How Network Structure Affects Epidemiological Indicators in ABMs

A Large Simulation Study Featuring Artificial and Real-world Networks

Sunbelt 2023
Portland, Oregon

George G. Vega Yon

Chong Zhang

Alun Thomas

Matthew Samore

Karim Khader

2023-06-28

Motivation

R\(_t\) in Small World Networks¹

Herd Immunity Threshold vs Reproductive Number (Wikipedia)

R\(_t\) > 1: Epidemic grows.
R\(_t\) < 1: Epidemic shrinks.

But…

The reproduction number in a SIR + small-world network is, on average, \(<\) 1!

Our goal

Agent-Based Models [ABMs] are an important research and policy tool in epidemiology.

ABMs usually feature random graphs, often using ‘simple models’ such as scale-free graphs.

But, real-world networks are not random.

Although we understand the latter, the question is: How much does it matter?

We aim to shed light on how network structure affects epidemiological measurements to inform ABMs better.

Simulation Study

Simulation study
Networks with different structures

Six different network models featuring almost (almost) the same density

We generated 1,000 networks for each model, using the ERGM as a baseline.

Simulation study
Outbreaks with different networks

We simulated 20,000 Susceptible-Exposed-Infected-Recovered [SEIR] outbreaks using the epiworldR package:

Sampled a network out of the six types.

For each network (with 534 nodes and avg. degree of ~14), we simulated an SEIR outbreak:

a. Starts with one exposed node.

b. Exposed nodes transmit the disease to their neighbors at a daily rate of 0.023.¹

c. Infected nodes recover at a daily rate of 1/7.

d. For 100 days.

Simulation study
Network and outbreak statistics

Epi measurements

For each network, we computed:

Edge count and density.

# of Balanced triads.

# of homophilic ties (grade and gender).

# of triangles.

# of two paths.

Avg. betweenness, closeness centrality, and eigenvector centrality.

Avg. path length.

Modularity.

Preliminary results

Network features

Network features (cont.)

Overall distribution of epidemiological measurements

Rt curve

Variance: Reproductive number

Using 1,000 bootstrap samples, we computed the variance of the reproductive number for each network type.

Variance: Peak time

Variance: Peak prevalence

Variance: Generation time

Predicting epidemiological measurements

Regression analysis

Regressed Rt, Gen time, Peak time, and Peak preval on network features (n = 18,015)

Avg. degree Peak time and Gen time.

Higher number of two-paths Peak time and Rt, but the Peak preval (more severe).

Transitivity Peak preval and Gen time.

Triangles the Peak time.

Higher balance significantly Rt.

Degree-sequence and Erdos-Renyi networks have consistently Peak preval and Peak time.

Scale-free networks have Peak time and Rt.

Discussion

In small networks, as \(p \to 0\), Rt\(\to\) 1.

High heterogeneity in the variance of the studied stats, especially in scale-free networks.

We captured part of the heterogeneity as a function of degree, two paths, transitivity, triangles, and balance.

Yet the model does not capture some properties of Erdos-Renyi and degree-sequence networks (higher prevalence and later peak-time).

Rt is hard to predict as a function of network structure.

Our study only looks at small networks. We need to dive deep into large network models.

Thanks!

george.vegayon at utah.edu

https://ggv.cl

@gvegayon

Bonus

	Peak preval	Peak time	Gen time	Rt
Fixed effects
Scale-free	0.78 (1.20)	-2.12 (0.37)^***	-0.03 (0.09)	0.40 (0.20)^*
Small-world (p=0.1)	-0.01 (1.88)	-4.05 (2.10)	-0.25 (0.14)	-0.18 (0.04)^***
Small-world (p=0.2)	1.58 (0.85)	-4.68 (1.43)^**	-0.13 (0.06)^*	-0.08 (0.03)^**
Degree-sequence	1.42 (0.14)^***	1.68 (0.55)^**	-0.00 (0.01)	-0.00 (0.02)
Erdos-Renyi	1.38 (0.15)^***	1.89 (0.57)^***	0.00 (0.01)	0.02 (0.02)
Network structure
Average degree	1.53 (0.34)^***		-0.23 (0.02)^***
Two-path	7.84 (2.44)^**	-19.16 (1.30)^***	0.34 (0.18)	-0.71 (0.35)^*
Transitivity	-11.90 (5.74)^*		0.85 (0.42)^*
Triangles		4.31 (1.00)^***
Balance				2.12 (0.74)^**
AIC	97427.02	113863.95	2992.63	77403.21
BIC	97505.01	113934.14	3070.62	77473.41
Log Likelihood	-48703.51	-56922.97	-1486.32	-38692.61
Deviance	235173.39	585716.77	1243.95	17454.99
Num. obs.	18015	18015	18014	18015
^*p < 0.001; ^p < 0.01; ^*p < 0.05. In the case of Rt, we used a negative binomial regression model.