Emergence of shield immunity during spatial contagions

Abstract

Contagions spreading across space—including epidemics, infodemics, and socio-economic turbulence — generate complex geo-spatial patterns shaped by contagion state and risk-driven population mobility. Distribution of resources for mitigating these contagions adds further complexity. We present a concise, generic framework to model various contagion types within a space characterized by bounded risk disposition parameters and generalized resource effectiveness. Specifically, we explore how (i) risk-averse behavior of “inoculated” individuals and (ii) resource effectiveness in reducing contagion “incidence” influence pattern formation and spread of infection, opinion polarization, social myths, and socio-economic disruptions. We show that “inoculated” individuals interacting with affected populations may help minimize contagion impact by curbing further transmission. We identify this as a generalized form of shield immunity and explain its emergence in terms of individual risk disposition. This shielding effect is strongest in socio-economic turbulence, moderate in epidemics, limited in social myth spreading, and not observed in polarization dynamics.

Introduction

Typically, contagions may take different forms: the spread of diseases, the dissemination of ideas and information, and civil unrest. These dynamic processes are influenced by external factors (e.g., pathogens and socio-economic shocks), as well as by individual human behavior and complex social interactions^1,2. Risk perception and response play a crucial role in shaping these behaviors. Risk-averse individuals tend to adopt preventive measures, while those with low-risk perception may engage in behaviors that contribute to further spread or escalation³.

As contagions spread across different locations, population mobility becomes important. Population mobility has been studied in the context of epidemics, natural disasters, opinion polarization, and conflicts, where individuals must navigate both the spread of contagion and the dynamics of social interactions^4–8. Individual decisions to change locations are shaped by the individual perceptions of risk and safety, and directly impact the contagion transmission patterns and the propagation speed^4,9. High levels of mobility can lead to a rapid transmission, while restricted movement can slow the spread and mitigate the impact of contagions¹⁰. The interplay between contagion dynamics and the risk- or benefit-driven population mobility creates intricate spatial patterns of the contagion spread across a network of locations^4,11–15.

To mitigate the diverse impacts of contagions, resources must be strategically allocated to affected areas and communities^16,17. Resources depend on the contagion type, and may take the form of vaccines, fact-checking campaigns and/or educational interventions. In general, a contagion mitigation strategy is characterized by prevention, or “inoculation”, which is a cornerstone strategy aimed at controlling contagions and mitigating their impacts^17–21. By inducing immunity in some individuals, inoculation may significantly reduce the contagion spread, contributing to (a generalized form of) “herd immunity”^22,23. However, the distribution of resources, along with changes in human behavior and population mobility, increases the complexity of modeling contagion dynamics.

For disease outbreaks, canonical epidemic models accounting for vaccination aim to identify the optimal resource (i.e., vaccine) allocation strategies^17,24. In information contagions, misinformation may rapidly propagate through the population, and the distribution of educational resources in the form of “refutation texts” among susceptible populations was found to significantly reduce misconceptions²⁵. Other educational resources may include anti-discrimination campaigns, psycho-social support programs, civil engagement training and financial literacy programs²⁶. In fact, peace education campaigns (e.g., human rights education, international education, conflict resolution education) has been argued to be an effective prevention strategy against collective violent tendencies that may lead to interpersonal and mass violence (i.e., peace education theory)^27,28.

The interplay between contagion spread, risk-driven population mobility, and resource distribution generates intricate dynamics that are difficult to model and predict. As a result, the combined impact of risk mitigation and resource distribution on complex contagion patterns remains an open question. Thus, this study is motivated by the need to develop a concise, generic framework for modeling contagion dynamics affected by risk-driven population mobility and resource distribution. This framework must be sufficiently versatile to accommodate several contagion types, from disease and information spread to civil unrest and socio-economic turbulence.

Several previous studies characterized a partial combination of these combined dynamics. Chen et al. ²⁹ presented a coupled model of resource allocation and disease spreading dynamics to examine the impact of self-awareness (which translates to a “self-protection” tendency) on resource distribution (in the form of donations from the unaffected population) and epidemic dynamics. A resource allocation model developed by Zhang et al. ³⁰ compared the performance of utilitarian and egalitarian resource allocation strategies with respect to interaction restriction (i.e., lockdown) levels. Dynamic resource allocation was also modeled by Papachristou et al. ³¹ to determine optimal mitigation strategies for contagions in financial networks. All these approaches considered a partial combination of contagion dynamics, resource distribution dynamics, and population mobility, but did not encompass all three.

Furthermore, the complexity of multiple interacting dynamics often produced counterintuitive effects. In general, for some contagion types, it appeared that strategically placing inoculated individuals among susceptible populations can create a protective barrier that suppresses new infections (i.e., new cases, in general). For example, “shield immunity”, introduced by Weitz et al. ³², was found to emerge when recovered agents (with some immunity) are strategically placed among susceptible populations to suppress new infections. The approach based on “shield immunity” suggested that (counterintuitively) sustaining, rather than preventing, interactions between (partially) immune agents and susceptible individuals may provide some protection for the susceptible population by substituting their interactions with the infected population, thereby reducing the likelihood of transmission:

The core idea is to leverage a mechanism of “interaction substitution” by identifying recovered individuals who have protective antibodies to SARS-CoV-2 and deploying them back into the community. Here, we assume that recovered individuals (virus-negative and antibody-positive) can safely interact with both susceptible and infectious individuals, in effect substituting interactions with susceptible and infectious individuals for interactions with recovered individuals.

In parallel, a somewhat similar strategy for curbing the spread of urban legends was proposed by Tambuscio and Ruffo¹⁶. In their study, several urban myth debunking strategies were ranked in a counterfactual analysis, revealing that covering the frontier of the “skeptic” (susceptible) community with “eternal fact-checkers” (i.e., individuals who are immune to the myth) provided significant protection, emerging as a powerful myth-debunking strategy. We hypothesize that similar shielding effects may be observed in different contagion phenomena, where “inoculated” individuals mix with “infectious” ones and replace the interactions between susceptible and infectious individuals. In other words, the concept of shield immunity may be generalized, with potential extensions going beyond epidemic scenarios and including information diffusion and other non-epidemic contagion dynamics.

Thus, the first objective of our study is to (i) map risk mitigation and resource distribution dynamics to several spatial contagion patterns and (ii) model and explain the emergence of generalized shield immunity across different contagion types.

Our second objective is to model how the attitudes of inoculated individuals towards risk influence the overall contagion dynamics and the resulting spatial patterns across the four identified contagion types: epidemics, opinion polarization, social myths and socio-economic turbulence. The behavior of inoculated individuals depends on their perception of residual, post-inoculation, risks. This, in turn, involves their personal beliefs, tendencies toward risk compensation, and changing awareness of the disease^33,34. Several studies reported on individual attitudes towards vaccination during epidemics (i.e., vaccine confidence, vaccine hesitancy, vaccine rejection)^35–37. However, less is known on how the risk attitude of the inoculated population impacts the general outcome of a contagion. In addition, the effectiveness of specific resources in mitigating each contagion type also significantly contributes to the contagion outcomes^18,38. Thus, our second objective is to systematically relate resultant contagion patterns across the four identified contagion types with respect to (i) the attitudes of inoculated individuals towards risk, and (ii) the effectiveness of resources in curbing the contagion spread (e.g., vaccine efficacy).

