Combining Mathematical And Statistical Models: A Disease .
Combining Mathematical and Statistical Models:a Disease Ecology PerspectiveJennifer HoetingDepartment of StatisticsColorado State Universitywww.stat.colostate.edu/ jahInternational Statistical Ecology ConferenceJuly 2018
Ecology of Infectious DiseasesExplores the relationship between123PathogenHost: animal and humanEnvironmentOverviewJ. Hoeting
Ecology of Infectious DiseasesTasmanian Devil Facial Tumor DiseaseOverviewJ. Hoeting
Ecology of Infectious DiseasesZika virusOverviewJ. Hoeting
Overview: MotivationFocus of talkChallenges of statistical parameter estimation and inferencefor dynamical models of infectious diseaseWhy are infectious diseases challenging to model?1Disease transmission is not observable2Much uncertainty about what is observed3Highly nonlinear dynamicsOverviewJ. Hoeting
Overview: MotivationWhy are infectious diseases challenging to model?You need to synthesize a broad range of ideas frombiology,mathematics,andstatisticsOverviewJ. Hoeting
Overview: Process versus patternClassification of modeling approachesMathematical versus StatisticalDeterministic versus StochasticTheoretical versus PhenomenologicalProcess versus PatternTraditional approach:choose between a mathematical or a statistical modelAlternative approach:combine the advantages of mathematical and statisticalmodelsOverviewJ. Hoeting
Overview: Process versus patternMathematical model (process)Model structure reflects explicit hypotheses about thebiological mechanisms that drive infection dynamicsExample: Susceptible-Infectious-Recovered (SIR) ModeldS βSIdtdI βSI γIdtdR γIdtAdvantages:Explicitly model nonlinearities in the processUseful for predictionDisadvantages:Represent the average behaviorOften focus is on model form,not parameter estimation for observed dataOverviewJ. Hoeting
Overview: Process versus patternStatistical model (phenomenological)Model describes the observed relationship betweenvariablesExample: Linear regression modelY β0 β1 X1 β2 X2 . . . βp Xp Advantages:Data can inform the modelRich characterization of different types of errorsDisadvantages:May only describe the observed dataGives little information about the mechanismInteractions not always well capturedOverviewJ. Hoeting
Overview: historyWhy not combine the advantages ofmathematical biology and statistics?Long history of combining mathematical and statisticalmodels (e.g., Berliner 1991)Many sessions at this conference include these ideasReferences and this talk:See the end of the talk for complete list of referencesThese slides are available online:www.stat.colostate.edu/ jahOverviewJ. Hoeting
Motivating example: CWD TransmissionChronic Wasting DiseaseDeer (female) withChronic Wasting DiseaseHealthy deer (male)OverviewJ. Hoeting
Motivating example: CWD TransmissionChronic wasting disease (CWD)100% fatal contagious disease that affects cervids (deer family)CWD is a prion diseaseImportant to understand the transmission mechanisms of CWDSeveral deterministic epidemic models were proposed byMiller, Hobbs & Tavener (2006)OverviewJ. Hoeting
Motivating example: CWD TransmissionMathematical model for disease transmissionSusceptible-Infectious-Recovered (SIR) viewDeadDeer# animalsSIR100 0090 10 080 10 1070 10 20N 100J. Hoeting
Motivating example: CWD TransmissionWe develop a type of Susceptible-Infectious-Recovered (SIR) modelfor disease transmission where the state variables are described by aset of differential equations.Consider the state vector X(t) (S(t), I(t), R(t))T , whereS is the number of susceptible animalsI is the number of infectious animalR is the number of deaths from CWDOverviewJ. Hoeting
Motivating example: : CWD TransmissionDirect transmission ODE model for CWDdS a S(βI m)dtdI βSI I(µ m)dtdR µIdtwhere β is the transmission rateunknownµ is the per capita CWD mortality rate a is the number of susceptible animals annually added to the population via births or importation knownm is the per capita natural mortality rateOverviewJ. Hoeting
8060SusceptibleInfectedDead02040# of animals100120Motivating example: CWD Transmission05101520TimeOverviewJ. Hoeting
