a minimal vaccination cost () and elevated vaccine effectiveness (), establishing a context conducive to vaccination. Each row analyzes a distinct outcome variable across time and fractional orders, with the rightmost subplots (designated (iii)) representing the state at the final time point ().

Fig. 9.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure aspiration () social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

The first row of Fig. 9 (infected outcome, a*) indicates that infection prevalence is significantly affected by . For low values (), infection stays negligible owing to memory-driven inertia and gradual strategy adaptation, which restricts exposure. As approaches around 0.6, infection rates peak, indicating a critical threshold where memory effects wane, but aspiration-driven responses have not entirely curtailed the transmission. Furthermore, the incidence of infection diminishes swiftly with increased , indicating that quickly responding populations with little memory significantly mitigate infections when vaccination is both available and dependable. The final-day curve () with increasing fractional order has a distinct bell-shaped profile, indicating a transition threshold approximately at . The second row of vaccinated figures (b*) indicates that vaccine adoption consistently rises with . At very low , populations adapt insufficiently to recognize the advantages of vaccination, leading to negligible uptake. Once surpasses 0.3, vaccination rates increase rapidly and ultimately approach complete coverage for . This change marks a pivotal moment when aspiration, bolstered by affordability and excellent effectiveness, drives widespread vaccination use, particularly among vulnerable groups. Subfigure b(iii) validates this monotonic rise and demonstrates that a high guarantees herd immunity via behavioral modification. The third row (c*) indicates that recovery trends correspond with infection patterns. Recovery escalates with until about , coinciding with the peak of infections, and thereafter diminishes beyond this threshold as vaccination curtails new infections. This dynamic underscores the inevitable outcome of successful vaccination: as the number of infections decreases, the number of recoveries also decreases. Consequently, the apex of recovered individuals signifies optimal epidemic infiltration, beyond which the synergistic impact of aspiration-driven learning and vaccination effectiveness results in disease regulation and a decline in recovery figures. The last row of Fig. 9 (d*) offers insight into the socioeconomic ramifications. At low α, payoffs are almost neutral due to modest infection and immunization efforts. As grows, the system transitions into an expensive phase (), characterized by escalating infection rates, vaccination expenditures, and recovery loads, which significantly reduce the average payoffs. Furthermore, when infection rates decline and most of the population embraces vaccination, the social cost recovers marginally. It stabilizes at a mild negative level, illustrating that while intermediate α is the most turbulent and expensive, high produces satisfactory long-term results, despite considerable immunization expenses. In recaptulation, Fig. 9 demonstrates that fractional memory () substantially influences epidemiological and behavioral outcomes in pure aspiration. The system operates optimally at high , where rapid behavioral adaptation, facilitated by high vaccine effectiveness and low costs, ensures maximum vaccination, minimal infections, and stabilized payoffs. Intermediate α values constitute a significant bottleneck, as the population exhibits sufficient reactivity to alter behavior but lacks the rapidity necessary to avert epidemic peaks, resulting in the highest social cost. These results underscore the importance of developing prompt behavioral response mechanisms in public health initiatives.

Figures 10, 11 and 12 explain the impact of three different combinations of vaccination cost () and vaccine efficacy () on epidemic and behavioral dynamics under moderate imitation and aspiration () protocol. In this context, vaccinators and non-vaccinators possess identical aspiration levels (), however, the fractional order varies between 0 and 1. Each row in Figs. 10, 11 and 12 illustrates a distinct output: infection (a), vaccination (b), recovery (c), and social payback (d), including both time-series and final-day cross-sectional perspectives.

Fig. 10.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 11.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 12.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

