Characterizing Infectious Disease
Outbreaks:
Traditional and Novel Approaches
Laura F White
15 October 2013
2009 Influenza A H1N1 Pandemic
• H1N1 pandemic first noticed in February in Mexico.
• Large outbreak early on in La Gloria-a small village
outside of Mexico City.
• Studied extensively in the first report on H1N1
(Fraser, Donelly et al. “Pandemic potential of a strain
of Influenza (H1N1): early findings”, Science Express,
11 May 2009.)
Example-H1N1 Outbreak
Example-H1N1 Outbreak
Edgar Hernandez (four years old): first confirmed case
Cases reported in La Gloria
Quantitative Issues
• How do we determine how fast the disease is
spreading?
– Reproductive number, serial interval
• How do we determine how severe the disease is?
– Attack rate, case fatality ratio
– A topic for another talk!
• How do we determine what interventions will be
most effective?
– Mathematical modeling, network models, etc.
– Estimates of severity and transmission by age group
Importance of parameter estimates
• Good information leads to good policy.
• School closure is expensive
– Important to determine if it will really help.
• If R0 < 2, some estimate that Influenza can be
controlled.
• Information on R0 and the serial interval can give a
good picture of how a disease might spread.
Source: Fraser et al (2004)
Impact of the serial interval
Some of the challenges in infectious
diseases
• Dependency in the data.
– Chain of infection.
• Undetected cases.
– Asymptomatic, but still infectious.
– Unable to detect with existing surveillance.
• Need to act fast with little information.
Approaches to estimation
• Classical: Mathematical models
• Network models
• Statistical approaches
Simple approach
• Assume exponential growth for the first part
of an epidemic.
• td is the doubling time of the epidemic, D is
the average serial interval. Then use the
following to solve for R0.
td  (ln 2) D / ( R0  1)
• Overly simplistic and sensitive.
Mathematical models
SIR Model
Susceptible
Recovered
Infected
(Contact
Rate)*(Transmission
Probability)Infected
1/(duration of
infectiousness)
R0=(attack rate)(contact rate)(duration of infectiousness)
Mathematical Models-Uses
• Modeling vaccination programs
• Determining optimal intervention strategies
for halt or control an epidemic
• HIV transmission routes
• Estimating parameters of disease
Mathematical Models: Limitations
• Make a lot of assumptions.
– Must plug in a lot of values in order to get estimates.
• Do not allow for randomness in processes-always
gives a number as the answer with no error bounds.
– Stochastic epidemic model.
• Can oversimplify the problem.
– Challenge to achieve balance between making the model
too simple and too complex.
References
• Hethcote
– The Mathematics of Infectious Diseases. Herbert
W. Hethcote. SIAM Review, Vol. 42, No. 4, 599653. Dec., 2000.
• Anderson and May
– Infectious Diseases of Humans: Dynamics and
Control, Oxford University Press, 1992.
Wallinga & Tuenis
• Network based method to estimate the
reproductive number each day of an
epidemic.
• Requires knowledge of the serial interval.
• Requires that all cases have been observed
and epidemic is over.
• Originated to analyze SARS.
American Journal of Epidemiology, 2004
= infected person
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
All possible infectors.
j
Day 1
p3
pt=probability of being infected
by a case that appeared t days
prior.
Day 2
p2
Day 3
p1
p1
Day 4
Day 5
Day 6
i
p1
p1
Wallinga & Teunis
• If g(t) is the distribution of the serial interval,
then, the relative probability that case i has
been infected by case j is:
pij 
g (ti  t j )
 g (t
i j
i
tj)
• The effective reproductive number for cases
on day j is then:
R j   pij
i
WT - SARS
White & Pagano
• Statistical method, using probability models to
estimate the serial interval and reproductive
number.
• Assume that we observe daily counts of new
cases: N1 , N2 ,, NT .
• Let Xij be the number of cases with symptoms
on day j that were infected by a case with
symptoms on day i.
Statistics in Medicine, 2008
White & Pagano
Method
• Using this scheme, we make some probabilistic
assumptions and get a likelihood equation:
e t tNt
L( R0 , p | N )  
Nt !
t 1
T
• Where
t  R0
min ( k ,t )

j 1
p j Nt  j .
