Potential Consequence of Interconnected Intervention against Systemic Risk (COVID-19) via a Model-Driven Network-Agent Dynamic

Tis study estimates the consequences of risk propagation, such as that of COVID-19, using network-agent dynamics. Given several scenarios, the network-agent model provides critical insights into infection risk using a model-driven approach to interconnected interventions. Te simulation results suggest that employing a nonevolutionary governing structure with evolutionary individual interaction parameters guided by testing can help suppress outbreaks to levels below the standard critical-care capacity. Furthermore, setting the protection level as the macroscale and the shrinking of individual interactions as the microscale, the efects of social distancing on transmission rates are refected in the disease model. In addition, the parameters that refect the best feasible scenarios can be determined. Tese fndings are relevant to COVID-19 pandemic policies wherein interconnected interventions reduce the socioeconomic costs of risk propagation.


Introduction
Pandemic contagions spread following the general principle of exponential growth [1]. Hence, as each agent is infected, risk propagation at least doubles. Although, initially, the cases increase slowly, an entire population can be infected within a few days [2]. Socioeconomically, COVID-19 (i.e., a systemic risk) has wreaked catastrophic global damage since early 2020. Te impact has driven fundamental social changes across the globe [3]. To understand how these efects occur and better identify them, their essential properties and structures have been widely investigated [4].
When an entire population is negatively afected, systemic failure of treatment and intervention is bound to occur [5]. Numerous studies have scrutinized this in terms of biology, fnance, medicine, and so on. Because various risks propagate diferently, the entire extent of any risk must be well-gauged to mitigate it efectively. Hence, progress has been made in using suitable approaches [6]. Recent studies have ofered clear recommendations about observed changes in risk phenomena [7]. Ideally, despite risks being immutable, they are mitigated and suppressed by the various interventions applied [8,9]. To elaborate on this argument, we consider the common interventions employed by several countries.

Lockdown.
Many studies have focused on the socioeconomic impact of COVID-19 [10]. Big data have been used to measure the percentage change in the propagation patterns (i.e., community mobility) [11]. Consequently, certain countries imposed quarantines, which reduced transmission by ∼90%. Others closed their borders to establish nationwide lockdowns [12]. However, many of these lockdowns were partially relaxed and later lifted prior to the risk being fully controlled. Mobility returned to normal in many places, and risk mitigation was relegated to individual choice [13]. Some countries imposed no lockdowns, and their mobility still decreased for similar reasons. Schools, cafes, restaurants, and shops remained open to avoid greatly disrupting the economy [14], and stay-at-home policies were only a recommendation. Nevertheless, citizens voluntarily reduced their mobility by 10-20% from the global baseline.
Tis raises questions about best-case scenarios. Countries that implemented strict lockdowns and experienced considerable socioeconomic losses recovered and now exhibit nearly normal patterns. Conversely, some countries that implemented strict lockdowns saw huge decreases in socioeconomic status and have not recovered [15]. Some countries did not impose strict restrictions, and they still sufer greatly. Other countries employed various responses based on their stronger socioeconomic status and fared consistently better, which implies that they have now also largely returned to normal [16].

Herd
Immunity. Another strategy suggests that herd immunity can be acquired against a pandemic, frst causing the exponential infection rate to fatten, followed by a steady drop [17]. Unfortunately, herd immunity does not apply well to scenarios in which the pathogen or virus mutates, resulting in new infectious strains. We investigate this further in this paper [18]. Herd immunity theoretically protects a considerable number of vulnerable people (e.g., children) from diseases [17,18], and ostensibly, it should allow the global economy to reopen safely; some believe that it will ultimately result in an end to pandemics. However, the path to herd immunity includes signifcant hurdles, and experts suggest that it can take years to achieve.
Widespread infections and vaccinations [19] are two scientifc approaches to reaching herd immunity. Experts estimate that fewer than 5% of people worldwide have contracted COVID-19 [20]. To reach herd immunity, that number would have to be 60-70%. For this, more than 4 billion people would have to be infected. Tis would result in tens of millions of people dying and hundreds of millions becoming gravely ill [19]. Tis would devastate global economies. Moreover, it is entirely unnecessary. Furthermore, there is no guarantee that a broad infection would lead to eradication. We do not have many examples to explore of complete immunity arising from natural conditions because the population continues to increase exponentially [21]. Tus, all new ofspring remain susceptible to infection.