The infection dynamics in all three Figures exhibit a bell-shaped pattern throughout . At low , infections are mitigated owing to prolonged memory and gradual behavioral adaptation. As escalates to intermediate levels (approximately 0.4–0.6), infections increase and reach a peak, indicating a tipping point where people begin to react to the disease but have not yet adopted effective immunization measures. Outside this range, infection rates decline significantly as behavior adjusts more swiftly. Contrasting several scenarios: Fig. 10 (a*) (): Illustrates the most pronounced infection peak, indicating that diminished effectiveness compromises behavioral regulation. Figure 11 (a*) (): Infection peaks are reduced and decline more rapidly owing to moderate vaccine efficacy and cost-effectiveness. Figure 12 (a*) (): Overall infection rates are minimal, even at intermediate , vaccination successfully curtails transmission. Vaccination rates have a positive correlation with . The combination of aspiration and imitation facilitates enhanced adoption, leading to a heightened response. Nonetheless, cost and effectiveness influence the steepness or breadth of this curve: Fig. 10 (b*): Elevated costs and diminished effectiveness hinder vaccination rates, even at high . The absorption levels stabilize at less than 50%. Figure 11 (b*): Moderate parameters enable uptake above 50%, especially for . Figure 12 (b*): Extensive vaccination is noted to increase with α sharply and stabilizes near a complete population at . The number of infections and vaccinations indirectly influences recovery patterns. The trajectory often aligns with infection: increasing towards intermediate , then decreasing as immunization ceases transmission. Figure 10 (c*): Enhanced recovery at mid α attributed to a substantial infection base. Figure 11 (c*): Equilibrium of ascension and descension, with recovery reaching a peak and thereafter declining smoothly. Figure 12 (c*): The recovery curve has a flatter trajectory and reduced infection rates, resulting from extensive early vaccination, which leads to fewer recoveries. Social payoff curves (d*) integrate health outcomes (infections and recoveries) with behavioral costs (vaccination efforts). These exhibit three distinctive regions: negligible reward at low (minimum alteration), a significant decrease at intermediate (elevated infection and strategic discord), and gradual recovery at elevated (disease mitigated, behavior steady). Figure 10 (d*): Most significant decline and slight recovery; elevated costs and ineffectiveness exacerbate social outcomes. Figure 11 (d*): Moderate decline; considerable social costs can be recuperated at a high . Figure 12 (d*): Optimal payout profile; early decline is recuperated, resulting in the most advantageous long-term outcome.

Finally, these three figures (Figs. 10, 11 and 12) demonstrate that fractional order () influences both vaccination cost () and effectiveness () in determining disease control, behavioral adoption, and total social benefit. Optimal results arise at elevated , especially when costs are minimal and vaccines demonstrate efficacy, facilitating the alignment of imitation-aspiration dynamics with public health objectives. Moderate values manifest as high-risk areas, where transitional conflicts result in the maximum infections and societal costs.

Figures 13, 14 and 15 illustrate the dynamics of epidemic transmission, vaccination behavior, recovery, and social reward under settings of pure imitation (), where people exclusively replicate the behaviors of others without aspiration-based assessment. Each figure represents a distinct scenario, varying vaccination cost () and vaccine effectiveness (), while accounting for negative aspiration (), illustrating the influence of fractional order (α, memory effect) on these results.

Fig. 13.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure imitation ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 14.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure imitation ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 15.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under pure imitation ( social learning with the symmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

In all three figures (Figs. 13, 14 and 15(a*)), the infection patterns are distinct: low infection frequency at small , abruptly increasing to a peak in the mid-range , followed by a decline at elevated values. Contrasting several scenarios: Fig. 13 (a*) () exhibits the greatest peak infection, indicating significant outbreaks attributable to elevated vaccination costs and reduced effectiveness. Even pure mimicry cannot adequately mitigate diseases, since vaccines are neither cost-effective nor protective. Figure 14 (a*) () illustrates mild infection peaks and a more rapid decline at higher fractional orders, indicating enhanced affordability and effectiveness, which aids in mitigating epidemics. Figure 15 (a*) () illustrates the minimal peak and rapid drop, emphasizing that high efficacy and cost-effectiveness significantly augment the impact of vaccination in infection control, even in scenarios of pure imitation. Vaccination rates under pure imitation have distinct peaks at intermediate fractional orders, followed by a minor decrease at high values. In several scenarios, Fig. 13 (b*) illustrates the minimal vaccination uptake attributable to expensive and inefficient vaccines, which diminishes the population’s motivation to adopt vaccination measures. Figure 14 (b*) illustrates a modest level of vaccine uptake, reaching a peak at mid- (about 0.7) when an improved equilibrium between cost and effectiveness makes imitation more appealing. Figure 15 (b*) demonstrates the best overall vaccine uptake, attributable to its low cost and high effectiveness, which effectively encourages people to engage in vaccination practices through imitation. Recovery graphs (c*) indicate the epidemic’s severity and the immunization patterns. Typically, recoveries increase swiftly at intermediate fractional orders, paralleling disease peaks, before stabilizing. Figure 13 (c*) exhibits the highest recovery rates at moderate , as an increase in infections inherently results in more recoveries. Figure 14 (c*) illustrates moderate recoveries, which peak and stabilize rapidly due to mild illnesses and enhanced vaccination efficacy. Figure 15 (c*) shows diminished recoveries overall, indicating fewer infections and effective immunization initiatives that are curbing disease transmission. Social payoffs (Figs. 13, 14 and 15 and d*) elucidate the aggregate social expenses of immunization and disease. Three unique regimes emerge across scenarios: Low fractional order, characterized by nearly neutral payoffs resulting from negligible behavioral or epidemiological alterations. Intermediate fractional order: significant declines indicate substantial societal costs stemming from extensive infections and vaccination expenditures, high fractional order: partial recovery or stabilization attributed to enhanced behavioral adaptation and diminished infection burden. In particular, Fig. 13 (d*) (high cost, low effectiveness) illustrates the most adverse outcomes, highlighting the socioeconomic strain of ineffective and costly immunization efforts. Figure 14 (d*) illustrates somewhat negative payoffs, emphasizing the advantages of enhanced vaccination efficacy and cost equilibrium. Figure 15 (d*) (cost-effective, high effectiveness) produces optimal payoff results, markedly reducing society expenses via effective disease management using imitation-based vaccination.

