Outbreak dynamics and population vulnerability in stochastic epidemic models on networks

TL;DR

The paper addresses how network structure and the distribution of prior immunity shape outbreak sizes in stochastic epidemic models across coupled subpopulations. It combines Gillespie-based stochastic simulations with branching-process analyses to compare networked and homogeneous populations, using mean outbreak size as a continuous vulnerability metric and examining random versus infection-acquired immunity. Key findings show that networked populations often have smaller mean outbreak sizes than homogeneous ones with the same ${\widehat{\mathcal{R}}_0}$, yet remain vulnerable post-extinction; random immunity can be more protective than infection-derived immunity in many networks, though this can reverse under strong coupling or high heterogeneity. The work has practical implications for outbreak control, highlighting the need to account for population network structure and immunity distribution when designing vaccination and mobility-intervention strategies.

Abstract

During infectious disease epidemics, pathogen transmission occurs in host populations made up of interacting subpopulations. Using stochastic simulation and analytical approximations, we examine how outbreak sizes in networked populations depend on network architecture, subpopulation sizes and the strength of coupling between subpopulations. We find, as expected, that mean outbreak sizes are frequently lower in networked populations than in homogeneous populations with the same basic reproduction number. However, after an outbreak ends, a networked population is often vulnerable to further outbreaks, and the ending of an outbreak may not imply herd immunity in any sense. Another key finding is that a relatively small amount of randomly distributed prior immunity can be more protective in a networked population than a homogeneous population, a phenomenon which can be reproduced analytically in certain cases. We also find that in networked populations, randomly distributed prior immunity is often more protective than infection-acquired immunity; but this conclusion can be reversed in populations with highly variable susceptibility. All of these conclusions have implications for designing outbreak control strategies that aim to reduce pathogen transmission during epidemics.

Figures (17)

• Figure 1: (a) A complete, symmetric network on $n=9$ vertices. (b) A (symmetric) Gamma network on $n=6$ vertices, with coupling strengths depicted as edge widths. (c) A ring-preferential network on $n=20$ vertices with initial ring size $n_r=6$, initial clique size $K=2$, and attachment degree $d_0=1$. (d) A random network on $n=12$ vertices and expected mean degree $\hat{d}=4$. (e) A small-world ring on $n=12$ vertices, with $K=4$. (f) A $5 \times 5$ small-world grid.
• Figure 2: An outbreak hitting all subpopulations in a complete, symmetric network with $n=5$, $N_c=1000$, and $\Theta N_c = 2$. During the outbreak we track $\widehat{\mathcal{R}}_t$, $\mathcal{R}_t$ and the mean secondary outbreak size ($n_{sim} = 50000$) scaled to have initial value $2$ for easy comparison with $\widehat{\mathcal{R}}_t$ and $\mathcal{R}_t$.
• Figure 3: Mean normalised outbreak sizes as we vary the leak. The normalisation is against the unconditional expected outbreak size in the homogeneous case, as in \ref{['lambertmain1']}. In each case, $n_{sim} = 20000$. Left. Complete, symmetric networks with $n=10$, and $N_c = 100$, $N_c=200$ or $N_c=1000$. The analytical approximation is based on results in Appendix \ref{['appcomplete']}, and matches simulations well in the case $N_c=1000$. For $N_c = 100$ and $N_c = 200$, the analytical estimates (not shown) are close to those for $N_c = 1000$ and overestimate outbreak sizes. Right. Various networks with $n=100$ and $N_c=10$. (1) a complete, symmetric network; (2) a random network with expected mean degree $\hat{d}=4$; (3) a $10 \times 10$ grid with no rewiring; and (4) a ring with clique size $4$ and no rewiring.
• Figure 4: Outbreak patterns in small-world grids as we increase the rewiring probability. In each case we have a $40 \times 30$ grid with $N_c = 30$ and $\Theta N_c=3.5$. Read lexicographically from top left, the rewiring probability increases from $0\%$, to $2\%$, to $5\%$, to $10\%$. The final population state at the end of typical outbreaks of approximately the same size are shown, with yellow indicating a fully susceptible population and darker colours indicating where infection has spread. Each plot is associated with a corresponding video on GitHub muradEpigithub.
• Figure 5: For each parameter value, random introductions to $100$ randomly chosen networks were simulated $n_{sim}=1000$ times each to determine the mean outbreak size (blue dots), with the global mean shown as a red dot. Top left. (Asymmetric) Gamma networks with $n=30, N_c=50$, $\Theta N_c = 2.5$ and variable shape parameter $\alpha$. Top right. Random networks with $n=100, N_c=50$, $\Theta N_c = 2.5$, and variable expected mean degree. Bottom left. Small-world rings with $n=100$, $K=4$, $N_c=50$, $\Theta N_c = 4$, and variable rewiring probability. Bottom right. A small-world $19 \times 13$ grid with $N_c = 30$, $\Theta N_c = 3.5$, and variable rewiring probability.