Vaccines.
Experts suggest that a more efcient approach to immunity is through vaccination [22]. Tis is how scientists (nearly) eradicated smallpox. Because of vaccines, childhood mortality has dropped dramatically for decades. Experts may not yet have sufcient information about COVID-19 to predict the likelihood of reaching herd immunity [23], and there are considerable barriers to achieving that goal. Even with a vaccine, scientists are unsure of how long the natural immunity may last [22]. Tere is a large diference between a 6-month immunity and being protected for 10+ years. For other known respiratory viruses (e.g., infuenza), antibodies tend to weaken over time. Scientists have already observed this in patients recovering from COVID-19; thus, it remains unclear how long the COVID-19 vaccine will remain efective, and the period of efectiveness will be measured in months or years instead of lifetimes [23]. Another challenge is that because not all vaccines work in the same manner (i.e., new COVID variant), their efectiveness will difer [24].
As cases surge worldwide, some experts are optimistic [25]. According to a recent vaccination report [26], it is estimated that the number of infected people could be reduced if the inoculation rates were fast enough, notwithstanding each vaccine's efcacy. Scientists have shown that the reproduction index is not fxed, and it is constantly refned by examining past experiences (i.e., Middle East Respiratory Syndrome (MERS)). Tis is similar to current COVID-19 mutations. Te reproductive index is very likely to repeat an increasing/decreasing pattern that includes infections after vaccinations [27]. Tus, health experts continue to recommend imposing appropriate interventions and measures at interconnected macro-and microscales to slow the spread [24,27]. Hence, it is necessary to continue rigorous controls [26]. More importantly, individuals with a tendency to maintain safe distancing from others and remaining isolated have a better chance of survival [25].

Gab Statement.
Although the abovementioned assumptions can help achieve anticontagion performance, detailed investigations of COVID-19 are still necessary [28]. Te future requires combining large repositories of data to predict trajectories that will provide realistic options [29]. Terefore, a computerized individualized model may serve as an essential tool that combines real circumstances with simulations [30]. Te results can be regarded as a sampled subset of the underlying social network [31]. For example, if a plausible model of the underlying network and its agent dynamics are found, we may be able to infer which contacts are likely to create a propagation route [32]. Indeed, communication among nearby individuals is usually more frequent than for long-range connections, thereby providing efcient paths for viral spreading, as observed in small-world networks. Furthermore, the mechanisms and serial algorithms that underpin our understanding of risk propagation by networked agents must be evaluated [33] to help us identify common grounds for integrating knowledge and strategies. We may then agree on defnitions and reconcile the approaches adopted in multiple felds for an interdisciplinary study of systemic risks.

1.5.
Purpose. Te purpose of this research is to analyze the potential consequences of risk propagation because pathogens play an integral role in regulating infection. In particular, by incorporating detailed macroscales (nonevolutionary) and microscales (evolutionary) of realworld network mechanisms (see the Materials and Methods section) [34], insightful information can be gained for a network-agent simulation to estimate parameter ranges from an extended disease model for susceptible, exposed, infectious, and recoverable (SEIR) simulation. Te advantage of this combined approach is its fexibility. Hence, we extend certain prototypical results of the disease model (e.g., population dynamics) into more realistic simulations (e.g., agent-based dynamics) that fully consider these potential 2 Complexity consequences under the assumption that most people would follow a reliable strategy for a long period.
1.6. Value. Pandemic risks are a signifcant concern in many countries, and they have been widely examined in various felds [35,36] because there are no specifc therapeutics licensed for treating the disease [37]. To understand the risk potential, we suggest intervention via appropriate contemporary dynamic model studies. Some results consider a direct relationship between infection risk and the level of intervention for mitigation. From this, a signifcant opportunity arises from which we can produce mitigating strategies to transform an entire population's systemic risk.

Materials and Methods
Risk is a property of system interconnectivity, and it can be described in terms of system instability, which is caused or exacerbated by idiosyncratic events that result in a potential catastrophe [38]. In particular, a connectivity pattern is a key to understanding risk and how elements of a framework communicate with each other [39]. Tus, it has been suggested that a strategic decision process can be used to explore the infuence of networked interaction among agents [40] and understand the architecture of artifcial systems from the network-property perspective [41,42]. In this paper, we develop a model for these processes with reference to the COVID-19 pandemic and its propagation. Te following model specifcation builds on an agent-based model of systemic risk with evolving strategies determining protective investments against cascading failures that has been originally developed by Ulf Dieckmann and analyzed together with Chulwook Park at the International Institute for Applied Systems Analysis (IIASA), Austria.

Network Property.
Many empirical graphs are modeled on small-world networks (e.g., Facebook and Wikipedia) [43].
Tere are individuals (nodes) and neighbors per node that denote the number of links, some of which can be rewired. Essentially, they can be attached to distant neighbors to model wide-area connections [44]. Te small-world aspect of our network model allows for a relatively efcient set of links that do not require many decentralized resources to maintain connectivity (see Supplementary Materials I for more details of the basic network property). Tis represents a range of degrees of distribution (random ← ⟶ regular) ( Figure 1). For example, we can infer from the structure which social contacts are favored as the infection route. Naturally, contacts between nearby people are usually more frequent than longrange connections, providing efcient routes for a virus to spread, as described by a small-world network. Te level of connectivity between a typical node and a local hub can be signifcant; however, it is not notably high. Hence, we can form network properties to help focus our analysis.

