Lecture 6: Stochastic models for small groups such as households

A Brief History of the Reed-Frost Model • PD En’ko (1889) Deterministic difference

equations• L Reed and WH Frost (1930) Marbles and shoots• M Greenwood (1931) Alternative formulation• H Abbey (1952) 1st analysis as a stochastic

process• L Elveback, JP Fox, E Ackerman (1960) 1st

computer program and lots of theory

Reed-Frost Model

Lowell Reed1886 - 1966

Wade Hampton Frost1880–1938

Both Former Deans: Johns Hopkins School of Public Health

Helen Abbey1915 - 2001

Eugene Ackerman 1920 - 2014

John P. FoxDied around 1989

Reed-Frost ModelStochastic process: discrete state space and time t0, t1, t2 ….

Infectious agent natural historyInfectious agent natural historyInfectious for one time unitInfectious for one time unit

Social contact structureSocial contact structureRandom mixingRandom mixingp = 1 p = 1 –– q,q, probability two people make contact probability two people make contact

sufficient to transmitsufficient to transmit

RR00 = = (n(n--1)p1)p

Reed-Frost Model

{ } chainMarkovaisISRPIPnSP

tnRISIRRISS

ISqqIS

ISIP

ttt

ttt

ttt

ttt

ttISIII

t

tttt

ttttt

,...1,0

1

11

1)(

11

,1]0)0([,1]1)0([,1]1)0([

,,,,

,,)1(),( 11

=

+

++

+

−

++

=====−=∀=++

+=−=

≥−⎟⎟⎠

⎞⎜⎜⎝

⎛= ++

See chain binomial chapter in the Encyclopedia Biostat., Vol 1, 593-7

Greenwood Model

{ } chainMarkovaisISRPIPnSP

tnRISIRRISS

ISqqIS

ISIP

ttt

ttt

ttt

ttt

ttISIII

t

tttt

ttttt

,...1,0

1

11

1)(

11

,1]0)0([,1]1)0([,1]1)0([

,,,,

,,)1(),( 11

=

+

++

+

−

++

=====−=∀=++

+=−=

≥−⎟⎟⎠

⎞⎜⎜⎝

⎛= ++

Estimated CPIs and SARs

Pre‐season titer RRAge

X < 8 8 ≤ X ≤ 64Low to high 

titer

Children CPI1 0.23 ± 0.03 CPI2 0.09 ± 0.03 2.5**

SAR1 36.6 ± 6.2 SAR2 3.4 ± 4.7 10.8**

Adults CPI3 0.13 ± 0.02 CPI4 0.09 ± 0.01 1.5

SAR3 18.2 ± 4.4 SAR4 1.6 ± 3.7 11.4**

Goodness of fit χ2 = 24 8 (df = 23) p = 0 36 **p < 0 05Goodness‐of‐fit  χ2 = 24.8 (df = 23), p = 0.36,   p < 0.05

2

/ www.sciencexpress.org / 10 September 2009 /Science 30 October 2009: 326, 729 - 33

3

E i i T i ibili

METHODS

Estimating Transmissibility

• Natural history of disease– Incubation period:

– Infectious period: from ILI onset to , and probability of

max

minPr( ) , subject to 1l ll

l

it it D

being infectious is for day .

• Transmission model– Daily transmission probabilities: (C2P) and (P2P)

d 1it d

b ( )p ty p ( ) ( )– Daily escape probability

– Likelihood

( )p

1( )

( ) (1 ) (1 ( ) )ji t t

j h i

e t b p t

( )j

min

1( ), ,

( , | )(1 ( )) ( )i

Ti it

i h t t

e t tL b p

e t e t t T

t

max 1

(1 ( )) ( ),ii

i i it tt te t e t t T

4

E i i T i ibili ( i )

METHODS

Estimating Transmissibility (continue)

• Accounting for missing data– Household sizes and some ILI onset dates are missing

– Multiple imputation (Schaffer, 1997).

• Calculating SAR and R0 for US data

11 (1 )D

ddSAR p

0 ( ) (1 )( )H C S H CR f R R R f R R

, , H H H S S S C HR N SAR R N SAR R R

• Calculating R0 for Mexican data– For large population, chain binomial becomes Poisson.

