Epidemic spreading in group-structured populations

TL;DR

It is shown that reshaping the organization of groups within a population can be used as an effective strategy to decrease the severity of an epidemic, and that outbreaks are longer but milder than for uncorrelated group structures.

Abstract

Individuals involved in common group activities/settings -- e.g., college students that are enrolled in the same class and/or live in the same dorm -- are exposed to recurrent contacts of physical proximity. These contacts are known to mediate the spread of an infectious disease, however, it is not obvious how the properties of the spreading process are determined by the structure of and the interrelation among the group settings that are at the root of those recurrent interactions. Here, we show that reshaping the organization of groups within a population can be used as an effective strategy to decrease the severity of an epidemic. Specifically, we show that when group structures are sufficiently correlated -- e.g., the likelihood for two students living in the same dorm to attend the same class is sufficiently high -- outbreaks are longer but milder than for uncorrelated group structures. Also, we show that the effectiveness of interventions for disease containment increases as the correlation among group structures increases. We demonstrate the practical relevance of our findings by taking advantage of data about housing and attendance of students at the Indiana University campus in Bloomington. By appropriately optimizing the assignment of students to dorms based on their enrollment, we are able to observe a two- to five-fold reduction in the severity of simulated epidemic processes.

Figures (17)

• Figure 1: Modeling framework and metrics of epidemic severity. (a) As a basic illustration of the modeling framework, we consider an edge-colored graph composed of $N=200$ nodes and two layers of interactions. Nodes are organized in four groups of size $q^{(\ell)} = 50$ for both layer $\ell=1$ and $\ell=2$. Colors of the nodes (i.e., orange, purple, blue, green) represent the community structure in layer $\ell=1$; shapes (i.e., circle, square, triangle, diamond) indicate the community memberships in layer $\ell=2$. Here, the community structures of the layers are maximally correlated (NMI = $1.0$, with NMI standing for normalized mutual information), meaning that there is a one-to-one map from shapes to colors. Connections among pairs of nodes are created such that the average degree is $\langle k^{(\ell)} \rangle = 3$; a fraction $\mu = 0.025$ of these edges connects a node to other nodes outside its own community. Using these parameters, we generate edges in layer $\ell=1$ (full gray) and in layer $\ell=2$ (dashed red). (b) Same as in (a), but for uncorrelated community structure (NMI = $0.0$), i.e., colors and shapes are assigned to nodes randomly. (c) We run $V=1,000,000$ simulations of the SIR model on top of the graphs of panels (a) and (b) by setting the spreading rate $\beta = 0.4$, i.e., setting the basic reproduction number $R_0 = \beta (\langle k^{(1)} \rangle+\langle k^{(2)} \rangle) = 2.4$. We display the fraction of infected nodes in the population as a function of time. We consider bins of size $0.05$ and only report the average values over at least $100$ surviving runs. (d) Same as in (c), but here we are displaying the fraction of recovered nodes in the population as a function of time.
• Figure 2: Epidemic spreading in synthetic group-structured populations. (a) We consider edge-colored graphs with $N=10,000$ nodes and parameters $\langle k^{(1)} \rangle = 3$, $q^{(1)} = 5$, $\langle k^{(2)} \rangle = 10$, $q^{(2)} = 25$, and $\mu = 0.025$. We tune the correlation among the community structure of the layers by swapping community memberships of nodes as explained in the Methods section. We simulate SIR dynamics, and measure the size of the outbreak. We plot it as a function of the NMI between the layers' partitions. Results are averaged over $V=5,000$ repetitions. Different colors/symbols refer to results valid for different choices of the spreading rate $\beta$, i.e., $\beta =0.2, 0.4$ and $0.6$, all corresponding to supercritical spreading rates [see inset in panel (c)]. The values of the reproduction number are $R_0= 2.6$, $5.2$, and $7.8$, respectively. (b) Same as in (a), but for the peak value of the fraction of infected. (c) Same as in (a), but for the duration of the epidemic. (d) We consider only the configurations corresponding to maximum (full curves, solid symbols) and minimum (dashed curves, transparent symbols) correlation among layers' partitions, and generate networks with variable mixing parameter $\mu$. We plot the size of the outbreak as function of $\mu$. Results represent averages over $V=5,000$ realizations of the model. (e) Same as in (d), but for the peak of the fraction of infected. (f) Same as in (d), but for the average duration of the epidemic.
• Figure 3: Immunization in synthetic group-structured populations. (a) We consider the same experimental setting as in the bottom row of Fig. \ref{['fig:3']}. We set the mixing parameter $\mu=0.025$ and consider three values of the spreading rate $\beta$, i.e., $0.2$, $0.4$, and $0.6$, corresponding to the reproduction number $R_0 = 2.6$, $5.2$, and $7.8$, respectively. We change, however, the initial condition of the dynamics, by immunizing a random fraction nodes. We then plot the size of the outbreak as a function of the fraction of immunized nodes. Results are obtained by averaging the outcome of $V=5,000$ repetitions of the epidemic process. (b) Same as in (a), but for the peak fraction of infected nodes. (c) We rescale the abscissa values of panel (a) by the outbreak size that is observed when a null fraction of nodes is immunized.
• Figure 4: Epidemic spreading in the student population of the Indiana University Bloomington. (a) We use data about housing and attendance for the Fall 2019 semester at the Indiana University Bloomington (IUB) campus to generate edge-colored graphs with block structure. The community partition in one layer reflects housing assignments; the partition in the other layer serves to group students based on their program and education level. Different graphs are generated depending on whether network partitions are (i) those directly observed from the data, (ii) randomized, or (iii) optimized for maximum correlation. We then simulate SIR dynamics on the graphs and measure the average size of the outbreak as a function of the spreading rate $\beta$. Results are averaged over $V=5,000$ repetitions. (b) Same as in (a), but for the peak fraction of infected. (c) Same as in (a), but for the duration of the spreading process. (d) We plot the size of the outbreak as a function of the fraction of individuals that are initially immunized. We consider three values of the spreading rate $\beta$, i.e., $0.2$, $0.4$, and $0.6$, corresponding to the reproduction number $R_0= 2$, $4$, and $6$, respectively. Different symbols correspond to different $\beta$ values; full curves and solid symbols indicate the optimized configuration considered in panel (a); dashed curves and transparent symbols refer to graphs created using ground-truth partitions. Results are averaged over $V=5,000$ repetitions. (e) Same as in (d), but for the peak fraction of infected. (f) Same as in (d), but with abscissa values rescaled by the outbreak size observed when zero individuals are initially immunized.
• Figure S1: Epidemic spreading in synthetic group-structured populations with variable group sizes. We consider edge-colored graphs with $N=10,000$ nodes and label shuffling probabilities $r=0$ (correlated community structure) and $r=1$ (uncorrelated community structure). Graphs with $\langle k^{(1)} \rangle = \langle k^{(2)} \rangle = 6$ and $\mu = 0.025$ are considered. We set the community sizes $q^{(1)} = q^{(2)} = q$, and consider $q = 20, 100,$ and $400$. The solid and dotted curves show the indicated epidemic properties as a function of the spreading rate $\beta$ for edge-colored graphs with correlated and uncorrelated community structures, respectively. We run $V=5,000$ simulations of the SIR dynamics, and display average values in the plot. (a) Outbreak size, (b) fraction of the infected population at the peak, and (c) total duration of the outbreak are plotted as functions the spreading rate $\beta$.