Statistical challenges in Disease Ecology

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1 Statistical challenges in Disease Ecology Jennifer Hoeting Department of Statistics Colorado State University February 2018

2 Statistics rocks!

3 Get thee to graduate school Colorado State University, Department of Statistics offers multiple options: Master of Applied Statistics (Distance and on Campus) including a new program in Data Science Master of Science PhD

4 How did I get here? Background: BS in Statistics and Psychology MS and PhD in Statistics University of Michigan University of Washington Research Develop statistical methods to solve problems in ecology Diverse areas of application: Animal abundance (prairie dogs to whales) Saving the remaining whopping cranes Stream and wetland ecology Understanding climate change Google Street View cars to detect natural gas leaks Estimating the of people who have died in the war in Syria

5 Ecology of infectious diseases Explores the relationships between 1. Diseases: Parasitic, bacterial, viral infectious 2. Hosts: animal and human 3. Their environment

6 Challenges in the ecology of infectious diseases Data can be messy and sparse You have to find the animals who are sick, they don t visit the nearest health clinic You need knowledge of mathematical biology You need to able to synthesize a broad range of statistical methods Sounds like fun!

7 Ecology Modeling Ecological models can be deterministic or stochastic Bolker, Ecological Models and Data in R, 2008 Classification of modeling approaches Mathematical Statistical Deterministic Stochastic Mechanistic Phenomenological Process Pattern

8 Mathematical Model for infectious diseases λ S I R β N I γ μ μ μ Susceptible-Infected-Recovered (SIR) model for disease transmission where the state variables are described by a set of differential equations.

9 Statistical Model for infectious diseases Logistic regression model Y i = 1 if deer i is disease positive, 0 otherwise Assume Y i Bernoulli(π i ) logit (π i )=β 0 + β 1 X 1i + β 2 X 2i where for animal i X 1i = animal age X 2i = level of exposure to disease

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11 April 2016: The first detection of CWD in Europe

12 Chronic wasting disease: mule deer in Colorado A wide variety of statistical methods and advances based on research on chronic wasting disease 1. Estimating parameters for stochastic differential equations 2. Selecting between different dynamical models 3. Spatial statistics and issues of scale 4. Models for animal movement and genetics 5. Functional data analysis

13 CWD Transmission Deer were held at the Colorado Division of Wildlife in Fort Collins, Colorado Annual observations of cumulative mortality from two CWD epidemics in captive mule deer Epidemic 1: 1974 to 1985 Epidemic 2: 1992 to 2001 (in a new deer herd) 21 observations over time The dataset also includes the annual number of new deer added to the herd and the per capita losses due to natural deaths and removals

14 CWD Transmission We develop a type of Susceptible-Infected-Recovered (SIR) model for disease transmission where the state variables are described by a set of differential equations. Consider the state vector X(t) =(S(t), I(t), C(t)) T, where S is the number of susceptible animals, I is the number of infected animals, C is the cumulative number of deaths from CWD over time. Only C(t) and N(t) =S(t) +I(t) +C(t) are observed for t = 1,...,21. The other two state variables, S(t) and I(t), are unobserved.

15 ODE model for CWD where ds =[a S(βI + m)] dt di =[βsi I(μ + m)] dt dc =μidt β is the CWD transmission coefficient μ is the per capita CWD mortality rate } unknown a is the number of susceptible animals annually added to the population via births or importation known m is the per capita natural mortality rate We assume X(0) =(S(0), I(0), C(0)) T are (un)known initial conditions.

16 Ordinary Differential Equation Model for CWD Cumulative deaths from CWD Deterministic dynamical models can be used to determine whether or not transmission will occur. Year

17 Stochastic Differential Equation Model for CWD Cumulative deaths from CWD Stochastic dynamical models Can be used to determine the probability of disease transmission between two individuals Allow more realistic description of the transmission of disease. Year

18 SDE Model for CWD A SDE model for direct transmission of CWD is given by where ds =[a S(βI + m)]dt + B 11 dw 1 + B 12 dw 2 + B 13 dw 3, di =[βsi I(μ + m)]dt + B 21 dw 1 + B 22 dw 2 + B 23 dw 3, dc =μidt + B 31 dw 1 + B 32 dw 2 + B 33 dw 3, initial condition X(0) =(S(0), I(0), C(0)) T assumed known W is a k-dimensional standard Wiener process. B =(B ij )= Σ with Σ= a + S(βI + m) βsi 0 βsi βsi + I(μ + m) μi. 0 μi μi

19 Method 1: Parameter estimation for SDEs Goal: Statistical inference for SDE model parameters Challenges: Multivariate state space Data are partially observed, discrete, sparse. The transition density between two observations is typically unknown. Likelihood functions involving SDEs are computationally expensive

20 Method 1: Parameter estimation for SDEs Sun, Lee, Hoeting 2015 Computational Statistics and Data Analysis Improved importance sampling: Automatic tuning of importance sampler via an auxiliary parameter Parameter estimation via penalized simulated maximum likelihood where we simultaneously estimate dynamical model parameters θ and optimize the importance sampler using parameter ρ to minimize the variation of the Monte Carlo approximation of the transition density.

