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Exponential-family random graph models

for within-household contact networks

Nele Goeyvaerts1,2,3, Gail Potter4, Kim Van Kerckhove1,2,

Lander Willem2,1, Philippe Beutels2, Niel Hens1,2

1 I-BioStat, Hasselt University, Belgium

2 CHERMID, Vaccine & Infectious Disease Institute, University of Antwerp, Belgium

3 Non-Clinical Statistics, Janssen Pharmaceutical Companies of Johnson & Johnson, Belgium

4 Statistics Department, California Polytechnic State University, USA

Interuniversity Institute for Biostatistics and statistical Bioinformatics

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Transmission within households

• Transmission of airborne infectious diseases driven bysocial contacts

• Households are important units: clusters characterized byintense contact

• Epidemic models often assume random mixing withinhouseholds:

• Households model: 2 levels of mixing (Ball et al., 1997)

• IBMs: random mixing in households, schools, workplaces(Halloran et al., 2002; Ferguson et al., 2006)

• No direct empirical evidence to support the assumption ofrandom mixing within households

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Flemish household survey

2010-2011

• POLYMOD-like diary-based contact survey in HHs

• Flemish geographical region including Brussels

• HHs with young children: at least 1 child aged ≤ 12 years

• Stratified sampling by age/gender youngest child, HHcomposition (1 or 2 parents), weekday-weekend, province

• HH member = living > 50% of the time in the same HH

• Contact = 2-way conversation (< 3m) or physical contact

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Sample sizes

• 1266 participants from 318 HHs who recorded 19685contacts in total

• HH sizes range from 2 to 7

• 3821 identified within-HH contacts with 98% reciprocity

• Assuming reciprocity of recorded within-HH contacts:

• 1946 distinct within-household contacts

• 96% involve skin-to-skin touching

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Reciprocity

Households represented by undirected contact networks

• Nodes: household members

• Edges: within-household contacts

Household Size 2 Household Size 2

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Within-HH contact networks

Focus on physical contacts:

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ERGMs to model within-HH

contact networks

• Exponential-family Random Graph Models (ERGMs)provide statistical framework to infer on processes driving

social interactions

• Y random adjacency matrix across all households:

Yij = Yji =

{1 if HH members i and j make physical contact

0 otherwise

• Ω support of Y i.e. the set of all obtainable networks

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Definition ERGM

• Exponential-family Random Graph Model (ERGM):

Pθ,Ω(Y = y) =exp{θTg(y,X)}

κ(θ,Ω), y ∈ Ω

• g(y,X) vector of network statistics

• θ corresponding vector of coefficients

• X additional covariate information

• κ(θ,Ω) normalizing factor

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Inference for ERGMs

• Approximate MLE using Markov Chain Monte Carlo in aNewton-Raphson type algorithm (Geyer and Thompson,JRSS-B 1992):

1 MCMC simulation of a distribution of random networks

from starting set of parameter values

2 Parameter values refined by comparing distribution of

networks against observed network

3 Process repeated until parameter estimates stabilize

• Software: ergm package in R, part of the statnet suite ofpackages for statistical network analysis (Hunter et al., J.

Stat. Soft. 2008)

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Goodness-of-fit

• Simulate within-HH contact networks from fitted ERGMusing MCMC

• Compare specific contact network characteristics to theobserved ones

• Proportion of complete networks: complete = every HHmember makes contact with every other HH member

• Household network density = # observed edges# potential edges

• Measure of clustering = # observed triangles# potential triangles

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Goodness-of-fit

Hunter et al. (J. Stat. Soft., 2008)

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Network statistics g(y,X)

• Dyad independent statistics:

Network statistic Legend

Edges Total no of edges

Within-household edges Total no of edges within households

Child-father mixing Total no of edges between children and fathers

Child-mother mixing Total no of edges between children and mothers

Father-mother mixing Total no of edges between partners

Boy-boy mixing Total no of edges between boys

Girl-girl mixing Total no of edges between girls

Age effect children Sum age(i) + age(j), ∀ (i, j) between siblings

Small households (≤ 3) Total no of edges within HHs of size ≤ 3Large households (≥ 5) Total no of edges within HHs of size ≥ 5

• Dyad dependent statistics: total number of isolates,2-stars, triangles, triangles in households of size ≥ 6

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ERGM: weekday

Network statistic Estimate s.e. p-value

Edges -29.48 0.26 0.00

Within-household edges 30.40 0.26

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ERGM: weekday

2 3 4 5 6−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Pro

port

ion

com

plet

e ne

twor

ks

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ERGM: weekday

2 3 4 5 6−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Mea

n ne

twor

k de

nsity

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ERGM: weekday

3 4 5 6−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Obs

erve

d vs

. pot

entia

l tria

ngle

s

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ERGM: weekend day

Network statistic Estimate s.e. p-value

Edges -21.23 0.76

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ERGM: weekend day

3 4 6−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Pro

port

ion

com

plet

e ne

twor

ks

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ERGM: weekend day

3 4 6−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Mea

n ne

twor

k de

nsity

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ERGM: weekend day

3 4 5−7 total

0.0

0.2

0.4

0.6

0.8

1.0

Simulated networks n = 500

Household size

Obs

erve

d vs

. pot

entia

l tria

ngle

s

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Epidemic simulation model

• Discrete-time chain binomial SIR model

• Closed, fully susceptible population of households

• Households model, 2 levels of mixing (Ball et al., 1997):• High-intensity mixing within households withβh = P(transmission) per physical contact, per time step