Our proposed approach uses a generalized Susceptible–Vaccinated–Infected–Recovered–Susceptible (SVIRS) epidemiological model to represent the progression of the contagion. It also applies the Maximum Entropy principle to determine dynamic mobility flows that adapt to the contagion state, the resource distribution dynamics, and the spatial population distribution. Additionally, we represent and parametrize the risk mitigation tendency of inoculated individuals and the resource distribution dynamics. These two factors are generalized across four identified contagion types in a concise conceptual modeling framework. We then analyze the resulting contagion state and spatial patterns that develop within the space formed by parameters quantifying the risk mitigation and the effectiveness of resources.

Our results show that resultant contagion patterns are similar to spatial Turing patterns, typically produced by reaction-diffusion systems (such as spots, labyrinth, and gaps)¹⁵. Moreover, we identify additional patterns (stripes, proto-stripes, proto-gaps, etc.), which characterize spatial configurations, separated by phase transitions in the space of the corresponding parameters within each contagion type. Crucially, our results show that “inoculated” individuals that are not avoiding the affected areas can drastically change contagion outcomes by providing varying degrees of (generalized) shield immunity across different contagion types. Specifically, this shielding effect is found to be most pronounced in socio-economic turbulence scenarios, moderate for epidemics, limited during the spread of social myths, and not observed in polarization dynamics.

Results

Risk disposition space

In this section, we introduce the main components of our model, as well as the high-dimensional space within which we characterize the dynamics of four contagion types: epidemics, polarization, social myths, and socio-economic turbulence. These contagion types are categorized in terms of the individual risk disposition and the preference to interact with the affected population.

Following the Boltzmann–Lotka–Volterra (BLV) methodology^39,40, our model has two main co-dependent components: a fast dynamic: population mobility, and a slow dynamic: the contagion spread. Importantly, these components are generalized across the four identified contagion types.

Our compartmental SVIRS epidemic model divides the population into four groups: susceptible (S), vaccinated (V), infected (I) and recovered (R)¹⁷. The resulting SVIRS dynamics are described by the transmission rate β, recovery rate γ, natural loss of immunity or susceptibility acquisition rate ζ, vaccination rate ν, vaccine waning rate η and the effectiveness in reduction of infection due to vaccination σ (see Section “Canonical SVIRS epidemic model”).

Population mobility is incorporated by expanding the SVIRS model into a generalized multi-city SVIRS-network model running on a lattice of locations. The fraction of individuals from each SVIRS compartment traveling across these locations is determined by population mobility flows denoted by ${\varphi }_{ij}^S$ for susceptible, ${\varphi }_{ij}^I$ for infected and ${\varphi }_{ij}^V$ for vaccinated. We do not consider ${\varphi }_{ij}^R$ for recovered individuals, as those who recover with immunity before becoming susceptible again do not contribute to the contagion state.

The optimal mobility flows ${\varphi }_{ij}^I$, ${\varphi }_{ij}^S$ and ${\varphi }_{ij}^V$ are derived from the BLV model (see Section “Generalized SVIRS-network model”, Eq. (16)–(18)), as functions of Lagrange multipliers α^I, α^S and α^V. We refer to these multipliers as the bounded risk disposition parameters, namely, bounded adaptive responsiveness α^I 15, bounded risk aversion α^S 15 and bounded risk mitigation α^V. Generally, these parameters quantify a preference to interact with or avoid affected (infected) individuals, by representing each compartment’s perception of a location benefit b_i, determined by $b_i=\frac{P_i-I_i}{P_i}$ (i.e., the unaffected population) (see Section “Generalized SVIRS-network model”). Importantly, parameters α^I, α^S and α^V may be positive or negative, representing attraction to or repulsion from a benefit, thus driving mobility towards or away from a location.

Each two-dimensional risk disposition combination (α^I, α^S) with varying signs can be categorized into four primary contagion types corresponding to a quadrant on the risk disposition plane (illustrated in Fig. 1)¹⁵. For example, the top-left quadrant represents the opinion polarization dynamics, where both susceptible and affected (infected) individuals prefer to mix within their own compartments (susceptible prefer to stay susceptible and infected prefer to stay infected). Conversely, the bottom-right quadrant represents the socio-economic turbulence dynamics, where both susceptible and affected groups seek to change their current state. In other words, susceptible individuals are attracted to locations with a higher fraction of affected (i.e., lower benefit), and affected individuals seek locations with a lower fraction of affected (i.e., higher benefit).

Fig. 1. The risk disposition space.

Each combination of the adaptive responsiveness (α^I, x-axis) of the affected population and risk aversion (α^S, y-axis) of the susceptible population characterizes specific contagion dynamics, where each group’s (S, I, and V) mixing preference is shown by their corresponding arrows. These (α^I, α^S) combinations correspond to four contagion types. Counter-clockwise from top right: (1) epidemic dynamics where both affected (I) and susceptible (S) groups avoid the contagion (indicated by both red and blue arrows pointing away from I towards S), because both risk aversion and adaptive responsiveness are positive; (2) opinion polarization dynamics where both groups seek to preserve their current state (positive risk aversion drives susceptible to stay unaffected and negative adaptive responsiveness drives affected to stay affected); (3) social myth spreading dynamics where both susceptible and affected gravitate towards individuals affected by the information contagion (driven by negative risk aversion and positive adaptive responsiveness); and (4) socio-economic turbulence where negative risk aversion and positive adaptive responsiveness draws individuals to opposite groups. The third dimension, risk mitigation (α^V, z-axis), refers to the tendency of inoculated individuals to seek additional measures to further reduce risk by avoiding areas of infection on top of being vaccinated or protected against the contagion. A positive risk mitigation (V in foreground) characterizes the inoculated group’s preference to mix with unaffected (susceptible) individuals to avoid contagion, as indicated by green arrow pointing to S, whereas a negative risk mitigation (V in background) indicates that inoculated individuals seek to interact with those affected by the contagion (green arrow pointing to I). Varying the signs of Lagrange multipliers α^I, α^S and α^V produces eight orthants (hyperoctants), with two risk mitigation scenarios for each of the four contagion types.

In this study, we expanded the risk disposition plane (composed of axes α^I and α^S) introduced by Jamerlan and Prokopenko¹⁵, to a three-dimensional risk disposition space formed by a third axis (α^V), which we labeled “risk mitigation” (see Fig. 1). We use the term “risk mitigation” to refer to the tendency of individuals to seek additional measures to further reduce the risk by avoiding areas of infection on top of being vaccinated or protected against the contagion^41,42. In contrast with the previous work, this study focuses on the role of the vaccinated (inoculated) population’s risk mitigation tendency in the development of spatial contagion patterns and their contribution to the overall contagion intensity.

Mapping inoculation and corresponding resources to contagion types

Importantly, each contagion type may correspond to real-world events and phenomena. For instance, epidemic scenarios capture events involving the spatial spread of infectious diseases, such as the COVID-19 pandemic^32,43. Polarization may correspond to population segregation⁴⁴, the formation of online echo chambers⁴⁵, or the emergence of highly polarized political landscapes⁴⁶. Scenarios mirroring the spread of social myths include the rapid dissemination of viral online content^16,47 or the formation of cults^48,49. Finally, socio-economic turbulence scenarios may reflect situations where war or civil unrest drives participation in violence^50,51, or where speculative bubbles in asset prices fuel increasingly risky speculative investment behavior^52,53.

