Advanced BAyesin Disase Mapping

Space-Time modeling IIInfectious diseasesBmtry 763Andrew B Lawson 20141Infectious Disease Modeling Andrew B Lawson 2014Infection: Mechanistic Models South Carolina County levelInfluenza C+ notificationsFlu season: 6 month period each yearBiweekly reports of counts Under-ascertained

ODC Files:ST_flumodel2_working.odc

Andrew B Lawson 20143Mechanistic ModelsFMD in Cumbria UK 2001Infected premises (IPs) are to be modeledWithin parishes we have counts of IPs and we also have a record of the total number of premises which changes over time.138 parishes and 13 half-monthly time periods (February 2001-August 2001)ODC file: FMD_model2a_UH_CARAndrew B Lawson 20144Basic SIR modelSusceptible population (S)Infective (I)Removed (R)

Also SEIR includes Exposed group alsoAndrew B Lawson 20145How can counts be modelled?Observe new incident infections (Infectives)We know previous infective numbersWe know the population (susceptibles)Removal?Can assume a given rate Andrew B Lawson 20146 Flu models: Model IModel assumes that the observed count is a proportion of the true infectivesThe true infectives depend on the previous true infective countAndrew B Lawson 2014

7Flu models: Model IIAccounting model Andrew B Lawson 2014

8Simpler version: Model 2

Andrew B Lawson 20149Model 2

How to parameterize the dependence on the previous infectives?Andrew B Lawson 2014

10Dependencies

Andrew B Lawson 201411Flu Model in WinBUGSAndrew B Lawson 2014ST_flumodel2_working

12Data: 4 counties Andrew B Lawson 2014

13Model IAndrew B Lawson 2014

14Model IIAndrew B Lawson 2014

15Results

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17Example StatisticsAndrew B Lawson 2014

18Andrew B Lawson 2014Summary

19Spatial heterogeneity

Andrew B Lawson 201420Results

Andrew B Lawson 201421Model CriticismPoisson (exact) assumptionCould use binomial Imputation of true count would be time-consumingBinomial model more realisticC+ is a subset of total flu?What to do about that ? Lawson and Song (2010) SSTE

Andrew B Lawson 201422FMD modelingSimilar to Flu example BUT with some differences alsoWe know the removal due to culling We have the (almost) fully ascertained infectives (IPs)Finite population of premises which varies in time.Also want to estimate terminationAndrew B Lawson 201423FMD Models Andrew B Lawson 2014

24FMD Model DependenciesWe will model the rate of infection:Model I: dependence on previous IPsModel II: dependence on IPs and countsModel III/IV: dependence on lagged neighbors

Note: the SIR accounting equation is fixed in this case:

Andrew B Lawson 2014

25Poisson SIRAndrew B Lawson 2014

26FMD data and Model I fittingAndrew B Lawson 2014

1410

27Parish 8000 & 8333Andrew B Lawson 2014

28Results for all FMD Models Andrew B Lawson 2014Modelrandom effectsDICDeviancepDModel 1UH+CAR1660.211580.5879.63UH only1665.711582.9982.72CAR only1665.511589.3776.14No RE1917.581914.52 3.059Model 2UH+CAR1625.531540.8084.74UH only1625.571537.7287.85CAR only1632.031550.2081.82No RE1910.791906.754.04Model 3UH+CAR1658.211577.2880.93UH only1662.751578.9783.78CAR only1664.601587.3977.21No RE1915.331911.274.061Model 4UH+CAR1625.471539.8085.67UH only1627.961539.5488.42CAR only1634.641552.1882.47No RE1906.701901.65 5.0529Model 2 resultsAndrew B Lawson 2014Parameter nameEstimated mean95% credible intervalLength of the credible intervalIntercept (time1)-4.26(-4.69,-3.87)0.82Intercept(time>1)-4.96(-5.86, -4.13)1.73coefficient log(case)0.19(0.16,0.21)0.05Coefficient log(pop)0.41(0.17,0.66)0.49Precisiontauv: 0.78tauW: 0.64(0.35,2.089)(0.18,1.78)1.741.6030Model running FMD_poisson_model2a_UH_CAR.odc UH and CAR componentsAndrew B Lawson 2014

31Model II: posterior Mean estimates Andrew B Lawson 2014

141032Detecting termination Can assess termination using monotone means ie continually descending mean estimates flag terminationAndrew B Lawson 2014

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34ReferencesLawson, A. B. and Song, H-R. (2010) Bayesian Hierarchical Modeling of the dynamics of Spatio-temporal influenza season outbreaks. Spatial and Spatio-temporal Epidemiology, 1, 2, 187-195Lawson, A. B., Onicescu, G., Ellerbe, C. (2011) Foot and mouth disease revisited: Re-analysis using Bayesian spatial susceptible-infectious-removed models, Spatial and Spatio-temporal Epidemiology, 2, 3, 185-194

Andrew B Lawson 2014