• Low-intensity random mixing in the community: socialcontact hypothesis βc(a, a

′) = qc · c(a, a′) with c(a, a′)physical contact rates (POLYMOD Belgium)

• 2 assumptions for within-household mixing:• Realistic mixing: contact networks simulated from ERGM

• Random mixing

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Probability of infection

• At each time step, each susceptible i acquires infectionwith probability:

pi1 = 1 − (1 − βh · δh)∑

j 6=i∈hi yijIj · (1 − βc,11)∑

j /∈hiIj1 · (1 − βc,12)

∑j /∈hi

Ij2

pi2 = 1 − (1 − βh · δh)∑

j 6=i∈hi yijIj · (1 − βc,21)∑

j /∈hiIj1 · (1 − βc,22)

∑j /∈hi

Ij2

• Index 1 = children ≤ 18 y, index 2 = adults > 18 y• δh = 1, for realistic mixingδh = network density, for random mixing

• hi: household of node i

• For random mixing: yij = 1,∀i 6= j ∈ same HH,• Ij : indicates whether node j is infected (1) or not (0)

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Epidemic model assumptions

• Susceptibility/infectiousness independent of age

• No latent period, thus infected = infectious

• Constant recovery probability for infectious individuals,mean infectious period ≈ 3.5 days

• βh and qc → explore several values based on literatureestimates

• Results from 500 stochastic epidemic simulations

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Results: incidence

0 20 40 60 80 100

010

2030

4050

Random mixing

Time

Tota

l inc

iden

ce

0 20 40 60 80 100

010

2030

4050

ERGM

TimeTo

tal i

ncid

ence

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Results: final size

Random mixing

Final size (individuals)

Fre

quen

cy

0 200 600 1000

020

4060

8010

0

ERGM

Final size (individuals)F

requ

ency

0 200 600 1000

020

4060

8010

012

0

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Results: final fraction

Random ERGM

0.2

0.4

0.6

0.8

Pro

port

ion

indi

vidu

als

affe

cted

Random ERGM

0.2

0.4

0.6

0.8

Pro

port

ion

hous

ehol

ds a

ffect

ed

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Results: HH attack rate

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Household size 2

Hou

seho

ld a

ttack

rat

e

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Household size 3

Hou

seho

ld a

ttack

rat

e

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Household size 4

Hou

seho

ld a

ttack

rat

e

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Results: HH attack rate

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Household size 5

Hou

seho

ld a

ttack

rat

e

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Household size >=6

Hou

seho

ld a

ttack

rat

e

Random ERGM

0.0

0.2

0.4

0.6

0.8

1.0

Total

Hou

seho

ld a

ttack

rat

e

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Summary

• First contact survey designed to measure socialinteractions within households

• ERGMs show high degree of clustering and decreasingconnectedness with increasing HH size on weekdays

• Epidemic simulation results seem to support assumptionof random mixing within households

• Further research: impact of control strategies,heterogeneity in duration of contact

• Assumption: physical contacts = good proxy

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Acknowledgements

• AXA Research Fund

• Gail Potter (California Polytechnic State University)

• Kim Van Kerckhove (Hasselt University and UAntwerp)

• Lander Willem (UAntwerp and Hasselt University)

• Philippe Beutels (UAntwerp)

• Niel Hens (Hasselt University and UAntwerp)

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References

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Epidemics with two levels of mixing.

The Annals of Applied Probability 7, 46–89.

Ferguson, N. M., D. A. T. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley, and D. S. Burke (2006).

Strategies for mitigating an influenza pandemic.

Nature Letters 442, 448–452.

Geyer, C. J. and E. A. Thompson (1992).

Constrained monte carlo maximum likelihood calculations.

Journal of the Royal Statistical Society B 54, 657–699.

Hunter, D. R., M. S. Handcock, C. T. Butts, S. M. Goodreau, and M. Morris (2008).

ergm: A package to fit, simulate and diagnose exponential-family models for networks.

Journal of Statistical Software 24, 1–29.

Keeling, M. J. and K. T. D. Eames (2005).

Networks and epidemic models.

Journal of the Royal Society Interface 2, 295–307.

Kolaczyk, E. D. (2009).

Statistical Analysis of Network Data: Methods and Models.

Springer, New York.

Longini, Jr., I. M. and J. S. Koopman (1982).

Household and community transmission parameters from final distributions of infections in

households.

Biometrics 38, 115–126.