Operating
Mechanisms. An operating network has a small-world appearance, and a relationship holds between the vertices (nodes) and the edges (lines). Hence, a mechanism is obtained as (Tables 1 and 2) Any node (n) can be linked to any other node. Because the collection of nodes infuences the probability of connections (p ∈ (0, 1)), our model investigates the distribution of network connections (i.e., degree probability). Additionally, network rewiring (β ∈ (0, 1)) will have unpredictable efects in relation to individual ties because this allows for diferent approaches to diverse risks (i.e., mobility and migration). Tis is because the nodes infuence each other in the same way as originating strategies and distributed security (e.g., frms, banks, food webs, supply chains, and germs).

Individual Property.
To observe the propagation process, the corresponding network-agent evolutionary dynamics [45] model uses an array (vector) to represent the probability of infection, p [∈ (0, 1)], with a given set of infuenced nodes (1 ≤ j ≤ N), denoted by (p j). Each node can persist in one of two states: uninfected or infected. All nodes initially exist without infection.
Te matrix no longer represents the adjacency matrix (A). It continues to be labeled as rows and columns with values of one and zero; the key diference is the possibility of showing each node's state (1 � infection; 0 � absence of infection) based on the time step (see Figure 2).

Immunodynamics as Antibodies to the Virus.
With the fundamental model characteristics, we stipulate that an elementary level of risk (i.e., cascades of infection) depends on the cooccurrence of the i and j nodes. In these dynamics, an agent is associated with each node and characterized by its immunity and strategy (the strategy values f p0 and f p1 evolve through social learning and strategy exploration; see section (vi) for more details). For each time step, each agent receives one unit of immunity, which is added to that agent's immune capacity, c, for which the fractions f m and f p are spent on maintenance and protection, respectively. Tus, the capacity value is updated to 1 + (1 − f m − f p )c.  probability, p n ∈ [0, 1], and it can propagate along each link with a probability, p l ∈ [0, 1], at each time step. Tis rate refects the transmission rate (β t ), which is the average number of individuals infected by one individual in a susceptible population. When this value is larger than one (i.e., β t > 1), it means that one infected individual transmits the disease or virus to two others (i.e., p l > 1), and the virus spreads exponentially. However, if this rate is less than one (i.e., β t < 1), the number of new infections decreases (i.e., p l < 0.1).

Susceptible Dynamics (Basic Transmission
To match this contextual spreading mechanism, let us denote the number and fraction of failed agents by N f and f, respectively, and consider two successive steps t and t + 1. Suppose that, at time t, there are N f and N − N f failed and not failed nodes, respectively. Terefore, the number of not failed nodes at t + 1 is N f + p p (N − N f ), and the number of failed nodes is Here, p p is the average protection probability among not failed nodes. Te above relation shows that the value of f is a decreasing function of p p . For p p � 1 (i.e., full protection), there will be no failed agents. We modifed the propagation probability, p l , because p p will need to be replaced by Tis rate gives us an idea of how quickly the infection spreads from one individual (p n ) to the next (i.e., the basic reproduction number: COVID-19 is estimated to be between 2 and 4). Tus, we apply this rate to COVID-19 and embed these potential infections that become infections with a probability of 1 − p p , depending on the agent's investment in protection. Hence, a possible choice is Here, p p,max denotes the designated protection maximum. c p, (1/2) denotes an allocated reference (in units of c p, (1/2) : simplifes the protection probability as p p � (p p,max /(1 + (1/(f p c))))), and f p c represents an evolutionary protection level multiplied by the updated immune capacity. Te infection lasts for one time step (default), resulting in the loss of an agent's immune capacity. For those strategies, each agent chooses its protection level according to the heuristic: For the initialization of strategy values, two arrays are added for vectorization: where w → i1 denotes the vectorization as the designated strategy of (f p0 ). w → ii represents a vectorization as the designated strategy of (f p1 ) multiplied by the eigenvector centrality from the graph (C), which indicates the centrality of the agent's node, as normalized to the interval (0, 1): Te eigenvector centrality (C) for node i is (Ax � λx), where the matrix denotes the network with eigenvalue λ.

Strategy Dynamics (as a Social Interaction).
Te strategy values, f p0 and f p1 , evolve through social learning and strategy exploration, refecting the individual interaction Table 1: Small-world network property.

