Partially Observed Stochastic Epidemics

Transcription

1 Institut de Mathématiques Ecole Polytechnique Fédérale de Lausanne

2 Stochastic Epidemics Stochastic Epidemic = stochastic model for evolution of epidemic { stochastic model = stochastic process (T = R+, Z Often + ) evolution = counting Purposes: Description (behaviour of epidemic) Prediction (outcome of epidemic) long history + rich literature

3 Origins + Recent Themes 1949: Bailey, 1950+: Bailey, Whittle, Bartlett, Kendall stochastic variants of Kermack-McKendrick deterministic model followed by explosion of literature... Recent Themes: Model approximation (Ball & Barbour [1990, JAP], Ball & Donnelly [1995, Stoc. Proc. App.]) Threshold theorems (Andersson & Djehiche [1998, JAP], von Bahr & Martin-Löf [1980, AAP]) Time to extinction (Barbour [1975, Biometrika], Nåsel [1999, JRSSB]) Random graph interpretations (Barbour & Mollison [1990, Springer Lec. Notes]) under variety of different assumptions (= models)

4 Outbreak of an Epidemic Offset of an epidemic: emergency control rules justified? Control rules: may decisively affect epidemic have grave cost Must decide on basis of initial observations: Typically look at P[E 0 ]/P[E ] Depends on what P is! (model) and interpretation of observations...

5 Offset in Large Populations Stochastic models of initial stages of epidemic? In practice have finite population: N = S + susceptibles I + infectious R removed Assuming large population of susceptibles Approximation by simple stochastic models: Birth and Death Processes (continuous time) Branching Processes (discrete time) Extinction/explosion duality: P[E 0 ] + P[E ] = 1 Both manifest phase transitions: Birth/Death rate ratio ρ 1 Offspring distribution mean µ 1

6 Incorporating Data Information Criticality results depend on general specifications Notation: Z t = #(infectious individuals at time t) Include history (data) up to present, t 0, via conditioning: P[E 0 {Z t } 0 t t0 ] = P[E 0 Z t0 ] (1) Markov property: current value is predictively sufficient QUESTION: Do we really completely observe {Z t }?

7 ? D.G.Kendall [1956, Proc. 3rd Berkeley Symp.]: In practice do not observe infections In deterministic models observe a proportion, but stochastic case more involved D.R. Cox [recently, personal communication]: Solvable perturbation of classical models to take partial observation into account? An approach: V.M. Panaretos [2007, J. Math. Biol.]

8 General Considerations In practice: cases recorded in discrete time (e.g. every day) some cases undetected observed cases are quarantined Suggests: T = Z + observations Y n = f (Z n, ξ n ), ξ n Z n interventions Z n+1 Z n d Zn+1 (Z n, Y n ) Want: tractability interpretable marginal model

9 A Simple Emission/Intervention Mechanism Emissions: Each one of Z n independently overlooked with probability θ Y n Z n Binom(1 θ, Z n ) Interaction: Observed individuals cannot spread disease. Each unobserved individual independently produces identically distributed offsprings Z n+1 d = Zn Y n k=1 ξ k,n, {ξ k,n } iid

10 Simple Coupling Construction Define a stochastic process {(Z n, Y n )} n 0 as follows: Z 0 = N 0, {ξ i,n } iid, B Bernoulli(1 θ) Z n Z n+1 = ξ i,n (1 B i,n ) n 1 i=1 Z n Y n := B in, n 0 i=1 (2) Branching process framework: framework includes discretely sampled birth/death processes as special case

11 Conditioning on Y n Instead of Z n Joint process {(Z n, Y n )} a Markov chain on Z 2 + Can obtain explicit k-step transition pgfs P n (r, s) = { 1 θ γ n(rθ + rs(1 θ)) 1 θ } z0 y 0 θ How do things differ when conditioning on the observable component Y n? E.G. two issues of basic interest: Conditional probability of extinction Distribution of conditional time to extinction Will need k-step prediction distribution (actually generating function): P[Z n+k = Y n = y n ]

12 Conditioning on the First Step Basic property {Z n } marginally a BRANCHING PROCESS Use self-similarity property for prediction distribution {Z n }-type branching process started at dist{z n Y n }? Need to be careful at first step:

13 Getting there... Following sequence of steps: Getting from Z n Y n to Z n+1 Requires G Zn Y n Hence obtain G Zn+1 Y n Use self-similarity Marginal Branching Binomial Thinning Conditional Independence Crucial in all steps.

