Optimizing information-driven awareness allocation for controlling activity-triggered epidemic spread

In the contemporary era, the advent of epidemics instigates a substantial upswing in relevant information dissemination, bolstering individuals’ resistance to infection by concurrently reducing activity contacts and reinforcing personal protective measures. To elucidate this intricate dynamics, we introduce a composite four-layer network model designed to capture the interplay among information-driven awareness, human activity, and epidemic spread, with a focus on the allocation of individuals’ limited attention in diminishing activity frequency and self-infection rates. One intriguing observation from our findings is an anomalous, concave non-monotonic relationship between awareness trade-off and epidemic spread, with a more pronounced prevalence at an intermediate least awareness efficacy. This underscores the inadvisability of relaxing self-protection through reduced activity frequency or compensating for increased activity frequency by enhancing self-protection. Especially noteworthy is the significance of enhancing self-protection in response to heightened information dissemination and inherent activity demands to curtail infection risk. However, in scenarios with increasing ancillary activity frequency, the emphasis should exclusively shift towards reducing activity exposure. The model establishes a theoretical threshold for accurately predicting awareness efficacy in epidemic outbreaks. Optimal awareness allocation consistently resides at the extremes—either completely avoiding unnecessary activity contact or adopting full self-protection. This guidance, contingent on information level and activity demand, offers valuable insights into the delicate balance between individual behaviors and epidemic prevention.


Introduction
The global impact of COVID-19, even in its early stages, has left an indelible mark on the world, creating wounds that are not easily healed due to its profound effects on both global life and the economy [1,2].Consequently, there has been a growing recognition of the paramount significance of research on models and methods closely associated with the real-world spread of epidemics [3][4][5][6].Within the domain of complex networks, the study of spread dynamics, as a crucial branch, employs mathematical models to depict the intricate interplay between individuals and the dynamic evolution of their states based on the structural connections formed through physical abstraction [7].This approach provides a promising framework for comprehending the underlying mechanisms of large-scale epidemics [8][9][10].Through both theoretical and experimental analyses [11][12][13], it becomes possible to effectively reveal the essence and characteristics of epidemic transmission within complex systems influenced by social interactions.This encompasses aspects such as propagation speed, infection range, and outbreak thresholds, thereby offering robust guidance for predicting and controlling real epidemic transmissions.[14,15].
The emergence of multi-layer network theory has broadened our understanding of the intricate factors influencing the spread of epidemics [16][17][18].Within this framework, each layer encapsulates a distinct dynamic process, with their interactions characterized by inter-layer connections [19,20].A notable application of this theory involves the examination of individuals with heightened awareness of epidemics, as they typically adopt self-protective behaviors to mitigate infection risks [21][22][23][24].Given that human consciousness is predominantly shaped by received information, certain studies delve deeper into understanding the properties of epidemic propagation under the heterogeneity of awareness diffusion stemming from differences in the attributes of information flow [25][26][27].In addition, the resource allocation induced by information-driven awareness dissemination has also been proven to play a pivotal role in the spread and suppression of epidemics [28,29].An intriguing outcome is the revelation that irrational acquisition and hoarding behavior can result in a shortage of resources for those in urgent need, thereby exacerbating the spread of diseases [24].Also, evolutionary game theory provides a powerful framework to study how individual behaviors, such as vaccination strategies and information dissemination, evolve over time and impact epidemic dynamics [30,31].In tandem with these investigations, the scope of work in this field has expanded to encompass real and complex epidemic spreading modes [20], awareness diffusion mechanisms [32], and multi-layer structures [33,34], elevating the coevolution dynamics of epidemics and awareness to a prominent topic of exploration.
In the unfolding of real epidemics, a logical interplay among information, awareness, activity, and the epidemic becomes evident, as depicted in figure 1(a).In particular, physical contacts generated through activities between individuals can trigger epidemic spread, while relevant epidemic information disseminated through multimedia is exchanged among individuals, propelling the diffusion of epidemic prevention awareness [35].An individual with heightened awareness significantly lowers their risk of infection through two primary avenues: fortifying self-protection measures against the epidemic, including the consistent use of masks and frequent disinfection, and curtailing the frequency of physical contact activities like outdoor exercise, shopping, and dining.Concurrently, an individual's infection status reciprocally shapes their awareness of the epidemic.This prompts a critical inquiry: when individuals, motivated by the pursuit of comfort and convenience or hampered by a diminishing endurance, experience a limited attention span towards epidemics, which specific aspects of protective measures should take precedence for heightened awareness?The necessity of establishing a procedural mechanism that systematically captures the awareness trade-off between day-to-day activities and potential epidemics becomes imperative.
Motivated by the above analysis, we have developed a composite four-layer network designed to model the intricate coevolution dynamics underlying information-driven awareness diffusion and activity-triggered epidemic spreading.In our framework, individuals with higher activity frequencies face an increased risk of infection, with their self-protection measures determining the probability of infection for each activity contact.The focal point of our model lies in the quantitative consideration of the trade-off of awareness, characterized by a fractional value of awareness efficacy.This trade-off encompasses the delicate balance between reducing activity frequency and enhancing self-protection to mitigate infection probability.Remarkably, our observations reveal that epidemic prevalence is most likely when individuals ease their self-protection due to reduced activity frequency or vice versa, and least likely when they either significantly curtail their activity or intensify their protective measures.The specific outcome depends on the information level and frequency of activity.Our research underscores, through quantitative allocation, the utmost importance of optimizing prevention measures dictated by epidemic-related awareness.This provides valuable guidance for information dissemination and governmental management to efficiently control the epidemic.