Finally, Figs. 13, 14 and 15 illustrate the essential interaction of fractional memory effects, vaccination effectiveness, cost, and imitation-driven behaviors in influencing epidemic dynamics. In pure imitation (), conduct is influenced mainly by observing others’ successful techniques. Elevated fractional orders, coupled with advantageous vaccination conditions (affordable pricing and high efficacy), yield optimal epidemiological and socioeconomic outcomes, highlighting the necessity of rendering vaccination appealing and accessible to effectively harness imitation-driven behaviors for disease control.

Figures 16, 17 and 18 scrutinize the synergistic effects of fractional-order dynamics, vaccination cost (), and vaccine effectiveness () under circumstances of moderate imitation and aspiration (), including an asymmetric aspiration scenario ( for vaccinators, for non-vaccinators). This asymmetry suggests that vaccinators have comparatively more positive ambitions (fewer negative ones), indicating a more hopeful outlook toward vaccination than non-vaccinators.

Fig. 16.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 17.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 18.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

In Figs. 16, 17 and 18, infection patterns exhibit notable similarities: minimal infections at lower fractional orders (), pronounced peaks at middle (roughly 0.4–0.6), and reductions at elevated fractional orders. In particular, Fig. 16 (a*) illustrates the most pronounced infection peak, indicating that expensive and low-efficacy immunization severely undermines behaviorally-driven epidemic management. The uneven aspiration somewhat reduces the effectiveness of vaccination because unfavorable views overshadow its attractiveness. Figure 17 (a*) illustrates a modest infection peak and a more rapid drop in infection with elevated , indicating enhanced but balanced cost-effectiveness. The somewhat elevated ambition of vaccinators facilitates quick uptake, hence decreasing infections. Figure 18 (a*) depicts the minimal infection peak, greatly influenced by increased vaccination effectiveness and reduced cost. The hopeful ambition of vaccinators significantly improves early vaccination uptake, markedly reducing disease transmission at elevated fractional levels. The fractional order and characteristics of the vaccine significantly influence the uptake of vaccination. Figure 16 (b*) vaccination scenario remains relatively low owing to elevated costs and inadequate effectiveness, despite the heightened ambitions of vaccinators. Peak uptake remained moderate (~ 0.35 at intermediate ), suggesting that aspiration asymmetry alone cannot entirely surmount significant immunization obstacles. Figure 17 (b*) shows a notable enhancement, with vaccination reaching a considerable peak at middle (~ 0.45 uptake). Equitable cost-effectiveness frameworks align with diverse aspirations to promote improved vaccination practices. Figure 18 (b*) illustrates the peak and most consistent vaccine uptake (~ 0.7 at elevated fractional orders). The combination of low cost, substantial effectiveness, and ambitious goals significantly motivates swift and extensive immunization efforts. The dynamics of recovery (c*) directly mirror infection patterns and are also affected by vaccination rates. Figure 16 (c*): Elevated recovery rates correspond with significant infection peaks, underscoring the repercussions of extensive infections in contexts of insufficient vaccination appeal. Figure 17 (c*): Moderate recoveries align with the balanced infection-vaccination scenario, signifying an intermediate disease load and regulated epidemic results. Figure 18 (c*): Depicts the lowest overall recovery levels, indicating a reduced infected population due to early and comprehensive vaccination efforts resulting from advantageous circumstances and optimistic expectations. Social payoffs (d*) illustrate the socioeconomic dynamics resulting from disease burdens and immunization expenses. Figure 16 (d*): The most adverse payoffs occur when widespread disease is combined with inefficient and expensive vaccination techniques, notwithstanding asymmetric aspirations. Figure 17 (d*): Moderate reward results; significant initial decreases at moderate , followed by a partial recovery at higher . Equitable cost-effectiveness and ambitions provide acceptable social loads. Figure 18 (d*) exhibits optimal payoff results, characterized by slight early declines and robust rebound towards negligible negative values. The optimal integration of low-cost, high-effectiveness, and favorable motivation among vaccinators reduces societal expenditures, clearly indicating the most advantageous societal outcome.

