inverse problems in epidemiology€¦ · inverse problems in epidemiology karyn sutton1,2 1 center...

Inverse Problems in Epidemiology

Karyn Sutton1,2

1 Center for Research in Scientific Computation &2 Center for Quantitative Studies in Biomedicine

North Carolina State University

3 Department of Mathematics and StatisticsArizona State University

Collaborators:H.T. Banks1,2

Carlos Castillo-Chavez3

Wednesday, October 29, 2008

Workshop on Inverse and Partial Information Problems: Methodology and Applications

INVERSE PROBLEMS IN EPIDEMIOLOGY

Public Health Challenges in Infectious Diseases

• Prescribing and implementing control strategies (prevention and/ortreatment)

• Collection and analyzing surveillance data

• One strategy likely not effective in all populations

– Heterogeneous populations– Drugs or vaccines may be inappropriate for population

Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008


Mathematical Approaches & Inverse Problems

• Physiological structure can be incorporated into population models• Determine appropriate level of detail• Determine if mechanisms/terms should be included in model• Calibrate a mathematical model to population of interest• Theoretically study prevention and/or treatment strategies• Assess impact/effectiveness of implemented prevention or treatment strategy• Improve surveillance data collection

– ‘Types’ of data– How many longitudinal data points– Frequency of longitudinal observations



Pneumococcal Diseases as ExampleInvasive infections caused by Streptococcus pneumoniae include pneumonia,meningitis, bacteremia, sepsis.

• Population heterogeneity plays a role in infection dynamics• Multiple serotypes complicate prevention and treatment; Vary by:

– Geographic region– Age groups affected– Ability to colonize individuals– Ability to cause infection in colonized individuals



Pneumococcal Diseases as Example

• Vaccine development active research area– Distinct in structural design → distinct in induced immunity

• Polysaccharide vaccine: licensed in 1983, effective in elderly• BUT most affected group is children,

– 1 million children under the age of 5 die from pneumococcal pneumoniaannually (WHO, 1999)

• Protein conjugate vaccine: licensed in 2001, effective in children• BUT may induce undesirable evolutionary changes in endemic pneumococci,

changing landscape of infections in unknown and potentially serious ways.



Key Assumptions in Mathematical Model ofPneumococcal Diseases

• Asymptomatic nasopharyngeal colonization (or ‘carriage’) results from casualcontacts

• Infection established only if colonies cannot be cleared• Seasonality in infection rates due to changes in host susceptibility (comorbidity)• Susceptible and colonized individuals vaccinated at same rate• Vaccines may induce protection against infection and colonization (conjugate)• Vaccine protection may be lost



Pneumococcal Disease Dynamics Model

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κ(t) = κ0(1 + κ1 cos[ω(t− τ)]



Surveillance Data from Australian NNDSObservations from 2001-2004, before conjugate vaccine

• Total Cases (monthly):Y

(1)j ∼

R tj+1tj

[κ(s)E(s) + δκ(s)EV (s)] ds; (t1, ..., t37) = (1, ..., 37).

• Unvaccinated Cases (annually):Y

(2)k ∼

R tk+1tk

[κ(s)E(s)] ds; (t1, ..., t4) = (1, 13, 25, 37).

• Vaccinated Cases (annually):Y

(3)k ∼

R tk+1tk

[δκ(s)EV (s)] ds; (t1, ..., t4) = (1, 13, 25, 37).

Observations after conjugate vaccine freely available

• Total Cases Jan 2005 - Jun 2007 (monthly):Y

(1)j ∼

R tj+1tj

[κ(s)E(s) + δκ(s)EV (s)] ds; (t1, ..., t31) = (1, ..., 31).

• Unvaccinated Cases 2005 (annually):Y

(2)k ∼

R tk+1tk

[κ(s)E(s)] ds; (t1, t2) = (1, 13).

• Vaccinated Cases 2005 (annually):Y

(3)k ∼

R tk+1tk

[δκ(s)EV (s)] ds; (t1, t2) = (1, 13).



Statistical Model

• Assume a statistical model

Y(i)

= f(i)

(t, θ0) + ε(i)

where Y (1) = {Y (1)j }36

j=1, Y (2) = {Y (2)k }3

k=1, Y (3) = {Y (3)k }3

k=1 and f (i), ε(i)

are defined similarly.• We further assume

1. There exist ‘true parameters’ θ0 which generated observations.2. ε(i)j are i.i.d. for fixed i.

3. mean E[ε(i)j ] = 0, and variance var[ε(i)j ] = σ2

0,i.4. σ0,2 = σ0,3; data likely arose by same counting process.



Inverse Problem Formulation

• θOLS(Y ) = arg minθ∈Θ J(θ, σ1, σ2)where Θ ⊂ Rp is the feasible parameter space.