Model
We employ a foundational four-layer multiplex network, illustrated in figure 1(b), with N individuals representing nodes in each layer.These layers encompass dynamics related to information exchange, awareness diffusion, activity contact, and epidemic spread.Given that information-driven awareness primarily occurs through online social networks, whereas activity-triggered epidemic spreading takes place in offline networks, we establish a unified network topology for the activity and epidemic layers.Additionally, we introduce additional random links, totaling M, to form a unified topology for the information and awareness layers.This configuration reflects the prevalence of online socializing over offline engagement.For analytical purposes, let A = a ij and B = b ij denote the adjacency matrices of the information (awareness) layer and the activity (epidemic) layer, respectively.
In the information layer, each individual acquires r pieces of the latest disease-related information per time step.Acknowledging information replicability, individuals probabilistically share each piece with their neighbors, governed by the parameter φ.In practice, this process can be effectively measured using digital technologies to monitor and quantify information exchanges on social media platforms, official health websites, and through public health campaigns.Due to timeliness considerations, information from the preceding moment is purged at the subsequent moment.Within the awareness layer, we integrate the standard unaware-aware-unaware (UAU) model [22] with information exchange dynamics to simulate awareness diffusion.Typically, individuals' attitudes toward disease-related information hinge on their current awareness status.An aware individual transmits one piece of information to an unaware neighbor, assimilating them into awareness with a probability of α.However, aware individuals may revert to unawareness at each time step with a probability of δ, attributed to factors like forgetfulness or waning patience.
To quantify activity contact, we introduce λ 0 and λ as the inherent and ancillary activity frequencies, representing necessary activities related to work or learning and optional activities like entertainment, shopping, and exercise, respectively.At each time step, individuals engage in each activity by randomly selecting a neighbor, thereby triggering potential epidemic propagation between them.In the epidemic layer, we apply the classical susceptible-infected-susceptible (SIS) model [14] to depict the epidemic spreading process.Without awareness interference, when every activity occurs between infected and susceptible individuals, the latter become infected with a probability β U = β.Simultaneously, assuming medical treatment or self-healing, infected individuals recover to a susceptible state with a probability of µ at each time step.
Indeed, an aware individual typically adjusts their behavior by either reducing ancillary activity frequency or enhancing self-protection during activity exposure to mitigate infection risk.Considering the limited attention, we introduce the parameter η (0 ⩽ η ⩽ 1) to characterize the efficacy of awareness in these two aspects.Specifically, if an individual is aware, they will decrease the ancillary activity frequency to λ A = ηλ and the infection probability to β A = (1 − η)β.Clearly, η quantifies the trade-off of aware individuals in measures to prevent epidemic infections.A higher η suggests that aware individuals are more inclined to bolster self-protection to reduce infection probability, whereas, conversely, they seek to minimize ancillary activity.Additionally, neglecting the presence of asymptomatic infections and the delay in symptom presentation, susceptible individuals immediately become aware of the epidemic upon infection.
For convenience, we have summarized all definitions used in our information-driven awareness-activityepidemic coevolution model, as illustrated in table 1.

