Inference for epidemic data using diffusion processes with small diffusion coefficient

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Inference for epidemic data using diffusion processes with small diffusion coeﬀicient

Romain Guy, Catherine Laredo, Elisabeta Vergu

To cite this version:

Romain Guy, Catherine Laredo, Elisabeta Vergu. Inference for epidemic data using diffusion processes

with small diffusion coeﬀicient. Rencontres des jeunes statisticiens Français 2011, Société Française

de Statistique (SFdS). FRA., Aug 2011, Aussois, France. pp.1. �hal-02803499�

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Back to the epidemics and simulations results

Inference for epidemic data using diusion processes with small diusion coecient

Romain GUY¹^,² with C. Laredo¹^,²and E. Vergu²

1Laboratoire de Probabilités et modèles aléatoires (Paris Diderot) 2Unité Mathématiques et Informatique Appliquées, INRA, Jouy-en-Josas

4èmes Rencontres Des Jeunes Statisticiens (Aussois) 7 septembre 2011

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Introduction

A work guided by data

Estimation of key parameters of the epidemic dynamics Often low frequency time series

Often aggregated and incomplete datasets.

Objective

Provide estimators with good properties for the diusion model approximating the epidemic dynamics

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Outline

1 Classical SIR epidemic models

2 Parametric inference for discretely observed diusion processes

3 Back to the epidemics and simulations results

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Notations and model assumptions

Notations

N : population size ; m : initial infectives

λ: transmission rate ;γ : recovery rate (=1/mean holding time in infective state)

R0=λ/γ: basic reproduction number (=mean number of secondary infections generated by an infective in an entirely susceptible population)

S(t),I(t): numbers of susceptibles, infectives ; s(t) = ^S_N⁽^t⁾,i(t) = ^I⁽_N^t⁾ : proportion of susceptibles, infectives

Assumptions

Homogenous mixing in closed population

Discrete observations of S and I on a xed interval[0,T], with sampling interval∆(T =n∆) (acceptable assumption as a rst attempt)

S ^λI/N I ^ɣ R

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Markov pure jump model and Inference

Let Xt= (St,It)and X0= (N−m,m). Transitions

(S,I)−→^λ^N^SI (S−1,I+1)and(S,I)−→^γI (S,I−1) Exponential holding times

Observations

All jumps are observed

Maximum Likelihood Estimators (MLE) and asymptotic normality (Andersson and Britton 2000)

ˆλ_MLE =N N−m−S(T) R_T

0 S(t)I(t)dt,ˆγ_MLE =N−S(T)−I(T) R_T

0 I(t)dt

√N

ˆλ_MLE−λ₀ ˆ

γ_MLE−γ₀

N−→→∞N

0,

var(λ₀) 0 0 var(γ₀)

with var(λ0),var(γ0)being explicit

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ODE model and Inference (as a rst approximation)

Let xλ,γ(t) = (s(t),i(t))and(s(0),i(0)) = (1−^m_N,^m_N).

Classical ODE system (for N large) ds

dt =−λsi and didt =λsi−γi, Observations

Discrete observations Xt_k =xλ,γ(tk) +_k at times tk=k∆(k=0, ...,n), with _k ∼

iidN₂(0,Σ)

Least Square Estimator (LSE) and asymptotic normality LSE(λ, γ) = ¹_n

n

X

k=0

(Xt_k−xλ,γ(tk))²,(ˆλ_LSE,γˆ_LSE) =argmin

(λ,γ)∈Θ

LSE(λ, γ)

√nλˆ_LSE−λ₀ ˆ γ_LSE−γ₀

n−→→∞N(0,Σ)

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Diusion approximation model

Let Xt= (st,it),(s0,i0) = (1−^m_N,^m_N)and B1,B2, two independent Brownians motions.

Stochastic Dierential Equation dst = −λstitdt+^√¹

N

√λstitdB1(t)

dit = (λstit−γit)dt−^√¹

N

√λstitdB1(t) +^√¹_N√

γitdB2(t)

Remarks

Classical approximation : before passing to the limit (N→ ∞) in the normalized system of the Markov jump process

MLE untractable when the path is discretely observed Framework

Multidimensionnal diusion processes Small noise∼^√¹_N in large population

Parameters(λ, γ)both in drift and diusion coecients

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General diusion model and existing results

Let Xtbe the unique strong solution of the SDE dX_t=b(α,X_t)dt+σ(β,X_t)dBt, X0=x0∈R^p

We observe X_tat times tk=k∆on a xed interval[0,T](T =n∆) σ(β,x)∈Mp(R),b(α,x)∈R^p,Σ(β,x) =σ(β,x)^tσ(β,x)invertible.

Existing estimation results for high-frequency data (Gloter and Sorensen (2009))

Under the condition∃ρ >0,_n¹ρ bounded, for a class of contrast processes, associated Minimum Contrast Estimators (MCEs) are consistent and

⁻¹( ˆα,n−α₀)

√n( ˆβ,n−β₀)

n→∞,→−→ 0N

0,I_b⁻¹(α₀, β₀) 0 0 Iσ⁻¹(α₀, β₀)

with I_b⁻¹(α₀, β₀)explicit and optimal (= asymptotic variance for continuous observations on[0,T]).

For epidemics :⁻¹=√ N

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Main idea of our inference approach (extension of Genon-Catalot(90))

Use of Taylor's stochastic expansion formula (Azencott (82)) X_t=xα(t) +gα,β(t) +²Rα,β (t)

where xα(t)is the deterministic solution ^dx^α_dt⁽^t⁾=b(α,xα(t)), x(0) =x0∈R^p, dgα,β(t) = ∂b

∂x(α,xα(t))gα,β(t)dt+σ(β,xα(t))dBt, gα,β(0) =0R^p

and Rα,β satises

t∈[sup0,T]

{kRα,β (t)k} −→

P,→00.

