Network Modeling for Epidemics with EpiModel Workshop @ UMN School of Public Health November 14, 2018

Samuel M. Jenness, PhD MPH EpiModel Research LabDepartment of Epidemiology // Rollins SPH // Emory University

samueljenness.org smjenness @SamuelJenness

Workshop Website

http://statnet.github.io/umn/

Moving Beyond the Individual

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Study Participant #1

Study Participant #2

Moving Beyond the Individual

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P1 has non-zero risk of infection as a result of his partner’s risk

Why Networks Matter

• Individual risk is function of partners’ partners’ partners’ … risk over time

• Risk is dynamic and heterogenous

• Dependencies between epidemiology and networks

• Minor changes to inputs result in major changes to outputs

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Network Connectivity

6Morris & Carnegie, 2012, PLoS One

Small changes in “individual” behavior result in large changes to the network structure

Intersection between Networks and Individual Risk

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Eaton et al., 2011, AIDS Behav Jenness et al., 2016, Sex Trans Inf

Concurrency & Acute Stage Infection Concurrency & Male Circumcision

Network composition amplifies role of “individual-level” biological attributes

Network Research

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Empirical and Intervention Questions • How does dynamic network structure of sexual partnerships (or other

contact patterns) drive infectious disease transmission over time?• What are the optimal network-based targets for prevention, and what

are their predicted impact and efficiency?

Sources of HIV Racial Disparities

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• Modeling Approaches to Racial Disparities in HIV in Atlanta MSM (R21; PI: Goodreau)

• Investigating network structures that could contribute to ongoing black/white incidence disparities among MSM

• Finding 1: Assortative mixing alone (the network “prevalence pool” hypothesis) cannot sustain disparities

• Paper currently under review

Impact of CDC PrEP Guidelines

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Used EpiModelHIV to estimate the epidemiological impact and efficiency of CDC’s PrEP clinical practice guidelines for MSM

Jenness et al., Journal of Infectious Diseases, 2016

Impact of CDC PrEP Guidelines

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• Main sensitivity analyses varied levels of coverage (% of MSM indicated for PrEP who received it) and adherence (% of MSM taking 4+ doses/week)

• Estimated 33% of HIV infections would be prevented over the next decade if we reach 40% coverage and 62% high-adherence (base model)

• Developed a Web Tool for local policymakers

http://prism.shinyapps.io/cdc-prep-guidelines/

Implications of PrEP for GC/CT Incidence

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Next we modeled how the same sort of PrEP scale-up could impact rectal and urogenital GC and CT infection among MSM given potentially counteracting forces of risk compensation (replacement of condoms) and increased STI screening. Presentation @ CROI 2017.

Implications of PrEP for GC/CT Incidence

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PrEP-related STI screening identifies a significant amount of STIs that would otherwise go undiagnosed (mainly asymptomatic rectal infections). CDC’s PrEP indications could be a high-impact targeting mechanism for STI prevention.

Statistical vs Mathematical Models

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Statistical Models • Start with data

• Choose functional framework for summarizing data

• Fit model to estimate parameters

• Infer population associations or casual effects

0 20 40 60 80 100

0100

200

300

400

500

x

y

Statistical vs Mathematical Models

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Mathematical Models • Start with the parameters

• Construct the processes to get from micro to macro- Micro: Individual-level biology,

behavior, demography- Macro: Population-level disease

incidence and prevalence0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

Prevalence

s.numi.numr.num

Dynamic

The Epidemic Feedback Loop

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Prevalence

Risk

Incidence

Force of Infection Transmission probability per contact ● Rate of contacts ●Probability contacting an infected

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

Prevalence

s.numi.numr.num

With biology and behavior fixed, risk depends on prevalence:

Ratio of susceptible, infected, and (potentially) immune in a population

Indirect Effects & Herd Immunity

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Observed

Example of Cholera Vaccine

Your direct protection = your exposure ● efficacy of exposure Your indirect protection = all of your contacts’ exposures ● efficacy

Counterfactuals: In Silico Lab

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Deterministic Compartmental Models

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• Prevailing method for modeling most diseases

• Population divided into groups by disease states

• System of differential equations solved for dynamic compartment size

• Benefits: analytical tractability and computational efficiency

SIR Model Diagramtime=25 | run=3

Susceptible n=119.4

Infected n=9.6

Recovered n=844.1

si.flow=1.6 ir.flow=3.2

ds.flow=1.2 di.flow=0.3 dr.flow=8.4

b.flow=10.8

Exponential Complexity of Compartments

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Cassels et al., 2009, AIDS

Each new population heterogeneity multiplies number of equations to solve

Standard Treatment of Partnerships within DCMs

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• Models with sexual partnerships typically define a per-partnership transmission rate