• pj describes the serial interval (i.e. probability of
having symptoms j days after infector).
• Use numerical methods to get MLEs of Ro and p.
H1N1 Example
• In April the public became aware of a novel strain of
Influenza that was affecting Mexico.
• Fraser, Donelly et al published initial report in Science
on 11 May 2009.
• Estimate the reproductive number to be between 1.4
and 1.6.
• Estimate the average serial interval to be 1.91 days.
H1N1 Example
• We obtained data from the CDC with
information on each confirmed and suspected
case (1368 cases) as of May 8.
• 750 had a date of symptom onset.
Influenza A/H1N1: Serial Interval
• Spanish work estimate average serial interval
to be 3.5 days, range=1-6 days.
– Use contact tracing data.
• Seasonal influenza (Cowling et al, 2009)
– 3.6 days, SD=1.6
– From a household contact study
Influenza A/H1N1: R0 estimates
•
•
•
•
•
Mexico: 1.3-1.4 (Cruz-Pacheco et al)
Mexico: less than 2.2-3.1 (Boelle et al)
Japan: 2.3 (Nishiura et al)
Netherlands: less than 1 (Hahne et al)
US: 1.7-1.8 (White et al)
Influenza A/H1N1: USA
Influenza A/H1N1: USA
• Missing dates of symptom onset
– All cases have report date but many lack date of
symptom onset.
– Calculate the distribution of time between
reported date and symptom onset for those with
both.
– Impute a date of symptom onset for those with
missing information from the observed
distribution.
Reporting delay distribution
Other issues in the data
• Imported cases
– Make an adjustment in the estimation method to account
for those who were known to have traveled to Mexico.
• Reporting delay
– The decline in cases as it gets closer to May 8 is likely due
to reporting delays, rather than a true drop off in case
numbers.
– Augment the data at the end, using the reporting delay
distribution.
Augmented data
Estimates in the USA
• Using the White & Pagano Method with the
modifications mentioned we get estimates for
R0 and the serial interval in the initial outbreak
in the US.
Serial interval estimate
Using data up to and including April 25, 2009.
Using data up to and including April 27, 2009.
HETEROGENEITY
Heterogeneity
• Variation in transmission between adults and
kids, geographically, etc.
• Can lead to better policy decisions
– Who gets vaccinated first?
– Social distancing measures that might be most
effective?
Overview
•
•
•
•
Social mixing matrices
Glass method
Modification of Wallinga and Teunis
Modification of White and Pagano
Social mixing
• To understand who is most culpable for
transmission, we typically need to understand
how people interact
• Many approaches to this, but we choose most
popular currently: social mixing matrices
PolyMod study
• Large European study
– Belgium, Finland, Great Britain, Germany, Italy,
Luxembourg, the Netherlands, and Poland
• 97,904 contacts among 7,290 participants
• Participants record number and nature of
contacts in a diary
• Contact matrices were created to describe all
close contacts and separately, close contacts
that involve physical touch
Table 1.
Mossong J, Hens N, Jit M, Beutels P, et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. PLoS Med
5(3): e74. doi:10.1371/journal.pmed.0050074
http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0050074
Figure 1. The Mean Proportion of Contacts That Involved Physical Contact, by Duration, Frequency, and Location
of Contact in All Countries
Mossong J, Hens N, Jit M, Beutels P, et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. PLoS Med
5(3): e74. doi:10.1371/journal.pmed.0050074
http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0050074
Figure 2. The Distribution by Location and by Country of (A) All Reported Contacts and (B) Physical Contacts Only
Mossong J, Hens N, Jit M, Beutels P, et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. PLoS Med
5(3): e74. doi:10.1371/journal.pmed.0050074
http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0050074
Figure 3. Smoothed Contact Matrices for Each Country Based on (A) All Reported Contacts and (B) Physical
Contacts Weighted by Sampling Weights
Mossong J, Hens N, Jit M, Beutels P, et al. (2008) Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. PLoS Med
5(3): e74. doi:10.1371/journal.pmed.0050074
http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.0050074
Other studies
• Similar studies have been conducted in South
Africa and Vietnam
• First of this nature in Netherlands (Wallinga et
al, 2006)
• Johnstone-Robertson et al (2011) carried out a
very similar study in a South African township
Approaches
• Glass et al, 2011
– Estimate R for adults and children
– Do not require transmission data
• Modify Wallinga and Teunis method
– Estimate Rt (and R0) across age groups.
