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BIOST/STAT 578 A Statistical Methods in Infectious Diseases Lecture 6 January 22, 2009 Estimating R 0 for emerging infectious diseases in real time The Problem Observed illness onset times Exposure information Is the disease

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BIOST/STAT 578 A Statistical Methods in Infectious Diseases Lecture 6 January 22, 2009

Estimating R0 for emerging infectious diseases in real time

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The Problem

  • Observed illness onset times

– Exposure information

  • Is the disease infectious?
  • If so, what are estimates of transmission

parameters?

– Secondary attack rates – Reproductive number

  • How effective are interventions?
  • Calibration of intervention models.
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Information Needed

  • Data

– Illness onset times – Crude population connectivity

  • Ri – Risk set for person i
  • Illness serial interval distribution

– ω(t│θ) – Mean ≈ incubation period + ½ infectious period – Incubation and infectious period information

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Estimation Methods

  • fitting an exponential growth curve to the data

– e.g., Lipsitch et al., Science 2003)

  • fitting a random graph to the data

– e.g., Wallinga and Teunis, Am J Epidemiol 2004

  • fitting a more detailed transmission model to the data

– e.g., Yang, Longini, Halloran, Ann Apply Stat 2007 – e.g., Ph.D. thesis of Gail Potter, Stat Dept

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Fitting Exponential Growth

Fitting Exponential Growth: Basic Idea

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Available information: epidemic curve

  • List with values ti , date of onset of

symptoms of the ith case

60 100 140 180 2 6 10 14 Singapore Time (days) Number of cases

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Random Graph

J Wallinga and P Teunis. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. American Journal of Epidemiology, 160: 509 – 516, 2004. S Cauchemez, , PY Boelle, G Thomas, and AJ Valleron. Estimating in real-time the efficacy of control measures in emerging communicable diseases. American Journal of Epidemiology, 164: 591 – 597, 2006.

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Available information: generation interval

  • the generation interval, ti - tj, is the duration

between onset of symptoms of a secondary case ti and its primary case tj

  • distribution of ti - tj can be described by a

Weibull distribution w(ti - tj |α,β)

5 10 15 20 0.02 0.04 0.06 0.08 0.10 0.12 Generation interval (days)

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1 5 2 4 3 p21 p51 p52 p25 p31 p41 p54 p45 p43 p34 p23 p32 p53 p35 8 7 6 8 total people n = 5 cases q =1 index cases red = case

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Methods: from epidemic curve to reproduction number (given the parameters α and β)

  • Likelihood of case i being

infected by case j

  • Probability of case i being

infected by case j

  • Number of secondary

cases attributable to case j

  • Reproductive number

( )

β α, |

j i ij

t t w L − =

∑

= ≠ =

=

n k i k k ik ij ij

L L p

, 1

∑

− =

=

q n i ij j

p R E

1

) (

n R E R E

n j j

∑

=

=

1

) ( ) (

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Results for SARS

60 100 140 180 2 6 10 14 60 100 140 180 2 6 10 14 60 100 140 180 2 6 10 14 60 100 140 180 2 6 10 14 60 100 140 180 20 60 100 140 60 100 140 180 2 6 10 14 60 100 140 180 2 6 10 14 60 100 140 180 2 6 10 14

Time (days) Reproduction number R Cases Hong Kong Singapore Vietnam Canada (a) (e) (b) (f) (c) (g) (h) (d)

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Results for SARS

Average daily effective reproduction numbers R with 95 percent CI for cases with symptom onset date before and after the first global alert against SARS on 12 March 2003 for regions where infection was introduced late February 2003. Hong Kong Vietnam Singapore Canada before alert 3.7 3.1, 4.2 2.5 1.8, 3.3 3.1 2.3, 4.0 2.7 1.8, 3.8 after alert 0.7 0.7, 0.8 0.3 0.1, 0.7 0.7 0.6, 0.9 1.1 0.9, 1.2

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Conclusions for SARS

  • SARS is a highly contagious disease

– R in uncontrolled situation is around 3 – R in controlled situation is around 0.7

  • R decreased markedly after WHO alert on

12 March 2003

– control measures have prevented 75% of potential secondary infections – which has been enough, but only barely, to stop the epidemics

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Likelihood-based estimation procedure for the reproduction number R

  • estimate of the reproduction number R is

based on integrated likelihood of case i being infected by case j

  • likelihood is integrated over all possible

infection trees (genealogies) for the

  • bserved cases
  • requires the assumption that all infectious

contacts are independent

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Likelihood-based estimation procedure for the reproduction number R

  • estimate which infection tree is most likely
  • Canadian SARS epidemic:

20 40 60 80 100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193

t (days)

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Model-based Estimation

Yang, Y., Longini, I.M. and Halloran, M.E.: A resampling- based test to detect person-to-person transmission of infectious disease, Annals of Applied Statistics 1, 211– 28 (2007). Yang, Y., Halloran, M.E., Sugimoto, J. and Longini, I.M.: Detecting human-to-human transmission of avian A (H5N1) influenza, Emerging Infectious Diseases 9, 1348-1353 (2007).