• Objective function

J(Y, θ, σ1, σ2) =1

σ21

36Xj=1

˛f

(1)j (t, θ)− Y

(1)j

˛2

+1

σ22

3Xk=1

»˛f

(2)k

(t, θ)− Y(2)k

˛2+

˛f

(3)k

(t, θ)− Y(3)k

˛2–

• Variance formulas

σ21 =

1

36− p

36Xj=1

˛f

(1)j (t, θ)− Y

(1)j

˛2

σ22 =

1

6− p

3Xk=1

»˛f

(2)k

(t, θ)− Y(2)k

˛2+

˛f

(3)k

(t, θ)− Y(3)k

˛2–



Inverse Problem Formulation

• Estimating θ = (β, κ0, κ1, δ)T requires simultaneous estimation of σ2

1, σ22 via

an iterative process.• Estimate ψ = (θ, σ2

1, σ22)T by:

1. Guess σ(0)1 , σ

(0)2 .

2. θ(0) = arg minθ∈Θ J(θ, σ(0)1 , σ

(0)2 )

3. Calculate σ21, σ

22 with θ(0).

4. Continue updating θ(k) and then σ21, σ

2 until˛||ψ(k)|| − ||ψ(k−1)||

˛≤ 10−q

where q is a pre-determined constant.• Obtain standard errors from estimated covariance matrix:

SE(θk) ≈q

Σkk

Σ =

"3Xi=1

1

σ2i

χTi (θ)χi(θ)

#−1

where the (j, k)th entry of χi(θ) is∂fij∂θk



Model Calibration Vaccine Assessment

0 5 10 15 20 25 30 35 400

100

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t (months)

Case

s

1 1.5 2 2.5 3600

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t (years)

Case

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1 1.5 2 2.5 3600

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1800Vaccinated Cases

t (years)

Case

s

0 5 10 15 20 25 300

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t (months)

case

s

Jan 05 thru Jun 07

Used calibrated model to show that vaccine becoming increasingly less effective ⇒suggests need for quantitative monitoring



Age-Specific Surveillance Data

• Age recorded with most infectious disease reports• Not reported frequently enough for reliable parameter estimation• Generated age-dependent data to explore following questions:

– Which ‘types’ of information should be collected to estimate certainparameters?

– How many longitudinal points are needed? How frequently? Over whatlength of time?

– How can we tell if a model is ‘over-specified’?



Age-Structured Model

• Age is a reasonable marker for physiological processes which govern infectiondynamics

• Analogous age-structured pneumococcal disease model with parameters/ratesfunctions of age

• Discretize PDE to system of ODE’s assuming stable age distribution• State variables represent age cohorts, possibly of different lengths

– Consider parameters constant within each age class– Use smaller lengths in younger and older age classes

• Facilitates computational studies and connection to surveillance data



Age-Structured Model

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SV(a,t) EV(a,t)

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"(a)

S(0, t) =R ∞

0f(a′)n(a′, t)da′



Inverse Problem for Age-Structured Data

• For m age classes, nk longitudinal observations, data is generated according tothe following statistical model

Y(k,i)j = f

(k,i)(tj, θ0) + ε

(k,i)j

i denotes age class, k denotes ‘type’ of observation, j denotes time.• ‘True’ parameters θ0 known!• ε

(k,i)j ∼ N (0, σ2

k,i)

– Variance independent of time, but scaled according to observation– σk,i = l

100 ∗ avgjf(k,i)(tj, θ0) for l% noisy data



Age-Structured Generated Data

• Total cases: f (1,i)j =

R tj+1tj

[κi(s)Ei(s) + δiκi(s)Ei(s)] ds wherej = 1, ..., n1 for each i = 1, ..., 5,

• Vaccinated Cases: f (2,i)j =

R tj+1tj

δiκi(s)EV i(s)ds where j = 1, .., n2 foreach i = 1, ..., 5,

• Colonization Prevalence: f (3,i)j =

Ei(tj)+EV i(tj)

Niwhere j = 1, ..., n4 for each

i = 1, ..., 5,

• Vaccinated Colonization Prevalence: f (4,i)j =

EV i(tj)

Niwhere j = 1, ..., n5 for

each i = 1, .., 5.

** Colonization prevalence data not currently collected, but public health officialshave considered the benefits of this information



Model Calibration Results

• Estimated mean infection rates κ0 from reports of total cases Y (1)

– ψ = (κ0,1, ..., κ0,m, σ21,1, ..., σ

21,m)

• Estimated mean infection rates κ0 and vaccine infection protection δ from totalY (1) and vaccinated cases Y (2)

• Estimated force of infection Λ from colonization prevalence data Y (3)

– Assuming mixing structure (proportionate), can estimate contact matrix ci,i′– Difficult to quantify ⇒ rarely, if ever, available in literature– Drives horizontal spread of infections



Annual, 6 yrs Monthly, 2 yrsψ0 ψ10 SE(θ10) ψ10 SE(θ10)