Therefore, Figs. 16, 17 and 18 illustrate that unequal ambition, especially elevated optimism among vaccinators, significantly influences vaccination rates and epidemic results. Advantageous vaccination circumstances (affordable pricing and good effectiveness) considerably enhance the beneficial effects of asymmetric desires, resulting in increased vaccination rates, a significant decrease in diseases and recovery burdens, and a substantial improvement in average societal benefits. Conversely, inadequate vaccination circumstances render even optimistic vaccinator intentions ineffective in mitigating elevated infection rates and societal consequences, highlighting the need to ensure vaccine accessibility and efficacy in conjunction with behavioral interventions.

Figures 19, 20 and 21 demonstrate the influence of epidemic and behavioral outcomes under moderate imitation and aspiration (), characterized by an asymmetric aspiration dynamic in which non-vaccinators () exhibit more optimism than vaccinators (). This shift in preference skews the population against immunization. The data fluctuates in vaccination cost () and vaccine efficacy (), examining the impact of fractional order () ranging from 0 to 1 on significant epidemiological and social consequences.

Fig. 19.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 20.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Fig. 21.

Dynamics (3D surface diagram, 2D phase diagram, line graph) of infected, vaccinated, recovered individuals, and average social payoff (ASP) analysis for fractional-order vaccination cost, and efficacy under symmetric imitation and aspiration ( social learning with the asymmetric vaccinated () and non-vaccinated () individuals’ aspiration level. The remaining parameters settings are .

Figure 19 (a*) (): Infection levels increase dramatically, reaching a maximum at , followed by a modest decrease. Due to the limited effectiveness and high expenses, individuals often choose not to vaccinate, despite initially intending to do so, driven by more optimistic expectations regarding vaccination. This results in a significant epidemic burden. Figure 20 (a*) (): The peak of the infection is diminished and has moved somewhat to the left (). Improved cost-effectiveness ratios enable imitation to reduce illnesses; yet, the disparity in aspirations continues to advantage non-vaccinators, hindering control efforts. Figure 21 (a*) (): Infections are very few overall. Despite non-vaccinators having more desires, the robust protective efficacy and affordability of vaccines facilitate imitation-driven vaccination, thereby reducing the incidence of illnesses. Regarding the vaccination scenarios, Fig. 19 (b*) shows that vaccination rates increase gradually with and stabilize below 40%. The diminished benefit of vaccines and increased costs, together with aspirational bias toward non-vaccinators, hinder adoption. Figure 20 (b*) demonstrates a more gradual, elevated vaccination trajectory, reaching around 75% at . Notwithstanding skewed ambitions, enhanced vaccination incentives promote some levels of uptake. Figure 21 (b*): Vaccination rises significantly and approaches near-universal levels (~ 95%) as grows. In this context, the significant effectiveness and low cost of vaccination supersede the behavioral bias against it. Figure 19 (c*): Recovery reflects infection levels, with significant peaks around . Widespread disease transmission requires natural immunity via recovery. Figure 20 (c*): Recovery increases substantially and stabilizes at as immunization mitigates infections. Reduced infection rates result in diminished recoveries. Figure 21 (c*): Recovery is negligible due to successful prophylaxis; most of the population entirely evades infection due to vaccination. Figure 19 (d*): Exhibits significantly adverse outcomes, reaching a maximum at moderate . The infection peak is substantial, and inadequate vaccination rates stemming from skewed aspirations result in significant social costs. Figure 20 (d*): Moderate adverse social payoffs. As vaccination rates increase and disease rates decrease, societal costs diminish markedly, particularly when exceeds 0.6. Figure 21 (d*): The optimal social result. Payoffs diminish somewhat before stabilizing at minimum negative values. Extensive vaccination and minor illness provide a socially viable equilibrium, notwithstanding preferences for non-vaccination.

