Dureau, Joseph

Kalogeropoulos, Konstantinos

English

LSE Research Online

Inference on epidemic models with time-varying parameters:methodology and preliminary applicationsJoseph Dureau and Konstantinos KalogeropoulosDepartment of Statistics, London School of Economics, UKEmail: j.dureau@lse.ac.uk1 Why have time-varying parameters?Inference on epidemic models is an active topic of research. Moti-vations are multiple: exploring mechanisms, testing theories, monitoringcontrol interventions, surveilling upcoming epidemics.A very typical compartmental model would be:S I R Suceptibles Infective Resistent Whith the following definitions:St : proportion of the population that is susceptible, that can be in-fectedIt : proportion of the population that is infected and infectiveRt : proportion of the population that is resistent, that is not infectiveany more and cannot be infected againβ : transmission rateγ : recovery rateUsually, inference is made for constant values of β and γ.However, there are many reasons for β to be time-varying:• Climate forcing is likely to have an impact on immunity and virustransmission• Contact patterns evolve according to holidays, school/work periods,seasonal migrations,...• Individual awareness to an epidemic can spontaneously decrease, orat the contrary increase under the influence of preventive measures• etc...2 How to model these time variations?Fully parametric models for time-varying transmission rates havebeen explored:• sinusoidal• low-dimensional polynoms, splines,...+ tractable inference with classic MCMC algorithms− limiting and arbitrary model choice”Semi-parametric” models, on the other hand, have been used:• random walk diffusion (Cazelles and Chau 1997, Mathematical Bio-sciences)+ very flexible model− inference implied gaussian approximations (Extended Kalman Filter)Our proposition• use a diffusion process for βt’s trajectory, typically a geometric Brow-nian motion to preserve positivity• apply novel MCMC algorithms to solve the inference problem, withlow-informative priors on the diffusion coefficientsClassic SIR model:dSt = −βStItdtdIt = (βStIt − γIt)dtdRt = γItdtBecomes⇒⇒⇒Time-varying β modeldSt = −βtStItdtdIt = (βtStIt − γIt)dtdRt = γItdtd log βt = σβdBtGoing further...• Try other diffusion processes (Ornstein-Uhlenbeck processes, inte-grated random walks, ...)• Chose model from expert knowledge and/or indicators as the Bayesfactor and the DIC.3 A challenging inference problemObjectiveWe want, under the following notations,Xt : dynamic vector of compartments populationsθ : static parametersβt : dynamic parametersg(.|y) : observation process modeln : number of observations (y1, .., yn)N : number of particlesto explore the posterior density p((Xt, βt, t ∈ [0, T ]), θ|y1:n).Difficulties• it is a high-dimensional density• the posterior density and the Kolmogorov forward equation are in-tractableEstimating time-varying parameterswith a Particle MCMC algorithm(Andrieu et al. 2010, JRSS.B)Initialize θSet Wj1 =1Nfor IndIt = 1 to NbIterations doSample θ∗ from Q(θ, .)L(θ∗) = 1for i = 1 to n− 1 dofor j = 1 to N doSample (Xji+1, βθ∗,ji+1 ) from p(., .|Xji , θ∗, βθ∗,ji )Noting Yji+1 = h((Xjt , t ∈ [0, ti+1])),set αj = g(Yji+1|yt) and Wji+1 ∝ αjend forL(θ∗) = L(θ∗) ∗ (∑Nj=1Wji αj)Resample (Xji+1, βθ∗,ji+1 ) according to (Wji+1), set Wji+1 =1mend forAccept θ∗ with probability 1 ∧L(θ∗)Q(θ∗,θ)L(θ)Q(θ,θ∗)Sample jrand from 1, .., NbParticulesKeep θ and βθ,jrand1:nend for4 Preliminary application: surveilling Influenza outbreaks from Google’s FluTrend dataGoogle FluTrend Data:Estimates of Influenza-Like Illnesses cases(Ginsberg et al. 2008, Nature)Google FluTrend estimates (computed with a 1-day delay) US surveillance data (gathered within 1 to 2 weeks) A simple model for Influenza:dSt = −βtStItdtdEt = (βtStItdt− kEt)dtdIt = (kEt − γIt)dtdRt = γItdtd log βt = σβdBtg(.|y) = N (y, σobsy)Note: E is the group of individuals who were infected but arenot infectious yet. k−1 is the referred to as the latency period.Informative priors were taken for k and γ, based on bibliography.Questions:•How transmittable is the upcoming strain of influenza?•Does the effective reproduction rate Rt =βtStγTotPop vary along time?•What is the population immunity to the upcoming strain of in-fluenza?a) Validating the algorithm on simulated datasep nov jan mar may jul04812ILI percentagesep nov jan mar may jul0123Rt0% 20% 40% 60% 80% 100%Initial immunity R(0)b) 2008-2009 epidemic in France, a ”classic” seasonal epidemicsep nov jan mar may jul04812ILI percentagesep nov jan mar may jul0123Rt0% 20% 40% 60% 80% 100%Initial immunity R(0)c) 2009-2010 epidemic in France, the H1N1 pandemicsep nov jan mar may jul04812ILI percentagesep nov jan mar may jul0123Rt0% 20% 40% 60% 80% 100%Initial immunity R(0)Data / simulation values Mean estimate 95% Confidence interval 50% Confidence interval Legend: 

Inference on epidemic models with time-varying parameters: methodology and preliminary applications

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