In general, the vaccinated population (V) may be represented in other types of spatial contagion as the “inoculated” population (we acknowledge the difference between “vaccination” and “inoculation”, but for the purpose of this study, we use “inoculation” as a general term referring to individuals who become (temporarily) immune to the contagion after being provided or subjected to contagion mitigation resources such as vaccines, learning materials, information campaigns, and educational programs or interventions). Inoculation or protection from information or behavioral contagions can be achieved via the distribution of educational resources such as anti-discrimination campaigns, counter-radicalization programs, violence prevention programs, civic engagement training, and speculative investment awareness and education campaigns. In opinion polarization, the inoculated compartment (V) represents individuals who are resistant to bias or prejudice. In the spread of social myths and online viral trends (as well as cult formation), protected or inoculated individuals are fact-checkers and critical thinkers who do not accept these widespread beliefs. Finally, during socio-economic turbulence, inoculation against the contagion implies the individual resistance to engaging in collective violence or speculative boom-and-bust investment cycles (see Table 1).

Table 1.

Contagion types, spatial patterns, and shield immunity

Contagion type	Immunity to/Inoculation against	Pattern types	Shield immunity
Epidemics	infection	proto-gaps	moderate
Polarization	bias/prejudice, tribalism	stripes, proto-stripes, labyrinth, proto-labyrinth, gaps, proto-gaps	not observed
Social Myths	belief in myths, influence of online viral phenomena	spots, proto-spots, labyrinth, gaps	limited
Socio-economic Turbulence	socio-economic contagion (participation in collective violence or risky speculative investment activities)	proto-checkerboard, checkerboard	strong

For all contagion types, α^V represents the inoculated individuals’ risk mitigation tendency, or their preference to mix or interact with affected individuals given their current protected state. When α^V > 0, inoculated individuals mitigate the risk by avoiding locations with higher affected populations despite already being protected (e.g., vaccinated individuals still avoiding infected during an epidemic). On the other hand, α^V < 0 suggests that inoculated individuals seek to interact with the affected population driven by non-risk-avoiding behavior (due to their protected state^34,54) or altruistic acts (i.e., fact-checkers seeking to interact with myth-believing individuals to involve them in educational campaigns).

The effectiveness in reduction of infection due to vaccination, denoted by σ, can be generalized for other contagion types as the effectiveness of the corresponding mitigation resources in preventing new infections (i.e., new contagion cases). In an epidemic scenario, it measures the effectiveness of a vaccine in preventing new infections. For non-epidemic contagions, we consider its generalized form, which we refer to as the “effectiveness of resources”, to represent the effectiveness of each resource in mitigating its corresponding contagion type. Thus, σ may represent the effectiveness of anti-discrimination campaigns in preventing individuals from having prejudiced beliefs (polarization), the effectiveness of fact-checking campaigns in suppressing the spread of a social myth, or the effectiveness of peace inoculation programs in safeguarding populations from engaging in forms of social turbulence. This parameter may be varied smoothly along a spectrum, along which we can identify notable shifts in contagion dynamics which correspond to phase transitions in contagion pattern formation. Specifically, the value of σ ranges from 0 to 1, with 0 indicating a completely effective resource or perfect inoculation (i.e., no new infections arise from inoculated individuals interacting with the affected population) and 1 indicating a useless resource (i.e., transmission rate stays at β for inoculated individuals who interact with the affected population).

Each combination of α^I, α^S and α^V produces a specific spatial contagion pattern on the lattice of locations, where darker areas represent locations with more severe relative contagion states (see Fig. 2). We fix α^I and α^S at specific values to represent each contagion type and vary α^V to investigate the effects of different risk mitigation tendencies on the contagion pattern formation. Additionally, the parameter for effectiveness of reduction in infection due to vaccination (σ) is also varied to analyze the impact of the effectiveness of the distributed mitigation resources (i.e., how well they prevent new infections/cases among the inoculated population) on the resultant patterns of contagion. Varying both α^V and σ across independent axes generates the α^V− σ phase space, allowing us to analyze transitions in pattern formation in response to parameter changes.

Fig. 2. Spatial contagion pattern types.

Sample proto-patterns and patterns: (a) proto-spots, produced by social myths spreading, (b) proto-gaps, produced by epidemics and polarization dynamics, (c) proto-labyrinth, produced by polarization dynamics, (d) proto-checkerboard, produced by socio-economic turbulence dynamics, (e) proto-stripes, produced by polarization dynamics, (f) spots, produced by social myths spreading, (g) gaps and (h) labyrinth, both produced by polarization dynamics and social myths spreading, (i) checkerboard, produced by socio-economic turbulence dynamics and (j) stripes, produced by polarization dynamics.

Effect of varying risk mitigation and effectiveness of resources on contagion pattern formation

We explored pattern formation across the α^V − σ phase space for each contagion type: (1) epidemics, (2) polarization, (3) social myths and (4) socio-economic turbulence. The observed patterns and proto-patterns are shown in more detail in Fig. 2. The different pattern types observed within the α^V − σ phase space of each contagion type are listed in Table 1. Notably, adjacent (α^V, σ) combinations may produce different pattern types within the same contagion type, implying that small changes in bounded risk mitigation and bounded effectiveness of resources produce transitions across pattern types.

For each contagion type, we show the resultant contagion states in two ways: (a) resultant infection (i.e., contagion “incidence”, in general) levels normalized per (α^V, σ) configuration (local maximum) to emphasize pattern formation, and (b) resultant infection ("incidence”) levels normalized across the α^V − σ space (global maximum) to highlight differences in the overall contagion severity.

For epidemics, the system’s response to variations in effectiveness of resources σ is more significant compared to risk mitigation α^V, as evidenced by larger shifts in pattern formation (from proto-gaps to a uniform state). In Fig. 3a (bottom row), the proto-gaps pattern for σ = 0 (perfect vaccine) remains unchanged as α^V is varied from left to right. At σ = 0.2, a structure-less pattern appears at α^V = −40, which develops more structure as α^V is increased. In terms of resultant contagion severity, Fig.3b shows that infection levels increase as σ increases (ineffective vaccines).

Fig. 3. Spatial patterns for epidemic dynamics.

Resultant contagion patterns for epidemics (α^I = 20, α^S = 50) are shown across the α^V − σ phase space. Darker areas indicate higher levels of relative infection. The resulting infection values are normalized for: (a) each α^V − σ configuration, and (b) the whole phase space.

In opinion polarization (Fig. 4a), configurations characterized by a “proto-stripes” pattern (combination of stripes and gaps) form at a high σ (0.4–1) and negative α^V (−40 to −20). For σ > 0, the gaps pattern can be found above the positive diagonal of the α^V − σ phase space. Below the diagonal, sections of the gaps patterns disintegrate to form labyrinthine and proto-labyrinthine patterns. Figure. 4b shows an expected increase in resultant infection levels as σ increases (less effective resources) and risk mitigation α^V decreases (inoculated V prefer to mix with the affected I).

Fig. 4. Spatial patterns for polarization dynamics.

Resultant contagion patterns for polarization (α^I = −30, α^S = 50) are shown across the α^V − σ phase space. Darker areas indicate higher levels of relative infection. The resulting infection values are normalized for: (a) each α^V − σ configuration, and (b) the whole phase space.

Figure 5 shows that for social myths, a higher α^V results in lower levels of resultant infection. The same lower resultant infection levels are achieved as resources become more effective (approaching σ = 0). Social myth spreading dynamics produces a range of spots, proto-spots, labyrinthine, proto-labyrinthine and gaps patterns. A uniform state is observed above the positive diagonal of the α^V − σ space, while spots are observed for all values of α^V when σ = 0 and for α^V≤ 0 when σ = 0.2. The spots patterns become labyrinthine patterns almost abruptly at α^V > 0 when σ ≤ 0.4. A “proto-spots” pattern characterized by enlarged gaps forms gradually at relatively high risk mitigation α^V = 20 and less effective resources (0.6 ≤ σ ≤ 1). This proto-spots pattern comprises larger but fewer safe havens from the infection. Additionally, Fig. 5b shows an expected increase in resultant contagion levels and fewer safe havens as σ increases (ineffective resources).

