How to predict an epidemic of Zika virus?coloquiomea/apresentacoes/cunha_2019.pdf · C. Manore and M. Hyman, Mathematical Models for Fighting Zika Virus, SIAM News, May 2016. 10

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How to predict an epidemic of Zika virus?A challenge in nonlinear stochastic dynamics

Americo Cunha Jr

Universidade do Estado do Rio de Janeiro – UERJ

NUMERICO – Nucleo de Modelagem e Experimentacao Computacional

http://numerico.ime.uerj.br

In collaboration with: Eber Dantas (UERJ)Michel Tosin (UERJ)

COLMEA 2019 - IM-UFRJMarch 28, 2019

Rio de Janeiro, Brazil

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Outline

1 Introduction

2 Dynamic Model

3 Inverse Problem

4 Sensitivity Analysis

5 Uncertainty Quantification

6 Ongoing

7 Final Remarks

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Section 1

Introduction

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Zika virus (ZIKV)

Member of Flaviviridae virus family

First isolated in 1947 at Uganda, Africa

Mainly spread by Aedes mosquitoes

W.H.O declared it a public healthemergency of international concern

More than 140,000 confirmed cases inBrazil since 2015

International consensus that ZIKV is acause of:

Guillain–Barre syndrome

Microcephaly

Zika virus

Aedes aegypti

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Global outbreak of Zika virus

World Map of Areas with Risk of Zika

Centers for Disease Control and Prevention, World Map of Areas with Risk of Zika, March 2018.

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Zika virus outbreak in Brazil

New cases in Brazil by epidemiological week of 2016

0

6

12

17

23

num

ber

of p

eopl

e

103

10 20 30 40 50

time (weeks)

Ministerio da Saude. Obtencao de numero de casos confirmados de zika, por municıpio e semana

epidemiologica. https: // bit. ly/ 2OVgGGt

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Dengue virus (DENV)

Member of Flaviviridae virus family

Mainly spread by Aedes mosquitoes, as inthe case for Zika virus

Probable cases in Brazil:

> 170,000 in 2018> 250,000 in 2017> 3 million in 2016 and 2015

Dengue virus

Aedes aegypti

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Other Arbovirus

ARthropod-BOrne virus

Yellow Fever: South America and Africa(261 deaths in Brazil in 2017)

Chikungunya: worldwide(> 204,000 confirmed cases in Brazil since 2015)

Rift Valley fever: Africa and Arabian Peninsula(ongoing outbreak in Kenya by June 2018)

Countries and territories where chikungunya cases have been reported* (as of May 29, 2018)

*Does not include countries or territories where only imported cases have been documented.

Data table: Countries and territories where chikungunya cases have been reported

AFRICA ASIA AMERICAS

Angola Bangladesh Anguilla Panama

Benin Bhutan Antigua and Barbuda Paraguay

Burundi Cambodia Argentina Peru

Cameroon China Aruba Puerto Rico

Central African Republic India Bahamas Saint Barthelemy

Comoros Indonesia Barbados Saint Kitts and Nevis

Cote d’Ivoire Laos Belize Saint Lucia

Dem. Republic of the Congo Malaysia Bolivia Saint Martin

Djibouti Maldives Brazil

Saint Vincent & the Grenadines

Equatorial Guinea Myanmar (Burma) British Virgin Islands Sint Maarten

Gabon Nepal Cayman Islands Suriname

Guinea Pakistan Colombia Trinidad and Tobago

Kenya Philippines Costa Rica Turks and Caicos Islands

Madagascar Saudi Arabia Cuba United States

Malawi Singapore Curacao US Virgin Islands

Mauritius Sri Lanka Dominica Venezuela

Mayotte Thailand Dominican Republic

Mozambique Timor-Leste Ecuador

Nigeria Vietnam El Salvador OCEANIA/PACIFIC ISLANDS

Republic of the Congo Yemen French Guiana American Samoa

Reunion Grenada Cook Islands

Senegal Guadeloupe Federal States of Micronesia

Seychelles EUROPE Guatemala Fiji

Sierra Leone France Guyana French Polynesia

Somalia Italy Haiti Kiribati

South Africa Spain Honduras Marshall Islands

Sudan Jamaica New Caledonia

Tanzania Martinique

Papua New Guinea

Uganda Mexico Samoa

Zimbabwe Montserrat Tokelau

Nicaragua Tonga

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Powered by TCPDF (www.tcpdf.org)Powered by TCPDF (www.tcpdf.org)Powered by TCPDF (www.tcpdf.org)

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Chikungunya cases (May 2018)

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Other Arbovirus

Japanese encephalitis: Southeast Asia, Western Pacific

West Nile virus: widely established from Canada to Venezuela

Both transmitted by the Culex mosquitoes

West Nile virus activity in USA (July 2018)

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Typical questions to be answered

How many people will the outbreak potentially infect?

