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HAL Id: hal-02746369

https://hal.inrae.fr/hal-02746369

Submitted on 3 Jun 2020

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Modeling and estimating parameters for epidemics using

diffusion processes. Application to influenza surveillance

Romain Guy, Catherine Laredo, Elisabeta Vergu

To cite this version:

Romain Guy, Catherine Laredo, Elisabeta Vergu. Modeling and estimating parameters for epidemics

using diffusion processes. Application to influenza surveillance. Dynstoch Meeting 2013, Apr 2013,

Copenhague, Denmark. pp.43. �hal-02746369�

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Characteristics of epidemic processes Diusion approximation for density dependent Markov Jump processes

Inference for epidemics using diusion processes

Modeling and estimating parameters for epidemics using diusion

processes.

Application to inuenza surveillance

Romain GUY1,2

Joint work with C. Larédo1,2_{and E. Vergu}1 1UR 341, MIA, INRA, Jouy-en-Josas 2 UMR 7599, LPMA, Université Paris Diderot

Dynstoch April 18th, 2013

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Inference for epidemics using diusion processes Outline

Main goal: provide a framework to analyze epidemic data, from modeling to parameter estimation

1 Characteristics of epidemic processes

Constraints related to the observation of the epidemic process Mechanistic model, mathematical formalism and statistical framework 2 Diusion approximation for density dependent Markov Jump processes

Diusion approximation for time dependent Markov jump processes Links with other approximations

Application to SIR and SIRS epidemic models 3 Inference for epidemics using diusion processes

Theoretical results for discrete observations on a nite time interval Partial observations of the epidemics

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Characteristics of epidemic processes

Diusion approximation for density dependent Markov Jump processes Inference for epidemics using diusion processes

Constraints related to the observation of the epidemic process Mechanistic model, mathematical formalism and statistical framework

Outline

2 Diusion approximation for density dependent Markov Jump processes

Application to SIR and SIRS epidemic models

3 Inference for epidemics using diusion processes

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Constraints related to the observation of the epidemic process

Mechanistic model, mathematical formalism and statistical framework

Incomplete Data

Imperfect data

Incomplete observations Temporally aggregated Sampling & reporting error Unobserved cases

Dierent dynamics

Ex.: Inuenza like illness cases (Sentinelles surveillance network)

One outbreak (3 months)

Recurrent oubreaks (27 years)

Main goal: key parameter estimation

Basic reproduction number, R0 (nb. of secondary cases generated by one primary case in an entirely susceptible population) Average infectious time period (d)

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First step (Mechanistic model): Compartmental representation of the population dynamics Dene: Nb. of health states, possible transitions and associated rates

Notations N population size, λ transmission rate, γ recovery rate (R0=λ_γ, d = 1_γ) S, I , R numbers of susceptible, infected, removed individuals

One outbreak: SIR model

Closed population ⇒ N=S + I + R Well-mixing population

⇒ (S, I )λSI /N→ (S − 1, I + 1)

Several outbreak: SIRS with seasonality

δ: waning immunity rate (years)−1 µ: demog. renewal rate (decades)−1 λ(t) = λ₀(1 + λ₁sin(2π_Tt

per))

λ0 basic trans. rate, λ1seasonal eect λ₁=0 ⇒ oscillations vanish

Observations: taken as new infecteds or new removeds (short infectious period)

Zt+∆ t λSIdt or

Zt+∆ t γIdt

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Second & Third steps: mathematical formalisms and statistical inference techniques

Stochastic modeling

Pure jump processes (State space Nd₎ d nb of dierent health states, Exponential holding times ⇒ Markov processes.

Deterministic modeling with noise

ODEs solutions on Rd

Statistical inference usually based on Least squares methods

Statistical inference for stochastic models

Observation of all the jumps: MLE,

⇒Infectious and recovery datesobserved for all individuals,

Incomplete observations: data augmentation methods (e.g. Breto et al (09), Toni et al (09)),

⇒Computer intensive simulations

⇒Computation times: too large especially for large population sizes Not addressed: parameter identiabilitygiven the data.

Consider statistics for diusion processes for an analytic approach (third step) ⇒_{Before the third step: need of a rigorous and user-friendly method to build diusion} processes (second step).