Defnition
Parameter Range Created small-world network (example) Number of individuals n ∈(0, ∞) Connection probability p ∈(0, 1) Rewiring probability β ∈(0, 1) Linear dimension (matrix) m * n row, col * Tree parameters. n: number of nodes. p: number of edges for a new node to attach to. β: rewiring probability to the edge.
where p i denotes the probability of the role model being accepted for imitation, π f represents the immune capacity of the focal individual, π r denotes the immune capacity of the role individual, "exp" denotes the exponential, and ω represents the intensity of the selection (ω < 1 � weak selection, ω ⟶ ∞ � strong selection). Te focal individual imitates the nearby role individual's strategy, comparing its new immunity (large Δπ � large immune capacity diference; small Δπ � small immune capacity diference). Ten, the focal individual chooses to imitate the strategy of the role individual (see Table 3 (code book) for the details of the mechanism). Additionally, at each time step, each agent having a probability of p e ∈ [0, 1] randomly chooses one of its two strategy values and alters it with a normally distributed increment with a mean of zero and a standard deviation where x denotes individual immunity, μ represents the mean (location), and σ 2 denotes the variance (square scale) (see Table 4 (code book) for the detail of the mechanism).

Recovery Dynamics (Delay of Isolation and Quarantine).
Given a recovery which, in this context, refers to the fraction of recovered individuals having immunity, this function (or infection) lasts for one time step and results in the loss of an agent's immunity according to the reset virus infection potential: where random(n) < rec � 1 denotes randomly chosen individuals having a certain probability (random(n)) and the infectious potential of any individual is (1 − p p ), which is selected based on a certain probability that approaches zero. At the end of each time step, all failed agents will recover.
By default, this recovery rate is implemented by resetting the infectious potential after every r t � 1 time steps. Simultaneously, to control this intervention, we allowed the number of time steps to be controlled by another parameter (t r ∈ [0, ∞]), which represents the delay of social distancing, infected isolation, and quarantine (until recovered) as the recovery time delay. Te parameterization of these macrovariables (t r) and microvariables (r t) is designed to provide insight into the more realistic conditions for recovery from the virus and the more sophisticated recovery mechanisms.

Performance Evaluation (Including Interpretation of the Results)
3.1. Assumption of the Network-Agent Dynamic from the Disease (SEIR) Dynamic. First, we assume a risk potential (virus) that is represented by the COVID-19 basic reproductive number (R t ) [46] to the network-agent dynamic's number of infections (f n ← p n ∈ [0, 1], p l ∈ [0, 1]). Tis epidemiological term can be derived from the average number of new ofspring generated by each infuenced individual and refects the transmission rate (β t ). Diferently ] expressed, the number of people currently infected in a region and the percentage of new infections are given. A high value implies that it propagates easily, and a low value indicates resistance. Te critical assumption here is that everyone is susceptible; no one is excluded.
Tere are three basic scenarios for the number of infections (f n ), which matches the reproductive number (R t ). It can be less than one, meaning that the number of new cases will decrease over time; it can also be equal to one, meaning that the case number is stable over time; or it can be greater than one, meaning that the infection is self-sustaining unless control measures are implemented. When the infectious potential (f � 1 − (1/(Np l (1 − p p )))) infuenced by this model's mechanism equals 1, one individual transmits the disease to another. Tese individuals will infect others either in a pessimistic-case scenario or in an optimistic-case scenario. Tus, the number of infected cases remains stable over time. If the infectious potential or reproduction number is less than one, then R t < 1 � f n < 1. Tus, over time the number of cases will decrease and will be dominated by the absence of infection, which is the ideal outcome. When the infectious potential or reproduction number is greater than one (R t > 1 � f n > 1), the protection equals 0.1. Hence, when each individual infects another (more than one) and the newly infected individuals then infect others (more than one), the number of infections increases exponentially over time (see Figure 3).
For the potential simulation (Figure 3), the population is set to N, which we have applied to South Korea (2019; 51.64 million); the results are represented as fractions of the population [45]. At each moment, the population is divided into four categories that add to a total of 1 (S � susceptible, E � exposed, I � infected, and R � recovered (or dead), as inspired by the disease model), which indicates fractions of the population that evolve as follows: where parameter c can be used for the rate per day at which an infected individual either infects or does not infect at a fxed parameter. For example, the COVID-19 case (c � 1/18) refects the estimated duration of the infuence over 18 days [47]. Te parameter σ represents the rate at which those exposed to the virus become infected; furthermore, it is considered a fxed parameter (σ � (1/4)) to refect the estimated incubation period. Tus, COVID-19 has an incubation period of 4 days in South Korea [48]. Here, β t is the transmission rate, similar to the infectious potential defned by Np l ←p l ∈ [0, 1] in the network-agent simulation. Of the people infected, a fraction S/N is susceptible. Tus, the transition occurs upon exposure. Te parameter governing the rate of infection varies over time (see Supplementary Materials II for more details of the SEIR Model mechanism).

