IntroductionTypically, 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 interactions1,2. 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 interactions4,5,6,7,8. 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 speed4,9. 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 locations4,11,12,13,14,15.To mitigate the diverse impacts of contagions, resources must be strategically allocated to affected areas and communities16,17. 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 impacts17,18,19,20,21. 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 strategies17,24. 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 disease33,34. Several studies reported on individual attitudes towards vaccination during epidemics (i.e., vaccine confidence, vaccine hesitancy, vaccine rejection)35,36,37. 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 outcomes18,38. 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.ResultsRisk disposition spaceIn 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) methodology39,40, 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 \({\phi }_{ij}^{S}\) for susceptible, \({\phi }_{ij}^{I}\) for infected and \({\phi }_{ij}^{V}\) for vaccinated. We do not consider \({\phi }_{ij}^{R}\) for recovered individuals, as those who recover with immunity before becoming susceptible again do not contribute to the contagion state.The optimal mobility flows \({\phi }_{ij}^{I}\), \({\phi }_{ij}^{S}\) and \({\phi }_{ij}^{V}\) are derived from the BLV model (see Section “Generalized SVIRS-network model”, Eq. (16)–(18)), as functions of Lagrange multipliers αI, αS and αV. We refer to these multipliers as the bounded risk disposition parameters, namely, bounded adaptive responsiveness αI 15, bounded risk aversion αS 15 and bounded risk mitigation αV. Generally, these parameters quantify a preference to interact with or avoid affected (infected) individuals, by representing each compartment’s perception of a location benefit bi, determined by \({b}_{i}=\frac{{P}_{i}-{I}_{i}}{{P}_{i}}\) (i.e., the unaffected population) (see Section “Generalized SVIRS-network model”). Importantly, parameters αI, αS and αV 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 (αI, αS) 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 (αI, x-axis) of the affected population and risk aversion (αS, 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 (αI, αS) 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 (αV, 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 αI, αS and αV produces eight orthants (hyperoctants), with two risk mitigation scenarios for each of the four contagion types.Full size imageIn this study, we expanded the risk disposition plane (composed of axes αI and αS) introduced by Jamerlan and Prokopenko15, to a three-dimensional risk disposition space formed by a third axis (αV), 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 contagion41,42. 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 typesImportantly, 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 pandemic32,43. 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 content16,47 or the formation of cults48,49. Finally, socio-economic turbulence scenarios may reflect situations where war or civil unrest drives participation in violence50,51, or where speculative bubbles in asset prices fuel increasingly risky speculative investment behavior52,53.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 immunityFull size tableFor all contagion types, αV represents the inoculated individuals’ risk mitigation tendency, or their preference to mix or interact with affected individuals given their current protected state. When αV > 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, αV