Motivating example: CWD TransmissionThere are always challenges with observed data:Complete data# of animalsTime SI Rt1100 0 0t290 10 0t380 10 10t470 10 20N 100OverviewObserved data# of animalsTime S I Rt10t20t310t420N 100J. Hoeting
120Motivating example: CWD Transmission 80 SusceptibleInfectedDead60 40# of animals100 020 0 5101520TimeOverviewJ. Hoeting
ChallengesWe explore challenges in1Observed data2Model development:II3Which class of dynamical model? (Mathematics)Which mechanisms to include? (Biology)Statistical inference:IIIChallengesWhich statistical model/paradigm?Which computational method?How to select models?J. Hoeting
Challenge 1: Data challengesChallenges in the observed data:1Missing dataII2Sparse dataI3Uncertainties about the testing procedure(false negatives and/or false positives)Initial conditions unknownIChallengesInterval between observations can be long and irregularUncertain dataI4You have to find the animals, they don’t visit thenearest health clinicSome states are unobservedUsually don’t observe the population beforethe disease outbreakJ. Hoeting
Challenge 2: Model developmentChallenges in model development:1Which class of dynamical model? (Mathematics)2Which mechanisms to include? (Biology)ChallengesJ. Hoeting
Challenge 2: Model developmentWhich class of dynamical model?What is a dynamical model?A dynamical model describes a system that evolves in time.A dynamical model includes:A description of the state(s) of the systemA time indexA rule by which the state(s) evolves forward in timeSome classifications of dynamical modelscontinuous or discretestochastic or deterministicChallengesJ. Hoeting
Challenge 2: Model developmentWhich class of dynamical model?Ordinary Differential Equation (ODE) model:ODE models can be used to determine whether or not diseasetransmission will occur.Dynamical model classifications: deterministic, continuoustime, continuous state space modelExample:dS a S(βI m)dtdI βSI I(µ m)dtdR µIdtChallengesJ. Hoeting
Challenge 2: Model developmentWhich class of dynamical model?Stochastic Differential Equation (SDE) model:SDE models be used to determine the probability of diseasetransmission between two individualsNatural extension of ODE modelsDynamical model classifications: stochastic, continuous time,continuous state space modelChallengesJ. Hoeting
Challenge 2: Model developmentWhich class dynamical model?A SDE model for direct transmission of CWD is given bydS [a S(βI m)]dt B11 dW1 B12 dW2 B13 dW3 ,dI [βSI I(µ m)]dt B21 dW1 B22 dW2 B23 dW3 ,dC µIdt B31 dW1 B32 dW2 B33 dW3 ,whereW is a k-dimensional standard Wiener process. B (Bij ) Σ with a S(βI m) βSI0 βSIβSI I(µ m) µI .Σ 0 µIµIChallengesJ. Hoeting
Challenge 2: Model developmentWhich class of dynamical model?Continuous time Markov chain (CTMC) model:CTMC models be used to determine the probability of diseasetransmission between two individualsMay be more complicated to derive than SDE modelsDynamical model classifications: stochastic, continuous time,discrete state space modelChallengesJ. Hoeting
Challenge 2: Model developmentWhich class of dynamical model?A CTMC model for direct transmission of CWD is given by S(t δ) i 1P I(t δ) jR(t δ) k S(t δ) i 1P I(t δ) jR(t δ) k S(t δ) i 1P I(t δ) j 1R(t δ) k S(t δ) iP I(t δ) j 1R(t δ) k S(t δ) iP I(t δ) j 1R(t δ) k 1 S(t) iI(t) j R(t) k S(t) iI(t) j R(t) k S(t) iI(t) j R(t) k S(t) iI(t) j R(t) k S(t) iI(t) j R(t) kaδ o(δ),imδ o(δ),βijδ o(δ),jmδ o(δ),jµδ o(δ),where o(δ) 0 as the time interval δ 0.Each probability statement in the CTMC model corresponds to acomponent of the ODE model.ChallengesJ. Hoeting
Challenge 2: Model developmentModel development:1Which class of dynamical model? (Mathematics)2Which mechanisms to include? (Biology)ChallengesJ. Hoeting
Challenge 2: Model developmentWhich mechanisms to include?ChallengesJ. Hoeting
Challenge 2: Model developmentWhich mechanisms to include?A two serotype model for dengue feverC. Leach, 2015ChallengesJ. Hoeting
Challenge 3: Statistical inferenceChallenges in statistical inference1Which statistical model/paradigm?2Which computational method for inference?3Which model do the data support?ChallengesJ. Hoeting
Challenge 3: Statistical inferenceStatistical model/paradigmThe statistical method you select for inference willdepend on theshortcomings of your datadynamical model you developedChallengesJ. Hoeting