– Require contact information.
• Moser and White, 2013 (in preparation)
– Bayesian approach to the problem
– Modify White & Pagano method to incorporate age
contact information
– Incorporate contact information as a prior distribution
Approach 1: Glass et al
• Modify Wallinga & Teunis and White & Pagano
methods to estimate R for children and adults
• Assume a form for a reproduction matrix:
mCC
mCA
mAC
mAA
M=
• mij describes the number of cases of type i
infected by cases of type j.
• Some pre-specified structure must be imposed
on the matrix M must be assumed to estimate
the mij.
Matrix constraints
Source: Glass et al, 2011
Modification of White & Pagano
• Let 𝑋𝑖 ~𝑃𝑜𝑖𝑠(𝑚𝐶𝐶 𝐶𝑖 + 𝑚𝐴𝐶 𝐴𝑖 ) and
𝑌𝑖 ~𝑃𝑜𝑖𝑠(𝑚𝐶𝐴 𝐶𝑖 + 𝑚𝐴𝐴 𝐴𝑖 )
where Ai and Ci are the incidence counts for
adults and children, respectively.
Xi and Yi are the total number of cases infected
by children and adults from day i, respectively.
Modification of White & Pagano
𝜇𝐴 and 𝜇𝐶 are the expected number of adults and children on day t.
Modification of White & Pagano
The likelihood used is:
Maximize this over the mij to obtain estimates.
Applying constraints to M, creates relationships
between the mij and they become identifiable.
Modification of Wallinga & Teunis
• Modify the pij to incorporate the probability children
infect each other and adults infect each other.
• 𝑞𝑎𝑖 is the probability that an individual of type ai was
infected by someone of the same type (ai=C or A).
Modification of Wallinga & Teunis
• To estimate qA and qC, define f to be the total
number of cases that are children.
– Then the elements of the matrix elements in
terms of f and R (the population reproductive
number).
– For separable matrix, 𝑞𝐶 = 1/(1 + 𝑥 2 ) and 𝑞𝐴 =
1−𝑓
2
2
𝑥 /(1 + 𝑥 ) where 𝑥 =
.
𝑓
Approach 1: simulation study
True RC=2.5 and true RA=1. L, M and U are 3rd, median and 98th
percentiles over 100 simulations.
Approach 1: Japanese influenza data
Wallinga & Teunis Approach
Approach 1: Japanese influenza data
White & Pagano Method
𝑅𝐶 = 3.51, 3.49, 3.52 and 3.57
𝑅𝐴 = 0.34, 0.58, 0.21 and 0.37, for each matrix
Heterogeneity
APPROACH 2: MODIFICATION OF
WALLINGA AND TEUNIS
Approach 2: modification of Wallinga
& Teunis
• Similar to Glass et al, allow the probability of
infection to be impacted by more than just
distance apart in time
where
is the probability of a serial interval
of length j-i and
is a similarity measure (similar
to the matrices used by Glass et al).
Source: White, Archer and Pagano
(submitted, 2013)
Approach 2: modification of Wallinga
& Teunis
• Similar to Glass et al, but we do not assume
any structure on a similarity matrix, D=(dij).
• We use available data to define this matrix
and are able to obtain estimates of Rj for a
large number of age groups (or spatial
locations, etc.)
Similarity measures
• Individuals who are “close” together are more
likely to infect each other have larger
similarity measures.
• Can be used to address probability of infection
between different geographical regions, age
groups, etc.
Similarity measures
Similarity Matrix
Use a matrix to define
the similarity
measure.
Xij describes the
amount of contact
individuals in group i
have with those in
group j.