Theoretical analysis
In order to achieve a comprehensive understanding of epidemic characteristics, we embark on a theoretical analysis of infection size and adeptly forecast the outbreak threshold.This endeavor relies on the formulation of a microscopic Markov chain approach (MMCA) specifically tailored to adapt to our coevolution model.

Coevolution dynamics
Considering the integration of awareness and epidemic status, all individuals can be categorized as either unaware and susceptible (US), aware and susceptible (AS), or aware and infected (AI).To analyze the diffusion of awareness, it is imperative to assess the information flow driving it.For any pair of connected nodes, the average number of information units received by individual i from neighbor j is represented by: r ji = rφ. (1) Subsequently, the probability that an unaware individual i will be influenced by neighbor j to become aware is defined as: where p A j (t) = p AS j (t) + p AI j (t) represents the probability that neighbor j is aware at time t.Expanding this to all neighbors, the probability of any unaware individual i becoming aware is: The spread of epidemics is propelled by activity contact, which is also disturbed by awareness.From a quantitative perspective, the total activity frequency between unaware individuals and aware individuals is distinguished.For an unaware individual j, the total frequency of their activities is λ U j = λ 0 + λ.When individual j becomes aware, they reduce the frequency of unnecessary ancillary activities from λ to ηλ, resulting in a total activity frequency of λ A j = λ 0 + ηλ.Let λ U ji and λ A ji denote the average number of activities flowing from neighbor j to individual i when neighbor j is unaware and aware, respectively.These can be calculated as follows: where k j represents the degree of neighbor j in the activity layer, respectively.Given that mutual physical contact occurs regardless of the initiator of the activity, providing a potential avenue for the spread of the epidemic, we express the probabilities β U ji (t) and β A ji (t) of being infected by neighbor j when susceptible individual i is unaware and aware as follows: Excluding the case of being infected by all neighbors, the probabilities β U i (t) and β A i (t) of individual i being infected when in an unaware state and aware state can be obtained as: Having established the probabilities of information-driven awareness diffusion and activity-driven epidemic spreading, we can perceive the transition of epidemic and awareness states as a continuous Markov process to predict the evolution dynamics of individual states.Defining the probability of an individual i transitioning from state XY (XY = US, AS or AI) to state WZ (WZ = US, AS or AI) at time t as q XY→WZ i (t), we can approximate it as q XY→WZ i (t) = q Y→Z i|X (t)q X→W i|Z (t) by neglecting the dynamical correlations of state evolution between awareness and epidemic.Here, q Y→Z i|X (t) (q X→W i|Z (t)) represents the probability of epidemic (awareness) state transitioning from Y (X) to Z (W) when the awareness (epidemic) state of individual i is known to be X (Z).As a result, the transition probability of an individual i among three states at time t can be obtained in figure 2.
Subsequently, the infinitesimal generator Q i (t) of the Markov chain among three awareness-epidemic coevolution states can be represented as: ) ) ) Building upon this, we derive the dynamical equations of state evolution as follows: These equations enable the accurate evaluation of awareness size p ) and epidemic size p within the network system.

