Impact of behavioral heterogeneity on epidemic outcome and its mapping into effective network topologies

TL;DR

The paper tackles how behavioral heterogeneity influences epidemic outcomes by introducing HeSIR, a bimodal extension of the SIR model with high- and low-risk groups that differ in susceptibility and infectivity through factors $\alpha_S$ and $\alpha_I$, producing an effective infectious pool $I_{\text{eff}} = I_L + \alpha_I I_H$. It derives a closed-form epidemic threshold on networks with arbitrary degree distributions and homophily, showing the threshold depends on a combined parameter $\theta(p,h,\alpha)= (1 - h)(1 - p + \alpha p) + h(\alpha + 1)$ with $\alpha = \alpha_I\alpha_S$, and provides a linearized form for small transmission-to-recovery ratios. The work demonstrates a resurgence regime beyond the classical threshold, and validates the theory with extensive simulations on degree-heterogeneous and homophilic networks, confirming robustness across topologies. Furthermore, it shows that HeSIR dynamics map onto standard SIR processes on effectively constructed networks (directed SBM or DC-SBM), enabling the transfer of results and intuition from classical SIR theory to heterogeneous behavioral contexts. The findings highlight how dual heterogeneity and homophily can amplify transmission and create hidden transmission potential, informing targeted interventions and risk communication in fragmented populations, with extensions to non-behavioral heterogeneity as well.

Abstract

Human behavior plays a critical role in shaping epidemic trajectories. During health crises, people respond in diverse ways in terms of self-protection and adherence to recommended measures, largely reflecting differences in how individuals assess risk. This behavioral variability induces effective heterogeneity into key epidemic parameters, such as infectivity and susceptibility. We introduce a minimal extension of the susceptible-infected-removed~(SIR) model, denoted HeSIR, that captures these effects through a simple bimodal scheme, where individuals may have higher or lower transmission--related traits. We derive a closed-form expression for the epidemic threshold in terms of the model parameters, and the network's degree distribution and homophily, defined as the tendency of like--risk individuals to preferentially interact. We identify a resurgence regime just beyond the classical threshold, where the number of infected individuals may initially decline before surging into large-scale transmission. Through simulations on homogeneous and heterogeneous network topologies we corroborate the analytical results and highlight how variations in susceptibility and infectivity influence the epidemic dynamics. We further show that, under suitable assumptions, the HeSIR model maps onto a standard SIR process on an appropriately modified contact network, providing a unified interpretation in terms of structural connectivity. Our findings quantify the effect of heterogeneous behavioral responses, especially in the presence of homophily, and caution against underestimating epidemic potential in fragmented populations, which may undermine timely containment efforts. The results also extend to heterogeneity arising from biological or other non-behavioral sources.

Figures (7)

• Figure 1: The value of $\theta(p,h,\alpha)$ for different values of $p$, $h$ and $\alpha$. $\theta(p,h,\alpha)$ can be interpreted as the ratio between the threshold in the baseline SIR case and the threshold in the HeSIR model. As expected, $\theta(p,h,\alpha)$ increases with $p$, $h$ and $\alpha$. For large $h$ (resp. $p$), the dependence on $p$ (resp. $h$) weakens.
• Figure 2: Curves of the mean number of infected under different configurations of degree heterogeneity and homophily, with $p_H=0.4$, $\alpha_I=2$, $\alpha_S=3$. Results averaged over $400$ simulations per setting, for networks of $N=5\times 10^4$ nodes with mean degree $\langle k \rangle = 20$, with 50 seed nodes chosen uniformly at random. The dashed lines represent the critical values of $\beta / \gamma$ maximizing the epidemic variability $V_{AR}$. The insets provide a close--up of the outbreak’s early dynamics, highlighting the resurgence zone observed in some super--critical regimes: an initial decline in the infected population followed by renewed growth toward a full outbreak.
• Figure 3: Epidemic threshold estimates. Yellow dashed line: analytical threshold, computed numerically; Green dashed line: approximation of the analytical threshold, as given by \ref{['eq:threshold_general']}. The $R_\infty$ curves report the median value, with shaded areas showing the central 80% of simulated outcomes. Results are averaged over 400 simulations per setting, for networks of $N = 5 \times 10^4$ nodes and mean degree $\langle k \rangle = 20$, with 50 seed nodes chosen uniformly at random. Other parameters set as $p_H = 0.4$, $\alpha_I = 2$, and $\alpha_S = 3$. Additional results in the Appendix confirm the robustness of the analysis across a range of values for $\alpha_S$, $\alpha_I$, $p_H$, $N$ and $\langle k \rangle$.
• Figure 4: Model equivalence on homogeneous networks. For $\beta/\gamma \in [0.01, 0.04]$, the average values of $R_\infty$ (panel a), the frequency of infected individuals at the peak$I_\text{max}$ (panel b), and the time of the peak$t_\text{max}$(panel c) are shown for both the HeSIR and the equivalent SIR model, exhibiting perfect agreement. Averages are computed over 200 realizations. Parameters are set to $\alpha_I=2$, $\alpha_S=3$, and $p_H=0.4$.
• Figure 5: Model equivalence on heterogeneous networks. For $\beta/\gamma \in [0.01, 0.04]$, the average values of $R_\infty$ (panel a), fraction of infected individuals at the peak$I_\text{max}$ (panel b), and the time of the peak$t_\text{max}$(panel c) are shown for both the HeSIR and the equivalent SIR model, exhibiting perfect agreement. Averages are computed over 200 realizations. Parameters are set to $\alpha_I=2$, $\alpha_S=3$, and $p_H=0.4$. Node degrees in the CM network follow a Pareto distribution with exponent $a=2$.