A disease-spread model on hypergraphs with distinct droplet and aerosol transmission modes

TL;DR

This work considers a susceptible--infected--susceptible (SIS) disease on a hypergraph, which allows it to incorporate the effects of both dyadic and polyadic interactions on disease propagation, and derives mean-field approximations of the model for two types of hypergraphs.

Abstract

We examine the spread of an infectious disease, such as one that is caused by a respiratory virus, with two distinct modes of transmission. To do this, we consider a susceptible--infected--susceptible (SIS) disease on a hypergraph, which allows us to incorporate the effects of both dyadic (i.e., pairwise) and polyadic (i.e., group) interactions on disease propagation. This disease can spread either via large droplets through direct social contacts, which we associate with edges (i.e., hyperedges of size 2), or via infected aerosols in the environment through hyperedges of size at least 3 (i.e., polyadic interactions). We derive mean-field approximations of our model for two types of hypergraphs, and we obtain threshold conditions that characterize whether the disease dies out or becomes endemic. Additionally, we numerically simulate our model and a mean-field approximation of it to examine the impact of various factors, such as hyperedge size (when the size is uniform), hyperedge-size distribution (when the sizes are nonuniform), and hyperedge-recovery rates (when the sizes are nonuniform) on the disease dynamics.

Figures (9)

• Figure 1: An illustration of the infection mechanisms in our stochastic model of disease spread. Infected nodes (red) infect susceptible nodes (blue) at a rate $\beta_d$ via edges. Infected nodes infect uncontaminated hyperedges (gray) that include them at a rate $\sigma$. Contaminated hyperedges (light orange) infect susceptible nodes that are attached to them at a rate $\beta_e$.
• Figure 2: Proportions of infected nodes (red) and contaminated hyperedges (blue), averaged over 10 simulations, in the individual-level stochastic model \ref{['eq:node_infected']}--\ref{['eq:h_contaminated']} (solid curves) and proportions of infected nodes (red) and contaminated hyperedges (blue) in the mean-field model \ref{['eqn:meanfield']} (dashed curves) on a regular $(2,s)$-uniform hypergraph. The recovery rates of the nodes and hyperedges are $\gamma = \delta = 1$. In (a), we show results for an endemic scenario. In (b), we show results for a disease-free scenario.
• Figure 3: Proportions of infected nodes (red) and contaminated hyperedges (blue), averaged over 10 simulations, from the individual-level stochastic model \ref{['eq:node_infected']}--\ref{['eq:h_contaminated']} (solid curves) and proportions of infected nodes (red) and contaminated hyperedges (blue) in the mean-field model \ref{['eqn:meanfield']} (dashed curves) on an ER hypergraph. The recovery rates of the nodes and hyperedges are $\gamma = \delta = 1$. In (a), we show results for an endemic scenario. In (b), we show results for a disease-free scenario.
• Figure 4: Proportions of infected nodes, averaged over 10 simulations, from the individual-level stochastic model \ref{['eq:node_infected']}--\ref{['eq:h_contaminated']} on an ER hypergraph when (a) the dyadic transmission mode is dominant and (b) the polyadic transmission mode is dominant. The recovery rates of the nodes and hyperedges are $\gamma = \delta = 1$. The initial proportion of infected nodes is $p_0 = 0.1$.
• Figure 5: Proportions of infected nodes, averaged over 10 simulations, in the individual-level stochastic model \ref{['eq:node_infected']}--\ref{['eq:h_contaminated']} on the hypergraphs S4 (red) and S8 (blue). For our simulations on hypergraph S4, we use the parameter values $\beta_d = 0.01$, $\beta_e = 0.1$, $\sigma = 0.5$, $\gamma = \delta = 1$, and $p_0 = 0.3$. (a) For our simulations on the hypergraph S8, we use the same parameter values as for hypergraph S4 except for the environmental infection rate $\beta_e$. From top to bottom, the values of $\beta_e$ are $0.1$, $0.075$, $0.05$, and $0.025$. (b) For our similations on the hypergraph S8, we use the same parameter values as for hypergraph S4 for all parameters except for the environmental recovery rate $\delta$. From top to bottem, the values of $\delta$ are $1$, $2$, $3$, and $4$.

Theorems & Definitions (12)

• Theorem 3.1
• proof
• Remark 3.2
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Theorem 3.6