Fig. 5. Spatial patterns for social myth spreading dynamics.

Resultant contagion patterns for social myth spreading (α^I = −50, α^S = −20) are shown across the α^V − σ phase space. Darker areas indicate higher levels of relative infection. The resulting infection values are normalized for: (a) each α^V − σ configuration, and (b) the whole phase space.

Figure 6 shows that for socio-economic turbulence, an almost counter-intuitive effect is observed, where a higher α^V results in higher resultant infection levels despite the inoculated individuals avoiding interactions with affected individuals. We argue that this effect is similar to shield immunity in epidemics³² where individuals with immunity are deployed for interaction substitution, aiming to reduce contacts between susceptible and affected individuals. This implies that the inoculated individuals (“prudent”, for the bottom-right quadrant of Fig. 1) who have undergone socio-economic awareness training, and thus are resistant to the pressures of engaging in violence or speculative boom-and-bust investment activities, act as a shield for the individuals who are susceptible and unaware. Arguably, this is an example of the shield immunity effect, and we will verify this conjecture in the following section.

Fig. 6. Spatial patterns for socio-economic turbulence dynamics.

Resultant contagion patterns for socio-economic turbulence (α^I = 30, α^S = −40) are shown across the α^V − σ phase space. Darker areas indicate higher levels of relative infection. The resulting infection values are normalized for: (a) each α^V − σ configuration, and (b) the whole phase space.

Measuring generalized shield immunity: Average cluster intensity

In the previous section, we observed the emergence of shield immunity in the space of α^V and σ for socio-economic turbulence (Fig. 6). In order to quantify this effect, and check the extent of possible shield immunity for other contagion types, we compute the average cluster intensity for each contagion type, defined as the average cluster size scaled by the mean contagion level (see Section “Average cluster intensity”). We intend to quantify a higher shield immunity with a lower average cluster intensity.

In epidemic scenarios (Fig. 7, upper right quadrant), shield immunity emerges moderately for σ = 0.2 (high resource effectiveness). This is evidenced by a progressive decrease in $\tilde{r}$ from 0.1689 to 0.0279 (i.e., progressive increase in shield immunity) as α^V decreases. In other words, as inoculated individuals increasingly prefer to interact with infectious (α^V < 0) rather than avoid them (α^V > 0), shield immunity increases. For less effective resources (σ > 0.2), no patterns are formed (uniform infection levels), rendering the shielding effect undetected in these scenarios. We conclude that shield immunity is moderately observed in epidemic scenarios.

Fig. 7. Average cluster intensities for each contagion type.

Average cluster intensity $\tilde{r}$ of each α^V − σ configuration for the considered contagion types. Each $\tilde{r}$ was calculated from the binarized contagion patterns in Figs. 3a, 4a, 5a, and 6a. Note that $\tilde{r}$ is undefined when there are no clusters (i.e., contagion levels reach a uniform state and no pattern is formed). Generalized shield immunity, characterized by decreasing intensity $\tilde{r}$ as α^V decrease, is most pronounced in socio-economic turbulence scenarios, while being moderate in epidemic scenarios and limited in social myth scenarios.

In contrast, for polarization scenarios (Fig. 7, upper left quadrant), decreasing α^V does not reduce average cluster intensity $\tilde{r}$. In fact, this intensity tends to grow with increasing interactions between inoculated and affected individuals. Interestingly, the configurations with the lowest average cluster intensity $\tilde{r}$ can be found at α^V = 0 (i.e., when the inoculated population is indifferent). The lack of a discernible trend in $\tilde{r}$, combined with the clear increase in resultant contagion levels as resources become less effective (i.e., σ increases) and risk mitigation α^V decreases (as shown in Fig. 4; discussed in Section “Effect of varying risk mitigation and effectiveness of resources on contagion pattern formation”), suggests that shield immunity is not observed in polarization scenarios. However, increasing resource effectiveness (reducing σ in the vertical dimension) reduces both average cluster intensity $\tilde{r}$ (see Fig. 7, upper left quadrant) and average contagion level (see Fig. 4).

In the spread of social myths (Fig. 7, lower left quadrant), a sharp decline in $\tilde{r}$ is observed for a high resource effectiveness (σ = 0.2) as α^V shifts from −20 to −40, corresponding to a stronger preference of inoculated individuals to interact with the affected population. However, for less effective resources (0.4 ≤ σ ≤ 1), this effect is not observed; instead, $\tilde{r}$ tends to increase as α decreases from +40 to +20. The sharp decrease in the average cluster intensity $\tilde{r}$ only occurs for social myth spreading dynamics within a limited range of the α^V − σ space (−40 ≤ α^V≤ −20, σ = 0.2). This is also confirmed by Fig. 5. These findings indicate that shield immunity is limited in this scenario, being observed only when the inoculated population is highly immune to the social myth.

In scenarios modeling socio-economic turbulence (Fig. 7, lower right quadrant), the average cluster intensity $\tilde{r}$ exhibits a consistent decline as α^V decreases for each level of resource effectiveness σ. Interestingly, the intensity also reduces in the vertical dimension as resources become more effective, for each risk mitigation level α^V. This indicates that shield immunity is clearly present in the entire α^V − σ phase space. This suggests that the generalized shield immunity is most pronounced in socio-economic turbulence scenarios in comparison to other contagion types.

Discussion

We have modeled four distinct contagion types in a unified framework comprising elements of risk disposition, resource distribution, effectiveness of resources and population mobility. Using a generalized SVIRS-network model and Boltzmann’s Maximum Entropy principle (MaxEnt), we related system constraints to model parameters representing varying levels of risk disposition (α^I, α^S, and α^V). The effects of inoculated individuals’ bounded risk mitigation on the resultant contagion state and pattern formation within each contagion type were then investigated.

The two risk disposition parameters, adaptive responsiveness α^I and risk aversion α^S (introduced in our previous work where susceptibility acquisition (ζ) was included as a third dimension¹⁵), were fixed at representative values. These values were selected to illustrate each of the four contagion types on the risk disposition plane. In this work, we introduced the third risk disposition parameter, risk mitigation α^V, as the third dimension. This allowed us to quantify the tendency of inoculated individuals to mitigate risk associated with the contagion.

The risk mitigation tendency is characterized by the preference to either avoid affected populations (α^V > 0, positive risk mitigation) or interact with them (α^V < 0, negative risk mitigation). Another parameter, denoted σ, traditionally used in epidemic modeling to represent effectiveness in reduction of infection due to vaccination, was generalized to represent the effectiveness of the distributed resources in mitigating each contagion type, with a value ranging from 0 (perfect inoculation) to 1 (no inoculation). Risk mitigation and effectiveness of resources were varied along two independent axes, forming the α^V– σ phase space within which we characterized resultant contagion patterns.