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Diusion approximation for density dependent Markov Jump processes

Outline

2 Diusion approximation for density dependent Markov Jump processes Diusion approximation for time dependent Markov jump processes Links with other approximations

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Diusion approximation for time dependent Markov jump processes

Links with other approximations

General result for time dependent Markov Jump processes

Results from (Guy et al (13)), extending (Ethier & Kurtz (05))

Density dependent Markov process

The state space is E = {0, .., N}d_{, E}−_{= {−}_{N, ..., N}}d

Dene non negative measurable functions: αl(.) :E → R+indexed by l ∈ E− (Z_N(t), t ≥ 0) Markov process with state space E, starting from Z(0) = Nz₀ Transition rates k → k + l : qk,k+l(t) = α_l(t, k)

Fundamental assumption for density dependent Markov process

(H): The functions αl(.)satisfy: ∀x ∈ [0, 1]d, ∀t ∈ [0, T ],_N1α_l(t, [Nx]) → β_l(t, x) as N → ∞

(for y = (x1, . . . ,x_d) ∈ Rd, [y] = ([y₁], . . . , [y_d])with [y_i]integer part of y_i

Main idea

The deterministic limit of ZN(t)

N , the Gaussian approximation ofZ(t)N −x(t) and the diusion approximation ofZN(t)

N are completely determined by the collection of functions (βl)_l∈E−

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Detailed result

Denition of drift and diusion functions

Dene b(t, x) = X l∈E− lβl(t, x), and Σ(t, x) = X l∈E− llτ_β l(t, x) Approximation result Deterministic limit: x(t) = z0+ Z t 0 b(s, x(s))ds satises, sup t∈[0,T ]k ZN(t) N −x(t)k −→_N→∞0 in probability.

CLT: Dene Φ(t, s) as the solution of dΦ(t,s)_dt =∂_∂b_x(t, x(t))Φ(t, s), Φ(s, s) = Id √

N(Z(t)_N −_{x(t)) −→}

N→∞g(t) centered Gaussian process with covariance Cov(g(t), g(r)) = R₀t∧rΦ(t ∧ r, s)Σ(s, x(s))Φ(t ∧ r, s)τ_ds

Di. Approx.: Dene σ(t, x(t)) a solution of σστ _{= Σ}_, Xt solution of dXt=b(t, Xt)dt +√1

Nσ(t, Xt)dBt,X0=z0: sup

t∈[0,T ]

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Other links between the processes

Taylor stochastic expansion of the diusion

Under regularity assumptions, Xt=x(t) +√1

Ng(t) + OP(_N1), ⇒_{g(t) =}

Zt

0 Φ(t, s)σ(x(s))dBs

Conclusion: Gaussian approximation and diusion approximation of the Markov Jump process are mainly equivalent as N → ∞

→_{Diusion approximation}

→_{Expansion in N of the process} →_{Taylor's stochastic expansion}

Characteristics of the approximation

Diusion with small diusion coecient b(t, x) = X l∈E− lβl(t, x), Σ(t, x) = X l∈E− llτ_β l(t, x) ⇒same parameter in the drift and diusion functions

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Application: diusion approximation of the SIR epidemics (1/2)

Collection functions αL(.)

Only two transitions: (S, I ) → (S − 1, I + 1) and (S, I ) → (S, I − 1),

L = (−1, +1) ⇒ αL(S, I ) = λS_NI, L = (0, −1) ⇒ αL(S, I ) = γI . Q-matrix of Z(t): qk,k+L= α_L(k).

Computation of the limit functions βL(x)

Let x = (s, i) ∈ [0, 1]2 1 Nα(−1,1)([Nx]) = _N1_Nλ[Ns][Ni] −→ N→+∞ β_(−1,1)(s, i) = λsi 1 Nα(0,−1)([Nx]) = _N1γ[Ni] −→ N→+∞ β(0,−1)(s, i) = γi. Functions b and Σ

depend on the two parameters (λ, γ) ⇒ b(λ,γ)(s, i), Σ(λ,γ)(s, i), b(λ,γ)(s, i) = −λsi −₁ 1 + γi 0 −1 = −λ_si λsi − γi . Σ(λ,γ)(s, i) = λsi −₁ 1 −_{1 1 + γi} 0 −₁ 0 −1 = λ_−λsi_si _λ_{si + γi}−λsi .

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Application: diusion approximation of the SIR epidemics (2/2)

Diusion approximation Assume (S0 N,IN0)_N→+∞−→ (s0,i0) =x0; Choose σ(s, i) = √ λsi 0 −√λsi √γi solution of σστ_{= Σ} ( dSt = −λS_tI_tdt +√1 N √ λS_tI_tdB₁(t) dIt = (λS_tI_t− γS_t)dt −√1 N √ λS_tI_tdB₁(t) +√1 N √ γI_tdB₂(t)

Small values of R0=λ_γ: appropriate signal/noise ratio in large population

Figure:Number of infected Itthrough time for N = 10000, R0=1.5,γ = 1/3,(s0,i0) = (0.9990.001) (signal/noise ratio :√It

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Signal/noise ratio and observations for Sentinelles surveillance network data