Age
group 1
Age
group 2
Age
group 3
Age
group 1
x11
x12
x13
Age
group 2
x21
x22
x23
Age
group 3
x31
x32
x33
Basic similarity measures
• Matrix of all 1’s: original estimator
– Implies that transmission is equally likely among
all individuals
• Diagonal matrix: transmission only occurs
within homogenous groups (no mixing)
– Comparable to applying original method to each
homogenous group separately
• Can also use matrix that describes contact
patterns
Example: Pandemic Influenza In South
Africa
• Between 6/15/2009 and 11/23/2009
there were 12,630 confirmed cases
Source: Archer et al (2009)
Age Analysis
• We restrict our attention to Gauteng Province
(the most populous) to limit geographic
effects
• Use two sources of information on contact
patterns between age groups:
– PolyMod Study (Mossong et al, 2009)
– Study in South African township (JohnstoneRobertson, 2011)
JSM 2012
PolyMod contact trace matrix
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70+
0-4
5-9
10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70+
1.92
0.65
0.41
0.24
0.46
0.73
0.67
0.83
0.24
0.22
0.36
0.2
0.2
0.26
0.13
0.95
6.64
1.09
0.73
0.61
0.75
0.95
1.39
0.9
0.16
0.3
0.22
0.5
0.48
0.2
0.48
1.31
6.85
1.52
0.27
0.31
0.48
0.76
1
0.69
0.32
0.44
0.27
0.41
0.33
0.33
0.34
1.03
6.71
1.58
0.73
0.42
0.56
0.85
1.16
0.7
0.3
0.2
0.48
0.63
0.45
0.3
0.22
0.93
2.59
1.49
0.75
0.63
0.77
0.87
0.88
0.61
0.53
0.37
0.33
0.79
0.66
0.44
0.74
1.29
1.83
0.97
0.71
0.74
0.85
0.88
0.87
0.67
0.74
0.33
0.97
1.07
0.62
0.5
0.88
1.19
1.67
0.89
1.02
0.91
0.92
0.61
0.76
0.63
0.27
1.02
0.98
1.26
1.09
0.76
0.95
1.53
1.5
1.32
1.09
0.83
0.69
1.02
0.96
0.2
0.55
1
1.14
0.94
0.73
0.88
0.82
1.23
1.35
1.27
0.89
0.67
0.94
0.81
0.8
0.29
0.54
0.57
0.77
0.97
0.93
0.57
0.8
1.32
1.87
0.61
0.8
0.61
0.59
0.57
0.33
0.38
0.4
0.41
0.44
0.85
0.6
0.61
0.71
0.95
0.74
1.06
0.59
0.56
0.57
0.31
0.21
0.25
0.33
0.39
0.53
0.68
0.53
0.55
0.51
0.82
1.17
0.85
0.85
0.33
0.26
0.25
0.19
0.24
0.19
0.34
0.4
0.39
0.47
0.55
0.41
0.78
0.65
0.85
0.57
0.09
0.11
0.12
0.2
0.19
0.22
0.13
0.3
0.23
0.13
0.21
0.28
0.36
0.7
0.6
0.14
0.15
0.21
0.1
0.24
0.17
0.15
0.41
0.5
0.71
0.53
0.76
0.47
0.74
1.47
Great Britain, all contacts
South African township contact matrix
Source: Johnstone-Robertson et al, AJE,
2011
Estimate of Rt
Epidemic curve
Estimate of Rt
(a) All contacts involving physical touch; (b) all close contacts
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45+
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45+
Age Group
Age Group
(a) All contacts involving physical touch; (b) all close contacts
800
1.0
Number of Cases
400
0.5
0.0
800
1.0
1200
1.5
1200
1.5
(a)
0
400
0.5
0
0.0
^
R0
Estimates of R0 by age group
(b)
Estimates of R0
Age group
N (%)
R0, close contacts
R0, all physical
contacts
0-4
484 (8.73)
0.94 (0.91-0.97)