Epidemic threshold
Advancing further, we delve deeper into the epidemic threshold, a pivotal performance indicator for assessing the system's resilience against epidemic outbreaks.To theoretically predict it, we analyze the system's state upon achieving stability, expressed as: for i = 1, . . ., N. As the probability β of epidemic transmission approaches the infinitesimally close threshold β c , only a marginal fraction of individuals in the system becomes infected.Assuming we combine them with equation ( 14) to rewrite equation (13) as With the Taylor approximation, the linearizations of (1 β A can be achieved.Integrated with equations ( 8) and ( 9), we have: Substituting them into equation (15), we can obtain the simplified expression as follows: where w ji is the Kronecker delta symbol, and To ascertain the critical outbreak, the epidemic threshold must satisfy: Here, Λ max (L) denotes the maximum eigenvalue of the matrix L, a square matrix composed of b ji h ji .According to equations ( 19) and ( 20), we observe that the epidemic threshold β c is intricately linked to activity frequency, the level of information-driven awareness diffusion, and the heterogeneous network structure.
To vividly illustrate the impact of the awareness trade-off, we consider any homogeneous network structures, such as random or random-regular networks, as the backbone of each layer to analyze the epidemic threshold theoretically.In this scenario, with k j = k i = ⟨k⟩ and p A i = p A introduced into equation ( 19), h ji can be simplified as: As a result integrated with equation (20), Focusing on the role of awareness efficacy, we find: Given that B is a real non-negative symmetric matrix, the Perron-Frobenius theorem assures that Λ max (B) > 0. It is evident that β c may evolve non-monotonically with η.As ∂βc ∂η = 0, the critical point, which corresponds to least awareness efficacy, can be explicitly expressed as: Here, β c decreases with η when η < η lea , and conversely, β c increases with η when η > η lea .Consequently, η lea represents the awareness trade-off at its least efficient, most likely leading to an epidemic outbreak.The most effective awareness efficacy in inhabiting epidemic outbreaks must lie at the two extremes.According to equation ( 22), the thresholds they produce are , respectively.Through comparison, we deduce: Clearly, η lea and η mos hinge on the proportion p A of aware individuals, determined by information level r and φ, inherent activity frequency λ 0 and ancillary activity frequency λ.To examine this, we use equation (20) to identify η lea and η mos that yield the minimum and maximum epidemic thresholds, as illustrated in figure 3.
From figures 3(a) and (c), it is evident that η lea decreases as both r and φ increase, resulting in a higher awareness level p A , which is similarly observed for λ 0 , while it increases with λ.A distinct boundary between the two extremes of most awareness efficacy is apparent in figures 3(b) and (d), where fully dedicating oneself to self-protection measures (η = 1) emerges as the optimal approach under conditions of high information exchange and inherent activity frequency, while entirely avoiding unnecessary physical exposure (η = 0) becomes more urgent for high ancillary activity demands.The qualitative alignment with equations ( 24) and ( 25) underscores the importance of adaptively tailoring awareness-driven strategies to counter epidemics, taking into account both the level of information exchange and the frequency of activities in real-world situations.

Numerical simulations and discussion
To empirically validate the efficacy of our theoretical framework in describing the coevolution state of individuals and the epidemic threshold of systems, extensive numerical experiments are conducted using the Monte Carlo (MC) simulation method.If not otherwise mentioned, we envision a social system composed of N = 5000 individuals, employing the uncorrelated configuration model [36] to connect them, forming a SF network with a power-law degree distribution for the structure of the activity and epidemic layers.Additionally, M = N random edges are added to create the structure of the information and awareness layers, where self-connections and multiple edges between layers are excluded.In each simulation, the initial state involves no epidemic, with all individuals being set as unaware and susceptible (US).Then, each individual generates λ 0 inherent activities and λ ancillary activities at each time step, and each activity randomly selects neighbors to engage in physical contact together.This process iterates over sufficient time steps until the system reaches stability.Subsequently, to model an epidemic outbreak, a randomly chosen proportion ρ = 0.1 of individuals are set as aware and infected (AI), which ensures that the initial state of the system is sufficiently random.From then on, at each time step, each individual receives r pieces of epidemic information and shares each piece of information with each neighbor with a probability of φ.Thereafter, the dissemination of awareness driven by information flow and the dynamics of awareness-activity-epidemic coevolution proceed synchronously within the multiplex system.Following a prolonged transition period, where micro-level interactions among individuals reach equilibrium, a macro stability of the system state is achieved.The results, including awareness and epidemic sizes, are averaged over this stable time period.Multiple realizations are conducted to obtain ensemble averages of ρ I 2 and ρ I .By calculating the susceptibility according to the expression and identifying the spreading probability β corresponding to its maximum value, we determine the epidemic threshold β c [11].

General results of coevolution dynamics
Considering the imperative concern for the stable dynamic characteristics of the system, a crucial statistical indicator reflecting this is the size distribution of individuals in each state.As shown in figure 4, we present the evolutions of the proportions p A and p I for aware and infected individuals, respectively, as the probability β of epidemic transmission varies for different probabilities α of awareness diffusion.Notably, an evident epidemic threshold emerges, beyond which the infection size becomes non-zero, signifying the prevalence of epidemics.Furthermore, as the probability of awareness diffusion increases, a greater number of individuals transition into the aware state, exerting a mild inhibitory effect on the epidemic.This is primarily manifested in a marginal elevation of the epidemic threshold.Importantly, the MC simulation results exhibit close agreement with the theoretical evaluations based on equations ( 11)-( 13) obtained by the MMCA, with an average relative error of 0.7%, 0.1%, 0.1%, and 0.1% for p A , and 1.8%, 0.2%, 1.7%, and 2.2% for p I in figures 4(a)-(d), respectively.This robust agreement provides compelling evidence for the accuracy of our theoretical calculations concerning the information-driven awareness-activity-epidemic coevolution dynamics.