Figure:Reported (ρ) new removed (ρZtk+1 tk

γ_I_t_{dt) through time for N = 6.10}7_{(signal/noise} ratio ∈ [0.1, 6])

Specicity of the SIR observations

Zt k+1 t_k ρλStItdt = ρ(Stk −_S_t k+1), and Zt k+1 t_k ργItdt = ρ(Rtk+1 −_R_t k)

⇒transform integrated observations in discrete observations of increments of other coordinates

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Diusion approximation of the SIRS model

Description of ZN(t) = (S(t), I (t)) Notations: (s, i) = (S N,NI) ∈ [0, 1]2 β(−1,+1)(s, i) = λs(i + η), β_(0,−1)(s, i) = (γ + µ)i, β_(−1,0)(s, i) = µs, β(1,0)(s, i) = µ + δ(1 − s − i)

ODE and Diusion approximation

b(t, (s, i)) = P_LLβL(t, (s, i)) =−λ(t)s(i + η) + δ(1 − s − i) + µ(1 − s) λ(t)s(i + η) − (γ + µ)i

Σ(t, (s, i)) = P_LβL(t, (s, i))LLτ=

λ(_{t)s(i + η) + δ(1 − s − i) + µ(1 + s)} −λ(_{t)s(i + η)} −λ(_{t)s(i + η)} λ(t)s(i + η) + (γ + µ)i

.

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A model appropriate for very large populations

Dierent behaviors depending on the parameters (λ1 bifurcation parameter)

Proportion of infected over time (N = 107_{, R}₀₌_1.5 1

γ =3(day), Tper=365(day), 1

δ =2(yrs), η = 10

−6 _{and λ}₁_{∈ {}_{0.05, 0.15} (λ(t) = λ}₀₍_{1 + λ}₁_{cos(2πt/T ))}

⇒_{Due to reporting rate ρ, signal/noise ratio is lower for Sentinelles data}

Observations for SIRS model

Zt_k+1 t_k ρλStItdt6=ρ(Stk−Stk+1) ⇒≈ ρλ∆StkItk Ito req.: b(t, x) ⇒ b(t, x) 1 ∆ Z t k+1 tk ργItdt ≈ ργIt_k

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Outline

2 Diusion approximation for density dependent Markov Jump processes

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Theoretical results for discrete observations on a nite time interval

Partial observations of the epidemics

Specicities of the statistical framework

Model: (Xt)Multidimensional diusion process on Rd, dXt=b(θ1,Xt)dt + σ(θ2,Xt)dBt,X0=x0∈ Rd.

Observations: Discrete observations with sampling ∆ on [0, T ] Xtk for tk=k∆, k ∈ {0, .., n}, tk∈ [0, T ] (n∆ = T ), T is xed

Dierent rates of convergence for parameters in the drift and in the diusion coecient⇒splitting the parameters θ1, θ2required

Continuous observation of the diusion on [0, T ] (Kutoyants (80))

MLE: −1 _θMLE

1 − θ01 → N 0, Ib(θ₁0, θ₂0)−1

Discrete observations of the diusion on [0, T ]

Minimum contrast approaches: estimation of θ2 at rate√n (⇒ ∆ = ∆n→_0).

Diusion approximation of density dependent processes and Epidemics

=√1

N; Identical parameters in the drift and diusion coecients: θ1= θ2. N >> n ⇒Focus on θ1for a more accurate estimation

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Case of Time dependent diusion

Existing results for multidim. autonomous di. with small di. coe., discretely obs.

For , ∆ → 0 (Sorensen & Uchida (03), Gloter & Sorensen (09), Guy et al. (12)): −1( ˆθ0₁− θ₁0) → N (0, I_b−1(θ₁0, θ0₂))

For → 0, ∆ xed (Guy et al. (2012)):−1_{( ˆ}_θ0

1− θ10) → N (0, I∆−1(θ01, θ02))

Contrast for discrete observations from (Guy et al. (13))

˜ U= n X k=1 log(det(Ck(θ1))) + 1 2∆ t_N k(θ1)C_k−1(θ₁)N_k(θ₁), with Ck(θ1) = 1∆Ztk tk−1Φ(tk, s)Σ(x(s))Φ(tk, s)τ ds, Nk(θ1) = Xtk − xθ1(tk) − Φ(tk, tk−1)(Xtk − xθ1(tk))

Fundamental property based on the Gaussian process g

g(tk) = Φ(tk,tk−1)g(tk−1) + Z t_k

t_k−1Φ(tk,s)σ(x(s))dBs

Extension

Results hold for dXt=b(t, Xt)dt + σ(t, Xt)dBt

Taylor Stoch. expansion (Azencott (82)) still valid for non autonomous di. Rem: Results do not hold for general model dXt=b(t, Xt)dt + σ(t, Xt)dBt ⇒_{But hold for b}(t, x) = b0(t, Xt) + 2b₁(t, Xt)(case of new infecteds)