0.74 (0.72-0.76)
5-9
927 (16.72)
1.20 (1.17-1.24)
1.29 (1.25-1.33)
10-14
1150 (20.75)
1.53 (1.49-1.58)
1.47 (1.44-1.51)
15-19
1026 (18.52)
1.36 (1.32-1.40)
1.47 (1.42-1.50)
20-24
556 (10.03)
1.06 (1.03-1.09)
1.03 (1.01-1.06)
25-29
389 (7.02)
0.98 (0.94-1.01)
0.97 (0.94-1.01)
30-34
229 (4.13)
0.92 (0.88-0.94)
0.86 (0.82-0.88)
35-39
246 (4.44)
0.85 (0.82-0.88)
0.75 (0.82-0.78)
40-44
171 (3.09)
0.86 (0.83-0.90)
0.83 (0.80-0.87)
45+
363 (6.55)
0.79 (0.75-0.85)
0.75 (0.71-0.81)
R0 by age group, by country used
(a) results for all close contacts and (b) for contacts involving physical touch
Estimates of R0 depending on
contact matrix used
Overall R0
R̂0
Method
Homogenous mixing
Overall
1.28 (1.26-1.31)
Contact matrix used
All close contacts
Contacts with physical touch
South Africa
1.27 (1.25-1.31)
1.27 (1.25-1.31)
Belgium
1.26 (1.24-1.31)
1.27 (1.24-1.31)
Finland
1.27 (1.25-1.32)
1.27 (1.25-1.32)
Great Britain
1.27 (1.25-1.32)
1.27 (1.25-1.31)
Germany
1.27 (1.25-1.32)
1.27 (1.25-1.32)
Italy
1.27 (1.25-1.31)
1.27 (1.24-1.31)
Luxembourg
1.27 (1.25-1.32)
1.27 (1.25-1.32)
Netherlands
1.27 (1.25-1.32)
1.27 (1.25-1.32)
Poland
1.27 (1.25-1.31)
1.27 (1.24-1.31)
Heterogeneity
APPROACH 3: MODIFICATION OF
WHITE AND PAGANO
Moser and White
• Modification of the White and Pagano method
to estimate R0 and incorporate heterogeneity
in the population
• Revise the likelihood to incorporate
heterogeneity in the reproductive numbers
• Consider the scenario where we look at adults
and kids only (2 group scenario)
– RA and RC are the reproductive numbers for adults
and children, respectively
Moser and White
• Reparameterize the problem to allow for
inclusion of contact matrix information
– qhg is the probability that individual of type h has
contact with individual of type g
– Example: RCA= qCA*RC
– RC=RCA+RCC
Day 0:
Day 1:
Day 2:
Day 3:
Day 4:
…….
Day T:
N0C
N0A
Derivation of Likelihood Function
Two Group Example
N1C = XC0C1 + XA0C1
N1A = XC0A1 + XA0A1
N2C = XC0C2 + XA0C2 + XC1C2 + XA1C2
N2A = XC0A2 + XA0A2 + XC1A2 + XA1A2
N3C = XC0C3 + XA0C3 + XC1C3 + XA1C3 + XC2C3 + XA2C3
N3A = XC0A3 + XA0A3 + XC1A3 + XA1A3 + XC2A3 + XA2A3
N4C =
+ XC1C4 + XA1C4 + XC2C4 + XA2C4 + XC3C4 + XA3C4
N4A =
+ XC1A4 + XA1A4 + XC2A4 + XA2A4 + XC3A4 + XA3A4
………….
NTC
NTA
Day 0:
Day 1:
Day 2:
Day 3:
Day 4:
…….
Day T:
N0C
3 Day Serial Interval
N0A
XC0A2 = Adults infected
on day 2 by a
child from day 0
N1C = XC0C1 + XA0C1
N1A = XC0A1 + XA0A1
N2C = XC0C2 + XA0C2 + XC1C2 + XA1C2
N2A = XC0A2 + XA0A2 + XC1A2 + XA1A2
N3C = XC0C3 + XA0C3 + XC1C3 + XA1C3 + XC2C3 + XA2C3
N3A = XC0A3 + XA0A3 + XC1A3 + XA1A3 + XC2A3 + XA2A3
N4C =
+ XC1C4 + XA1C4 + XC2C4 + XA2C4 + XC3C4 + XA3C4
N4A =
+ XC1A4 + XA1A4 + XC2A4 + XA2A4 + XC3A4 + XA3A4
………….
NTC
NTA
Day 0:
Day 1:
Day 2:
Day 3:
Day 4:
…….