Anomalous role of awareness efficacy
Awareness, functioning as individual decision-making, necessitates thorough exploration for controlling and optimizing the spread of epidemics through quantitative evaluation.As shown in figure 5, we plot the evolutions of proportions p I representing infected individuals concerning awareness efficacy η for various probabilities β of epidemic transmission.Strikingly, a unified qualitative upward trend emerges, where p I initially increases and subsequently decreases with η.Of particular note is the presence of a critical intermediate awareness efficacy that maximizes the epidemic size, while at the extremes of awareness efficacy, the prevalence of the epidemic diminishes.From a quantitative standpoint, the non-monotonic trend between p I and η exhibits a discernible shift towards larger values of η with increasing β.As a result, the critical awareness efficacy that maximizes p I decreases as β rises.On the other hand, a higher η proves more advantageous in significantly alleviating p I in the context of a larger β.This observation underscores the critical importance of optimizing awareness allocation, especially as the infectivity of the epidemic intensifies.In such circumstances, reinforcing self-protection emerges as notably more effective in suppressing epidemic spread than simply reducing activity frequency.These findings accentuate the nuanced dynamics of awareness trade-offs and their pivotal role in epidemic control, especially in scenarios of heightened contagion.
To elucidate the impact of awareness allocation on epidemic outbreak, figure 6 illustrates the epidemic threshold β c as a function of η for different epidemic recovery probabilities µ under two different network sizes N.While variations in µ can lead to significant changes in β c quantitatively, as indicated by the direct ratio in equation ( 20), the more crucial observation is the consistent non-monotonic trend, decreasing first and then increasing, across all µ and N.This discernible trend becomes more pronounced with higher values of µ.Additionally, a decreasing trend of β c with increasing N for different µ is evident, consistent with previous findings on classical epidemic processes [11].Remarkably, an intermediate least awareness efficacy exists, leading to the minimum epidemic threshold, while the most awareness efficacy associated with the maximum epidemic threshold is situated at the two extremes of the awareness trade-off.The practical significance of this intriguing finding lies in the recognition that neither an increase in activity frequency resulting from strengthened self-protection nor a relaxation of self-protection due to reduced activity frequency is conducive to curtailing the spread of epidemics.In contrast, prioritizing one of these measures exclusively may be more effective in suppressing epidemic outbreaks.The threshold predicted by the theoretical approach aligns qualitatively with the simulation results.Notably, the slightly reduced differences between the theoretical and numerical values with increasing network size N indicate that our analytical method becomes more accurate for larger network systems [35,37].Any quantitative deviations between the two can be attributed to the higher-order approximation in the solving process and the neglect of coevolutionary dynamic correlations in the multiplex network system.

