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Prologues Estimating the Reproduction Number Epilogue References Inferring Reproductive Numbers from Final Size Epidemic Data on Hepatitis A in Flanders, Belgium: Truncation, Censoring and Heterogeneity Niel Hens InFER - Warwick 2011 Hens et al. InFER - Warwick 2011 1 Prologues Estimating the Reproduction Number Epilogue References Outline 1 Prologues Prologue I: Immunological Assessment Prologue II: Serological Assessment 2 Estimating the Reproduction Number Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Truncation Censoring Heterogeneity 3 Epilogue Hens et al. InFER - Warwick 2011 2 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Introduction Case Study on Hepatitis A ‘Federal’ knowledge centre: If any, targeted or universal vaccination? Five steps: Immunological assessment Serological assessment Outbreak modelling Mathematical modelling Cost-effectiveness analysis Hens et al. InFER - Warwick 2011 3 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Immunology Hepatitis A vaccine study: post-vaccination antibody levels follow-up over 10 years (113 individuals) Consider long and short living plasma cells producing antibodies dPs dt dPl dt dA dt = −µs Ps , = −µl Pl , = φs Ps + φl Pl − µA, where µ. are mortality rates and φ. are production rates. Analytical solution Inference using nonlinear mixed models (SAS & Monolix) Hens et al. InFER - Warwick 2011 4 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Immunology Results: antibody levels: lifespan of 1 month short-lived plasma cells: average lifespan of 7 months long-lived plasma cells persist for life Assumption of lifelong immunity seems reasonable Hens et al. InFER - Warwick 2011 5 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Serological samples from 1993 and 2002 Serial seroprevalence Ades and Nokes [1993], Batter et al. [1994], Marschner [1997], Nagelkerke et al. [1999]: multiplicative age & time specific model λ(a, t) = exp(µ0 + I X µi ai + i=1 J X θj tj ), I, J = 1, 2, ..., j=1 assuming differential selection and a (semi-)parametric model. Focus: hepatitis A: SIR non-differential selection general framework Hens et al. InFER - Warwick 2011 6 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Nagelkerke et al. [1999]: proportional hazards model for two calendar times t1 and t2 : λ(a, t2 ) = exp(β(t2 − t1 ))λ(a, t1 ). (1) a cohort born at calendar time b has age a at calendar time a + b: λb (a) = λ(a, a + b). the proportional hazards assumption: λb (a) = exp(β(b − b0 ))λb0 (a), (2) where λb0 (a) = λ(a, a + b0 ) is the baseline force of infection MM-algorithm: maximization-maximization with isotonic stepwise estimate to achieve λb0 (a) ≥ 0 Hens et al. InFER - Warwick 2011 7 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence fully nonparametric proportional hazards model: λ(a, t2 ) = exp(g(t2 − t1 ))λ(a, t1 ) (3) where g(t) is a smooth function with the constraint that g(0) = 0. In terms of the proportion susceptible xb (a) = x(a, b + a) = exp{exp(g(b − b0 )) log xb0 (a)}, (4) where, with xb0 (a) = x(a, b0 + a), Z xb0 (a) = exp{− a λb0 (s)ds}. 0 Hens et al. InFER - Warwick 2011 8 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Denote πb (a) = 1 − xb (a) the proportion subjects infected at age a or before: log(− log(1 − πb (a))) = s1 (a) + s2 (b), (5) where s1 (a) = log(− log(1 − πb0 (a))). s2 (b) = g(b − b0 ), Cohort-specific force of infection: λb (a) = πb0 (a) = exp(s2 (b))λb0 (a), 1 − πb (a) (6) πb0 0 (a) d = exp(s1 (a)). 1 − πb0 (a) da (7) with λb0 (a) = Extensions are possible Hens et al. InFER - Warwick 2011 9 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Standard statistical software: ‘mgcv’-package in R: thin plate regression splines Test for cohort-effect: approximate F-test Results: Model Components Model Model Model Model Model 1: 2: 3: 4: 5: a+b s(a) + b a + s(b) s(a) + s(b) s(a, b) edf AIC rank BIC rank 3 10.03 10.55 13.85 19.04 6062.03 5976.90 5958.68 5934.30 5936.16 5 4 3 1 2 6082.07 6043.94 6082.07 6026.83 6063.34 4 2 5 1 3 Table: Results for different models: parametric model 1, semi-parametric models 2 and 3, non-parametric models 4 and 5. Note: TB example of Nagelkerke et al. [1999]: Model 5 - best model Hens et al. InFER - Warwick 2011 10 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Ensuring positivity for the cohort-specific FOI: maximization-maximization age: monotone P-spline time: P-spline Backfitting algorithm Iterate until convergence Two-dimensional tuning parameter selection Hens et al. InFER - Warwick 2011 11 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment 0 0.20 ● ● ● ● 0.6 ● ● 0.2 ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● 80 ● ● ● ●● ● ● ● ● 0 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 20 age 40 0.15 ● ● 0.10 ● ● ● ● 60 ● ● ● ● 40 ● ● ● 20 ● ● 0.05 ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● 0.00 ● ●● ● ● ● 0.8 ● ● 0.0 ● ● ● ●● ● 0.4 0.6 0.4 ● ● ●●● ● ● ● 0.0 ● ● ● ●● 0.2 seroprevalence 0.8 ●● ●●●●●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ●● ● seroprevalence ● ● ● proportion immunized between 1993 and 2002 ● ●●●●●●●●●● ● ●● 1.0 1.0 Hepatitis A Serial Seroprevalence 60 80 age 1940 1950 1960 1970 1980 1990 year of birth Figure: HAV-seroprevalence in Flanders based on serological sample from 1993 (left panel) and 2002 (middle panel) together with the proportion infected in the period 1993-2002 (right panel). Model-based estimates (solid lines) and 95% bootstrap-based percentile confidence intervals (dashed lines) are based on the analysis reported by Hens et al. [2011]. Hens et al. InFER - Warwick 2011 12 Prologues Estimating the Reproduction Number Epilogue References Prologue I: Immunological Assessment Prologue II: Serological Assessment Hepatitis A Serial Seroprevalence Seroprevalence data 2002 for Belgium Odds ratios for the spatial effect (tensor-product spline for age) [0,0.75) [0.75,1) [1,1.25) [1.25,1.5) [1.5,3) [3,9] Hens et al. InFER - Warwick 2011 13 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Hepatitis A Outbreak Data 80 113 outbreaks (provincial health agencies) available information: 20 final sizes 40 Frequency 60 5 foodborne outbreaks: excluded from the analysis 0 5 Flemish provinces auxiliary information: school, family, children, . . . 0 10 20 30 40 observed size reporting issues: number of initial cases mostly unknown outbreaks with size at least 2 underreporting due to long incubation period: 15-45 days Hens et al. InFER - Warwick 2011 14 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Objectives Estimate the reproduction number from final size data Dealing with reporting issues: marginalization truncation censoring heterogeneity Input for a mathematical model with demography migration rates → confirmation of no transmission, only importation Hens et al. InFER - Warwick 2011 15 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Estimating the Reproduction Number Mortal branching process Assume we know the number of initial cases Final size: total number of infected persons with links Becker (1974): final size distribution P (X = x; s) = b(x, s) θx−s , A(θ)x where S = s initial cases X = x is the final size θ is the reproductive ratio b(x, s) is constant. A(θ)x is a normalizing factor Hens et al. InFER - Warwick 2011 16 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Estimating the Reproduction Number Offspring distribution: generalized power series distribution Poisson offspring distribution: Borel-Tanner distribution [Haight and Breuer, 1960] P (X = x; s) = sxx−s−1 θx−s e−xθ , (x − s)! x = s, s + 1, . . . . Geometric offspring distribution: s θx−s 2x − s P (X = x; s) = , x−s 2x − s (1 + θ)2x−s x = s, s + 1, . . . . Defined for X < ∞ and θ ≤ 1 Hens et al. InFER - Warwick 2011 17 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Applied to the Data −300 Assume 1 source case for each outbreak (ignoring truncation): 0.4 0.6 0.8 −400 profile log−likelihood 0.2 x −500 −350 −450 −550 profile log−likelihood x 1.0 0.2 0.4 R 0.6 0.8 1.0 R profile likelihood confidence intervals Borel-Tanner: R̂ = 0.76 (0.68, 0.84) Geometric Offspring: R̂ = 0.76 (0.66; 0.88) Hens et al. InFER - Warwick 2011 18 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Unknown Number of Initial Cases Crucial assumption: conditioning S = s assume initial case distribution derive the joint and marginal likelihood {x ≥ s} Initial case distribution one-parameter distributions degenerate: S = 1 discrete cases: S = 1, 2 truncated Poisson two-parameter distributions discrete cases: S = 1, 2, 3 truncated negative binomial Hens et al. InFER - Warwick 2011 19 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Applied to the Data Unknown initial cases (ignoring truncation): Illustration: truncated Poisson [ = 2.41 Borel-Tanner: R̂ = 0.37, E(S) 4.0 4.0 [ = 2.41 Geometric offspring: R̂ = 0.42, E(S) −3 0 0 −3 00 50 −3 3.0 ms ms 3.0 −250 ● 2.0 2.0 ● 0.0 −3 50 −30 1.0 1.0 −250 −50 0 0 0.2 0.4 0.6 0.8 1.0 −5 00 0.0 −35 0 0.2 R −300 0.4 0.6 0.8 1.0 R Hens et al. InFER - Warwick 2011 20 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Known and Unknown Number of Initial Cases Marginalization induces assumption dependent