Challenge 3: Statistical inferenceStatistical model/paradigmExample: For the Chronic Wasting Disease dataIt is reasonable to allow for errors in theobserved number of deathsWe use a hierarchical model with adynamical model at the process levelHierarchical model consists ofStage 1: Data modelStage 2: Process modelStage 3: Parameter modelChallengesJ. Hoeting
Challenge 3: Statistical inferenceComputational methods for inferenceSome possible computational methods to enable statistical inferencefor dynamical models1Bayesian approachesII2Maximum likelihood approachesII3Markov chain Monte Carlo (MCMC)Approximate Bayesian Computation (ABC)Iterated filteringPenalized simulated maximum likelihoodLeast squares approachesIIGradient matchingTrajectory matchingMany other options available. See talk references.ChallengesJ. Hoeting
Challenge 3: Statistical inferenceComputational methods for inferenceMost computational methods can’t be used ‘out of thebox’ for modeling infectious diseases due toThe sparse nature of the dataSmall changes in the parameters can lead to verydifferent dynamic behaviorFinding good starting values for the computationalstatistical algorithms can be particularly challenging.Latin hypercube sampling (Marino et al., 2008) canbe usefulChallengesJ. Hoeting
Challenge 3: Statistical inferenceModel selectionChoice of model selection methods will depend on the inferenceparadigm you choose.Some options:1BayesianIICompare models via their posterior model probabilities.For model Mk the posterior model probability is given by P(Mk D).Compare models using Bayes factors (Kass & Raftery 1995)2Maximum likelihood: AIC3Other options for both paradigms: see referencesJust to be precise: model selection isn’t statistical inferenceChallengesJ. Hoeting
Example: Chronic Wasting DiseasePutting all the pieces together:Parameter inference and model selection in deterministic andstochastic dynamical models via approximate Bayesian computation(Sun, Lee, Hoeting, 2015)Challenges in1Observed data2Model development: We’ll consider several disease modelsStatistical inference:3IIIExampleWhich model/paradigm? Bayesian hierarchical modelWhich computational method? ABCHow to select models? Posterior model probabilities and BayesfactorsJ. Hoeting
Motivating example: CWD TransmissionThe observed data:Annual observations of cumulative mortality from two CWDepidemics in captive mule deerNo live-animal test, vaccine, or treatment for CWD existed prior to2008.Epidemic 1: 1974 to 1985Epidemic 2: 1992 to 2001 (in a new deer herd)21 observations over timeThe dataset also includesIIExampleannual number of new deer added to the herdper capita losses due to natural deaths and removalsJ. Hoeting
Example: Chronic Wasting Disease6040200Cumulative # of CWD DeathsObserved data from two CWD earJ. Hoeting
Example: Chronic Wasting DiseaseFor our chronic wasting disease example:Stage 1: Data ModelAt time t let R̃(t) observed cumulative number of deaths fromCWD where R(t)R̃(t) Binomial N(t);N(t)whereN(t) S(t) I(t) R(t) is the total # of animals at time tOnly R̃(t) and N(t) observed at discrete time t t0 , t1 , . . . , tnExampleJ. Hoeting
Example: model set-upStage 2: Process Model: Direct transmission ODEdS a S(βI m)dtdI βSI I(µ m)dtdR µIdtStage 3: Parameter ModelPrior distributions for all model parametersInference: We can’t write the likelihood in closed form so wecertainly don’t have the posterior distribution in closed formExampleJ. Hoeting
Example: Model selectionBiologists and statisticians have proposed multiple reasonable models.Which model should we use?Goal 1: Choose a data modelWe consider Binomial or PoissonGoal 2: Choose the disease transmission model1Direct (basic SIR)2Indirect (environmental transmission)3Both direct and indirect disease transmissionGoal 3: Choose a class of dynamical model1Ordinary differential equation (ODE) model2Stochastic differential equation (SDE) model3Continuous time Markov chain (CTMC) modelExampleJ. Hoeting
Example: Results for CWDGoal: compare models for Chronic Wasting Disease.Posterior model probabilities for each model P(M D)DataCWDProcessInformative prior setModel Transmission ModelP(M D) Bayes factorBinom Direct/Indirect SDE0.211.00Binom DirectSDE0.181.15Binom DirectODE0.131.55Binom DirectCTMC0.111.87Binom Direct/Indirect ODE0.092.43PoisDirect/Indirect .87PoisDirect/Indirect ODE0.044.63PoisDirectCTMC0.036.17ExampleJ. Hoeting
