Inverse Problems in Epidemiology Karyn Sutton 1 , 2 1 Center for Research in Scientific Computation & 2 Center for Quantitative Studies in Biomedicine North Carolina State University 3 Department of Mathematics and Statistics Arizona State University Collaborators: H.T. Banks 1 , 2 avez 3 Carlos Castillo-Ch´ Wednesday, October 29, 2008 Workshop on Inverse and Partial Information Problems: Methodology and Applications
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Public Health Challenges in Infectious Diseases • Prescribing and implementing control strategies (prevention and/or treatment) • 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
I NVERSE P ROBLEMS IN E PIDEMIOLOGY 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 Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Pneumococcal Diseases as Example Invasive 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 Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY 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 pneumonia annually (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. Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Key Assumptions in Mathematical Model of Pneumococcal Diseases • Asymptomatic nasopharyngeal colonization (or ‘carriage’) results from casual contacts • 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 Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Pneumococcal Disease Dynamics Model � I µ E µ I µ S � S E + E V + I + I V � � � � � N � � I � S E I l � ( t ) E � E � S � E � S V � E V � � E + E V + I + I V �� S V � � � N � �� ( t ) E V � I V S V E V I V � E V µ I V µ S V µ E V � I V κ ( t ) = κ 0 (1 + κ 1 cos[ ω ( t − τ )] Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Surveillance Data from Australian NNDS Observations from 2001-2004, before conjugate vaccine • Total Cases (monthly): R tj +1 Y (1) ∼ [ κ ( s ) E ( s ) + δκ ( s ) E V ( s )] ds ; ( t 1 , ..., t 37 ) = (1 , ..., 37) . tj j • Unvaccinated Cases (annually): R tk +1 Y (2) ∼ [ κ ( s ) E ( s )] ds ; ( t 1 , ..., t 4 ) = (1 , 13 , 25 , 37) . tk k • Vaccinated Cases (annually): R tk +1 Y (3) ∼ [ δκ ( s ) E V ( s )] ds ; ( t 1 , ..., t 4 ) = (1 , 13 , 25 , 37) . tk k Observations after conjugate vaccine freely available • Total Cases Jan 2005 - Jun 2007 (monthly): R tj +1 Y (1) ∼ [ κ ( s ) E ( s ) + δκ ( s ) E V ( s )] ds ; ( t 1 , ..., t 31 ) = (1 , ..., 31) . j tj • Unvaccinated Cases 2005 (annually): R tk +1 Y (2) ∼ [ κ ( s ) E ( s )] ds ; ( t 1 , t 2 ) = (1 , 13) . tk k • Vaccinated Cases 2005 (annually): R tk +1 Y (3) ∼ [ δκ ( s ) E V ( s )] ds ; ( t 1 , t 2 ) = (1 , 13) . tk k Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Statistical Model • Assume a statistical model Y ( i ) = f ( i ) ( t, θ 0 ) + ǫ ( i ) where Y (1) = { Y (1) j =1 , Y (2) = { Y (2) k =1 , Y (3) = { Y (3) } 36 } 3 } 3 k =1 and f ( i ) , ǫ ( i ) j k k are defined similarly. • We further assume 1. There exist ‘true parameters’ θ 0 which generated observations. 2. ǫ ( i ) are i.i.d. for fixed i . j 3. mean E [ ǫ ( i ) j ] = 0 , and variance var [ ǫ ( i ) j ] = σ 2 0 ,i . 4. σ 0 , 2 = σ 0 , 3 ; data likely arose by same counting process. Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Inverse Problem Formulation • ˆ θ OLS ( Y ) = arg min θ ∈ Θ J ( θ, σ 1 , σ 2 ) where Θ ⊂ R p is the feasible parameter space. • Objective function 36 J ( Y, θ, σ 1 , σ 2 ) = 1 2 ˛ ˛ f (1) ( t, θ ) − Y (1) ˛ X ˛ ˛ σ 2 j j ˛ 1 j =1 3 2 2 – + 1 »˛ ˛ f (2) ( t, θ ) − Y (2) ˛ ˛ f (3) ˛ ( t, θ ) − Y (3) ˛ X + ˛ ˛ ˛ ˛ σ 2 k k k k ˛ ˛ 2 k =1 • Variance formulas 36 1 2 ˛ ˛ σ 2 ˛ f (1) θ ) − Y (1) X ( t, ˆ ˆ 1 = ˛ ˛ j j 36 − p ˛ j =1 3 2 2 – 1 »˛ ˛ f (2) θ ) − Y (2) ˛ ˛ f (3) ˛ θ ) − Y (3) ˛ σ 2 X ( t, ˆ ( t, ˆ ˆ 2 = + ˛ ˛ ˛ ˛ k k k k 6 − p ˛ ˛ k =1 Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Inverse Problem Formulation • Estimating θ = ( β, κ 0 , κ 1 , δ ) T requires simultaneous estimation of σ 2 1 , σ 2 2 via an iterative process. 2 ) T by: • Estimate ˆ ψ = (ˆ σ 2 σ 2 θ, ˆ 1 , ˆ σ (0) σ (0) 1. Guess ˆ 1 , ˆ 2 . θ (0) = arg min θ ∈ Θ J ( θ, ˆ σ (0) σ (0) 2. ˆ 1 , ˆ 2 ) 2 with ˆ σ 2 σ 2 θ (0) . 3. Calculate ˆ 1 , ˆ ˛ ˛ θ ( k ) and then ˆ σ 2 until 4. Continue updating ˆ ˛ || ˆ ψ ( k ) || − || ˆ ψ ( k − 1) || ˛ ≤ 10 − q σ 2 1 , ˆ ˛ ˛ where q is a pre-determined constant. • Obtain standard errors from estimated covariance matrix: q SE (ˆ ˆ θ k ) ≈ Σ kk # − 1 3 " 1 ˆ X χ T i (ˆ θ ) χ i (ˆ Σ = θ ) σ 2 ˆ i i =1 ∂fi where the ( j, k ) th entry of χ i (ˆ j θ ) is ∂θk Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Model Calibration Vaccine Assessment New Infections 400 300 Jan 05 thru Jun 07 250 Cases 200 100 200 0 0 5 10 15 20 25 30 35 40 t (months) 150 Unvaccinated Cases Vaccinated Cases cases 1800 1800 1600 1600 100 1400 1400 Cases Cases 1200 1200 1000 1000 50 800 800 600 600 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 0 5 10 15 20 25 30 t (years) t (years) t (months) Used calibrated model to show that vaccine becoming increasingly less effective ⇒ suggests need for quantitative monitoring Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY 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 certain parameters? – How many longitudinal points are needed? How frequently? Over what length of time? – How can we tell if a model is ‘over-specified’? Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
I NVERSE P ROBLEMS IN E PIDEMIOLOGY Age-Structured Model • Age is a reasonable marker for physiological processes which govern infection dynamics • Analogous age-structured pneumococcal disease model with parameters/rates functions 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 Workshop on Inverse and Partial Information Problems: Methodology and Applications October 29, 2008
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