LetΦαbe the invertible matrix solution of ^d_dt^Φ^α(t,t0) = ^∂b_∂_x(α,xα(t))Φα(t,t0), withΦα(t0,t0) =Ip.

Properties of gα,β

gα,β is a Gaussian process for which we can obtain the analytic expression.

gα,β(tk) = Φα(tk,tk−1)gα,β(tk−1) +√

∆Z_k^α,β Z_k^α,β are independentN

0,S_k^α,β

variables.

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Main idea of our inference approach (extension of Genon-Catalot(90))

Contrast process derived from(Z_k^α,β)_k-likelihood U∆,(α, β) = ²

n

X

k=1

logh det

S_k^α,βi + _∆¹

n

X

k=1

tNk(α)(S_k^α,β)⁻¹Nk(α) with Nk(α) = Xt_k−xα(tk)−Φα(tk,tk−1)h

Xt_k−1−xα(tk−1)i . ( ˆα,∆,βˆ,∆) =argmin

(α,β)∈ΘU∆,(α, β)

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Results for low frequency data ( ∆ and n xed)

No asymptotic results forβˆ,∆

βknown

We only considerαˆ,∆(β₀) =argmin

α∈Θ_a U∆,(α, β₀)and then ⁻¹( ˆα,∆(β₀)−α₀)−→

→0N(0,I_∆⁻¹(α₀, β₀)),with I∆(α₀, β₀) −→

∆→0I_b(α₀, β₀)

βunknown

We modify the contrast process in a conditional least square contrast U˜,∆ α,(Xt_k)_k∈{1,..,n}

= 1

∆

n

X

k=1

tNk(α)Nk(α).

Thenαˆ=argmin

α∈Θ_a

U˜,∆(α) satises⁻¹( ˆα−α₀)−→

→0N(0,˜I∆⁻¹(α₀, β₀)).

For epidemics :α= (λ, γ) =β⇒Special case, results for knownβhold if we replace eachβoccurence withα.

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Results for high frequency data ( ∆ → 0)

Contrast process

UsingkS_k^α⁰^,β⁰−Σ(β₀,Xt_k−1)k −→

,∆→00, we consider : U∆,(α, β)) = ²

n

X

k=1

logh det

Σ(β,Xt_k−1)i + _∆¹

n

X

k=1

tNk(α)Σ⁻¹(β,Xt_k−1)Nk(α)

Asymptotic Normality Under the condition²n −→

,∆→00 ⁻¹( ˆα,∆−α₀)

√n( ˆβ,∆−β₀)

n→∞,→−→ 0N

0,

I_b⁻¹(α₀, β₀) 0 0 Iσ⁻¹(α₀, β₀)

.

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Simulation study (using Matlab)

Algorithm

1 Exact simulation of an epidemic with Markov pure jump process (Gillespie algorithm with choice of N,m, λ, γ)

2 Calculation ofˆλ_MLE,γˆ_MLE (observation of the whole path of the process)

3 Discrete observations on a xed interval

4 Estimation phase for LSE, Gloter and Sorensen (2009) contrast, our MCE forα=β, and for unknownβ(conditionnaly least square contrast).

Numerical optimisation using fminsearch.

Results

Mean of the point estimation on 1000 runs, theoretical condence intervals for MCEs, empirical condence interval for LSE, for each scenario and for each of the four estimators (+ MLE as a reference)

Simulated values

N∈[1000;10000],∆∈[T/10;1;T/100],γ=1/3, R0∈[1.2;2]

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Example of trajectories : proportion of infectives over time (N = 1000, R

0

= 2)

Figure:Trajectories of the three processes for N=1000 R0=2,γ=1/3, T=40 and 1% of initial infectives

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Simulation results for N = 1000, R

₀

= 2, γ = 1 / 3, T = 40 and n = 10 , 40 , 100

Notations

green arrows = best ponctual estimator for one scenario

For each scenario, and each parameter in order : LSE, GS09(α), MCE(α=β), MCE(βunknown)

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Simulation results for N = 10000, R

₀

= 2, γ = 1 / 3, T = 40 and

n = 10 , 40 , 100

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Another Case : R

₀

= 1 . 2

Figure:Trajectories of the three processes for N=1000, R0=1.2,γ=1/3, T=65 and 1% of initial infectives

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Simulation results for N = 1000 , 10000, T = 65 and n = 10 , 65 , 100

N=1000

N=10000

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Limits and perspectives

To be rened

1 Limits of the SIR model : acceptable as a rst attempt, possible to be rened to integrate more realistic assumptions (e.g. increase the state space dimension)

2 Idealized statistical framework : here the two system coordinates(st,it)are assumed observed (instead of a function of it, a more realistic assumption)

Next directions (work in progress)

1 Complexify the model : no diculty if the new system is still an autonomous system, regardless to its size

2 Modify the statistical framework : observe integrated diusion

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References

H. Andersson and T. Britton.

Stochastic epidemic models and their statistical analysis.

Springer, 2000.

S.N. Ethier and T.G. Kurtz.

Markov processes : characterization and convergence.

Wiley, 1986.

V. Genon-Catalot.

Maximum contrast estimation for diusion processes from discrete observations.

Statistics, 1990.

A. Gloter and M. Sorensen.

Estimation for stochastic dierential equations with a small diusion coecient.

Stochastic Processes and their Applications, 2009.

Thank you for your attention !