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

1

Introduction

2

Modelling approach Univariate Multivariate

3

Examples Measles in Lower Saxony, Germany Influenza in USA

4

Summary and Outlook

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

• f 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. Susceptible-Infectious-Recovered 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¨

• hle, 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

yt ∼ Po(µt) µt = νt + λyt−1 log(νt) = α +

S

• s=1
• γs sin(ωst) + δs cos(ωst)
• and ωs = 2π

p s with period p

Endemic component: log(νt) 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(µt) by NegBin(µt, ψ)-Likelihood Y ∼ NegBin(µ, ψ) : E(Y ) = µ Var(Y ) = µ + µ2

ψ

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Inference

Maximum likelihood estimators obtained by numerical optimisation

• f 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 S ˆ λ (se) ˆ ψ (se) log L(y, θ) |θ| AIC 0.99 (0.01)

• 4050.9

2 8105.9 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 S ˆ λ (se) ˆ ψ (se) log L(y, θ) |θ| AIC 0.50 (0.04)

• 919.2

2 1842.4 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 yi,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 yi,t ∼ NegBin(µi,t, ψ) µi,t = νt + λyi,t−1 + φyj,t−i where j = i Note: ψ, νt, 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 λflu 0.74 (0.05) 0.74 (0.05) 0.74 (0.05) λmen 0.16 (0.06) 0.10 (0.06) 0.10 (0.06) φflu

• 0.00 (0.00)
• φmen
• 0.01 (0.00)

0.01 (0.00) ψflu 3.39 (0.41) 3.40 (0.41) 3.39 (0.41) ψmen 20.34 (4.83) 25.32 (6.98) 25.32 (6.98)

log L(y, θ)

• 1889.7
• 1881.0
• 1881.0

|θ|

14 16 15

AIC

3807.5 3793.9 3791.9 Sflu = 3 and Smen = 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: µi,t = νt + λyi,t−1 + φ

• j=i

wjiyj,t−1 Note: νt, λ and φ may also depend on i

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Choice of weights wji

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

50 100 150 20 40 60 two−week Number of cases

WST AUR CLP DEL EMD EL FRI NOH LER OLL OLS OSL OSS VEC BRA WTM Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Results

S ˆ λ (se) φ ˆ 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

Yearly incidence ˆ φ, wji =

1 |k∼j|1(j ∼ i)

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Fitted values

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Example: Influenza in USA, 1996 − 2006

Data on weekly mortality from pneumonia and influenza in 9 geographical regions obtained from the CDC 121 Cities Mortality Reporting system Data on yearly number of passengers travelling by air obtained from TranStats database, U.S. Department of Transportation

Brownstein, J.S., Wolfe, C.J. and Mandl, K.D. (2006). Empirical Evidence for the effect of airline travel on inter-regional influenza spread in the United States, PLOS Medicine 3(10): 1826–1835 Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Data

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Introduction Modelling approach Examples Summary and Outlook

Air travel data, 1996 − 2006

Average yearly number of passengers per 100 000

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Parameter Estimates

weights ˆ λ (se) ˆ ψ (se) log L(y, θ) |θ| AIC

• 0.34 (0.01)

31.60 (0.92)

• 19827.8

19 39693.6 geographical 0.30 (0.01) 32.36 (0.95)

• 19787.8

28 39631.6 pji (average) 0.23 (0.02) 32.39 (0.95)

• 19784.8

28 39625.7 pji (yearly) 0.28 (0.01) 32.71 (0.96)

• 19768.7

28 39593.5

S = 4 pji relative proportion of persons travelling from region j to region i

Michaela Paul University of Zurich

Introduction Modelling approach Examples Summary and Outlook

Summary and Outlook

Choice of weights wji Identification problems Validation through out-of sample predictions Modelling of seasonal variation

Michaela Paul University of Zurich