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SAR2 SAR1 b

Statistical Model

Common Source

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THE END

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Important parameters

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Cumulative 397 reported cases, 249 deaths, 63% case fatality ratio in 15 countries 31 documented family clusters

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H5N1 Influenza in Family Cluster in North Sumatra, May 2006

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Overview

  • Suspected human-to-human of H5N1 in a familial cluster
  • late April – May 2006
  • Location

– Kubu Sembilang and Kabanjahe – North Sumatra – ~80 km from Medan (largest city in the area)

  • Situation

– Group of 3 adjacent houses in the Kubu Sembilang – 1 house in Kabanjahe (~ 10 km from Kubu Sembilang) – 8 cases among 21 members of this extended family

  • 1 suspected – index case
  • 7 confirmed by WHO (PCR+)

– 54 non-index family members and close contacts

  • Limited information
  • Received prophylactic Tamiflu (oseltamivir), unless contra-indicated
  • May have included some of the above 21 members
  • Outcome

– 7 Deaths – 1 Survivor

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Courtesy of Google Earth

~ 80 km

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With Mother With Mother Cared for Index Case Cared for Index Case

April 23 April 30 May 7 May 14 May 21

House 1 House 2 House 4 House 4 Father Son Daughter 32 yr 10 yr 6 yr Mother Son 29 yr 6 yr Index’s Brother ?? yr

Kubu Sembilang Kabanjahe

Index Case Mother 35 yr PCR- Case 2 Son 15 yr PCR+ Case 3 Son 17 yr PCR+ Case 4 Mother 29 yr PCR+ Case 5 Daughter 18 mo PCR+ Case 6 Father 32 yr PCR+ Case 7 Son 10 yr PCR+ Case 8 Father 25 yr PCR+ 5 mo Mother Son Son ?? yr 3 yr Son G’mother Daughter 10 yr 55 yr 21 yr Fiancé ?? yr

Family Gathering – April 29

Cared for Sick Son Cared for Sick Son Frequently Visited House 1 Frequently Visited House 1 Exposed to Husband Exposed to Husband

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Index Case

Handling Dying Chickens and Chicken Feces

Case 6

Secondary Transmission Or Exposure to a Non- Human Source Dead Dead Dead Alive Dead Dead Dead Dead

Case 4 Case 5 Case 7

Single Exposures to Index Case

7 non-cases Case 2 6 non-cases Case 3 Case 8

Family Gathering (Sleeping in Room with Index Case)

88% CFR

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Tests and estimates

  • Statistical evidence of person-to-person

transmission (p = 0.009)

  • Household SAR: 29% (95% CI, 15-51%)
  • Lower bound on the local R₀: 1.14 (95%

CI, 0.61-2.14)

– 12% chance of no further spread

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Epidemiology of H5N1 Influenza Cluster in Dogubayazit, December 2005 - January 2006

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Epidemiology of Turkey H5N1 Cluster

  • Time: December 18, 2005 –

January 15, 2006

  • Place:

– 2 villages

  • Sagdic
  • Sulucan

– Dogubayazit, Agri province – Bordering Iran and Armenia

  • People (N = 21):

– 3 Households

  • Family 1: n = 4
  • Family 2: n = 11
  • Family 3: n = 6

– 700 – 1500 meters apart

  • Outcome

– 7 confirmed cases

  • 4 deaths
  • 3 survivors

– 14 non-cases – CFR = 57%

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Turkey

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  • Dec. 18
  • Dec. 25
  • Jan. 1

Jan 8 Jan 15

House 1 House 2 House 3 Son 15 yr Daughter 15 yr Son 11 yr Child 1 yr Child ?? yr

Dogubayazit

Father ?? yr Case 2 Mother ?? yr PCR+ Mother 33 yr Daughter 9 yr Case 4 PCR+ Case 5 Son 3 yr PCR+ Son 13 yr Son 6 yr Father ?? yr Case 7 PCR+ 5 yr Son Mother ?? yr Case 6 PCR+ Daughter 14 yr Daughter 15 yr PCR+ Case 3 Daughter 11 yr Son 6 yr Father ?? yr

Dinner at Household 1

PCR+ Son 14 yr Index Case

Exposed to diseased poultry or corpses

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Tests and estimates

  • No statistical evidence of person-to-person

transmission (p = 0.114)

  • Estimate of daily common source

probability of infection, b: 0.011 (95% CI, 0.005 - 0.025)

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TranStat I

https://www.epimodels.org/midas/about.do

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TranStat II

  • Covariate adjustment
  • Simulating epidemics with given

parameters

  • Multiple statistical methods