Disease Transmission on Random Graphs Using Edge-Based Percolation

TL;DR

An accompanying R package is presented that takes epidemic and network parameters as input and generates estimates of the epidemic trajectory and final size and was developed to help researchers easily understand and use network models to investigate the interaction between different community structures and disease transmission.

Abstract

Edge-based percolation methods can be used to analyze disease transmission on complex social networks. This allows us to include complex social heterogeneity in our models while maintaining tractability. Here we review the seminal works on this field by Newman et al (2001); Newman (2002, 2003), and Miller et al (2012). We present a systematic discussion of the theoretical background behind these models, including an extensive derivation of the major results. We also connect these results relate back to the classical literature in random graph theory Molloy and Reed (1995, 1998). Finally, we also present an accompanying R package that takes epidemic and network parameters as input and generates estimates of the epidemic trajectory and final size. This manuscript and the R package was developed to help researchers easily understand and use network models to investigate the interaction between different community structures and disease transmission.

Figures (10)

• Figure 1: The MA-SIR model. Here $S$, $I$ and $R$ are the susceptible, infectious and recovered/removed compartments respectively. Under the mass action (MA) assumption, the rate of flow from $S$ to $I$ is equal to their product times a constant per-infected transmission rate of $\hat{\beta}$. Infectious individuals are assumed to recover at a per capita rate of $\hat{\gamma}$. Figure was created using Latex Tikz Picture.
• Figure 2: Epidemic curves of an MA-SIR model and a network-SIR model by MillerSlimVolz:2012 with the same basic reproduction numbers. Figure was created using Matlab.
• Figure 3: This figure is based on the explanation and figure presented by NewmanStrogatzWatts:2001 to explain the derivation of fixed point equation equivalent to our equation in Corollary \ref{['cor: ueqn']}. Here we use the figure as an illustration of \ref{['eq: decomp']} in our proof of Theorem \ref{['thm: PGFHq']} which we prove first before deriving the equations in Corollary \ref{['cor: ueqn']}. The square represents an unknown tree cluster and the circle is the vertex connected to the randomly chosen initial edge below. Figure was using MS Office.
• Figure 4: Site percolation a) focus on whether vertices (sites) are occupied (solid circles) or not (open circles). Bond percolation b) focus on whether the edges (bonds) are occupied (thick lines) or not (thin lines). Figure was created using MS Office.
• Figure 5: Probability density function $P(\tau)$ with respect to recovery time $\tau$ with the same expectation $\mathbb{E}_{i,j}[\tau]=\frac{1}{\hat{\gamma}}$. The black is curve is the concentrated distribution in Newman:2010. The red curve is the exponential distribution in (A8a). The blue line is the constant recovery time $\tau=\frac{1}{\hat{\gamma}}$ in (A8d). Figure was created using R

Theorems & Definitions (72)

• Definition 2.1: BondyMurty:2008
• Theorem 2.2
• proof
• Definition 2.3: BondyMurty:2008
• Definition 2.4: Adapted from FriezeKaronski:2016
• Definition 2.5: Lovasz:2012
• Definition 2.6: FriezeKaronski:2016
• Definition 2.7
• Example 2.8: Erdös-Rényi Graphs
• Definition 2.9: Adapted from FriezeKaronski:2016