Example: Parameter estimates442.01.00.0P(γ D)3.0The marginal posterior distribution for 2 of the parameters of theindirect transmission SDE model based on the CWD epidemic data.0.60.00.20.40.60.8420P(µ D)6γ0.60.80.00.20.40.60.8µExampleJ. Hoeting
Example: Parameter estimatesMarginal posterior modes and 95% HPD intervals of the parameters ofthe indirect transmission SDE process model with the Binomial datamodel based on the CWD epidemic data.Parameterγ Indirect transmission rate (mass 1 yr 1 )µ CWD mortality rate (yr 1 ) Per capita rate of excretion of infectious agent (yr 1 )τ Rate of loss of infectious agent (yr 1 )S(0) of the first epidemicI(0)E(0)S(0) of the second epidemicI(0)E(0)ExampleInformative prior setMode95% HPD0.05(0.01, 0.36)0.20(0.10, 0.59)0.47(0.15, 0.91)0.88(0.01, 4.52)18(10,26)10(5,18)1.73 (0.97,5.84)48(24,50)2(0,5)3.47 (0.24,4.85)J. Hoeting
50 40 30 20Cumulative deaths from CWD6070Example: Fitted SDE Model for CWD 10 0 510 1520YearExampleJ. Hoeting
Future workMuch work left to do in development of statistical methods and modelsfor the ecology of infectious disease:As data complexity and model complexity increase, the currentmethods often fail.More developments needed in:1Develop efficient computational algorithms for estimation2Inference for data from multiple sources and across multiplescalesDid this session pique your interest?Attend the session on Disease Ecology today at 17:00ConclusionsJ. Hoeting
AcknowledgmentsA special thank you toIIColleen Webb, Colorado State UniversityN. Thompson Hobbs, Colorado State Universityfor teaching me much about mathematical biology and ecologyThank you to my main collaborators on my work on diseasedynamicsIIIILibo Sun, Johnson and JohnsonChihoon Lee, Stevens Institute of TechnologyN. Thompson Hobbs, Colorado State UniversityMike Miller, Colorado Division of Parks and WildlifeThank you to CSU colleagues (faculty, students, post-docs)IIIIIMevin HootenClint LeachJoshua HewittErin GorsichFaustina France-NkansahAcknowledgmentsJ. Hoeting
Thank you to the ISEC2018 organizers!Scientific CommitteeRachel McCrea, University of Kent, UK (Chair)Res Altwegg, University of Cape Town, SARichard Barker, University of Otago, NZSimon Bonner, U. of Western Ontario, CANRachel Fewster, University of Auckland, NZMarc Kry, Swiss Ornithological Institute, SUIJose Lahoz-Monfort, U. of Melbourne, AUSBrett McClintock, NOAA, USAShirley Pledger, Victoria U of Wellington, NZEric Rexstad, University of St Andrews, UKDavid Warton, U. of New South Wales, AUSConference ambassadors:Rick CampCharles PaxtonCatriona HarrisLindesay Scott-HaywardLocal Organising CommitteeSteve Buckland, Univ. St Andrews (co-chair)Eric Rexstad, Univ. St Andrews (co-chair)David Borchers, Univ. St AndrewsGui Bortolotto, Univ. St AndrewsMark Brewer, BioSSLouise Burt, Univ. St AndrewsClaudia Faustino, Univ. St AndrewsDanielle Harris, Univ. St AndrewsJanine Illian, Univ. St AndrewsRuth King, Univ. EdinburghTiago Marques, Univ. St AndrewsMichail Papathomas, Univ. St AndrewsCharles Paxton, Univ. St AndrewsRhona Rodger, Univ. St AndrewsLindesay Scott-Hayward, Univ. St AndrewsLen Thomas, Univ. St AndrewsProjectionists:Louise Burt, Rick Camp, Fanny Empacher,Claudia Faustino, Chrissy Fell, Andy SeatonAcknowledgmentsJ. Hoeting
References ITalk references (web links in blue)Slide 9: Ray Hilborn & M. Mangel (2013)The Ecological Detective Confronting Models with DataPrinceton University PressSlide 10: Classic references by L. M. Berliner:“Statistics, Probability and Chaos,” Statistical Science (1992)“Likelihood and Bayesian Prediction of Chaotic Systems”JASA (1991).Slide 22:Stochastic Epidemic Models and Their Statistical Analysis (2010), H.Andersson & T. Britton, SpringerStatistical Science special issue on stochastic models for infectious diseasedynamics (2018) Vol. 33, No. 1, T. Kypraios and V. Minin, editors.J. Cresson and S Sonner (2018)A note on a derivation method for SDE models: Applications in biology andviability criteriaStochastic Analysis and ApplicationsAcknowledgmentsJ. Hoeting