21 Method 1: Penalized simulated maximum likelihood (PSML) The PSML estimator (ˆθ, ˆρ) is defined by n (ˆθ, ˆρ) =arg max log 1 J h ρ subject to J i=1 j=1 n ĉv (h ρ ) s, i=1 where s 0 is a tuning parameter h ρ is the importance sampling ratio where ρ is an auxiliary parameter used to tune the importance sampling function. ĉv(h ρ ) is the sample coefficient of variation of h ρ

22 Estimated parameters for CWD 90% confidence intervals: Rate of direct transmission: ˆβ =[0.03, 0.10] Per capita CWD mortality rate: ˆμ =[0.16, 0.34]

23 Basic reproductive number R 0 The basic reproduction number R 0, is the average number of secondary infections that occur when one infected individual is introduced into a completely susceptible population. For deterministic models, if R0 > 1 = the infection persists in the population if R 0 1 = the infection will eventually die out. For stochastic models, SDE models predict the infection will die out eventually, regardless of R 0 value If R0 > 1, the infection may die out quickly, with probability ( 1 ) I(0) R 0 where I(0) is the initial number of infected animals, the infection persists with probability 1 (R 0 ) I(0)

24 Basic reproductive number R 0 For example, assuming a natural mortality rate of m = 0.15 (Miller et al., 2006) and closed susceptible population, the corresponding estimate for the basic reproductive number R 0 equals βnˆ 0 ˆμ + m 0.09N 0 with 90% confidence interval [0.07N 0 ; 0.32N 0 ], where N 0 is the initial population or susceptible size. Hence, we would expect that CWD will spread if a few infected animals are introduced to a closed susceptible population with of at least 1/ animals.

25 CWD real data example Simulated sample paths of the SDE CWD direct transmission model with estimated parameters. CWD Death Estimates via penalized importance sampling successfully capture the pattern of the CWD death data over time, especially for such a small sample size. Year

26 Method 2: Model selection for dynamical models Hierarchical models Stage 1: Observation model Stage 2: Process model (dynamical model) Stage 3: Parameter model Goal: Choose between 3 dynamical models used at stage 2 1. Ordinary differential equation (ODE) model 2. Stochastic differential equation (SDE) model 3. Continuous time Markov chain (CTMC) model Libo, Sun, Hoeting (Envirometrics, 2015) use an Approximate Bayesian Computation method (ABC-SMC)

27 Method 2: Model selection for dynamical models Approximate Bayesian Computation (ABC): Method to estimate the model parameters when the likelihood is difficult to compute Basic idea: Simulate data from the model and compute a distance function between simulated data and the observed data ABC-SMC (sequential Monte Carlo): Improve ABC by simulating data through a sequence of intermediate distributions Model selection: Compare models using Bayes factors (Kass & Raftery 1995)

28 Method 3: Spatial statistical models Linking Chronic Wasting Disease to Mule Deer Movement Scales Goal: identify scales of mule deer movement and mixing that exerted the greatest influence on the spatial pattern of CWD in northcentral Colorado. We hypothesized that three scales of mixing might control spatial variation in disease prevalence individual winter subpopulation or summer subpopulation

29 Method 3: Spatial statistical models Background: Chronic Wasting Disease (CWD): a deadly disease in deer and elk in Colorado (and elsewhere) Of keen interest because Data: CWD is a prion disease (like Mad Cow) CWD could wipe out deer population ($) Transmission to humans or other wildlife populations response (y): 3855 mule deer tested for presence of CWD from in Colorado predictors (x): age, sex, % private land, and two other GIS covariates Data are analyzed on a grid cell level because precise locations are unknown and to investigate issues related to scale

30 Method 3: Modeling Chronic Wasting Disease Study Site Colorado Kilometers

31 Method 3: Spatial statistical models Analysis: Bayesian hierarchical model Let y ij be the CWD status for the jth deer in the ith grid cell y ij π ij Bernoulli(π ij ) iid logit(π ij )=μ + xij T β + γ i + δ i where μ = overall mean (intercept) x ij = vector of covariates for the jth deer in the ith grid cell β = regression parameters γ i = spatial random effect for ith grid cell δ i = independent random effect for ith grid cell Next level of hierarchy: priors on parameters

32 Method 3: Spatial statistical models Output Parameter estimates Maps Model comparisons: which scale is appropriate? Predictions for unobserved locations For more details see: M. L. Farnsworth, J. A. Hoeting, N. T. Hobbs, M. W. Miller (2006) Linking mule deer movement scales to the spatial distribution of chronic wasting disease: a hierarchical Bayesian approach, Ecological Applications, 16(3),

33 Method 3: Spatial statistical models Model selection results to identify the candidate models best explaining observed spatial patterns of chronic wasting disease (CWD) prevalence in mule deer in north-central Colorado, USA. Model Scale km2 Model pd DIC w DIC 1 9 Demo + Space Demo + Env + Space + Het Demo + Env + Space Demo + Space + Het Demo + Env + Space Demo + Env + Het Demo + Env + Het Demo + Env + Space + Het Demo + Env Demo + Space + Het

34 Method 3: Spatial statistical models Univariate parameter estimates from Model 1 and Model 2. Estimates for individual-level covariates SEX and AGE are from Model 1, the top DIC model, which did not contain environmental covariates, with environmental covariate effects from Model 2, the best model containing these effects Variable Model rank Mean Std.dev. 2.5% C.I. 97.5% C.I. SEX AGE %PRIV %HAB DISP

35 Method 3: Spatial statistical models CWD Spatial Random Effects

36 Method 3: Spatial statistical models We developed a Bayesian spatial hierarchical model for CWD presence/absence. Findings: Strong evidence that the finest mixing scale corresponded best to the spatial distribution of CWD infection. Evidence that land ownership and habitat use play a role in exacerbating the disease, along with the known effects of sex and age.

37 Summary Studying wildlife diseases can lead to the development of new statistical methods. The study of diseases in wildlife can lead to new scientific discoveries. Webpage: Jennifer Hoeting jah