Assume each case corresponds to K 1 uninfected people– Assume each case corresponds to K-1 uninfected people.

5

Household SARRESULTS

Household SAR

6

Household SAR: SensitivityRESULTS

Household SAR: Sensitivity

7RESULTS: Historic Influenza SAR Estimates

8RESULTS: Historic Influenza SAR Estimates

Estimates of Household Secondary Attack Rates of COVID-19

Updated: Madewell CJ, Yang Y, Longini IM, Halloran ME, Dean NE. Factors Associated with Household Transmission of SARS-CoV-2: An Updated Systematic Review and Meta-analysis, JAMA Network Open 3(12) (2021). DOI: 10.1001/jamanetworkopen.2020.31756

Geographic distribution of household transmission studies

0 0.25 0.5 0.75 1

Secondary Attack Rate

Lopez Bernal et al., U.K.Wu et al., Zhuhai, ChinaSun et al.*, Zhejiang Province, ChinaBöhmer et al., Bavaria, GermanyDong et al.*, Tianjin, ChinaHua et al.*, Zhejiang Province, ChinaChen et al.*, Ningbo, ChinaXin et al., Qingdao Municipal, ChinaWu et al., Hangzhou, ChinaHu et al., Guangzhou, ChinaJing et al., Guangzhou, ChinaZhang et al., ChinaLi et al., Wuhan, ChinaWang et al., Beijing, ChinaPark et al., Seoul, South KoreaLiu et al.*, Guangdong Province, ChinaZhang et al.*, Liaocheng, ChinaPark et al., South KoreaBi et al., Shenzhen, ChinaBurke et al., USALuo et al., Guangzhou, ChinaHu et al., Hunan, ChinaKorea CDC, South KoreaZhuang et al.*, Guangdong Province, ChinaWu et al., ChinaCheng et al., TaiwanNg et al., SingaporeKim et al., South Korea

Tak et al., IndiaPatel et al., London, UKBoscolo−Rizzo et al., Treviso Province, ItalyRosenberg et al., New York, USATrunfio et al., Turin, ItalyDattner et al., Bnei Brak, IsraelCarazo et al., Quebec, CanadaBender et al., Southern GermanyLewis et al., Utah & Wisconsin, USAvan der Hoek et al.*, NetherlandsDawson et al., Wisconsin, USAWang et al., Beijing, ChinaSeto et al., Yamagata, JapanHan et al., South KoreaMiyahara et al.*, JapanBae et al., Cheonan, KoreaDoung−ngern et al., ThailandPett et al., Northern IrelandWilkinson et al., Winnipeg, CanadaFateh−Moghadam et al., Trento, ItalyKuba et al., Okinawa, JapanPhiriyasart et al., Pattani Province, ThailandArnedo−Pena et al., Castellon, SpainMalheiro et al., Eastern Porto, PortugalChaw et al., BruneiMetlay et al., Boston, USASon et al., Busan, South KoreaYung et al., SingaporeLee et al., Busan, South KoreaDraper et al., Northern Territory, Australia

Grijalva et al., Tennessee & Wisconsin, USAVallès et al., Barcelona, SpainKoureas et al., Thessaly, GreeceJashaninejad et al., Hamadan, IranDemko et al., Baltimore, USATeherani et al., Atlanta, USAAreekal et al., Kerala, IndiaLyngse et al., DenmarkCharbonnier et al., Paris, FranceIslam et al., Chattogram, BangladeshAdamik et al., PolandLaxminarayan et al., TN & AP, IndiaShah et al., Gujarat, IndiaSemakula et al., Rwanda

Gomaa et al., EgyptTanaka et al., Los Angeles, USACerami et al., North Carolina, USAHsu et al., TaiwanSundar et al., Chennai, IndiaLoenenbach et al., GermanyReid et al., San Francisco, USAPeng et al., San Francisco, USALyngse et al., DenmarkTelle et al., NorwayAwang et al., Terengganu, MalaysiaTibebu et al., Ontario, CanadaVerberk et al., Netherlands and BelgiumAkaishi et al., Miyagi Prefecture, JapanHarris et al., England, UK

