Uncertainty quantification in a nonlinear transmission model for Zika virus

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https://hal.archives-ouvertes.fr/hal-02190071

Submitted on 21 Jul 2019

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Uncertainty quantification in a nonlinear transmission model for Zika virus

Eber Dantas, Michel Tosin, Americo Cunha Jr

To cite this version:

Eber Dantas, Michel Tosin, Americo Cunha Jr. Uncertainty quantification in a nonlinear transmission model for Zika virus. Conference on Perspectives on Nonlinear Dynamics 2019 (PNLD 2019), Jul 2019, São Paulo, Brazil. 2018. �hal-02190071�

(2)

Uncertainty quantification in a nonlinear transmission model for Zika virus

Eber Dantas Michel Tosin Americo Cunha Jr

[email protected] [email protected] [email protected] NUMERICO – Nucleus of Modeling and Experimentation with Computers

Introduction

• Zika virus: global widespread and con- nection with congenital diseases

• 2016: Zika becomes a public health emergency of international concern

• Main vector: Aedes mosquitoes Aedes aegypti

• 30 countries in the last 20 years

• 140,000 confirmed cases in Brazil

• 3,000 confirmed cases of related birth de- fects and growth disorders in Brazil

Zika virus

Objectives

• Incorporate a general uncertainty quantification framework

• Perform sensitivity analysis and construct confidence bands

• Generate more robust predictions and diverse statistics

Computational Model

Compartmental Model

v

h h h h

v v

γ

(βvI_h)/N_h

α

v

(βhIv)/Nv

α

_h

Human population

Vector population δ ^N

^v

δ

δ δ

Dynamical system

,

dR

_h

dt = γ I

_h

, dC

dt = α

_h

E

_h

. + Initial Conditions

Quantities of interest (QoI)

• Cumulative cases of infectious: C (t) = R

_t

τ=0

α

_h

E

_h

(τ ) dτ

• New cases per week: N

^w

= C

_w

− C

_w₋₁

, w = 1 . . . 52 , N

¹

= C

₁

UQ Framework

Stochastic modeling

Computational Model

Y

_t

= M (X, t)

Input Parameters

X ∼ F_X

Output QoI

Y ∼ F_Y_t

₁

· · · X

_n

) ,

M⁰ = E[Y_t], Mⁱ(X_i) = E

Y_t|X_i

−M⁰, M^ij(X_i, X_j) = E

Y_t|X_i, X_j

−M⁰−Mⁱ−M^j.

Sobol’ Indices: interaction effect of inputs in u S

_u

= Var

M

^u

(X

_u

) Var

M (X)

Metamodelling: Polynomial Chaos

The Polynomial Chaos Expansion of model Y = M (X), for a multivari- ate orthonormal polynomial family Φ

_α

with coefficients y

_α

,

Y

_t

= X

α∈N^k

y

_α

(t) Φ

_α

(X) , enables analytic computation of Sobol Indices:

S

_u

= X

α∈ A^u

y

_α²

,

X

α∈ A\0

y

_α²

, A

^u

= { α ∈ A : i ∈ u ⇐⇒ α

_i

6 = 0 }

Maximum entropy principle

The most unbiased distribution of X maximizes the entropy

ε ^p

X

(X)

= − Z

Sⁿ

p

_X

^(x) ln p

_X

^{(x) dx} ^,

while abiding to µ + 1 restrictions Z

Sⁿ

p x ^{(x) dx} ^{= 1} ^,

Z

Sⁿ

g (x) p x ^{(x) dx} ⁼ ^b ^,

where g(x) : R

ⁿ

→ R

^µ

and b ∈ R

^µ

compiles the available information MaxEnt distribution with µ + 1 restrictions

p

_X

^{(x) =} ₁

_S_n

^{(x) exp(} ₋ ^λ

₀

^{) exp}



 −

µ

X

i=1

λ

_i

g

_i

^(x)





Results

Sobol’ Indices

Calibration tuning

10 20 30 40 50

time (weeks)

0 100 200 300

number of people

10³

Cumulative infectious

0 6 13 19 25

number of people

10³

10 20 30 40 50

time (weeks)

data first new

New cases

First probabilistic model

time (weeks)

0.0 0.8 1.5 2.3 3.0

number of people

10⁴

95% prob.

mean nominal data

N

^w

– CV : 10%

Third probabilistic model

R.V.: β

_h

, β

_v

, CV ∼ MaxEnt. b: support and mean for β , support for CV.

