Epidemic reproduction numbers in spatial networks

Abstract

The basic and effective reproduction numbers are widely used metrics for characterizing the dynamics of infectious disease epidemics. However, the interpretation of these numbers is based on the assumption of homogeneous mixing and may not hold in real-world populations where the contact patterns deviate from that assumption. In this paper, we present a network-based framework to compare reproduction numbers in populations with and without spatial structure, while other parameters of the disease remain fixed. Using this framework, we show that in homogeneously mixed populations, in the absence of external interventions, the effective reproduction number decreases exponentially as the susceptible population declines. In contrast, in spatially structured populations, the basic reproduction number is smaller, and the effective reproduction number initially decreases faster but eventually converges to unity. We show that the reproduction number is determined by the level of competition between infectious nodes, which is governed by the network structure. Our results suggest that without knowledge of the network structure, reproduction numbers may not be informative for parameterizing the contagiousness of the disease or predicting the behavior of epidemic spreading.

Figures (7)

• Figure 1: The basic reproduction number in a random geometric graph. (A) Visualization of a random geometric graph. The nodes relevant for the computation of the basic reproduction number $R_0$, defined as the expected number of neighbors (in blue) infected directly by the initially infected node (in red), are shown in the circle. (B) In the presence of an EPN edge between neighbors, the same EPN configuration can lead to two different numbers of nodes in the first generation, $n_1$. (C) An initially infected node (in red) with the same degree can have different neighborhood isomorphic classes $A$, each occurring with probability $p(A)$ and resulting in the expected number of secondary cases, $R_0(A)$.
• Figure 2: The distribution of the generation $g$ of the neighbors of the initially infected node in Erdős-Rényi graphs (ERGs) and random geometric graphs (RGGs). The network size is $N=10^4$ and the average degree is $\langle k\rangle =12$. The transmission rate is $\beta=1$, and the recovery rate is $\alpha=0.001$. The result is an average over $200$ runs.
• Figure 3: The basic reproduction number $R_0$ as a function of average degree $\langle k \rangle$ and transmission probability $T$. A) Shows $R_0$ as a function of $\langle k \rangle$ for Erdős-Rényi graphs (ERGs) and B) for random geometric graphs (RGGs). C) Shows the ratio between $R_0$ in RGG to $R_0$ in ERG as a function of $T\langle k \rangle$. The transmission probability is $T\in\{0.25, 0.5, 0.99\}$. Each simulation and analytical result (points and lines, respectively) is an average over $500$ runs on a network of size $N=10000$. For ERGs, the analytical lines follow the tree-like assumption of Eq. \ref{['eq:rnot_tree']}. For RGGs, the analytical lines are calculated from Eq. \ref{['eq: r_0_rgg']}.
• Figure 4: The reproduction number per generation $R_g$. Results are shown for the SI dynamics on Erdős-Rényi graphs (ERGs) and random geometric graphs (RGGs). The inner plot illustrates the normalized number of nodes per generation $n_g$. Points indicate simulations and lines show analytical results. All networks have size $N=10^4$ and average degree $\langle k\rangle=6$. Results are averaged over $500$ runs.
• Figure 5: The reproduction number per generation $R_g$ and competition between infected nodes. (A-C) Points indicate $R_g$, and lines correspond to the right-hand side of Eq. \ref{['eq: n_g_r_g']} as a function of generation $g$. Insets show $n_\mathrm{b}(g)$. A) Results for Erdős-Rényi graphs (ERGs) and random geometric graphs (RGGs). B) Results for HSNs with negative binomial degree distributions with dispersion parameter $r\in[2, 4, 16]$ and spatiality $\tau=0.01$. C) Results for HSNs with negative binomial degree distributions with dispersion parameter $r=16$ and spatiality $\tau\in [0.33, 0.2, 0.01]$. D) Shows the total number of generations $g_{max}=\max\{g|R_g>0\}$. Results are shown for the temperatures $\tau\in(10^{-2}, 10)$ and the dispersion parameters $r\in[2, 4, 8, 16]$. All Results correspond to SI dynamics. All HSNs have size $N=5000$, while ERGs and RGGs have size $N=10^4$. All networks have average degree $\langle k\rangle=6$. Results are averaged over $10000$ simulation runs.

Theorems & Definitions (1)

• Theorem 1