Part 1: Nonevolutionary Scale Efect with Rewiring.
Based on the aforementioned premise, a protection dynamic can be implemented against the spreading virus. Te network-agent dynamic model allows an agent to make a costly investment for protection. We assume that the risk is the    infectious potential with a probability of 1 − p p ; p p � (p p,max /(1 + c p,1/2 /(f p c))) when an infection lasts for one time step. Figure 4 presents diferent observations parameterized by the nonevolutionary macroparameter (p p,max ) as the main protection variable with the created matrix of the network (λx � eigenvector centrality). Te patterns observed from the four parameterizations provide diferent evolutionary patterns based on the time series. For small connections (λx low ← connection p k � 0.1) among individuals, a weakly spreading pattern is observed in which the centralities are widely distributed with strong connections rather than extensive links (λx high ← connection p k � 0.9) being observed, wherein the centralities are tightly distributed. Furthermore, we see that when the protection thickness is driven, a robust regime emerges in the sense that a sufcient degree of immunity produces a symmetric pattern between the centrality and the protection level and weak protection does not. Te results provide insights into the possible patterns available with the artifcially designated parameters plausibly being the protective factors against virus difusion.

Imposing Rewiring.
Even in the observed scenarios that match the disease model, we propose that the contagion is likely to be intensifed with a rewiring probability on the nonrevolutionary scale (p β ), thereby causing additional infection costs (i.e., mobility and immigration). We determine that intervention provides associated values with the propagation criteria on a macroscale (p p,max ) for each node. Tis observation enables us to gain a sense of the varying nature of propagation within systems. Tis type of infection is limited to connections and is caused by regulation. Put diferently, there is another rate that continues to increase as the protection regime causes cascading infections, even when individuals continue to invest their potential. Figure 5 represents individuals that difered in their underlying contact rates (p β ), assuming a lower rate from the rewiring interactions among the less vulnerable simultaneously with their behavioral responses to public nonevolutionary interventions. Tis shows that the infectiousness difered among individuals even within groups, owing to nonuniform characteristics in the population. We also show that the individuals can vary because of their acceptance, ability, or the availability of a macroscale intervention (p p,max � 1) which seems to protect against propagation.
Furthermore, nodes that are controlled by rewiring seem to maintain their potential only if they can avoid losing their protection, p p,max � 1. We extend this observation until the process properties do not change (i.e., attain stationarity). Hence, they neither increase nor decay, and the process converges to a stationary probability (see Supplementary Materials III for the mathematical mechanism of stationarity).
In terms of stationarity, we observe diferent levels of protection potential between the parameters (p p,max , p β ) ( Figure 6). Tus, to reduce the ramifcations of additional losses, a weak connection and rewiring may be preferred to the potential damage (f) from individual infections, but it may guarantee that it is strategically possible for a large insolvent number of individuals to become uninsured connectors (see Supplementary Materials IV for more detail on the mathematical mechanism).

Part 2: Evolutionary Scale Efect with Recovery.
Inspired by COVID-19, to discover the nonevolutionary scale efect underlying the basic reproduction number (R t ≈ f n ), we present a network-agent analytical characterization. Tis is to determine whether an alternative strategy could infuence the infection trend that evolves via cultural evolution (i.e., social learning and exploration). We begin this simulation with the assumption that the environment has high protection (p p,max � 1) and moderate centrality (p k , p β � 0.25); however, an agent does (not) have sufcient opportunity to interact with other strategies (f p0 and f p1 C) with values (p r , p e � 0.1) that refect the evolutionary microscale that matches the social distance (η) of the disease model. Te imitation and exploration probability causes diferent protection levels, f p , according to the virus infection mechanisms: Te parameter set assumes that diferent values of p r ∈ [0, 1] and p e ∈ [0, 1] are incapable of the dynamics, resulting in a given protection level. Hence, even if we set high protections at a macroscale, each agent will (not) choose another as a role model and imitate that agents' strategy values (immunity) at each time step with the applied function (p i � [1 + exp − s(π r − π f )] − 1 ). Furthermore, each agent can (rarely) alter its strategic value because there is no opportunity to explore another. Figures 7 and 8 indicate diferent strategies for diferent network infection evolution. As with most real-world networks, simulation results that use imitation and exploration have parameters that expand nonlinearly over time. Te densifcation suggests that the existing structure may constrain occurrences at the next time step. Te evidence indicates that infections between parameters occur during very early stages, demonstrating that the difusion can be exaggerated by interactions among individuals. Hence, the probability of social learning may be another crucial factor that leads to novel trends having a signifcant impact on socioeconomic status (see the following mathematical mechanism for more detail).  (c) (d) Figure 7: Described protection dynamics against virus infection with imitation and exploration rate. In each section, plots of the left show the matrix (horizontal axis � time step from 1 to 100; vertical axis � individuals from 0 to 100; and color of the matrix � infectious state between infected (red) and absence of infected (blue)). Te plot on the right represents the individuals' dynamics with a network; node color � states (infection (red) ⟵⟶ (blue) absence of infection; green � protection potential; and yellow � initial structure of the state without infection and protection). Te initialized infectious potentials are p n � 0.1; p l � 0.1.