• Standard method in DCMs:

Per-act transmission probability raised to frequency of acts1� (1� ⌧)↵

10% • • • 10% • • • 10%

t1 t3

Reality

10%10%10%

t1

Math

A (Crude) Model of Epidemic Models

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Static/Markov Models DCMs

Dynamic Models

ABMs

Network Models

Epidemic modeling methods are differentiated by their treatment of time and persons over time

Agent-based models track individuals over time (vs groups in DCMs)

Network models are a form of ABMs that explicitly incorporate partnerships

From Data to Inference (…on the mechanism generating the data)

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• What determines global network structure over time?

• What are the predictors of partnership (edge) formation and dissolution?

• What is most relevant for disease transmission?

Can We Observe the Data (i.e., the Network)?

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• Phylogenetic networks- Probabilistic transmission clusters- Transmission directionality cannot

be inferred without external (clinical) data

- Answering a different (but still very important) set of questions

- Ongoing work to link phylo and contact networks

• Network “census”- Likoma Island partnership network- Around 50% missing data

Missing Network Data

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With Independent Data • If we don’t sample person A, it doesn’t affect the sampling of person B• If person A doesn’t respond to a survey question, it doesn’t affect B’s

response

With Dependent Data • If you don’t observe A, it affects the properties of other persons, and

the network structure• Missing edges: a highly problematic form of missing data

Network Sampling Methods

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Network Census • Data on every node and every link

Adaptive Sampling • Variations on link tracing, snowball sampling, contact tracing• Lots of interesting work on statistical inference in this area

Egocentric Sample • Sample persons, ask them to report on partners• Use a statistical model to estimate population parameters from sample• Fully specified model available for simulation

Mathematical Representation of Networks

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Consider a set of n persons and a partnership (edge) between each dyad.

Define a graph Y with 1, …, i, j, …, n persons.

Then:

Exponential Random Graph Models

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• Probability of network Y is a function of network statistics with associated parameter values

• Can be reexpressed as a conditional logit for Yij

Network Parameters

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Degree

Network Parameters

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Preferential Mixing

Network Parameters

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An Arbitrarily Complex Set of Interactions

Egocentric Data to Models to Simulation

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1. Sample Egos

Egocentric Data to Models to Simulation

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2. Query on Alters

Egocentric Data to Models to Simulation

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3. Estimate Sufficient Statistics

Stat ValueEdges 4

Isolates 1Concurrent 1

Age Diff 5 (years)Nodematch(“shape”) 2Nodematch(“color”) 0

Egocentric Data to Models to Simulation

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3. Fit an ERGM with Target Statistics

Stat ValueEdges 4

Isolates 1Concurrent 1

Mean Age Diff 5 (years)Nodematch(“shape”) 2Nodematch(“color”) 0

nw ~ edges + isolates + concurrent + absdiff(“age”) + nodematch(“shape”) + nodematch(“color”)

targets = (4, 1, 1, 4*5, 2, 0)*100

Egocentric Data to Models to Simulation

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4. Simulate from the Model

• MCMC algorithm similar to that used in estimation

• Summary of network stats consistent on average with summary target statistics

• Targets met even if changing network size and composition

Egocentric Data to Models to Simulation

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5. Add Time

• TERGM = temporal ERGMs• Separate ERGMs for formation and

dissolution• Dissolution parameters also

estimated by cross-sectional data

Dynamic Networks + Epidemic Models

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InfectedSusceptible Recovered

• Arbitrarily complex dynamic network model placed on top of an agent-based model for disease

• Need to account for some tricky dependencies between changes in network via epi/demog mechanics

EpiModel

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• Open-source software platform for epidemic modeling

• Tools for building, simulating, and analyzing models

• Three model classes- Deterministic compartmental models- Stochastic ABMs- Stochastic network models

• http://epimodel.org/

The Statnet Family Tree

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network

networkDynamic

sna

ergm

tergm

EpiModel

EpiModelHIV

tsna

ndtv

https://CRAN.R-project.org/package=statnet

http://statnet.org/

A suite of R packages for analysis, visualization, statistical modeling, (and now) epidemic modeling with networks

Workshop Structure

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• Estimating and simulating a TERGM with no infectious disease• Modeling a “basic” epidemic model over a closed network • Modeling a “complex” network over an open network• Methods for extending EpiModel for new research questions