results What if we do have some information on initial cases (7 × 1)? ignorance leads to inefficiency provided a correct model is specified missing data approach? direct likelihood 0 0 −33 0 −31 0 −37 −4 −4 00 0 −3 30 −3 60 90 −3 −4 60 −360 20 −27 ms 3 0.0 −4 8 0 −4 20 −300 −260 2 70 −3 −4 40 ● 20 0.2 −250 −2 80 0.4 0.6 20 80 0 −280 1 2 −3 −40 0 −240 ● −2 −3 10 −380 −340 −320 80 4 −260 −4 −3 −4 1 ms 4 50 6 −34 −29 5 5 −32 0 −300 −280 3 6 Illustration: truncated Poisson & Borel-Tanner 0.8 1.0 0.0 0.2 R 0.4 0.6 0.8 1.0 R Hens et al. InFER - Warwick 2011 21 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Truncation: Known Number of Initial Cases −300 −260 −240 −280 −220 x profile log−likelihood −200 x −320 profile log−likelihood Assume 1 source case for each outbreak: 0.2 0.4 0.6 0.8 1.0 0.2 0.4 R 0.6 0.8 1.0 R profile likelihood confidence intervals Truncated Borel-Tanner: R̂ = 0.58 (0.51, 0.66) Truncated Geometric Offspring: R̂ = 0.52 (0.44; 0.61) Hens et al. InFER - Warwick 2011 22 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Truncation: Unknown Number of Initial Cases Illustration: truncated Poisson [ = 1.00 Truncated Borel-Tanner: R̂ = 0.58, E(S) −3 0 00 4.0 −2 8 20 60 40 −3 −2 −3 4.0 [ = 1.01 Truncated Geometric offspring: R̂ = 0.52, E(S) 60 0 −3 0 0 −3 20 3.0 −220 ms 3.0 ms 2.0 2.0 −220 −2 0.4 1.0 0.6 0.8 1.0 20 0.2 −3 ● 0.0 −200 60 −200 60 −3 20 −2 1.0 −2 8 −2 −240 −240 0.0 ● 0.2 R 0.4 0.6 0.8 1.0 R Hens et al. InFER - Warwick 2011 23 Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Prologues Estimating the Reproduction Number Epilogue References Truncation: Known and Unknown Number of Initial Cases Illustration: truncated Poisson 3.0 ms 3.0 ms 2.0 2.0 1.0 1.0 −220 0.6 0.8 1.0 0 0.4 −3 40 −36 60 0.2 20 −240 −200 ● 0.0 −3 −260 −320 −2 −3 20 00 60 −220 −3 −280 −3 00 −240 −300 −3 0 4.0 −2 8 20 60 40 −3 −2 −3 4.0 [ = 1.00 Truncated Borel-Tanner: R̂ = 0.58, E(S) −200 ● 0.0 0.2 R 0.4 0.6 0.8 1.0 R Hens et al. InFER - Warwick 2011 24 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Censoring & Conditioning on Extinction Farrington et al. [2003]: X = ∞: P (X = ∞) = 1 − q(θ)s Censored likelihood: 1−ci x −1 n i Y X ci L(θ; s, x, c) = 1− P (X = j, si ) P (X = xi ; si ) j=si i=1 [ = 1.00 Only two clusters affected by censoring: R̂ = 0.59, E(S) Assumption: non-informative censoring Hens et al. InFER - Warwick 2011 25 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Heterogeneity Spread depends on the setting calendar time school vs non-school ... Auxiliary information can be used: regression approach: additive effect multiplicative effect random effects approach: ωR where ω ∼ Γ(θ, 1/θ) empirical Bayes estimates for individual effects Hens et al. InFER - Warwick 2011 26 Prologues Estimating the Reproduction Number Epilogue References Data Objectives Known Number of Initial Cases Unknown Number of Initial Cases Known and Unknown Number of Initial Cases Handling Reporting Issues Applied to the Data Illustration: regression approach & marginal truncated Borel-Tanner Provinces (significant effect due to Antwerp): Antwerp: R̂ = 0.84 East-Flanders: R̂ = 0.43 Flemish-Brabant: R̂ = 0.45 Limburg: R̂ = 0.56 West-Flanders: R̂ = 0.57 School vs Non-school: School: R̂ = 0.47 Non-school: R̂ = 0.72 Hens et al. InFER - Warwick 2011 27 Prologues Estimating the Reproduction Number Epilogue References Epilogue Results were used as input parameters for the mathematical model We used WAIFW and contact data Mathematical model with demography, including migration → confirms no transmission only migration Cost-effectiveness analysis → vaccination of second generation children Difficult to ‘sell’ that story Hens et al. InFER - Warwick 2011 28 Prologues Estimating the Reproduction Number Epilogue References Acknowledgements Co-authors: Christel Faes, Joke Bilcke, Steven Abrams, Stijn Jaspers, Yannick Vandendijck, Marc Aerts and Philippe Beutels Prologue I: Olivier Lejeune, Mathieu Andraud, Zebedee Musoro Epilogue: Chellafe Ensoy This work has been funded by “SIMID”, a strategic basic research project funded by the institute for the Promotion of Innovation by Science and Technology in Flanders (IWT), project number 060081. Hens et al. InFER - Warwick 2011 29 Prologues Estimating the Reproduction Number Epilogue References References A. E. Ades and D. J. Nokes. Modeling age- and time-specific incidence from seroprevalence:toxoplasmosis. Am J Epidemiol, 137(9): 1022–1034, May 1993. V. Batter, B. Matela, M. Nsuami, M. T., M. Kamenga, F. Behets, R. Ryder, W. Heyward, J. Karon, and M. St Louis. 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