Effect of information level
Information exchange, as the driving force behind awareness-activity-epidemic coevolution, can significantly influence the interplay between epidemic spread and awareness trade-off.To reveal this, we investigate the variations in awareness size p A and epidemic size p I with awareness efficacy η under different combinations of information generation rate r and transmission probability φ.As depicted in figure 7, a larger β leads to a higher proportion of infected individuals, resulting in larger p A and p I .Additionally, for various epidemic probabilities β, the trend in p A aligns with its corresponding trend in p I , both initially increasing and subsequently decreasing with η.An intermediate η, where individuals reduce a portion of their activity frequency while increasing their self-protection, peaks the epidemic, which is not advisable.The results based on the MMCA equations ( 11)-( 13) continue to align well with the simulation results, validating the efficacy of our analytical approach across different information levels.
Focusing on epidemic outbreak dynamics, we compare the evolution of the epidemic threshold β c with awareness efficacy η across two information levels and recovery probabilities of individual awareness in figure 8.The non-monotonic relationship between β c and η persists across various combinations of r, φ and δ, a consistency evident in both theoretical predictions and numerical simulations.Notably, the epidemic threshold achieves its minimum value, β min c , at the least awareness efficacy η lea -a value contingent on the information level parameters r, φ and δ.A higher epidemic threshold enhances the system's resilience against epidemic outbreaks, prompting an exploration of the optimization of η to improve β c .The results reveal that  ) for the epidemic threshold.Each data point corresponds to the MC numerical epidemic threshold, and the corresponding color curve denotes the theoretical prediction based on equation (20).Other parameters are α = 0.2, λ0 = 2, λ = 6 and µ = 0.8. the threshold reaches its maximum value, β max c , at a specific most awareness efficacy η mos .Intriguingly, η mos consistently resides at either extreme, η mos = 0 or η mos = 1, contingent on the information level and awareness recovery.A comparison between figures 8(a)-(d) highlights that, for a fixed η, an increase in the information level or an decrease in individual awareness restoration suppresses epidemic outbreaks by elevating the epidemic threshold.
For a nuanced assessment of awareness efficacy, we scrutinize the variations in η lea , η mos , and their corresponding epidemic thresholds β min c , β min c with varying r for both low φ = 0.2 and high φ = 0.4, as depicted in figure 9. Notably, η lea exhibits a monotonic decrease with increasing r, whereas η mos transitions Figure Least most awareness efficacies η lea ηmos, alongside their corresponding minimum and maximum epidemic thresholds β min and βmax, as functions of information generation rate r for different information transmission probabilities φ: (a) η lea and β min vs r when φ = 0.2; (b) ηmos and βmax vs r when φ = 0.2; (c) η lea and β min vs r when φ = 0.4; (d) ηmos and βmax vs r when φ = 0.4.Each data point is obtained through MC numerical experiments, and the corresponding color curve represents a theoretical prediction based on equation (20).Other parameters include α = 0.2, δ = 0.4, λ0 = 2, λ = 6 and µ = 0.8.from 0 to 1 with rising r.This shift is attributed to the heightened awareness among individuals, resulting in a larger p A and a subsequent reduction in the least awareness efficacy η lea as per equation ( 8) and the transition of η mos as per equation ( 9).This observation underscores that, with an escalation in information dissemination, individuals should redirect their focus from reducing activity frequency to enhancing self-protection, thereby more effectively curbing epidemic prevalence.Moreover, an escalation in both information generating rate r and transmission probability φ contributes to an enhanced epidemic threshold for the system-be it β min c or β max c .This implies that reinforcing information dissemination during an epidemic outbreak proves highly beneficial.The commendable consistency between simulation results and theoretical predictions affirms the rationality of our evaluation regarding the impact of information level.

Effect of activity frequency
Activity, while being disturbed by awareness, also plays a role in driving the spread of epidemics, making its frequency a crucial determinant of the effectiveness of awareness efficacy.In this regard, figure 10 reports the variation of awareness size p A and epidemic size p I with awareness efficacy η for different combinations of inherent activity frequency λ 0 and ancillary activity frequency λ, where the theoretical predictions have also been well validated by simulation results.Obviously, the non-monotonic impact of awareness efficacy on epidemics still exists for different transmission probabilities β, and such a non-monotonic amplitude varies with the two frequencies of activity.It is worth noting a special case where, with λ 0 = 3 and λ = 3, both p A and p I exhibit a monotonically decreasing trend with η.This reflects, to some extent, that the criticality and optimality of awareness efficiency are significantly constrained by the frequency of activities.
Turning the attention to the epidemic threshold, we explore the relationship between β c and η for different λ 0 and λ.As shown in figure 11, one can find that for different combinations of inherent activity frequency λ 0 and ancillary activity frequency λ, there are still minimum β min c and maximum β max c of epidemic threshold.These extremal values are attained at the least efficient awareness efficacy η lea and most efficient awareness efficacy η mos .While η lea is influenced by λ 0 and λ, η mos consistently resides at both ends of the awareness efficacy spectrum.Theoretical analysis has also demonstrated accurate predictions for epidemic thresholds, affirming the intertwined relationship between awareness efficiency and activity frequency.
Moving forward, we delve deeper into the influence of the two activity frequencies on critical awareness efficiency and the corresponding extremum of the epidemic threshold.Figures 12(a .The results reveal that with an increase in λ 0 , η lea consistently decreases, while η mos transitions from 0 to 1. Conversely, the evolution of η lea and η mos with λ  opposes that with λ 0 , aligning with the analysis of equations ( 24) and (25).As depicted in figures 12(c) and (d), η lea exhibits an increasing trend that eventually stabilizes with λ, prompting η mos to shift from 1 to 0. However, for both λ 0 and λ, their increase leads to a decrease in the two threshold extremes β min c and β max c .This aligns with the understanding that elevated activity frequency corresponds to increased physical contact and a heightened risk of infection.The theoretical predictions exhibit excellent agreement with numerical simulations, showcasing the adaptability of our analytical approach to activity frequency.These findings emphasize that reducing activity frequency is pivotal in resisting epidemics, and optimizing awareness efficacy on this foundation can further impede the spread of epidemics.In particular, when individuals have a high inherent frequency of activity, reinforcing self-protection becomes crucial.Simultaneously, with an increase in the frequency of ancillary activities, a focus on reducing unnecessary activity exposure becomes more imperative.