By extending the canonical SVIRS model into an SVIRS-network model with spatial considerations, we captured how individual movement and contagion dynamics resulted in spatial configurations similar to Turing patterns found in reaction-diffusion (RD) systems^55,56. In canonical RD systems, Turing patterns are formed by an imbalance between local self-enhancement (reaction) and lateral inhibition (diffusion) mechanisms, creating stable spatial patterns in the form of spots, gaps, and labyrinthine structures⁵⁷. Although not a canonical RD system, our SVIRS-network model incorporates elements of reaction (interactions S_iI_k and V_iI_k) and diffusion (flows ϕ^S, ϕ^I, and ϕ^V), enabling the emergence of these spatial patterns^4,15,55. Specifically, we modeled (slow) SVIRS-network dynamics using Lotka-Volterra predator-prey interactions, augmented with (fast) risk-driven mobility flows ϕ^S, ϕ^I and ϕ^V across lattice locations. These flows were updated at each iteration using the Maximum Entropy (MaxEnt) principle, based on perceived location benefits b_j and bounded risk disposition (α_S and α_I) and risk mitigation (α^V) parameters. This combination of Lotka-Volterra dynamics and MaxEnt produced varying spatiotemporal patterns under different risk disposition and risk mitigation attitudes. These patterns exhibited a degree of robustness, remaining observable within the bounds of the parameter space where small changes in parameters led to significant shifts in pattern structure. We also explored how these distinct contagion patterns evolved across the space of effectiveness of resources in contagion mitigation.

For each contagion type, we identified notable transitions across the α^V − σ space between distinct pattern types, such as spots, gaps, labyrinth and checkerboard. In addition to these patterns, several pattern types not observed in network SIS⁴, SIR⁵⁶ and SIRS¹⁵ models were identified, such as proto-spots, proto-gaps, proto-labyrinth, proto-checkerboard, stripes, and proto-stripes. The existence of several pattern types implies that the addition of resource distribution dynamics, represented by the generalized effectiveness of resources and the inoculated population’s risk mitigation tendency, has a different effect for each contagion type.

In epidemics, the formation of gaps and proto-gaps patterns was observed across the α^V − σ space. Changes along the effectiveness of resources parameter σ resulted in larger or more pronounced steps in pattern formation, indicating a stronger impact on epidemic dynamics compared to the risk mitigation parameter α^V. Thus, the effectiveness of vaccines in preventing new infections had a more significant role in mitigating the contagion than the vaccinated population’s risk mitigation tendency. This highlights the importance of implementing vaccination programs alongside non-pharmaceutical interventions (NPIs), as opposed to relying solely on NPIs without vaccination to effectively manage and reduce the spread of infection^38,58–60. Additionally, for the case with high resource effectiveness (low σ), the emergence of shield immunity is shown by the progressive decrease in average cluster intensity as inoculated individuals increasingly prefer to interact with the infected population. This behavior aligns well with Weitz et al.’s ³² characterization of shield immunity, whereby strategically placed recovered agents (i.e., effective inoculation) effectively substitute interactions of the susceptible population with the infected population and create a moderate shielding effect.

For polarization, patterns were found to be sensitive to changes in both risk mitigation and effectiveness of resources, resulting in transitions across pattern types between adjacent (α^V and σ) configurations. These pattern types include gaps, proto-gaps, labyrinth, proto-labyrinth, stripes and proto-stripes. Stripes were possibly produced by a form of axial spatiotemporal synchronization promoted by α^V. Some implications arise from the presence of stripes and proto-stripes. Firstly, the formation of stripes may suggest that the contagion oscillates spatially, with some locations experiencing peaks while others recover from the contagion (possibly alternating over time). Secondly, the boundaries between stripes may act as buffer zones where contagion spread slows, leading to lower connectivity between alternating regions. Thirdly, proto-stripes may indicate that the system has not fully synchronized globally but has achieved local synchronization within subsets of the lattice. In general, targeting interventions at the stripe boundaries may be particularly effective, as these regions could represent zones of disease propagation between synchronized clusters. In terms of the overall resultant contagion state for polarization dynamics, a combination of high resource effectiveness (low σ) and risk-mitigating inoculated individuals resulted in lower levels of contagion. Furthermore, average cluster intensity was observed to (predominantly) increase when the inoculated population are non-risk-mitigating, particularly when inoculation resources are less effective. In cases with moderate to high resource effectiveness, lower contagion intensities result when the inoculated population is neither risk-mitigating nor non-risk-mitigating (i.e., indifferent). These findings suggest that generalized shield immunity is not observed in polarization dynamics. The lack of generalized shield immunity therefore implies that individuals who exhibit some degree of resistance to bias or prejudice after being subjected to anti-discrimination campaigns may still benefit from non-engagement with prejudiced groups. This result is consistent with findings of how prejudice leverages on language-use-in-social-context, where prejudiced language is often used to gain and maintain specific relationships, rather than directly promote prejudice, resulting in more influential discourses⁶¹.

Social myth spreading dynamics produces a range of spots, proto-spots, labyrinthine and gaps patterns. Similar to polarization dynamics, lower levels of contagion are achieved in social myth spreading when high resource effectiveness (low σ) is coupled with risk-mitigating inoculated individuals. In other words, even when individuals are subjected to fact-checking campaigns and critical thinking programs, if these interventions are ineffective, they may still benefit from avoiding interaction with groups who accept a social myth. This risk-mitigating behavior reduces opportunities for repetitive exposure to misinformation (i.e., “illusory truth effect”⁶²). However, when the inoculated group is highly immune to the contagion (i.e., the intervention programs are highly effective), generalized shield immunity emerges. We characterize this effect as a limited shield immunity as it only occurs within a specific range of the risk-mitigation and resource-effectiveness space. Furthermore, this observation aligns with the findings of Tambuscio and Ruffo¹⁶, where a shielding effect was associated with “eternal fact-checkers” who are highly immune to the spreading myth (i.e., in our case, inoculated by resources with high effectiveness).

Socio-economic turbulence dynamics produced checkerboard patterns, which were previously characterized as checkerboard patterns with overlaid strands and rings¹⁵. However, a more precise characterization of this pattern is that the strands and rings are domain walls⁶³ separating different checkerboard sub-regions. The presence of domain walls in socio-economic turbulence dynamics may indicate persistence (i.e., continuous reintroduction) of the contagion at the boundaries. This highlights the importance of targeted interventions or surveillance in inter-city regions with high levels of violence or investment activity. Additionally, the domain walls separate regions with the same but discordant (i.e., temporally mismatched) checkerboard pattern. This indicates a potential amplification of contagion levels near the boundary if susceptible populations are available in alternating cycles. Additionally, generalized shield immunity was most pronounced in socio-economic turbulence dynamics, remaining clearly observable throughout the α^V − σ space. This implies that the proactive deployment of targeted interventions — such as financial awareness campaigns or peace education initiatives — when combined with interactions between inoculated individuals and the affected population, may serve as an effective strategy for reducing the impacts of boom-and-bust investment activities or civil unrest.

In general, we observed varying degrees of shield immunity across different contagion types. In the case of epidemics and socio-economic turbulence, where inoculated (V) and affected (I) individuals seek to mix and interact, generalized shield immunity emerges when (altruistic) inoculated individuals (e.g., peacemakers, volunteers) interact with affected individuals and indirectly provide protection for the susceptible population (via a form of interaction substitution³²). For polarization dynamics, generalized shield immunity was not observed. For social myth spreading dynamics, shield immunity was observed only for a limited region of the parameter space. In general, both polarization and social myths dynamics are relatively volatile as effectiveness of resources changes, however, making the resources more effective by even a small fraction may lead to more favorable outcomes.