16148

189553

151491950469310

844711134

33012

1248772

105999

27610410

1051

4579

1211311028732718

1252471677293

1473738741

5002112838328

1809161312

10210895

3144031

221371246

35533803418

6620098182834

83928341334030

1416404

44144

96898

4721485982425983527210628026754262

52822714225244193

1059268619

10151021119369715161511779208

6118529634328927169096421881746433513214775200230442793546174106745780264

179171962002351

191223246989130108849222618446

320234065386615

98316176396492

2381866

1661219443

6884125

2651144

960765

0.34 [0.30, 0.38]0.32 [0.25, 0.40]0.32 [0.28, 0.35]0.21 [0.07, 0.40]0.20 [0.16, 0.26]0.18 [0.16, 0.21]0.18 [0.14, 0.23]0.18 [0.11, 0.26]0.18 [0.14, 0.23]0.17 [0.13, 0.22]0.17 [0.14, 0.20]0.16 [0.08, 0.26]0.16 [0.16, 0.16]0.16 [0.13, 0.18]0.15 [0.11, 0.20]0.14 [0.12, 0.15]0.13 [0.07, 0.21]0.12 [0.11, 0.12]0.11 [0.09, 0.14]0.11 [0.00, 0.29]0.10 [0.09, 0.12]0.10 [0.08, 0.12]0.08 [0.03, 0.13]0.07 [0.07, 0.08]0.07 [0.06, 0.08]0.07 [0.03, 0.11]0.06 [0.05, 0.07]0.00 [0.00, 0.02]

0.74 [0.62, 0.84]0.43 [0.36, 0.50]0.41 [0.35, 0.47]0.38 [0.33, 0.43]0.35 [0.30, 0.41]0.32 [0.30, 0.34]0.30 [0.29, 0.31]0.29 [0.16, 0.43]0.28 [0.21, 0.34]0.27 [0.21, 0.34]0.25 [0.15, 0.36]0.23 [0.19, 0.28]0.22 [0.15, 0.29]0.21 [0.03, 0.47]0.19 [0.16, 0.22]0.18 [0.13, 0.24]0.17 [0.12, 0.22]0.16 [0.06, 0.28]0.15 [0.11, 0.19]0.14 [0.13, 0.15]0.12 [0.08, 0.17]0.11 [0.06, 0.18]0.11 [0.09, 0.14]0.11 [0.09, 0.13]0.11 [0.07, 0.15]0.10 [0.10, 0.11]0.08 [0.05, 0.12]0.06 [0.03, 0.10]0.04 [0.00, 0.18]0.04 [0.00, 0.11]

0.53 [0.46, 0.60]0.48 [0.42, 0.55]0.39 [0.33, 0.45]0.32 [0.29, 0.35]0.31 [0.23, 0.39]0.29 [0.21, 0.38]0.26 [0.23, 0.29]0.17 [0.15, 0.18]0.13 [0.09, 0.18]0.13 [0.05, 0.25]0.11 [0.11, 0.11]0.09 [0.08, 0.10]0.09 [0.06, 0.12]0.03 [0.02, 0.04]

0.67 [0.58, 0.76]0.63 [0.58, 0.69]0.56 [0.48, 0.63]0.46 [0.31, 0.62]0.44 [0.32, 0.56]0.37 [0.27, 0.47]0.35 [0.33, 0.37]0.33 [0.30, 0.36]0.25 [0.24, 0.26]0.21 [0.20, 0.21]0.21 [0.12, 0.31]0.19 [0.19, 0.20]0.17 [0.12, 0.21]0.13 [0.11, 0.15]0.10 [0.10, 0.10]

0.189 (0.162, 0.220)Combined Estimate

Subgroup Estimate 0.134 (0.107, 0.167)

Subgroup Estimate 0.194 (0.152, 0.245)

Subgroup Estimate 0.199 (0.130, 0.293)

Subgroup Estimate 0.311 (0.226, 0.411)

Author, Location Infected Total Estimates SAR (95% CI)

January − February, 2020

March − April, 2020

May − June, 2020

July, 2020 − March, 2021