Hence, Figs. 19, 20 and 21 illustrate that when aspiration asymmetry benefits non-vaccinators, managing an epidemic becomes much more challenging unless vaccinations are both efficacious and economically accessible. Unadulterated imitation (lacking personal assessment) compels individuals to emulate others, and if those others are non-vaccinators with elevated ambitions, the uptake of vaccines is impeded. Nonetheless, when vaccinations are affordable and efficacious (Fig. 21), imitation alone may overcome aspirational bias, resulting in superior health and social results. Consequently, vaccination promotion methods must either narrow aspiration gaps or create sufficiently favorable settings to enable imitation processes to foster protective behavior.

Conclusion

This study formulates a Caputo fractional-order vaccination model that integrates imitation and aspiration-driven decision-making behaviors into the dynamic process of epidemic transmission. By introducing a fractional-order derivative, the model effectively captures the memory-dependent nature of human behavior, allowing individuals’ past experiences and social influences to shape present vaccination choices. The analytical and numerical results demonstrate that the fractional order parameter serves as a critical measure of the population’s collective memory: higher α values indicate strong memory retention that moderates epidemic peaks but prolongs the persistence of infection, whereas lower α values reflect short-term behavioral responses that accelerate transitions yet can destabilize the epidemic equilibrium. Moreover, the study reveals that aspiration asymmetry, the mismatch between individuals’ expected and actual outcomes, significantly affects vaccination motivation. Populations with high aspiration gaps tend to exhibit reduced vaccination uptake, reflecting dissatisfaction or skepticism toward vaccine efficacy or accessibility. In contrast, imitation dynamics, where individuals mimic the behaviors of successful peers, promote conformity and amplify prevailing social norms. This effect highlights the importance of early and widespread vaccination adoption, as initial positive behaviors can spread through the population via social learning. The model offers crucial policy-relevant insights, emphasizing that vaccination campaigns should focus not only on vaccination efficacy but also on behavioral and psychological determinants. Strategies such as reducing vaccination costs, ensuring consistent vaccine effectiveness, and addressing aspiration imbalances through transparent communication are essential to sustaining public trust. Additionally, harnessing social imitation mechanisms, such as influencer-led campaigns or peer endorsement, can accelerate the diffusion of vaccination. Finally, incorporating memory-aware policies that recognize the population’s long-term behavioral inertia can enhance both vaccine coverage and the effectiveness of epidemic control.

Despite these insights, several limitations remain. The current model assumes a homogeneous contact structure and a single-pathogen framework, which may overlook the complexities of real-world social interactions and multi-strain epidemic processes. Furthermore, empirical estimation of fractional-order parameters, imitation strength, and aspiration thresholds remains difficult due to the scarcity of behavioral datasets that capture long-term memory and motivation dynamics. Social heterogeneity, misinformation, and cultural variability also pose additional challenges to accurate calibration.

Future research should focus on integrating empirical behavioral and epidemiological data to validate the behavioral dynamics and memory parameters of fractional-order systems. Extending the model to heterogeneous network topologies and multi-strain or co-infection settings will allow for more realistic characterization of population interactions and disease competition. Moreover, developing data-driven calibration techniques and hybrid approaches that combine fractional calculus with agent-based or network-based simulations will significantly enhance the model’s predictive accuracy and policy relevance. These efforts will strengthen the empirical foundation and broaden the applicability of fractional-order behavioral epidemic models as robust tools for forecasting and controlling epidemics.

Author contributions

Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation: Writing-original draft preparation: Mohammad Sharif Ullah; Conceptualization: K M Ariful Kabir; Writing-review and editing: Mohammad Sharif Ullah, K M Ariful Kabir, and Jin Wang; Supervision: Jin Wang. All authors have read and agreed to the submitted version of the manuscript.

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.