Mathematical Mechanisms of the Evolutionary Microscale Parameter Efect.
Let us now consider the numerical insight behind the observation to understand how the evolutionary microscale parameter results can be interpreted. We assume that the applied model is obtained from the most straightforward network algorithm that consists of n inputs and outputs, where n is the number of features of our dataset (i.e., evolutionary parameters). Te process of passing data through the network is defned as propagation and is carried out with a systemic risk (virus) as follows.
For the input, multiply the input value x → i with the potential w → i and sum the multiplied values. Te potential corresponding to the model mechanisms is represented by the strength of the interaction (imitation and exploration) protection between individuals, and it is used to determine the extent of the infuence that will be assigned to the input on the individual's output. If the potential w → 1 has a higher value than the potential w → 2 , input x → i will have a more signifcant infuence on the output than w → 2 .
Te row vectors of the inputs and potentials are x → , respectively, and their product is given by Hence, the summation is equal to the product of vectors x and w: Next, let us add a risk (virus) b to the summation of the multiplied values, and let us call this expression z. Risk (which is also known as the ofset) is necessary to move the entire activation function to the left or right to generate the required output values.
Ten, we pass the value of z to a nonlinear function as we observed nonlinearity in the individual's output, which is the applied network function. Moreover, the output has a signifcant efect on the spreading speed of the network in that the network property has a role (p i � [1 + exp(− ωΔπ)] − 1 ) as their fundamental mechanism, which is defned here as where σ denotes the function, and the output we obtain after the projection is known as the predicted value y. Tis logic comprises an algorithm for computing the rate of change for the potentials. Te rate of change is calculated as the square of the diference between the actual (y i ) and predicted (y i ) values, which is calculated for the entire dataset, and the average was taken.
To fnd the best potential and risk for our model, we need to determine how the function changes based on potentials and risks. Tis is achieved with the help of the rate of change. In our case, we need to determine the rate of the function's change with respect to the potential and risk. We calculate the rate of change of the function C for potential w i using partial diferentiation. Because the function (C) is not directly related to the potential w i , we use the chain rule (probability) as zC To determine the following three rates of changes (zC/zy), (zy/zz), and (zz/zw i ), we calculate with the rate of change of the function (C) related to the predicted value (y). Tat is, when y � actual vectors and y � predicted vectors, the equation is Now, we fnd the rate of change of the predicted value for z.
. (21) For simplifcation, z zz Te rate of change of z to the weight w i is Because z equals to Tus, we get Here, the risk is conceptually considered to have a constant input value of 1. Hence, Finally, we optimize the best element from a set of available alternatives, which, in our case, is the selection of the best potential and risk. Let us select the rate of change descent as our optimization algorithm, which changes the potential and risk, and it is proportional to the negative rate of change of the function C, concerning the corresponding potential or risk. Te rate (α) is a hyperparameter used to control the change in the potential and risk. Potential (w i ) and risk (b) are updated as follows, and the rate of change is repeated until we achieve stationarity (divergence or convergence).
We suggest that this mathematical mechanism behind the simulations is a fundamental concept to obtain all types of results (with some modifcations for ftting our model). Despite the logarithmic expansion of our small-world network, if we explore contact individuals or acquaintances in the network, the map of the network creates propagation linked by their connections among neighbors corresponding to the numerical explanations.

Imposing Recovery.
Te assessment process was to make macroscale (e.g., institutional competition and central intervention) observations, and microscale (e.g., individual behavior and relative gain) evaluations were simultaneously obtained. To address the central social distancing and isolation intervention methods as confrmed by the simulation, we extended the model to ft an estimation of recovery (microlevel individual immunity recovery) and recovery delays (macrolevel intervention delay), thus obtaining more detailed insights into the network structure.
Te simulations displayed in Figure 9 imply that virus difusion is likely to be intensifed by both recovery and recovery time delays (Figure 10), resulting in a signifcant social cost. We also observe that immediate or delayed temporal interventions (e.g., social distancing, isolation, and quarantining) are associated with micro-and macroscale propagation criteria at each node. Te nodes controlled by t r � 2 seem to lose their ability to protect, despite their strong protection level (p p,max � 1, r t � 1). Tus, infections are not limited by protection or strategy (r t � recovered individuals). However, the rate is afected by regulation over time (t r � time delay), owing to the propagation of infection. Te results having immediate recovery (plots in the left-hand set) appear to have the potential to protect against infection being propagated. Conversely, the results with malfunctions (plots in the right-hand set) do not have this potential, even when the individuals maintain an immune capacity.