We thus argue that generalized shield immunity is amplified by the combination of behaviors during epidemics and socio-economic turbulence where affected individuals (I) seek areas with lower affected populations and inoculated (V) individuals seek locations with higher affected populations (vacated by dispersing affected individuals). This results in inoculated individuals substituting the otherwise susceptible (vulnerable) population, facilitating the rapid development of herd immunity, which accelerates the eradication of contagion. Additionally, the susceptible-to-inoculated conversion via resource distribution is higher due to this uninfected ("unaffected”) majority. In contrast, during opinion polarization and the spread of social myths, the affected individuals prefer to stay in locations with higher levels of contagion, resulting in depleted resources as a larger fraction of the susceptible population becomes affected (compared to scenarios of socio-economic turbulence).

The emergence of (limited) shield immunity across different contagion types implies that, in some cases, the altruistic behavior of inoculated individuals (where they seek to interact with the affected population) may reduce the system’s total contagion state. This effect was particularly pronounced in socio-economic turbulence dynamics, moderately observable in epidemic scenarios, limited during the spread of social myths, and not observed in polarization dynamics.

Our study examined the impact of the adaptive behavior of inoculated individuals and the effectiveness of resources on resultant contagion patterns, and showed how slight changes in each of both factors can lead to drastically different outcomes. The calibration and empirical validation of our model using real data remains a subject of future research. Another limitation is that the adopted lattice topology and simplified mobility dynamics do not fully capture the heterogeneity and complexity of real-world contagions.

Overall, our unifying contagion modeling framework promotes the integrated study of contagion dynamics across several types within a coherent behavioral space, and contributes to the understanding of socio-economic and socio-political impacts associated with these dynamics. Additionally, the evidence of shield immunity emergence in specific contagion types may guide policy-makers to actively develop and implement targeted intervention programs and educational campaigns to combat epidemics, infodemics (misinformation, disinformation, etc.), civil unrest and boom-and-bust investment activities.

Methods

Canonical SVIRS epidemic model

The SVIRS epidemic model is defined by the following differential equations¹⁷:

$$\stackrel{°}{S}=-\frac{\beta }{N}SI-\nu S+\eta V+\zeta R$$ 1

$$\stackrel{°}{V}=\nu S-\sigma \frac{\beta }{N}VI-\eta V$$ 2

$$\stackrel{°}{I}=-\gamma I+\frac{\beta }{N}SI+\sigma \frac{\beta }{N}VI$$ 3

$$\stackrel{°}{R}=\gamma I-\zeta R$$ 4

where S, V, I, R are susceptible, vaccinated (inoculated), infected, and recovered compartments, respectively. The total population is denoted by N = S + V + I + R, while β is the transmission rate, γ is the recovery rate, ν is the vaccination (inoculation) rate, η is the vaccination (inoculation) waning rate, ζ is the loss of immunity, and σ is the effectiveness of resources in reduction of new contagion cases. These variables are listed in Table 2.

Table 2.

Simulation variables

Variable	Definition	Value or Range	Dimensions
L	lattice side	40	Scalar
M	number of locations	L × L	Scalar
P	population per location	1000	Scalar
β	infection rate	10	Scalar
γ	recovery rate	5	Scalar
ω	Lagrange multiplier of cost	1	Scalar
ν	vaccination (inoculation) rate	1	Scalar
ζ	rate of natural immunity loss	5	Scalar
η	vaccine (inoculation) waning rate	0.01	Scalar
σ	effectiveness of resources	[0, 1] in increments of 0.2	Scalar
αS	bounded risk aversion of susceptible individuals	[+50, +50, −20, −40]	Scalar
αI	bounded adaptive responsiveness of affected individuals	[+20, −30, −50, +40]	Scalar
αV	bounded risk mitigation of inoculated individuals	[−40, + 40] in increments of 20	Scalar
I	vector of infection values for all locations i: 1 → M	initialized with a random value between 0 and 5% of P	1 × M
bj	benefit of moving to location j	$\frac{P-I}{P}$	1 × M
cij	cost of moving from location i to j	1 if i and j are neighbors, ∞ otherwise	M × M
${\varphi }_{ij}^S$	fraction of susceptible individuals moving from i to j	$\exp \left({\alpha }^S{b}_j-\omega c_{ij}\right)$ if i and j are neighbors, 0 otherwise	M × M
${\varphi }_{ij}^I$	fraction of affected individuals moving from i to j	$\exp \left({\alpha }^I{b}_j-\omega c_{ij}\right)$ if i and j are neighbors, 0 otherwise	M × M
${\varphi }_{ij}^V$	fraction of inoculated individuals moving from i to j	$\exp \left({\alpha }^V{b}_j-\omega c_{ij}\right)$ if i and j are neighbors, 0 otherwise	M × M

Generalized SVIRS-network model

Our model is a generalized SVIRS-network (multi-city) contagion model (derived from the canonical SVIRS model) running on an L × L lattice of M locations. Each location i (i ∈ 1, . . . , M) has S_i susceptible individuals, I_i affected (infected) individuals, V_i vaccinated individuals and R_i recovered individuals, adding up to the total sub-population P_i = S_i + V_i + I_i+ R_i. We refer to the set of all I_i, denoted as I, as the state of contagion of the system (across all M locations).

We set the cost of mobility c_ij from location i to location j, denoted by the adjacency matrix C. Each element c_ij of C keeps mobility flows from each location i limited to within itself and its four adjacent neighbors (top, bottom, left, right). That is, c_ii = c_i+1i = c_i−1i = c_ii+1= c_ii−1 = 1 while all other c_ij values are set to ∞. Each element on C may be adjusted to represent general accessibility measures or actual cost of commute.

The generalized SVIRS multi-city network model is formalized as follows:

$$\frac{d{S}_i}{dt}=-\beta \sum _{j,k}{\varphi }_{ij}^S\left(\mathbf{I},\mathbf{C}\right){\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right)\frac{S_i{I}_k}{{\widehat{N}}_j\left(\mathbf{I},\mathbf{C}\right)}-\nu S+\eta V+\zeta R$$ 5

$$\begin{array}{c}\frac{d{I}_i}{dt}=-\gamma I_i+\beta \sum _{j,k}{\varphi }_{ij}^S\left(\mathbf{I},\mathbf{C}\right){\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right)\frac{S_i{I}_k}{{\widehat{N}}_j\left(\mathbf{I},\mathbf{C}\right)}\hfill \\ +\sigma \beta \sum _{j,k}{\varphi }_{ij}^V\left(\mathbf{I},\mathbf{C}\right){\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right)\frac{V_i{I}_k}{{\widehat{N}}_j^{\prime }\left(\mathbf{I},\mathbf{C}\right)}\hfill \end{array}$$ 6

$$\frac{d{V}_i}{dt}=\nu S-\sigma \beta \sum _{j,k}{\varphi }_{ij}^V\left(\mathbf{I},\mathbf{C}\right){\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right)\frac{V_i{I}_k}{{\widehat{N}}_j^{\prime }\left(\mathbf{I},\mathbf{C}\right)}-\eta V$$ 7

$$\frac{d{R}_i}{dt}=\gamma I_i-\zeta R_i$$ 8

where

$${\widehat{N}}_j\left(\mathbf{I},\mathbf{C}\right)=\sum _k{S}_k{\varphi }_{kj}^S\left(\mathbf{I},\mathbf{C}\right)+I_k{\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right).$$ 9

and

$${\widehat{N}}_j^{\prime }\left(\mathbf{I},\mathbf{C}\right)=\sum _k{V}_k{\varphi }_{kj}^V\left(\mathbf{I},\mathbf{C}\right)+I_k{\varphi }_{kj}^I\left(\mathbf{I},\mathbf{C}\right).$$ 10

The normalization term ${\widehat{N}}_j\left(\mathbf{I},\mathbf{C}\right)$ represents the number of susceptible and affected individuals mixing at a given location j, while ${\widehat{N}}_j^{\prime }\left(\mathbf{I},\mathbf{C}\right)$ refers to the number of vaccinated and affected individuals mixing at location j.

