The impact of heterogeneity on the co-evolution of cooperation and epidemic spreading in complex networks

TL;DR

This work investigates how heterogeneity in social structure and infection costs shapes the coupled evolution of cooperation and epidemic spreading by modeling a Public Goods Game (PGG) with a Susceptible–Infected–Susceptible (SIS) process on complex networks. Using Monte Carlo simulations on ER, BA, and square lattices, multiplex networks, and real-world contact data, it reveals a structural-heterogeneity lever: hubs facing higher infection risk promote self-interested cooperation that disproportionately reduces spread, effectively lowering $R_0 = (\alpha_0/\mu) \, (\langle k^2 \rangle / \langle k \rangle)$ in uncorrelated networks. In contrast, cost-heterogeneity in infection costs, with $c_I$ drawn from Uniform$[0,10]$, triggers a weakest-link effect where low-cost individuals defect or under-protect, sustaining transmission and reducing overall cooperation. The results suggest policy: prioritize hubs for protection to exploit leverage points and homogenize incentives to mitigate weakest links, with multiplex coupling requiring alignment between social influence and contact networks for effective interventions.

Abstract

The dynamics of herd immunity depend crucially on the interaction between collective social behavior and disease transmission, but the role of heterogeneity in this context frequently remains unclear. Here, we dissect this co-evolutionary feedback by coupling a public goods game with an epidemic model on complex networks, including multiplex and real-world networks. Our results reveals a dichotomy in how heterogeneity shapes outcomes. We demonstrate that structural heterogeneity in social networks acts as a powerful catalyst for cooperation and disease suppression. This emergent effect is driven by highly connected hubs who, facing amplified personal risk, adopt protective strategies out of self-interest. In contrast, heterogeneity in individual infection costs proves detrimental, undermining cooperation and amplifying the epidemic. This creates a ``weakest link'' problem, where individuals with low perceived risk act as persistent free-riders and disease reservoirs, degrading the collective response. Our findings establish that heterogeneity is a double-edged sword: its impact is determined by whether it creates an asymmetry of influence (leverage points) or an asymmetry of motivation (weakest links), recommending disease intervention policies that facilitate cooperative transition in hubs (strengthening the leverage point) and homogenize incentives to weakest links.

Figures (8)

• Figure 1: Time evolution of the fraction of cooperators $\rho_C$ and infected individuals $\rho_I$ on a random network with average degree $\langle k \rangle = 4$, over some realizations. The enhancement factor is $r = 1.5$ in (a)-(b) and $r = 2.6$ in (c)-(d). For the larger $r$, some of the realizations fall into the absorbing state with vanishing densities. Model parameters are set to $\alpha_0 = 0.5$, $\alpha_r = \alpha_t = 0.01$, $c_I = 10$, $c_G = 1$ and $\tau = 0.01$.
• Figure 2: The coevolutionary dynamics of cooperation and epidemics on the ER random network (a), BA network (b), and SQ lattice (c). The stationary values for the fraction of cooperators and infected individuals as functions of the enhancement factor, $r$, with a fixed disease transmission probability $\alpha_0 = 0.5$ (i), and the infection transmission probability $\alpha_0$, with a fixed enhancement factor $r=1.5$ (ii), are plotted. Bottom panels show $n_{0C}$ and $n_{0I}$, the fraction of simulations where the zero steady-state solutions (absorbing states) are reached. In the bottom right panel, we also plot the average disease lifetime of epidemics, $t^*$. The green-shaded regions denote the bistable region. Across all networks, increasing the enhancement factor reduces disease spreading. However, the epidemics are eradicated for a smaller enhancement factor on more heterogeneous networks, indicating the beneficial effect of heterogeneity for curbing epidemics by cooperation. Similarly, increasing the disease transmission rate, $\alpha_0$, reduces infection due to the evolution of higher cooperation. Results are averaged over 100 simulations, each simulated for $10^6$ time steps, with the time averages taken over the last $10^3$ time steps. The ER and BA networks are of size $N=1000$ with the average degree of $\langle k \rangle = 4$. The size of the SQ lattice is $N=1024$, with Von Neumann connectivity.
• Figure 3: The fraction of cooperators $\rho_C$ (a), infected individuals $\rho_I$ (b), absorbing states for the density of cooperators $n_{0C}$ and infected individuals $n_{0I}$, respectively (c)-(d) and the average disease lifetime $t^*$ (e) as functions of the normalized enhancement factor $r'$ in random ER network, for three different average degrees. Increasing the mean network degree ($\langle k \rangle$) promotes cooperation by amplifying the risk of the epidemics. This increases the health incentive for individuals to cooperate, driving the system toward disease eradication earlier than in sparse networks. Model parameters are $\alpha_0=0.5$, $\alpha_r = \alpha_t = 0.01$, $c_I=10$, $c_G=1$, and $\tau=0.01$. The network size is $N = 1000$. Results are averaged across 100 realizations, each run for $10^6$ time steps, with time averages calculated over the final $10^3$ steps.
• Figure 4: The fraction of cooperators $\rho_C$ (a), infected individuals $\rho_I$ (b), absorbing states for the density of cooperators $n_{0C}$ and infected individuals $n_{0I}$, respectively (c)-(d) and the average disease lifetime $t^*$ (e) as functions of disease spreading probability $\alpha_0$ in random ER network, for three different average degrees. As the baseline probability of infection ($\alpha_0$) rises, the population responds with a surge in cooperation. This adaptive behavioral response acts as a social immune system, effectively curbing the epidemic prevalence even as the biological transmissibility of the disease increases. Model parameters are $r=1.5$, $\alpha_r = \alpha_t = 0.01$, $c_I=10$, $c_G=1$, and $\tau=0.01$. The network size is $N = 1000$. Results are averaged across 100 realizations, each run for $10^6$ time steps, with time averages calculated over the final $10^3$ steps.
• Figure 5: The fraction of cooperators $\rho_C$ (a), and infected individuals $\rho_I$ (b), absorbing states for the density of cooperators $n_{0C}$ and infected individuals $n_{0I}$, respectively (c)-(d) and the average disease lifetime $t^*$ (e) as functions of disease spreading probability $\alpha_0$ in BA scale-free network, for three different average degrees. In scale-free networks, highly connected nodes face the greatest risk and thus lead the charge in cooperation. Increasing connectivity in heterogeneous networks leads to a precipitous drop in infection, as hubs transform from super-spreaders into super-blockers. Model parameters are $r=1.5$, $\alpha_r = \alpha_t = 0.01$, $c_I=10$, $c_G=1$, and $\tau=0.01$. The network size is $N = 1000$. Results are averaged across 100 realizations, each run for $10^6$ time steps, with time averages calculated over the final $10^3$ steps.