Fit the Network-Agent Dynamic (Evolutionary Microscale
Efect) to the Disease Population Dynamic. We suggest that the value depends on three factors. First is the duration of infectiousness of the disease. How long can an infected individual (f n ) cause infections to others? Te answer is not simple because this value varies. Infectiousness seems inherently and genetically associated with individuals, and it depends on the state of protection potential (infection) experienced by that individual. At diferent stages, an infuenced individual sheds varying amounts of infectious potential. Second is the probability of infection between infected and susceptible individuals. Tis depends on the type of contact and the method of propagation. Tus, the reproduction number (R t ) will be diferent, accounting for virus risk with limited propagation and transmission. Te third is the average rate of individual interaction controlled by social interaction (η) between infected and susceptible individuals. Tis variable depends on the types of structure, interaction, social activities, and so on, in which the individual is involved.
Te frst two factors are afected by nonevolutionary macroscale characteristics (i.e., regime, governance, and supervision), as determined by the system's centrality. Te third is afected by evolutionary microscale characteristics, where activities such as intervention (i.e., social distancing) play a signifcant role [49]. In Figure 11, we parameterize this rate to allow for an initial period of intense application of intervention measures (i.e., social distancing � η) followed by a relaxation of those measures to enable the socioeconomic activity to resume: therefore, Here, β 0 � ((β 1,0 + β 2,0 )/2) denotes the initial value of β t , which represents the spread of the virus during its initial phase. Te parameters β i for i � 1, 2 indicate that the longrun values of β 1t converge. Tus, eventually, β t converges to ((β 2 + β 2 )/2). To obtain a U-shaped pattern for β t , we make β 1t a rapidly declining function and β 2t a slowly rising function. Te parameter η 1 governs the rate at which β 1t infects β 1 . Te parameter η 2 governs the rate at which β 2t rises towards β 2 . For each simulation, we obtained slightly diferent results, which allowed us to provide mitigating scale parameters.
We performed the simulation to obtain the image shown in Figure 11. From the simulations, we can detect the efects of various types of interventions (i.e., distancing

