Multivariate modelling of time series of infectious disease counts - PowerPoint PPT Presentation

Multivariate modelling of time series of infectious disease counts Michaela Paul Leonhard Held Biostatistics Unit Institute of Social and Preventive Medicine University of Zurich Reisensburg, September 28, 2007

Introduction Modelling approach Examples Summary and Outlook Outline Introduction 1 Modelling approach 2 Univariate Multivariate Examples 3 Measles in Lower Saxony, Germany Influenza in USA Summary and Outlook 4 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Introduction Aim Development of a realistic model for the statistical analysis of surveillance data of infectious disease counts Features of surveillance data: Low number of disease cases Underreporting and reporting delays Seasonality Presence of past outbreaks Often no information about number of susceptibles Dependencies between time series Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Example: Influenza and meningococcal disease Several studies describe an association between influenza and meningococcal disease (Hubert et al., 1992; Jensen et al., 2004) Analysis of routinely collected surveillance data from Germany Weekly number of laboratory confirmed influenza cases and meningococcal disease cases obtained from the Robert Koch Institute ( http://www3.rki.de/SurvStat ) Hubert, B., Waitier, L., Garnerin, P. and Richardson, S. (1992). Meningococcal disease and influenza-like syndrome: a new approach to an old question, Journal of infectious diseases 166 : 542–545 Jensen, E., Lundbye-Christensen, S., Samuelson, S., Sørensen, H. and Schønheyder, H. (2004). A 20-year ecological study of the temporal association between influenza and meningococcal disease, European Journal of Epidemiology 19 : 181–187 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Influenza in Germany, 2001 − 2006 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Meningococcal disease in Germany, 2001 − 2006 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Models for infectious diseases Mechanistic models Directly model the infection process of the spread from person to person on an individual level e.g. S usceptible- I nfectious- R ecovered model ⇒ require to observe the complete infection process (exact infection time and duration, number of susceptibles) Empirical models Describe and predict the disease based on observed data e.g. log-linear Poisson model, GLMM Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Approach of Held et al. (2005) Idea Decomposition of incidence into an epidemic and an endemic component Modelling based on a generalised branching process with immigration Note: Branching process is an approximation of SIR-models in the absence of information on susceptibles ⇒ Compromise between mechanistic and empirical modelling Held, L., H¨ ohle, M. and Hofmann, M. (2005). A Statistical framework for the analysis of multivariate infectious disease surveillance counts, Statistical Modelling 5 : 187–199 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Model y t ∼ Po( µ t ) µ t = ν t + λ y t − 1 S � � � log( ν t ) = α + γ s sin( ω s t ) + δ s cos( ω s t ) s =1 and ω s = 2 π p s with period p Endemic component: log( ν t ) includes terms for seasonality, modelled parametrically as in log-linear Poisson regression Epidemic component: past counts act additively on disease incidence Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Overdispersion Underreporting Unobserved covariates that affect disease incidence . . . ⇒ overdispersed data Adjustment Replace Po( µ t ) by NegBin( µ t , ψ )-Likelihood Y ∼ NegBin( µ, ψ ) : E( Y ) = µ Var( Y ) = µ + µ 2 ψ Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Inference Maximum likelihood estimators obtained by numerical optimisation of log-likelihood Quasi-Newton method BFGS Autoregressive parameters λ , φ and dispersion parameter ψ are optimised on log-scale Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Example: Influenza infections Parameter estimates ˆ ˆ λ (se) ψ (se) log L ( y , θ ) | θ | AIC S 0 0.99 (0.01) - -4050.9 2 8105.9 0 0.98 (0.05) 2.41 (0.27) -1080.2 3 2166.5 1 0.86 (0.05) 2.74 (0.31) -1064.1 5 2138.2 2 0.76 (0.05) 3.12 (0.37) -1053.3 7 2120.6 3 0.74 (0.05) 3.39 (0.41) -1044.1 9 2106.3 4 0.74 (0.05) 3.44 (0.42) -1042.2 11 2106.3 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Fitted values and residuals Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Example: Meningococcal disease Parameter estimates ˆ ˆ S λ (se) ψ (se) log L ( y , θ ) | θ | AIC 0 0.50 (0.04) - -919.2 2 1842.4 0 0.48 (0.05) 11.80 (2.09) -880.5 3 1767.0 1 0.16 (0.06) 20.34 (4.83) -845.6 5 1701.2 2 0.16 (0.06) 20.41 (4.86) -845.5 7 1705.0 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Fitted values and residuals Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Multivariate modelling Suppose now multiple time series i = 1 , . . . , m are available y i , t : # cases from the i -th time series at time t = 1 , . . . , T Examples: Incidence of related diseases Incidence in different geographical regions Incidence in different age groups Idea Include also the number of cases from other time series as autoregressive covariates → Multi-type branching process Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Bivariate analysis Joint analysis of two related time series y i , t ∼ NegBin( µ i , t , ψ ) µ i , t = ν t + λ y i , t − 1 + φ y j , t − i where j � = i Note: ψ , ν t , λ and φ may also depend on i Example: Influenza and meningococcal disease ”Outbreaks” of meningococcal disease regularly occur at the end of influenza outbreaks → Include preceding influenza cases as covariate for meningococcal disease Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Parameter estimates univariate flu and men men λ flu 0.74 (0.05) 0.74 (0.05) 0.74 (0.05) λ men 0.16 (0.06) 0.10 (0.06) 0.10 (0.06) - 0.00 (0.00) - φ flu - 0.01 (0.00) 0.01 (0.00) φ men 3.39 (0.41) 3.40 (0.41) 3.39 (0.41) ψ flu 20.34 (4.83) 25.32 (6.98) 25.32 (6.98) ψ men -1889.7 -1881.0 -1881.0 log L ( y , θ ) 14 16 15 | θ | 3807.5 3793.9 3791.9 AIC S flu = 3 and S men = 1 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Fitted values for meningococcal disease Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Spatio-temporal models Suppose now data on the same pathogen are available for several geographical locations i = 1 , . . . , m Possible model extension: � µ i , t = ν t + λ y i , t − 1 + φ w ji y j , t − 1 j � = i Note: ν t , λ and φ may also depend on i Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Choice of weights w ji Geographical weights: 1 ( j ∼ i ) sum of counts in adjacent regions 1 | k ∼ j | 1 ( j ∼ i ) counts in adjacent regions weighted by the number of neighbours of region j Alternative: Include travel information (if available) SARS epidemic (Hufnagel et al., 2004) Influenza epidemics (Colizza et al., 2006) Hufnagel, L., Brockmann, D. and Geisel, T. (2004). Forecast and control of epidemic in a globalized world, Proceedings of the National Academy of Sciences 101 (42): 15124–15129 Collizza, V., Barrat, A., Barth´ elemy, M. and Vespignani, A. (2006). The role of the airline transportation network in the prediction and predictability of global epidemics, Proceedings of the National Academy of Sciences 19 : 181–187 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Measles in Lower Saxony, Germany, 2001 − 2006 Weekly number of measles cases in the administrative district ”Weser-Ems”, Lower Saxony, Germany, obtained from the Robert Koch Institute Latent period of measles is 6 − 9 days Infectious period is 6 − 7 days Analysis of biweekly counts in 17 areas Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Data WST OLL AUR OLS CLP OSL 60 DEL OSS EMD VEC EL BRA FRI WTM Number of cases NOH LER 40 20 0 0 50 100 150 two−week Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook Results ˆ ˆ S λ (se) φ ψ (se) log L ( y , θ ) | θ | AIC 1 0.73 (0.10) no 0.34 (0.05) -961.8 21 1965.7 1 0.49 (0.07) yes 0.51 (0.07) -897.6 38 1871.3 ˆ 1 Yearly incidence φ , w ji = | k ∼ j | 1 ( j ∼ i ) Michaela Paul University of Zurich