κ0,1 3e−4 3.22e−4 4.9e−6 3.00e−4 2.4e−6

κ0,2 2.5e−5 2.56e−4 1.1e−6 2.47e−5 6.4e−7

κ0,3 4e−5 3.89e−4 2.7e−6 3.96e−5 1.4e−6

κ0,4 6e−5 5.79e−5 5.6e−6 5.86e−5 2.8e−6

κ0,5 1.7e−4 1.82e−4 1.9e−5 1.74e−4 6.8e−6

σ101,1 149 148.6287σ10

1,2 51 51.2244σ10

1,3 96 96.2737σ10

1,4 54 54.4548σ10

1,5 127 127.2959σ10

1,1 5.5415 5.30σ10

1,2 2.0388 1.90σ10

1,3 4.8541 5.08σ10

1,4 2.8422 3.34σ10

1,5 5.4788 5.62



Model Calibration Results

0 5 10 15 200

50

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t (Mos)

Cas

es

Total Cases, Age Groups 1,2data 1model 1data 2model 2

0 5 10 15 200

50

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t (Mos)

Cas

es

Total Cases, Age Groups 3,4,5data 3model 3data 4model 4data 5model 5

0 5 10 15 200

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t (Mos)

Cas

es

Vaccinated Cases, Age Group 4data 4model 4

0 5 10 15 200

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t (Mos)

Cas

es

Vaccinated Cases, Age Group 5data 5model 5



Vaccine Impact Assessment

• Estimated vaccine infection protection δ in vaccinated classes from vaccinateddata Y (2)

• Estimated vaccine colonization protection ε in younger age classes fromvaccinated colonization prevalence Y (4)

– Studies attempting to quantify this parameter are controversial– Further motivation to collect colonization data as protein-based vaccines are

implemented• Simultaneously estimated vaccine protection parameters ε and δ in relevant age

classes from vaccinated infection Y (2) and colonization prevalence data Y (4)

• Unable to estimate ε or δ without vaccine information (Y (1), Y (3))



Age Class Refinement

• Age is commonly reported, but often unclear how to meaningfully aggregateinformation

• Consider models of this type with fewer age classes as special cases of modelswith more age classes– Model 1: (0,10], (10,20], (20,∞); Model 2: (0,20], (20,∞)– Age-dependent parameter α(a)

Model 1: α1, α2, α3; Model 2: α1 = α2, α3

– Estimating α(a) from data will likely result in a lower residual sum of squares(RSS) for Model 1 as compared to Model 2.

• Models with more age classes will always give a lower residual when fit to data,more degrees of freedom

• Employ the use of a RSS-based statistic– Tells when improvement of fit/reduction of RSS is statistically significant– If improvement is significant, increased level of detail is warranted in the

model



Model Comparison Statistic

• Consider the estimator

θOLS(Y ) = arg minθ∈Θ

Jn(Y, θ)

where Θ ⊂ Rp is the feasible parameter space.• Statistical model and corresponding assumptions unchanged

Yj = f(tj, θ0) + εj

• Define constrained parameter space

ΘH = {θ ∈ Θ|Hθ = c}

where H is an r × p matrix and c is a known constant. (r is the difference indegrees of freedom)

• Let θH(Y ) denote the OLS estimator over ΘH.



Model Comparison Statistic

• Testing the null hypothesis H0 : θ0 ∈ ΘH.

• Define RSS-based statistic

Urn(Y ) =

n`Jn(Y, θ(Y ))− Jn(Y, θ

H(Y ))´

Jn(Y, θ(Y ))

• If H0 true, Un → U(r) in dist’n as n→∞, U(r) ∼ χ2(r).• Choose significance level α, then Prob{U > τ} = α for threshold τ .• If Un > τ , reject H0. Otherwise, do not reject H0.• Back to example of model 1 vs. model 2:

– H =`1 −1 0

´, c = 0 and r = 1.

– Compare Un to χ2(1) at some pre-determined α.– If Un > τ reject H0 ⇒ considering additional age classes warranted by

observations.– If Un < τ , additional age classes do not explain observations better, using

the ‘simpler’ model, (model 2) is reasonable.



Model Comparison ResultsGenerated total case reports Y (1) with 14-age class model: (0,2mos], (2,4 mos],(4,6 mos], (6,24 mos], (2,5 yrs], (5,10], (10,15], (15,50], (50,65], (65,70], (70,75],(75,80], (80,85], 85+.

Aggregated (0,2 y] (2,15] (2,15] (65,∞) (65, ∞)age groups (65,∞)

r: χ2(r) 9 6 4τ 29.67 24.10 20.00J60 9,475.1 3,093.7 2,244.1Ur60 212.8 29.08 4.62

• Differences among older age classes not important ⇒ reasonable to groupthese into 1 age class

• Considering smaller age groups between 2 and 15 years of age provides abetter fit.

• Discretizing the youngest age classes also provides a significantly better fit.



0 20 40 6010

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data 1model 1

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data 2model 2

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data 3model 3data 4model 4

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data 5model 5

Model with 5 age classes fit to infection data grouped into 5 age classes.



0 20 40 600

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data 1model 1data 2model 2data 3model 3data 4model 4

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data 5model 5data 6model 6data 7model 7

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data 8model 8data 9model 9

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data 10model 10data 11model 11data 12model 12data 13model 13data 14model 14

Model with 5 age classes fit to infection data grouped into 14 age classes.


inverse problems in epidemiology€¦ · inverse problems in epidemiology karyn sutton1,2 1 center...

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