Day T:
N0C
RC
XC0A2 = Adults infected
on day 2 by a
child from day 0
RA
N0A
N1C = XC0C1 + XA0C1
N1A =
XC0A1
+
XA
RC
RA
0A1
N2C = XC0C2 + XA0C2 + XC1C2 + XA1C2
N2A = XC0A2 + XA0A2 + XC1A2 + XA1A2
RC
RA
N3C = XC0C3 + XA0C3 + XC1C3 + XA1C3 + XC2C3 + XA2C3
RC
N3A = XC0A3 + XA0A3 + XC1A3 + XA1A3 + XC2A3 + XA2A3
RA
N4C =
+ XC1C4 + XA1C4 + XC2C4 + XA2C4 + XC3C4 + XA3C4
N4A =
+ XC1A4 + XA1A4 + XC2A4 + XA2A4 + XC3A4 + XA3A4
………….
NTC
NTA
Is Mixing Assortative?
Day 0:
N0C
N0A
RCC
Day 1:
Day 2:
Day 3:
Day 4:
…….
Day T:
RCA
XC0A2 = Adults infected
on day 2 by a
child from day 0
N1C = XC0C1
N1A = XC0A1
N2C = XC0C2
N2A = XC0A2
N3C = XC0C3
N3A = XC0A3
N4C =
N4A =
………….
NTC
NTA
Is Mixing Assortative?
Updated Likelihood
• The likelihood can be written as:
where Ntg is the number of cases on day t from
group g.
• How do we maximize this likelihood?
Estimation
• We could try a frequentist approach, but there
are issues with identifiability
– We have four parameters to estimate and, similar
to Glass et al, would need to impose constraints
on the q’s in order to get estimates.
• Alternative approach: MCMC with prior
information
– Use contact frequency matrices from survey data
to inform the priors of the q’s
Epidemic curves by age in South Africa
Results from South Africa pandemic
Age = 20
𝑅𝐶
N (%) 1589 (66)
Age = 18
𝑅𝐴
801 (34)
𝑅𝐶
1340 (56)
Age = 15
𝑅𝐴
1050 (44)
𝑅𝐶
901 (38)
𝑅𝐴
1489 (62)
Mossong
Prior 1
Prior 2
Prior 3
1.34 (1.11- 1.51 (1.121.32
1.58)
1.91)
(1.12-1.53)
1.37 (1.16- 1.45 (1.071.36
1.60)
1.85)
(1.15-1.57)
1.67 (1.47- 0.94 (0.671.55
1.88)
1.23)
(1.35-1.76)
1.45
(1.21-1.71)
1.41
(1.17-1.67)
1.20
(0.98-1.43)
1.47
(1.07-1.90)
1.51
(1.10-1.93)
1.61
(1.22-2.00)
1.37
(1.11-1.64)
1.34
(1.08-1.61)
1.28
(1.04-1.54)
1.33
1.52 (1.131.31
(1.11-1.57)
1.92)
(1.11-1.53)
1.37 (1.16- 1.45 (1.081.35
1.60)
1.83)
(1.14-1.56)
1.65 (1.47- 0.97 (0.711.54
1.85)
1.25)
(1.35-1.74)
1.46
(1.21-1.72)
1.43
(1.18-1.69)
1.20
(1.00-1.41)
1.44
(1.04-1.88)
1.48
(1.09-1.90)
1.58
(1.23-1.93)
1.38
(1.11-1.65)
1.36
(1.10-1.62)
1.30
(1.09-1.53)
JohnstonRobertson
Prior 1
Prior 2
Prior 3
Glass
Separable
1.78
0.90
1.65
1.29
1.06
1.75
HiC2C
1.73
1.00
1.63
1.31
0.87
1.86
Contact Freq.
1.90
0.64
1.91
0.95
1.63
1.41
Proportional
1.89
0.67
1.97
0.87
1.92
1.24
Issues
• Reporting differences across age groups
– How might this impact our results?
– Example: kids are much more likely to show up at
the clinic and have their cases reported. Adults are
more likely to stay home.
• Non-uniformity of contact patterns globally?
• Other issues?
Final thoughts
• Quantitative methods are essential to informing
policy decisions in a disease outbreak
• Issues we want to address:
–
–
–
–
Severity
Transmissibility
Heterogeneity
Uncertainty
• Challenges with dependency in the data,
unobserved events, etc.
Thanks!
• Funding source: National Institute Of General
Medical Sciences of the National Institutes of
Health under Award Number U54GM088558.