10 20 30 40 50

time (weeks)

0.0 1.5 3.0 4.5 6.0

number of people

10⁵

C (t) – CV ∼ U (5%, 10%)

10 20 30 40 50

time (weeks)

0.0 0.8 1.5 2.3 3.0

number of people

10⁴

95% prob.

mean nominal data

N

^w

– CV ∼ U (5%, 10%)

Statistics and predictions

Histograms for the time average of cumulative infectious

10^-5

1.0 1.8 2.5 3.3 4.0

number of people 0.0

0.5 1.0 1.5 2.0

probability density function

10⁵

PDF mean mean std 95% prob.

C (t) – CV : 0%

10^-5

1.0 1.8 2.5 3.3 4.0

number of people 0.0

0.5 1.0 1.5 2.0

10⁵

PDF mean mean std 95% prob.

C (t) – CV ∼ U (5%, 10%) Evolution of QoI histograms per epidemiological week

0.0 1.5 3.0 4.5 6.0

number of people

0.0 0.4 0.7 1.1 1.5

10^-4

10⁵ 5

7 10 15 20

25 30 35 45 52

C (t) – CV ∼ U (5%, 10%)

0.0 0.8 1.5 2.3 3.0

number of people

0.0 0.8 1.5 2.3 3.0

10^-3

10⁴ 5

7 10 15 20

25 30 35 45 52

N

^w

– CV ∼ U (5%, 10%) Cumulative distribution function for the time average until EW 20

1.0 1.3 1.5 1.8 2.0

number of people

0.0 0.3 0.5 0.8 1.0

probability

10⁰

10⁵

C (t) – CV ∼ U (5%, 10%)

0.8 1.0 1.2 1.5 1.7

number of people

0.0 0.3 0.5 0.8 1.0

probability

10⁰

10⁴

N

_w

– CV ∼ U (5%, 10%)

Final Remarks

• Implementation of a UQ framework with robust determination of pa- rameter distributions in the epidemiological context via MaxEnt

• Observation of general parametric behavior exposed via SA

• Investigation of dispersion influence, changes in skewness, evolution of stochastic QoIs and statistical simulations

Acknowledgements

References

[1] E. Dantas, M. Tosin and A. Cunha Jr, Calibration of a SEIR–SEI epidemic model to describe Zika virus outbreak in Brazil. Applied Mathematics and Computation, 338: 249-259, 2018.

doi.org/10.1016/j.amc.2018.06.024

[2] E. Dantas, M. Tosin and A. Cunha Jr. Uncertainty quantification in the nonlinear dynamics of Zika virus, 2019. hal.archives-ouvertes.fr/hal-02005320

[3] C. Soize. Uncertainty Quantification: an Accelerated Course with Advanced Applications in Compu- tational Engineering, Springer, 2017.

Uncertainty quantification in a nonlinear transmission model for Zika virus

Uncertainty quantification in a nonlinear transmission model for Zika virus

Eber Dantas Michel Tosin Americo Cunha Jr

[email protected] [email protected] [email protected] NUMERICO – Nucleus of Modeling and Experimentation with Computers

Introduction

• Zika virus: global widespread and con- nection with congenital diseases

• 2016: Zika becomes a public health emergency of international concern

• Main vector: Aedes mosquitoes Aedes aegypti

• 30 countries in the last 20 years

• 140,000 confirmed cases in Brazil

• 3,000 confirmed cases of related birth de- fects and growth disorders in Brazil

Zika virus

Objectives

• Incorporate a general uncertainty quantification framework

• Perform sensitivity analysis and construct confidence bands

• Generate more robust predictions and diverse statistics

Computational Model

Compartmental Model

v

h h h h

v v

γ

α

α

Human population

Vector population δ N

δ

δ δ

Dynamical system

dS

dt = − β

S

I

N

, dS

dt = δ N

− β

S

I

N

− δ S

,

dE

dt = β

S

I

N

− α

E

, dE

dt = β

S

I

N

− (α

+ δ) E

,

dI

dt = α

E

− γ I

, dI

dt = α

E

− δ I

,

dR

dt = γ I

, dC

dt = α

E

. + Initial Conditions

Quantities of interest (QoI)

• Cumulative cases of infectious: C (t) = R

α

E

(τ ) dτ

• New cases per week: N

= C

− C

Vector population δ ^N

ε ^p

^(x) ln p

^{(x) dx} ^,

p x ^{(x) dx} ^{= 1} ^,

g (x) p x ^{(x) dx} ⁼ ^b ^,

^{(x) =} ₁

^{(x) exp(} ₋ ^λ

^{) exp}