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Application of the time dependent case: estimation on simulations of the SIRS model

Figure:Mean point estimation (on 1000 runs) and theoretical condence ellipsoid for a simulation case close to inuenza dynamics (N = 106_{, R}

0=_{1.5, d = 3(days), λ}₁=_0.15 ,1/δ = 2(years)) for ∆ = 1 and 7 days

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The Sentinelles surveillance network data

The data

Reported cases over time during 22 years (1990-2011, 1obs/day)

SIR model

Each year, starting point s0,i0not known

Estimation of i0dicult even with complete data

SIRS model

Bifurcation

Deterministic limit can not completely t the data

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Theoretical results in the case of diusion with small diusion coecient Additional assumption: Only the rst component is discretely observed

Existing results for partial observations

(Kutoyants (94)): dXt=ft(θ)Ytdt + dBt1,X0=0; dYt=b_t(θ)Y_tdt + σ_t(θ)dB_t2,Y₀=y₀; Continuous observation of Xt, functions ft,bt, σtknown:

consistence and asymptotic normality (at rate −1_{) for MLE of parameter θ.} Generalization by linearization of the drift function around its deterministic limit Identiability assumption based on identiability of the deterministic limit of ft(θ)E Y_t|X_s≤t

(James & Le Gland (95)): Consistence result for the MLE estimator in the general case of bidimensionnal process (Xt,Yt)with only the coordinate Xtobserved ⇒identiability of parameters based on x(t) deterministic limit of X_t ⇒_{If we observe only the increments of ργI}_t

kfor the SIRS model: the deterministic

identiability is ensured for θ1= (ρ, λ0, λ1, γ, δ)with η, µ, s0,i0 xed.

Our result (ongoing work)

Consistency and CLT, in the asymptotics → 0, ∆ → 0: −1_{( ˆ}_{α − α}₀₎

√

n( ˆβ − β0)

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Application to the estimation from real data (Sentinelles surveillance network)(1/2)

Figure: Deterministic trajectories associated to estimated values α R₀ d (days) 1_δ(years) λ₁ ρ

ˆ

α₁ 1.43 2.73 1.8404 0.1304 27.84

ˆ

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Application to the estimation from real data (Sentinelles surveillance network)(2/2)

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References

(Guy et al (13)) Approximation of epidemic models by diusion processes and their statistical inference. submitted

(Guy et al (12)) Parametric inference for discretely observed multidimensional diusions with small diusion coecient. to be published in S.P.A.

(Ethier & Kurtz (05)) Markov Processes: Characterization and Convergence (Kutoyants (94)) Identication of Dynamical Systems with Small Noise (James & Le Gland (95)) Consistent parameter estimation for partially observed diusions with small noise

(Sorensen & Uchida (03)) Small-diusion asymptotics for discretely sampled stochastic dierential equations

(Sorensen & Gloter (09)) Estimation for stochastic dierential equations with a small diusion coecient

(Azencott (82)) Formule de Taylor stochastique et developpement asymptotique d'integrales de Feynmann

(Breto et al (09)) Time series analysis via mechanistic models

(Toni et al (09)) Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

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Hints for proof of the diusion approximation

Complete proof by application of Theorems from Jacod & Shirayev

Representation Theorem (Ethier & Kurtz (05) Ch. 6.4)

Z(t) = Z(0) + Pl∈E−lY_l(

Rt

0αl(Z(s))ds) , where (Y_l)is a familly of independent Poisson process

Theorem: Representation of X (t) with random time changes

X (t) = X (0) + P_lalWl(Rt

0βl(X (s))ds) + R₀t X l∈E−

lβl(X (s))ds, where (W_l)is a familly of independent Brownian motions

en fr

HAL Id: hal-02746369

https://hal.inrae.fr/hal-02746369

Submitted on 3 Jun 2020

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Modeling and estimating parameters for epidemics using

diffusion processes. Application to influenza surveillance

Romain Guy, Catherine Laredo, Elisabeta Vergu

To cite this version:

Romain Guy, Catherine Laredo, Elisabeta Vergu. Modeling and estimating parameters for epidemics

using diffusion processes. Application to influenza surveillance. Dynstoch Meeting 2013, Apr 2013,

Copenhague, Denmark. pp.43. �hal-02746369�

Modeling and estimating parameters for epidemics using diusion

processes.

Application to inuenza surveillance

Modeling and estimating parameters for epidemics using diusion

Application to inuenza surveillance