Results on random multiplex networks
To promote the generalizability of our findings, we replace the heterogeneous SF layer structure with a homogeneous random layer structure generated the Erdős-Rényi (ER) model [38] with an average degree of 4 in our proposed information-driven awareness-activity-epidemic coevolution dynamics.Firstly, we compare the MC simulation results with the MMCA concerning the proportion p I of infected individuals in the combined plane of epidemic transmission probability β and awareness efficacy η.As depicted in figures 13(a) and (b), not only do the results exhibit visually consistent performance, but we also conduct quantitative testing, revealing a mere 0.6% relative error between the two.This reaffirms the accuracy and robustness of our MMCA prediction even when accounting for variations in network topology.Building on this, we employ the theoretical expression in equation (20) to analyze the effects of awareness efficacy on the epidemic threshold for ER multiplex networks.Figures 13(c) and (d) illustrate β c as a function of η under different information levels.Notably, increases in information generation rate r and transmission probability φ prove beneficial for suppressing epidemic outbreaks.For small r and φ, β c gradually decreases with η until reaching stability.However, with the increase of r and φ, β c exhibits a more pronounced non-monotonic trend, reaching its minimum at a certain least awareness efficacy, which decreases with r and φ.Regarding activity frequency, a reduction significantly enhances the system's ability to resist epidemics, as reflected in higher epidemic thresholds.As shown in figures 13(e) and (f), similar non-monotonic behaviors are observed between the epidemic threshold and awareness efficacy for different inherent activity frequencies λ 0 and ancillary activity frequencies λ.The least awareness efficacy decreases with λ 0 , while increasing with λ.Similarly, the most awareness efficacy η mos remains situated at the two extremes of awareness allocation.These consistent results across SF and ER multiplex networks indicate that while improving epidemic-related information dissemination and reducing activity frequency are crucial, optimizing awareness allocation can further enhance the system's resilience against epidemic prevalence.

Conclusion
In the real world, the impact of information awareness on epidemic spread involves two key aspects: direct influence on individual self-protection and indirect effects on epidemic transmission by shaping individual activities that lead to physical contact.This paper introduces an information-driven awareness-activity-epidemic coevolution dynamic model operating on a composite four-layer multiplex network.By quantifying awareness efficacy in individual physical contact and self-protection, reflected through reductions in activity frequency and lower infection probabilities, respectively, the study focuses on optimizing epidemic control through the awareness trade-off.
The results yield intriguing insights.Firstly, a non-monotonic relationship emerges between epidemic spread and the awareness trade-off.Slight relaxation of self-protection awareness or increased activity frequency can lead to heightened epidemic prevalence.In contrast, channeling attention exclusively to either reducing activity frequency or enhancing self-protection significantly suppresses the epidemic.Secondly, the study delves into the interplay of information level and activity frequency on the awareness trade-off.Qualitatively, heightened information exchange and inherent activity frequency call for prioritizing self-protection, while increased ancillary activity frequency demands a focus on reducing unnecessary physical contact.Quantitatively, the evolution of least and most efficient awareness allocations in resisting epidemics is determined, providing crucial guidance for government information dissemination and individual behavior during epidemics.Thirdly, leveraging the MMCA, the study accurately predicts the system's infection size.Based on this, it derives a theoretical expression for the epidemic threshold, validated through extensive MC simulations, supporting the evaluation of awareness efficacy on both heterogeneous SF multiplex networks and homogeneous ER multiplex networks.
The simplicity and effectiveness of our proposed model enable it to capture the essence of mutual interference between individual information exchange, awareness dissemination, activity contact, and epidemic spread, further shedding some light on the prediction and analysis of epidemic transmission dynamics in reality.The findings highlight the importance of directing awareness toward a specific aspect-either activity contact or self-protection.This insight holds valuable implications in navigating the delicate balance between personal life and public health.