14 Exact Expressions Probability of Extinction: P µ [E 0 Y n = y] = P µ [E 0 Z n+1 = z, Y n = y]p µ [Z n+1 Y n = y] = z=0 (s 0 ) z P µ [Z n+1 Y n = y] z=0 = G Zn+1 Y n=y = λ(y) n (θg ξ (s 0 )) λ (y) n (θ) Time to Extinction P µ [T 0 n + k Y n = y] = P µ [Z n+k = 0 Y n = y] = G Zn+k Y n=y (0) = λ(y) n (θg ξ (γ k 1 (0))) λ (y) n (θ)

15 Conditioning Further into the Past via Bootstrapping Saw what happened when replacing: Z n the true current value by Y n what we really observe BUT: Y n is predictively insufficient (Markov breaks down ). Should condition on whole history Y 1,..., Y n Can show: (A) G Zn+1 (Y n=y n,...,y 1 =y 1 )(s) = G Zn (Yn,...,Y 1 )(G ξ (s)) (G ξ (s)) yn (B) G Zn (Y n=y n,...,y 1 =y 1 )(s) = G (yn) Zn (Y n 1,...,Y 1 ) (sθ) G (yn) Zn (Y n 1,...,Y 1 ) (θ) Start from time origin, bootstrap upwards by iteratively switching: (B) (A) (B) (A) (B) G Z2 Y 1 G Z1 (Y 2,Y 1 ) G Z3 (Y 2,Y 1 ) G Z3 (Y 3,Y 2,Y 1 ) G Z4 (Y 3,Y 2,Y 1 )..

16 A More General Model Z 0 = N 0, ξ, ζ iid, B Bernoulli(1 θ) Z n Z n Z n+1 = ξ i,n (1 B i,n ) + ζ i,n B i,n, i=1 i=1 Z n Y n := B in, (3) Analysis via conditional independence (simple convolution with simpler model ): Z n ξ i,n (1 B i,n ) Y Z n n ζ i,n B i,n Y n (4) i=1 i=1 i=1

17 Some Remarks Small perturbation increases complication Inherent non-linearities make generalisations intractable Took advantage of solvable marginal model given emission and interaction mechanism Conditioning on the whole past σ(y 1, Y 2,..., Y n ) Loss of information Inference given observable data (pseudo-likelihood?) - Crucial Elements: Marginal Model Observation Mechanism ( potential function ) Feynman-Kac Path measures? Geometric case completely tractable (discretely observed birth-death process...)

18 A Couple of References D.G. Kendall (1956). Deterministic and Stochastic Epidemics in Closed Populations. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (Jerzy Neyman, Ed.), University of California Press: Berkeley and Los Angeles. V.M. Panaretos (2007). Partially Observed Branching Processes for Stochastic Epidemics. Journal of Mathematical Biology 54:

19 More References 1. Andersson, H. & Djehiche, B. (1998). A threshold limit theorem for the stochastic logistic epidemic. Journal of Applied Probability, 35: Ball, F. & Barbour, A.D. (1990). Poisson approximation for some epidemic models. Journal of Applied Probability, 49: Ball, F. & Donnelly, P. (1995). Strong approximations for epidemic models. Stochastic Processes and their Applications, 55: von Bahr, B. & Martin-Löf, A. (1980). Threshold limit theorems for some epidemic processes. Advances in Applied Probability, 12: Barbour, A.D. (1975). The duration of the closed stochastic epidemic. Biometrika, 62: Barbour, A.D. & Mollison, D. (1990). Epidemics and random graphs. In: Stochastic processes in epidemic theory (J.P. Gabriel, C. Lefevre & P. Picard, Eds), Lecture Notes in Biomathematics 86: 86-89, Springer. 7. Nåsel, I. (1999). On the time to extinction in recurrent epidemics. Journal of the Royal Statistical Society B, 61: