Statistical inference for epidemic models approximated by diffusion processes

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Statistical inference for epidemic models approximated

by diffusion processes

Romain Guy, Catherine Laredo, Elisabeta Vergu

To cite this version:

Romain Guy, Catherine Laredo, Elisabeta Vergu. Statistical inference for epidemic models

approxi-mated by diffusion processes. Journées de l’ANR MANEGE, Jan 2013, Université Paris 13

Villeta-neuse, France. 33 diapos. �hal-02807182�

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Statistical Inference for epidemic models approximated by

diusion processes

Romain GUY

Joint work with C. Larédo

_{and E. Vergu}

1UR 341, MIA, INRA, Jouy-en-Josas 2 UMR 7599, LPMA, Université Paris Diderot

ANR MANEGE, Université Paris 13

30 janvier 2013

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Outline

Provide a framework for estimating key parameters of epidemics

1

Characteristics of the epidemic process

Constraints imposed by the observation of the epidemic process

Simple mechanistic models

2

Various mathematical approaches for epidemic spread

Natural approach: Markov jump process

First approximation by ODEs

Gaussian approximation of the Markov jump process

Diusion approximation of the Markov jump process

3

Inference for discrete observations of diusion or Gaussian processes with small

diusion coecient

Contrast processes for xed or large number of observations

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

4

Epidemics incompletely observed: partially and integrated diusion processes (Work

in progress)

Back to epidemic data

Inference approach: Work in progress

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Simple mechanistic models

Outline

1

Characteristics of the epidemic process

Constraints imposed by the observation of the epidemic process

Simple mechanistic models

2

Various mathematical approaches for epidemic spread

Natural approach: Markov jump process

First approximation by ODEs

Gaussian approximation of the Markov jump process

Diusion approximation of the Markov jump process

3

Inference for discrete observations of diusion or Gaussian processes with small

diusion coecient

Contrast processes for xed or large number of observations

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

4

Epidemics incompletely observed: partially and integrated diusion processes (Work

in progress)

Back to epidemic data

Inference approach: Work in progress

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Simple mechanistic models

Incomplete Data

Imperfect data

Incomplete observations

Temporally aggregated

Sampling & reporting error

Unobserved cases

Dierent dynamics framework

Ex.: Inuenza like illness cases (Sentinelles

surveillance network)

One outbreak study

Recurrent oubreaks study

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)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Simple mechanistic models

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

S, I , R : numbers of susceptible, infected, removed individuals

One of the simplest model: SIR

Closed population ⇒ N=S + I + R

Well-mixing population

⇒ (

_{S, I )}

→

(

S − 1, I + 1)

Convenient to study one epidemics

Key parameters: R0

=

, d =

Summary: coecients α

(

S, I ) → (S − 1, I + 1) = (S, I ) + (−1, 1) at rate α(−1,1)

(

S, I ) = λS

and

(

S, I ) → (S, I − 1) = (S, I ) + (0, −1) at rate α(

(

S, I ) = γI

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Simple mechanistic models

Natural extensions of the SIR model

Increase the number of health states : e.g. Exposed Class ⇒ SEIR model

Additionnal transitions: e.g. (S, I ) → (S − 1, I ) (vaccination)

Temporal dependence: SIRS with seasonality in transmission and demography

δ

: waning immunity rate (years)

µ

: demographic renewal rate (decades)

λ(t) = λ0

(1 + λ1sin(2π

Tper

))

λ

=

0 ⇒ oscillations vanishes

Suited to study recurrent epidemics

Key parameters: R

=

, d =

Summary:

α

(

t

,

S, I ) = λ(

t

)

S

,

α

(

S, I ) = Nµ + δ(N − S − I ),

α

(

S, I ) = µS

α

(

S, I ) = (µ + γ)I

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Outline

1

Characteristics of the epidemic process

Constraints imposed by the observation of the epidemic process

Simple mechanistic models

2

Various mathematical approaches for epidemic spread

Natural approach: Markov jump process

First approximation by ODEs

Gaussian approximation of the Markov jump process

Diusion approximation of the Markov jump process

3

Inference for discrete observations of diusion or Gaussian processes with small

diusion coecient

Contrast processes for xed or large number of observations

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

4

Epidemics incompletely observed: partially and integrated diusion processes (Work

in progress)

Back to epidemic data

Inference approach: Work in progress

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Markov jump process

α(−

(

S, I ) = λS

, α(

(

S, I ) = γI

Notations:

E = {0, .., N}

∀

L ∈ E

_{= {−}

_{N, .., N}}

_{, we dene α}

L

(·) :

E → [0, +∞[

We dene (Z

)

the Markov jump process on E with Q-matrix: q

= α

(

X )

Assume α(X ) =

X

L∈E−

α

(

X ) < +∞ ⇒ Sojourn time Exp(α(X ))

Easily simulated (using Gillespie algorithm)

3 realizations for N = 10000, λ = 0.5, γ = 1/3, (S0

,

I0

) = (

9990, 10)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Interest of the deterministic approach

λ(

t) = λ0

(

1 + λ1sin(2πt/Tper

)

Some trajectories of the SIRS Markov proc. :

N = 10

_,

_R

0

=

_{1.5, d = 3,}

=

_2,

=

₅₀

Drawbacks of the Markov jump approach

N = 10

_{:more than 10}

_{events in one}

week (MLE: observation of all the jumps

required)

Extinction probability non negligible

SIRS ODE solution

= µ(

1 − s) + δ(1 − s − i) − λ(t)si

= λ(

t)si − (µ + γ)i

λ(

t) = λ0

(

1 + λ1sin(2πt/T )

(

s(0), i(0)) =

ODE trajectory

Link between the two approaches

As N → +∞ we have

N _N→∞

−→

x(t), where

x(t) is the deterministic solution of the ODE:

dx(t)

dt

=

b(x(t))

Function b is explicit

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Beyond deterministic limit : Gaussian process

Additionnal assumption: smooth version of αL, βL

We have α

:

E → (0, +∞) transition rate : X

→

X + l

Assume β

: [

0, 1]

→ [

0, +∞[ well dene and regular :

∀

x ∈ [0, 1]

_,

α

(b

Nxc) −→

β

(

x)

SIR: α

(

S, I ) = λS

⇒ β

(

x) = λx

x

, α

(

S, I ) = γI ⇒ β

(

x) = γx

Denition of function b (

=

b(x(t)))

b(x) =

X

Lβ

(

_x)

_{, SIR: b((λ, γ), (s, i)) =}

−

1

1



λ

_{si +}



0

1



γ

_{i =}



−λ

si

λ

_{si + γi}



ODE approximation : no longer dependance w.r.t. N

Asymptotic expansion w.r.t. N : Gaussian process

√

N



−

x(t)



−→

N→∞

g(t) where g(t) centered Gaussian process:

dg(t) =

∂x

(

x(t))g(t)dt + σ(x(t))dB

, where σ

σ(

x) =

Σ(

x) =

X

L∈E−

L

_Lβ

(

x)

SIR: Σ((λ, γ), (s, i)) = λsi

−

₁

1



−

1

1
+ γi



0

₁



0

1
=

 λ

_−λ

si

_si

_λ

−λ

_{si + γi}

si



Cholesky algorithm ⇒ σ(x) =



√

λ

_si

₀

−

√

λ

si

√

γ

i



Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Innitesimal generator approach: diusion approximation

Renormalized version of the Markov jump process (˜Zt

)

:

( ˜

Z

)

Markov jump process on E with transition rates q

=

Nβ

(

X )

¯

Z

=

normalized process

Asymptotic development of the innitesimal generator of Z

(Ethier & Kurtz (86))

Generator of ˜

Z

: Af (x) =

X

NLβ

(

x) (f (x + L) − f (x))

⇒

Generator of ¯

Z

: ¯

Af (x) = b(x). 5 f (x) +

X

Σ

(

x)

∂

_f

∂

x

∂

x

(

x) + O(

1

N

)

Dropping negligible terms leads to the generator of a diusion process:

dX

=

b(X

)

dt +

σ(

X

)

dB

, where b(x) =

X

Lβ

(

x), Σ(x) =

X

L

_Lβ

(

x)

Temporal dependence βL

(

t

,

x): generator approach no longer available

Decomposition of the diusion process using Gaussian process ( =

N

)

Taylor's stochastic formula (Wentzell-Freidlin(79), Azencott (82))

Let dX

=

b(X

)

dt + σ(X

)

dB

,

X

=

x

Then, under regularity assumptions, X

=

x(t) + g(t) + O

(

)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Gaussian approximation of the Markov jump processDiusion approximation of the Markov jump process

Links between approximations

Zt

Markov jump process on E with transition qX ,Y

= α

(

X )

¯

Zt

Markov jump process on E/N with transition qx,y

=

Nβb

(

x)

¯

Zt

∼

_{x(t) +}

N

g(t)

with x(·) the ODE solution and g a Gaussian process

Xt

: dXt

=

b(x(t))dt +

N

σ(

x(t))dBt

diusion with small diusion coecient

and

P{ sup

k ¯

_Z

−

_X

k >

_C

} −→

0

X

=

x(t) +

g(t) + O

(

)

→

_{Diusion approximation}

→

Expansion in N of the process

→

Taylor's stochastic expansion

Important : All mathematical representations completely dened by (αL

)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Outline

1

Characteristics of the epidemic process

Constraints imposed by the observation of the epidemic process

Simple mechanistic models

2

Various mathematical approaches for epidemic spread

Natural approach: Markov jump process

First approximation by ODEs

Gaussian approximation of the Markov jump process

Diusion approximation of the Markov jump process

3

Inference for discrete observations of diusion or Gaussian processes with small

diusion coecient

Contrast processes for xed or large number of observations

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

4

Epidemics incompletely observed: partially and integrated diusion processes (Work

in progress)

Back to epidemic data

Inference approach: Work in progress

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Classical Estimators

Maximum likelihood estimation for the SIR Markov Jump process (Andersson &

Britton (00))

Observation of all jumps

Analytic expression of the estimators for SIR model:

ˆ

λ =

N

Z

S(t)I (t)dt

, ˆ

γ =

Z

I (t)dt

√

N(ˆθ

− θ

) −→

N

0, I

(θ

)

, where

I

((λ

, γ

)) =





0

0





Maximum likelihood estimation for homoscedastic observations of the ODE

n observations at n discrete times tk

of xθ(

tk

) + ξ

, with ξk

∼ N (

_{0, CN}

(θ

)

Id

)

MLE=LSE

√

n

 ˆ

_θ

_{− θ}



−→

n→∞

N (

0, I

(θ

))

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Specicities of the statistical framework

Model:

We dene X

: dX

=

b(θ

,

X

)

dt + σ(θ

,

X

)

dB

,

X

=

x

_{∈ R}

⇒

separation of the parameters θ

, θ

required

(not estimated at the same rate)

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

MLE : 



θ

− θ



→ N

0, I

(θ

, θ

)

Existing discrete observation results : estimation of θ

at rate

√

n

Observations:

We observe X

for t

=

k∆, k ∈ {0, .., n}, t

∈ [

0, T ] (n∆ = T ), T is xed

Two dierent asymptotics:  → 0 & n (∆) is xed //  → 0 & n → ∞ (∆ → 0)

Notation: η = (θ

, θ

)

SIR:  =

N

, θ

= θ

= θ = (λ, γ)

, and I

(θ

, θ

)

equals the Markov jump process Fisher

Information matrix

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Main idea: study of g

(

t) (Multidimensionnal generalization of Genon-Catalot(90))

Gaussian process: Yt

=

xθ

(

t) + g

t), n obs. at regular time intervals tk

=

k∆, for

k = 1, .., n.

Denition : Resolvent matrix of the linearized ODE system Φθ

Let Φθ

be the invertible matrix solution of

dΦ_θ1

dt

(

t, t0

) =

(

xθ

(

t))Φθ

(

t, t0

),

with Φθ

(

t0

,

t0

) =

Id

.

Important property of g

gη(

tk

) = Φθ

(

tk

,

tk−1

)

gη(

tk−1

) +

√

∆

Z

(

Z

)

independent Gaussian vectors with covriance matrix S

S

=

Z

tk−1

Φ

(

_t

,

_s)Σ(θ

,

_x

(

_s))

Φ

(

_t

,

_s)ds

Function of the observations

Let y ∈ C [0, T ], R

Nk

(θ

,

y) = y(tk

) −

xθ

(

tk

) − Φθ

(

tk

,

tk−1

)



y(tk−1

) −

xθ

(

tk−1

) (= 

√

∆

Z

)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Back to the diusion process: dX

=

b(θ

,

X

)

dt + σ(θ

,

X

)

dB

(

Z

)

Gaussian familly ⇒ Likelihood tractable:

−

_{L∆,(η) = }

X

log[det(S

)] +

_∆

1

X

_Nk

_(θ

,

Y )(S

)

Nk

(θ

,

Y )

ˆ

θ

has good properties as  → 0, only if ∆ → 0 (n → +∞)

1. n xed,  → 0: General case (low frequency contrast with θ

unknown)

¯

U(θ

) =

X

k=1

t

_Nk

_(θ

_,

_{X )Nk}

_(θ

_,

_{X ) ⇒ Associated MCE ¯θ}

₌

_argmin

¯

U(θ

)

2. n xed,  → 0 : Case θ

=

f (θ

)

(low frequency contrast with information on θ

)

˜

U

) =

X

_Nk

_(θ

,

X )(˜S

)

Nk

(θ

,

X ) ⇒ ˜θ1,

=

argmin

˜

U

)

3. n → ∞,  → 0 (high frequency contrast)

ˇ

U

(θ

, θ

) = 

X

log[det(Σ(θ

,

X

))] +

1

∆

X

_N

(θ

,

X )Σ

(θ

,

X

)

N

(θ

,

X )

⇒ ˇ

θ

, ˇ

θ

=

argmin

¯

U,∆(η)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

What kind of distance is minimized: comparison with Least squares

Nk

(θ

,

y) = y(tk

) −

xθ

(

tk

) − Φθ

(

tk

,

tk−1

)



y(tk−1

) −

xθ

(

tk−1

)



N = 1000, R0

=

1.5, d = 3 days, 1 obs/day, T = 50 days

Figure:

Diusion (blue), x

(

t) (green)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

What kind of distance is minimized: comparison with Least squares

Zoom between t = 5 and t = 6

g

tk

) = Φθ

(

tk

,

tk−1

)

g

tk−1

) +

√

∆

Z

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

What kind of distance is minimized: comparison with Least squares

Zoom between t = 5 and t = 6

g

tk

) = Φθ

(

tk

,

tk−1

)

g

tk−1

) +

√

∆

Z

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

What kind of distance is minimized: comparison with Least square

Figure:

Distance to the deterministic model

Figure:

Comparison: N

(

X , θ

)

_{(blue) and X}

−

x

(

t

)

(green)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Results about dX

=

b(θ

,

X

)

dt + σ(θ

,

X

)

dB

Under classical regularity assumptions on b and σ, we proved

n xed,  → 0, low frequency contrast

Identiability assumption: θ1

6= θ

1

⇒ {∃

k, 1 ≤ k ≤ n, xθ

(

tk

) 6=

xθ

(

tk

)}.

1. General case (no information on θ

)



_θ

¯

− θ

−→

→0

N (

0, J

(θ

, θ

))

2. case θ

=

f (θ

)

(with information on θ

)



 ˜

θ

− θ



−→

→0

N (

0, I

(θ

, θ

))

3. n → ∞  → 0: high frequency contrast



_{( ˇ}

_θ

− θ

)

√

n( ˇ

θ

− θ

)



−→

N



0,



I

(θ

, θ

)

0

0

I

(θ

, θ

)



Remarks

Epidemics:  =

N

, θ = θ

= θ

is the same as for the

Markov jump process (all jumps observed)

J∆

is not optimal, but I∆

is, in the sense that I∆(θ

, θ

) −→

∆→0

Ib

(θ

, θ

)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Results on SIR for N = 10000, empirical mean estimators on 1000 runs and 95% theoretical CI

(R

=

1.2, d = 3)

Figure:

0: MLE, 1: ¯

θ

low frequency MCE (general case), 2: ˜

θ

low frequency MCE

(θ

= θ

), 3: ˇ

θ

high frequency MCE

About unpresented results

Good results even for N = 100 (our methods seem more robust than MLE)

Similar performance (w.r.t. MLE) on more sophisticated models

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

About ˜

θ

(Low frequency with information on θ

)

θ

unknown

1. ¯

U

(θ

) =

X

_N

(θ

,

X )N

(θ

,

X )

Figure:

Comparison between Data and ODE

(x

(

t))

Not good t of the data

With information on θ

2. ˜

U

(θ

) =

X

_N

(θ

,

X )

(˜

S

)

N

(θ

,

X )

Figure:

Evolution of det



Σ

(θ

,

x

1

(

t))



Too much weight on the boundaries

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

About ˜

θ

(Low frequency with information on θ

)

Comparing to high frequency contrast

3. ˇ

U

(θ

, θ

))

=



X

log [det (Σ

)]

+

X

_N

(θ

)Σ

N

(θ

)

where

Σ

= Σ(θ

,

X

)

Figure:

Previous results

Corrected contrast with information on θ

2

_{. ˜}

_U

(α)

=



X

log

hdet(˜S

)

i

+

X

_N

(θ

)

 ˜

S



N

(θ

)

Figure:

Corrected results

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Dierent shapes of the condence ellipsoids

SIR Model condence ellipsoids for the corrected contrast

N = 1000, (R0

,d) = {(1.5, 3), (1.5, 7), (5, 3), (5, 7)}

Number of observations:

n = 10 (blue)

,

n = 1obs/day ≈40 d.(green)

, n = 2000

(black),

MLE CR (red) (Theoretical limit)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Bias of MLE (N = 400; R

=

1.5; d = {3, 7}; (s

,

i

) = (

0.99, 0.01))

Too much variability

Figure:

Trajectories for d=3

Need of an empiric threshold: 5% of infected

Estimation results (d = 3)

Other values of d were investigated

Other thresholds: time of extinction, number max of

infected

Results (d = 7, time of extinction)

Zoom on the red trajectory (see Figure)

(

R0

,

d)MLE

= (

0.8749, 2.9945)

(

R0

,

d)LSE

= (

0.94, 2.5794)

(

R0

,

d)cont

= (

1.01, 3.0973)

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

Temporal dependence (SIRS) : λ

dicult to estimate

SIRS with constant immigration in I class

λ(

t) = λ0

(

1 + λ1sin(2πt/T ))

Values

R0

=

1.5; d = 3d;

per

=

2y,

λ

= {

0.05, 0.15}

Fixed: Tper

=

365, µ = 1/50Tper, ζ =

,

N = 10

_{, 1 obs/day(week) for 20 years}

Term for λ1

in Ib

(θ

)

very small ⇒

N > 10

_{for satisfactory CI}

λ

bifurcation parameter for the ODE

λ

=

0.05 (weak seasonality)

λ

=

0.15 (stronger seasonality)

Detailed results not shown

Main idea: R0

,d, δ well estimated

λ

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Inference approach: Work in progress

Outline

1

Characteristics of the epidemic process

Constraints imposed by the observation of the epidemic process

Simple mechanistic models

2

Various mathematical approaches for epidemic spread

Natural approach: Markov jump process

First approximation by ODEs

Gaussian approximation of the Markov jump process

Diusion approximation of the Markov jump process

3

Inference for discrete observations of diusion or Gaussian processes with small

diusion coecient

Contrast processes for xed or large number of observations

Correction of a non asymptotic bias

Comparison of estimators on simulated epidemics

4

Epidemics incompletely observed: partially and integrated diusion processes (Work

in progress)

Back to epidemic data

Inference approach: Work in progress

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Inference approach: Work in progress

Incidence for SIR models

Discrete observation of all the coordinates

Conned studies

Childhood diseases

SIR Model:

Incidence at t

:

Z

λ

S(t)

I (t)

N

dt

High frequency data ≈ λS(t

)

d small : new infected ≈ new removed,

Z

t1

γ

I (t)dt = R(t

) −

R(t

)

Diusion perspectives

Partial and discrete obs. of the diusion

process (Itô's formula)

Partial and Integrated discrete obs.

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Inference approach: Work in progress

Partially observed diusion process: initial idea

Previous main idea not directly applicable:

i(t) s(t) S_5 S_5 S_6 I_5 I_6 εgi(t5) εgi(t6) εgs(t6) εgs(t5) Φi , i(t6,t5)εgi(t5) Φs ,i(t6,t5)εgs(t5) ε√ΔZ6i

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Inference approach: Work in progress

Partially observed diusion process: initial idea

No good properties for n xed,  → 0.

Inference for discrete observations of diusion or Gaussian processes with small diusion coecient

Epidemics incompletely observed: partially and integrated diusion processes (Work in progress)Inference approach: Work in progress

Partially observed diusion process: initial idea

Use that Φθ

(

t + ∆, t) ≈ Id

as ∆ → 0

Function of the observations : It

−

_i(tk

) −

It

+

i(tk−1

)

Integrated diusion process

Integration of the relation : gη(

tk

) = Φθ

(

tk

,

tk−1

)

gη(

tk−1

) +

√

∆

Z

⇒

link between g and the integrated process : similar to Kalman ltering techniques