Figure 1 .
Figure 1.Schematic illustration of epidemic spread.(a) Logic of mutual interference between information, awareness, activity, and epidemic: Information drives awareness diffusion, activity triggers epidemic spread, awareness acts on both activity and epidemic, and the epidemic also affects the awareness state of individuals.(b) A four-layer multiplex network model characterizing the dynamic logic interference mentioned above: Layer 1-Information exchange, Layer 2-Awareness diffusion, Layer 3-Activity contact, and Layer 4-Epidemic spread.

Figure 2 .
Figure 2. Schematic representation of the transitions between individual states and their associated rates.

Figure 3 .
Figure 3. Phase diagrams depicting (a) the least awareness efficacy η lea and (b) the most awareness efficacy ηmos in the plane defined by the information generation rate r and transmission probability φ.Additionally, phase diagrams illustrating (c) the least awareness efficacy η lea and (d) the most awareness efficacy ηmos in the plane defined by the inherent activity frequency λ0 and ancillary activity frequency λ.It is important to note that the color gradient observed in the boundaries of regions in (b) and (d) is a result of varying fillings and does not accurately represent the true values.This gradient will disappear as data points continue to refine, causing ηmos to transition abruptly from 0 to 1.The same scale-free (SF) multiplex networks used in subsequent numerical experiments are adopted, with parameters, including λ0 = 2 and λ = 6 in (a) and (b), and r = 2 and φ = 0.5 in (c) and (d).The values of α = 0.2, δ = 0.4, and µ = 0.8 remain consistent across all figures.

7 .
Proportions p A and p I of aware individuals and infected individuals as functions of awareness efficacy η for different probabilities β of epidemic transmission under two different information levels: (a) p A vs η when r = 1 and φ = 0.2; (b) p I vs η when r = 1 and φ = 0.2; (c) p A vs η when r = 2 and φ = 0.4; (d) p I vs η when r = 2 and φ = 0.4.Each data point is obtained by averaging the results of 30 MC realizations, and the corresponding color curve represents the theoretical prediction of the MMCA.Other parameters include α = 0.2, δ = 0.4, λ0 = 2, λ = 6, and µ = 0.8.

Figure 10 .
Figure 10.Proportions p A and p I of aware individuals and infected individuals as functions of awareness efficacy η for different combinations of inherent activity frequency λ0 and ancillary activity frequency probabilities λ under two different probabilities β of epidemic transmission: (a) p A vs η when β = 0.2; (b) p I vs η when β = 0.2; (c) p A vs η when β = 0.3; (d) p I vs η when β = 0.3.Each data point is obtained by averaging the results of 30 MC realizations, and the corresponding color curve represents the theoretical prediction of the MMCA.Other parameters include r = 2, φ = 0.5, α = 0.2, δ = 0.4 and µ = 0.8.

Figure 13 .
Figure 13.Results on the ER multiplex networks: (a) Phase diagram of epidemic size p I obtained by MC simulations in the plane of epidemic transmission probability β and awareness efficacy η, where the number of points is 20 × 20, and each point is the result averaged over 30 realizations; (b) Phase diagram of p I obtained by the MMCA in the plane of epidemic transmission probability β and awareness efficacy η; (c) Epidemic threshold βc vs awareness efficacy η for different information generation rates r when information transmission probability φ = 0.4 ; (d) Epidemic threshold βc vs awareness efficacy η for different information transmission probabilities φ when information generation rate r = 2; (e) Epidemic threshold βc vs awareness efficacy η for different inherent activity frequencies λ0 when ancillary activity frequency λ = 6; (f) Epidemic threshold βc vs awareness efficacy η for different ancillary activity frequencies λ when inherent activity frequency λ0 = 2.In (a) and (b), r = 2, φ = 0.5, λ0 = 2 and λ = 6.In (c) and (d), λ0 = 2 and λ = 6.In (e) and (f), r = 2 and φ = 0.5.(c)-(f) are the theoretical results obtained by equation (20), where the solid point indicated by the arrow is the least awareness efficacy η lea that produces the minimum β min c of epidemic threshold.Other parameters include α = 0.2, δ = 0.4 and µ = 0.8.

Table 1 .
The probabilities of individual i (i = 1, . . ., N) being in US, AS, and AI states at time t are denoted as p US Summary of definitions in the model.
i (t), p AS i (t), and p AI i (t), respectively, with the constraint that p US i (t) + p AS i (t) + p AI i (t) = 1.