The mobility flows ${\varphi }_{ij}^S$, ${\varphi }_{ij}^I$ and ${\varphi }_{ij}^V$ represent the fraction of individuals from each compartment (S, I, V) moving from locations i to j and contributing to the contagion state I. Additionally, γ is the recovery rate, β is the probability of transmission or infection per susceptible–affected interaction (see sensitivity analysis in Section “Sensitivity analysis: transmission and recovery rates”), ν is the vaccination rate for susceptible individuals, η is the vaccine waning rate, ζ is the rate of natural immunity loss or susceptibility acquisition for recovered individuals, and σ is the effectiveness of reduction in infection due to vaccination¹⁷. Specifically, σ = 0 represents a perfect vaccine, with zero vaccinated individuals getting infected (“affected”) among all vaccinated–affected interactions. On the other hand, σ = 1 represents no vaccine (or a completely ineffective vaccine), where all vaccinated–affected interactions are subjected to the baseline infection transmission probability β. Hence, the last two terms of Eq. (6) denote the incidence of new infections among both susceptible and vaccinated populations of location i due to interactions with the affected population of location k, specifically occurring in (mixing) location j.

We emphasize that the mobility flows ${\varphi }_{ij}^S$, ${\varphi }_{ij}^I$ and ${\varphi }_{ij}^V$ are not fixed, but rather change in response to I, closing the feedback loop between contagion state and mobility. To determine the mobility flows at every time step, we first consider a benefit b_j defined by the state of contagion in j as $b_j=\frac{P_j-I_j}{P_j}$ (i.e., the fraction of unaffected individuals per location). Given both mobility cost and benefits, we utilize the MaxEnt approach to calculate flows ${\varphi }_{ij}^S$, ${\varphi }_{ij}^I$, and ${\varphi }_{ij}^V$ (see full derivation in Section “Maximum entropy method”), by selecting the probability distribution that maximizes Shannon entropy

$$H=-\sum _j{\varphi }_{ij}^X\log {\varphi }_{ij}^X$$ 11

while meeting the following constraints for susceptible, affected, and vaccinated individuals:

$$B^S=\sum _{i,j}{S}_i{\varphi }_{ij}^S\left(\mathbf{I},\mathbf{C}\right)b_j/\sum _i{S}_i,$$ 12

$$B^I=\sum _{i,j}{I}_i{\varphi }_{ij}^I\left(\mathbf{I},\mathbf{C}\right)b_j/\sum _i{I}_i,$$ 13

$$B^V=\sum _{i,j}{V}_i{\varphi }_{ij}^V\left(\mathbf{I},\mathbf{C}\right)b_j/\sum _i{V}_i,$$ 14

and

$$C=\sum _{i,j}\left(S_i{\varphi }_{ij}^S\left(\mathbf{I},\mathbf{C}\right)+I_i{\varphi }_{ij}^I\left(\mathbf{I},\mathbf{C}\right)+V_i{\varphi }_{ij}^V\left(\mathbf{I},\mathbf{C}\right)\right)c_{ij}/\sum _i\left(S_i+I_i+V_i\right).$$ 15

Given that recovered individuals who recover with immunity do not contribute to the infection spread (i.e., do not affect mixing interactions), we do not consider a benefit B^R nor a mobility flow ${\varphi }_{ij}^R$. Additionally, recovered individuals who lose natural immunity return to the Susceptible compartment. The MaxEnt method yields the least biased solutions for ${\varphi }_{ij}^S$, ${\varphi }_{ij}^I$ and ${\varphi }_{ij}^V$:

$${\varphi }_{ij}^S\left(\mathbf{I},\mathbf{C}\mid {\alpha }^S,\omega \right)=Z_{S,i}^{-1}\exp \left({\alpha }^S{b}_j-\omega c_{ij}\right),$$ 16

$${\varphi }_{ij}^I\left(\mathbf{I},\mathbf{C}\mid {\alpha }^I,\omega \right)=Z_{I,i}^{-1}\exp \left({\alpha }^I{b}_j-\omega c_{ij}\right),$$ 17

$${\varphi }_{ij}^V\left(\mathbf{I},\mathbf{C}\mid {\alpha }^V,\omega \right)=Z_{V,i}^{-1}\exp \left({\alpha }^V{b}_j-\omega c_{ij}\right),$$ 18

where the normalization terms are as follows: $Z_{S,i}={\sum }_j\exp \left({\alpha }^S{b}_j-\omega c_{ij}\right)$, $Z_{I,i}={\sum }_j\exp \left({\alpha }^I{b}_j-\omega c_{ij}\right)$ and $Z_{V,i}={\sum }_j\exp \left({\alpha }^V{b}_j-\omega c_{ij}\right)$. These solutions depend on the Lagrange multipliers α^I, α^S , α^V and ω.

All mobility flows equilibrate instantly at every time step in response to incremental changes in contagion state I. In contrast, I equilibrates at a slower pace over multiple time steps. This difference in time scales defines the fast and slow dynamics characterizing our BLV-based framework³⁹.

Each location i on the lattice is initialized with a random affected population I_i ranging from 0 to 5% of the total sub-population P_i. Integrating Eq. (5)–(8) with Δt = 0.001 over 10,000 time steps while varying risk mitigation α^V and effectiveness of resources σ (the generalized effectiveness in reduction of infection due to vaccination) produced distinct spatial patterns descriptive of the resulting contagion state for each α^V − σ combination.

Average cluster intensity

In percolation theory, spatial patterns within a lattice are characterized by connected clusters^4,64. We identify clusters by binarizing contagion patterns using an adjusted mean threshold of (1 + 10⁷) × 〈I〉 to ensure numerical stability in floating-point comparisons⁶⁵. Locations are considered part of the same cluster if they are adjacent either horizontally, vertically, or diagonally. Using the probability p(r) that a selected location belongs to a cluster of size r, we consider a normalized quantity $\widehat{r}=\frac{r}{L^2}$⁶⁶. The mean value, $⟨\widehat{r}⟩$, or the average cluster size (ACS), allows us to discriminate the resultant spatial patterns. Following Jamerlan and Prokopenko¹⁵, in computing the ACS, we do not exclude the largest cluster.

We define the average cluster intensity as follows:

$$\tilde{r}=⟨\mathbf{I}⟩\cdot ⟨\widehat{r}⟩$$ 19

where $⟨\mathbf{I}⟩=\frac{1}{L^2}{\sum }_i^{L^2}\frac{I_i}{P_i}$ is the mean contagion level. This measure quantifies how contagious the average cluster is in terms of the underlying contagion by scaling the ACS with the mean contagion level.

Sensitivity analysis: transmission and recovery rates

To establish robustness of the model, we performed a sensitivity analysis by varying the rates of transmission (β) and recovery (γ). Our focus was on modeling socio-economic turbulence scenarios where the shield immunity effect was most prominent. We compared the key model outcomes (i.e., resultant spatial patterns and the average cluster intensity) for two different combinations (β, γ): more contagious (scenario A) and less contagious (scenario B) than the combination used in the baseline model (β = 10, γ = 5).