Discussion
Te proposed infection probability model was created to run with many simulations based on the time steps of virus risk propagation throughout the network and eventually provide the fnal number of infections. Te results indicate that individuals are more afected when they are highly linked to the network [49]. Tus, the probability of infection is determined by the number of connections and the rewiring from the node of the specifcation scaled at the macrolevel. Te rationale behind these observations is that higher-degree agents are more exposed to infection than lower-degree ones, which increases the potential for cascading [50]. From these observations, we attempted to identify what stabilizes the propagation rate. In this regard, we know that there are delays in requesting and receiving tests, obtaining disease results, and sending the results to health authorities [51]. We have a variable β t �p l that refects the potential of infection in a designated population. Meanwhile, the variable is not constant in each trial, owing to interventions, which are (not) made up of (in)dependent trials [52]. In our results, this trend is observed in the model's (non)evolutionary macro/microscale parameters. However, their infuences are not identically weighted [53]. Te intervention applied to the risk potential can persist briefy, and we can cite this as a virus infection. Tus, to reduce the potential ramifcations (R t � (f n )), interconnected and immediate interventions are preferred to determine the potential damage from individual infections to guarantee that a strategy to recover from losses to uninsured connectors is possible [54].
Inspired by the plausible scenarios presented above, we focused on the evolutionary parameter (f p c) having a more individualized perspective with time steps. Te results yield the important insight that weakly centralized regimes allowing high numbers of interactions among individuals should not be recommended [55] in the current model. Moreover, we should shift away from the extant dominant potentials, norms, and routines that can lead to novel trends that have a high impact ( Figure 12).
Te evidence shows that strictly confned individual movements are key to difusion and that contagions occur at early stages (see Figure 8). Hence, individual interaction (e.g., social distancing) probabilities and rates are crucial factors that lead to violated expectations and novel trends having a high impact [56]. Following these observations, we can assume that successful outcomes require a delicate balance across these essential components (macro-and microscale). Individuals and organizations may fnd value in reducing their perceived virus risk potential by following this suggestion. Although many questions remain regarding how other network structures, governance schemes, and content evolve and interact over time and new COVID variants, resources should be used to help gauge the underlying potential. Te observations from our simulations may lead to benefcial network structures and connectivity supported by joint governance [57].
Tis interpretation helps estimate how countries have tackled the pandemic by connecting the governmentalindividual levels of efort. For example, some countries responded swiftly to initial cases [45]. Tey rolled out a "test, trace, and track (3T)" approach, fast-tracked the mass production of test kits, and set up drive-thru and walk-in test centers (i.e., robust macroscale parameters) that provided results within 24 h. Tey tested many people per day (i.e., ∼20,000), globally the highest at that time [58]. In the early cases, the authorities attempted to trace everyone who had encountered infected individuals and tested them to prevent the virus from spreading. Tose who were confrmed positive were asked to isolate themselves voluntarily (weak microscale interaction) and to download an app that alerted authorities if they left their home. People in the vicinity of an infected individual also received alerts (i.e., phone) from the government. Tis testing approach allowed the government to avoid a nationwide lockdown; however, large public gatherings were banned, and schools were shut down.
At the macroscale, all arrivals into the country should be quarantined (i.e., for two weeks). Countries that did this contained the virus' early spread [59]. Hence, for all positive tests, their government should test and trace all contacts to break the transmission chain. Furthermore, countries should prepare contact-tracing measures. Laws should be adapted to allow the government to collect patient data and security footage during an outbreak to increase their macroscale intervention [60]. At the microscale, all steps should be logged and shared to alert people to avoid the path to infection (i.e., decrease the number of connections). Websites and private apps could compile this information to allow everyone to see where infected persons are or have been and when. Citizens should actively check and avoid locales of infection. Such microlevel voluntary eforts would be extremely helpful in controlling the spread. Tracing people's movements is often considered controversial; however, governments should prioritize public safety over privacy during an outbreak [61]. Such eforts will allow countries to test hundreds of thousands of people quickly and efectively, thereby making it easier to contain the virus. By doing so, aggressive lockdowns could be avoided. Furthermore, it would proactively help fatten the outbreak curve from steep to level. Currently, the world has turned a corner in the pandemic; however, we must continue to be prepared. Levels of vigilance have set some countries apart [62] who, in this manner, demonstrated the benefts of widespread testing [63]. More testing is required, but aggressive strategies may be difcult to implement in countries with much larger populations.

Conclusion
Contagions and persistence patterns should not be viewed as having causal links; rather, they should be seen as driven by interconnectedness [64]. Te potential is caused by macro-and microscale parameters [65]. Delayed response times cause increased risk propagation, which is likely to allow widespread infections, thus leading to reductions in network (social) welfare. Governance and personal misbehavior (e.g., under-or overestimating the situation) may, at times, stretch to many networks, thus increasing the risk [66]. Given that the evolutionary mechanisms implemented in this simulation model match the real disease model, we often observed evolutionary responses to obtain a critical value for a plausible protection potential [67]. Although the structures have high infectious potential in terms of virus risk, the interconnected macroscale-microscale intervention measures become weak amplifers as protection increases [68].
According to the World Health Organization (2020), the COVID-19 pandemic and the associated economic crisis pose huge global and local challenges. Te health, social, and economic impacts have afected all segments of the population; however, it is particularly detrimental to vulnerable social groups, including people living in poverty, older persons, persons with disabilities, and Indigenous peoples. Epidemics and economic crises have a disproportionate impact on these vulnerable groups, which can trigger worsening inequality and poverty. Te global crisis requires the coordination of stakeholders, a global solution, local implementation of efective socioeconomic and public health policies, and solidarity [69]. Te idea is simultaneously to starve the virus everywhere through a combination of quarantine, social distancing, and restricted travel. Te critical factor is the synchronization of responses [70]. In a typical pandemic, when one country peaks, another may be observing its frst cases (even variants). Instead of leaders responding only to what is happening in their jurisdictions, the world should be treated as a giant interconnected system. If coordinated properly, global eforts can quickly end such pandemics, thus ensuring reduced loss of life. However, unless the virus is completely eradicated, which is highly unlikely, propagation risks will persist [71].
Tus, fnding a global response strategy for the pandemic is key. Breakthroughs in the treatment and prevention of symptoms can also make viruses far less dangerous while requiring fewer extreme containment measures [72]. Ultimately, the breakthroughs, social services, and systems we develop can be used for a better future for everyone.

Data Availability
Te data used to support the fndings of this study are included in the article.

Ethical Approval
Tis study was approved by the local ethics committee (SNUIRB No. 1509/002-002), conforming to the ethical standards of the 1964 Declaration of Helsinki (Collaborative Institutional Training Initiative Program, report ID 20481572).

Consent
All participants in the study provided written informed consent.

Conflicts of Interest
Te author declares that there are no conficts of interest regarding the publication of this paper.