Spreading processes on heterogeneous active systems: spreading threshold, immunization strategies, and vaccination noise

TL;DR

We study spreading processes in two-dimensional heterogeneous active Brownian particle systems with a speed distribution $p(v)$. Using a synthesis of kinetic theory and percolation-based generating functions, we derive an epidemic threshold that depends on the first and second moments, $\langle v \rangle$ and $\langle v^2 \rangle$, and show giant outbreaks can occur even when $\langle v \rangle$ vanishes for fat-tailed $p(v)$. Vaccination is treated as a heterogeneity in transmission; random vaccination is ineffective in heterogeneous populations, whereas directed vaccination targeting the fastest agents substantially reduces outbreak size; vaccination acts as quenched noise that can both decrease and increase the final outbreak size depending on realization. The results bridge kinetic theory and complex-network/epidemic tools to provide predictions for information spreading in heterogeneous active systems and highlight the crucial role of $p(v)$'s moments in determining thresholds.

Abstract

We study spreading processes in two-dimensional systems of heterogeneous active agents that exhibit different individual active speeds. We obtain, combining kinetic and complex network theory, an analytical expression for the spreading threshold that depends not only on the first but also second moment of the speed distribution. Moreover, we prove that spreading can even occur for vanishing average active speed. Furthermore, we find that random vaccination strategies are ineffective in heterogeneous active systems, whereas targeted ones are effective. We also show that vaccination acts as (quenched) noise: it can decrease or increase the outbreak size. Our results offer insights into how information propagates in heterogeneous populations of active agents.

Figures (2)

• Figure 1: (a) Contagion networks for three distinct active speed distribution $p(v)$that display exactly the same $\langle v \rangle=0.05$. Mean outbreak size $\langle n_R \rangle$ vs $\langle v \rangle$ in simulations (b) and from Eq. (\ref{['eq:n_R']}) in (c). Recall that in simulations $N$ is finite, while the theory assumes $N\to\infty$. Vertical lines indicate epidemic thresholds; in (c) correspond to Eq. (\ref{['eq:threshold_generic']}). (d) Epidemic threshold for the power-law distribution vs exponent $q$ in simulations and theory [Eq. (\ref{['eq:threshold_speed_Power']})], where $\langle \tilde{v} \rangle^P_c = \langle v \rangle^P_c / \langle v \rangle^{SV}_c$. Parameters: $\rho\!=\!0.044$, $\beta\!=\!0.005$, $\lambda\!=\!0.01$, $N\!=\!1024$, $P_{\text{inf}}\!=\!1$. $q\!=\!4$ in (b), (c), with averages computed over $3000$ simulations.
• Figure 2: (a) Normalized mean epidemic size, $[\langle n_R \rangle -N_v]/N$, average over $3000$ simulations, vs fraction of vaccinated population, $f=N_v/N$ for random (RVS) and directed (DVS) vaccination strategies. Mean epidemic size, $\langle n_R \rangle - N_v$ vs $N_v$ for the single-valued (b), exponential (c) and power-law (d) distributions. Insets show the probability that an increase of $\Delta N_v$ vaccinated agents leads to an increase in epidemic size. Parameters: as in Fig. \ref{['fig:EpiThreshold']} and $\langle v \rangle = 0.05$; $S=[0.01, 4]$.