Socio-economic turbulence scenario A (Fig. 8a) used β = 10 and γ = 2.5. It showed that the expected checkerboard patterns with domain walls produced by the baseline model (cf. Fig. 6a) only appeared at a high resource effectiveness (0 ≤ σ ≤ 0.4). Pattern formation at σ = 0.4 was limited to negative risk mitigation values (−40 ≤ α^V≤ −20), where the inoculated group prefers to mix with the affected group.

Fig. 8. Socio-economic Turbulence Scenario A (more contagious than baseline model).

Contagion patterns and average cluster intensity for socio-economic turbulence (α^I = 30, α^S = −40, β = 10 and γ = 2.5). Sub-figure (a) should be compared with Fig. 6a, sub-figure (b) with Fig. 6b, and sub-figure (c) with Fig. 7 (lower right quadrant). Darker areas indicate higher levels of relative contagion or average cluster intensity.

The resulting average cluster intensity ($\tilde{r}$), shown in Fig. 8c, indicated the presence of shield immunity for higher resource effectiveness (i.e., low values of σ) and negative risk mitigation values −40 ≤ α^V≤ 0. However, the shield immunity effect observed for this more contagious combination of transmission and recovery rates was less pronounced than the shield immunity produced by the baseline model.

For socio-economic turbulence scenario B we used β = 5.5 and γ = 5. Figure 9a shows that the expected checkerboard patterns with domain walls (cf. Fig. 6a) remain observable across the entire α^V − σ parameter space. A consistent decrease in the average contagion level can also be observed as α^V decreases (see Fig. 9b), in concordance with the baseline model (cf Fig. 6b). Finally, similar to the baseline model (cf. Fig. 7, lower right quadrant), the average cluster intensity $\tilde{r}$ also decreases with the decreasing risk mitigation level. This confirms the presence of shield immunity when inoculated individuals interact with the affected population in this less contagious scenario (cf. Fig. 9c).

Fig. 9. Socio-economic Turbulence Scenario B (less contagious than baseline model).

Contagion patterns and average cluster intensity for socio-economic turbulence (α^I = 30, α^S = −40, β = 5.5, γ = 5). Sub-figure (a) should be compared with Fig. 6a, sub-figure (b) with Fig. 6b and sub-figure (c) with Fig. 7 (lower right quadrant). Darker areas indicate higher levels of relative contagion or average cluster intensity.

Overall, the comparison of the baseline model with both more contagious and less contagious scenarios of socio-economic turbulence suggests that the model is robust to variations in the transmission (β) and recovery (γ) rates. We also note that shield immunity can be observed for different values of resource effectiveness (σ).

Maximum entropy method

To determine the flows ${\varphi }_{ij}^S$, ${\varphi }_{ij}^I$, and ${\varphi }_{ij}^V$ at each time step, we apply the Maximum Entropy (MaxEnt) principle to derive the least biased distributions consistent with known constraints. For each origin location i, we seek the distribution ${\varphi }_{ij}^X$ over destinations j (where X ∈ {S, I, V}) that maximizes the Shannon entropy

$$H=-\sum _j{\varphi }_{ij}^X\log {\varphi }_{ij}^X$$

subject to benefit and mobility cost constraints. Specifically, we impose

$$\sum _j{\varphi }_{ij}^X=1,\sum _j{\varphi }_{ij}^X{b}_j=B_i^X,\sum _j{\varphi }_{ij}^X{c}_{ij}=C_i^X$$

where b_j is the benefit at location j, and c_ij is the cost of moving from i to j. Using Lagrange multipliers α^X, ω, and λ, the Lagrangian becomes

$$\begin{array}{c}\mathcal{L}=-\sum _j{\varphi }_{ij}^X\log {\varphi }_{ij}^X+{\alpha }^X\left(\sum _j{\varphi }_{ij}^X{b}_j-B_i^X\right)\hfill \\ -\omega \left(\sum _j{\varphi }_{ij}^X{c}_{ij}-C_i^X\right)+\lambda \left(\sum _j{\varphi }_{ij}^X-1\right).\hfill \end{array}$$

Differentiating with respect to ${\varphi }_{ij}^X$ and solving yields the MaxEnt solution:

$${\varphi }_{ij}^X=\frac{1}{Z_{X,i}}\exp \left({\alpha }^X{b}_j-\omega c_{ij}\right),$$

where the partition function Z_X,i ensures normalization:

$$Z_{X,i}=\sum _j\exp \left({\alpha }^X{b}_j-\omega c_{ij}\right).$$

Thus, the full expressions for each flow type are

$${\varphi }_{ij}^S=\frac{1}{Z_{S,i}}\exp \left({\alpha }^S{b}_j-\omega c_{ij}\right),{\varphi }_{ij}^I=\frac{1}{Z_{I,i}}\exp \left({\alpha }^I{b}_j-\omega c_{ij}\right),{\varphi }_{ij}^V=\frac{1}{Z_{V,i}}\exp \left({\alpha }^V{b}_j-\omega c_{ij}\right).$$

Here, α^X captures the perception of benefit for each modeling compartment X where X ∈ {S, I, V} (translating to risk aversion, adaptive responsiveness and risk mitigation parameters respectively), while ω captures a global perception of cost (in our case, ω = 1 to simplify analysis). These multipliers are determined such that the derived flows satisfy the global constraints on total benefit and total mobility cost.

Model variables

The variables and their actual values used in the experimental runs are defined and listed in Table 2.

Model algorithm

The model algorithm is an extended version of our previous work on a multi-city SIRS network model¹⁵. To this, we have added the vaccinated compartment V_i, along with its corresponding mobility flow ϕ^V.

The main loop contains two processes: Algorithm 1 calculates ϕ^S, ϕ^I and ϕ^V, and Algorithm 2 calculates the contagion state I in response to these changes, and vice versa, in a closed feedback loop.

Algorithm 1

FAST DYNAMIC (updates ϕ^S, ϕ^I and ϕ^V)

Algorithm 2

SLOW DYNAMIC (updates S, I, R, and V)

The Maximum Entropy method allows the mobility functions to reach equilibrium instantly per time step. In contrast, infection I updates slowly over time and reaches equilibrium over multiple time steps.

Equation (6) (Section “Generalized SVIRS network model”) can be optimized for a lattice of M = L² locations using the Hadamard element-wise product ⚪, applied to M × M matrices ϕ^S, ϕ^I and ϕ^V, and 1 × M vectors S, I, and V (see Algorithm 2):

$$\begin{array}{c}\frac{d{I}_i}{dt}=-\gamma I_i+\beta S_i\sum _{j=1}^M{\left[{\mathbf{N}}^{-1}\cdot {\left[{\varphi }_{i*}^S\circ \left({\varphi }^I\circ {\mathbf{I}}^T\right)\right]}^T\right]}_j\hfill \\ +\sigma \beta V_i\sum _{j=1}^M{\left[{\mathbf{N}}^{{\prime }^{-1}}\cdot {\left[{\varphi }_{i*}^V\circ \left({\varphi }^I\circ {\mathbf{I}}^T\right)\right]}^T\right]}_j\hfill \end{array}$$ 20

Author contributions

M.P. and C.M.J. designed the framework and computational experiments. C.M.J. performed the computational experiments and prepared figures. M.P. supervised the study. All authors contributed to drafting, reviewing and editing of the article and approved the final manuscript.

Data Availability

All data presented in this study are reproducible through an execution of the model source code available on Zenodo⁶⁷.

Code availability

The model source code (for Python 3.9.15, running on a high-performance computing cluster) is available on Zenodo⁶⁷.

Competing interests

The authors declare no competing interests.