References IISlide 23: Linda Allen (2008)An introduction to stochastic epidemic models,Chapter 3 in Mathematical epidemiology, p 81–130.Slide 34: ComputationD He, EL Ionides, AA King (2009)Plug-and-play inference for disease dynamics: measles in large and smallpopulations as a case studyJournal of the Royal Society InterfaceEL Ionides, C Bret, AA King (2006)Inference for nonlinear dynamical systemsProceedings of the National Academy of SciencesL. Sun, C. Lee, and J. A. Hoeting (2015)A penalized simulated maximum likelihood approach in parameter estimationfor stochastic differential equations Computational Statistics and DataAnalysis, 84: 54–67J. Ramsay and G. Hooker Dynamic Data AnalysisChapman & HallAcknowledgmentsJ. Hoeting
References IIIApproximate Bayesian Computation OverviewDavid Nott (2018)Talk and slides available at SIAM Uncertainty Quantification Conference 2018M. Fasiolo, N. Pya, S. Wood (2016)A comparison of inferential methods for highly nonlinear state space modelsin ecology and epidemiologyStatistical ScienceSlide 35: S. Marino, I. Hogue, C. Ray, D. Kirschner (2008)A methodology for performing global uncertainty and sensitivity analysis in systemsbiology, Journal of Theoretical BiologySlide 36: Model selectionJA Hoeting, D Madigan, AE Raftery, CT Volinsky (1999)Bayesian model averaging: a tutorialStatistical ScienceRE Kass, AE Raftery (1995)Bayes factorsJournal of the American Statistical AssociationAcknowledgmentsJ. Hoeting
References IVMB Hooten, NT Hobbs - Ecological Monographs (2015)A guide to Bayesian model selection for ecologistsEcological MonographsGJ Gibson, G Streftaris, D Thong (2018)Comparison and Assessment of Epidemic ModelsStatistical ScienceSlide 37:L. Sun, C. Lee, and J. A. Hoeting (2015)Parameter inference and model selection in deterministic and stochasticdynamical models via approximate Bayesian computation:modeling a wildlife epidemicEnvironmetrics, 26: 451–462.Two other perspectives on the same disease system:1AcknowledgmentsC Geremia, MW Miller, JA Hoeting, MF Antolin, NT HobbsBayesian Modeling of Prion Disease Dynamics in Mule Deer Using PopulationMonitoring and Capture-Recapture DataPloS One 10 (10)J. Hoeting
References V2L. Sun, C. Lee, and J. A. Hoeting (2015)A penalized simulated maximum likelihood approach in parameter estimation forstochastic differential equations Computational Statistics and Data Analysis, 84:54–67Other useful references:Recent Review (theory): K. McGoff, S. Mukherjee, N. Pillai (2015)“Statistical inference for dynamical systems: A review”Statistics Surveys, 9: 209–252.K. Newman (2018) Population Demography for Ecologyin Handbook of Environmental and Ecological StatisticsChapman & Hall, to appear.AcknowledgmentsJ. Hoeting
Photo acknowledgmentsWeb links in blue.Slide 3: Save the Tasmanian Devil Program, Tasmanian Government, Dept ofWildlife Management‘Nice’ Tasmanian Devil, FodorsSnarling Tasmanian Devil, Animal SoundsSlide 4: Zika virus, Hollywood GazetteSlide 29: Types of disease transmissionAcknowledgmentsJ. Hoeting
Example: ODE model for direct/indirecttransmission of CWDAn ODE model for the direct and indirect transmission of CWD (Milleret al. 2006) Sa S(γE m) I γSE I(µ m) dt,d E I τ ECµIwhereγ is the indirect transmission coefficient is the per capita rate of excretion of infectious material byinfectious animalsτ is the mass-specific rate of loss of infectious material from theenvironmentThe unknown quantities to be estimated are (γ, µ, , τ, S(t0 ), I(t0 ), E(t0 )).AcknowledgmentsJ. Hoeting
Example: Parameter estimates2.0P(γ D)00.01.042P(γ D)683.0The marginal posterior distribution for the parameters of the indirecttransmission SDE model based on the CWD epidemic data.Left column is prior set 1 and right column is prior set 20.00.20.40.60.80.00.20.40.60.8γ4P(µ D)232001P(µ .00.20.41230.81.045670.0 0.1 0.2 0.3P(τ D)00.6ε0.0 0.1 0.2 0.3 0.4εP(τ D)0.81.0P(ε D)2.0P(ε D)1.00.00.0Acknowledgments0.6µ2.0µ01234567J. Hoeting
Chronic wasting disease (CWD) 100% fatal contagious disease that affects cervids (deer family) CWD is a prion disease Important to understand the transmission mechanisms of CWD Several deterministic epidemic models were proposed
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