How far and how quickly will the disease spread?

What areas and people are at highest risk, and when are theymost at risk?

How can we best make use of limited resources?

How can we best slow or prevent the outbreak and protectvulnerable populations?

C. Manore and M. Hyman, Mathematical Models for Fighting Zika Virus, SIAM News, May 2016.

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Typical questions to be answered

How many people will the outbreak potentially infect?

How far and how quickly will the disease spread?

What areas and people are at highest risk, and when are theymost at risk?

How can we best make use of limited resources?

How can we best slow or prevent the outbreak and protectvulnerable populations?

Mathematical models tosimulate Zika virus spread

can provide importantguidance and insight to

these questions.

C. Manore and M. Hyman, Mathematical Models for Fighting Zika Virus, SIAM News, May 2016.

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Research objectives

Develop an epidemic model to describe the recent outbreak ofZika virus in Brazil

Verify (qualitatively and quantitatively) the epidemic modelcapacity of prediction

Calibrate this epidemic model with real data to obtain reliablepredictions

Construct a stochastic model to deal with data uncertaintiesand made more robust predictions

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Section 2

Dynamic Model

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

Rate of change of S = Input of S - Output of S

dS

dt=

bN︸︷︷︸Births

+ γI︸︷︷︸Recovery

− β

I

NS︸ ︷︷ ︸

Infections

+ µS︸︷︷︸Mortality

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

Rate of change of S = Input of S - Output of S

dS

dt=

bN︸︷︷︸Births

+ γI︸︷︷︸Recovery

− β

I

NS︸ ︷︷ ︸

Infections

+ µS︸︷︷︸Mortality

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

Rate of change of S = Input of S - Output of S

dS

dt=

bN︸︷︷︸Births

+ γI︸︷︷︸Recovery

− β

I

NS︸ ︷︷ ︸

Infections

+ µS︸︷︷︸Mortality

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

dS

dt=

bN︸︷︷︸Births

+ γI︸︷︷︸Recovery

− β

I

NS︸ ︷︷ ︸

Infections

+ µS︸︷︷︸Mortality

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

Rate of change of I = Input of I - Output of I

dS

dt=

bN︸︷︷︸Births

+ γI︸︷︷︸Recovery

− β

I

NS︸ ︷︷ ︸

Infections

+ µS︸︷︷︸Mortality

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SIS model

bN Sµ

βI/NIµ

γ

S - Population of susceptibleI - Population of infectedN - Total populationβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

Rate of change of I = Input of I - Output of I

dI

dt= β

S

NI︸︷︷︸

Infections

−

γI︸︷︷︸Recovery

+ µI︸︷︷︸Mortality

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SIS model dynamical system

dS

dt= bN + γ I −

(β

I

N+ µ

)S

dI

dt= β

I

NS − (γ + µ) I

+ initial conditions

S - Population of susceptibleI - Population of infectedN - Total population

β - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

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SIR model

bN Sµ

βI/NIµ

γRµ

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SIR model dynamical system

dS

dt= bN + γ I −

(β

I

N+ µ

)S

dI

dt= β

I

NS − (γ + µ) I

dR

dt= γ I − µR

+ initial conditions

S - Population of susceptibleI - Population of infectedR - Population of recoveredN - Total population

β - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

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SEIR model

bN Sµ

βI/NEµ

αIµ

γRµ

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SEIR model dynamical system

dS

dt= bN − β I

NS − µ S

dE

dt= β

I

NS − (α + µ)E

dI

dt= αE − (γ + µ) I

dR

dt= γ I − µR

+ initial conditions

S - Population of susceptibleE - Population of exposedI - Population of infectiousR - Population of recoveredN - Total population

α - Incubation ratioβ - Transmission rateγ - Recovery rateb - Birth rateµ - Mortality rate

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SEIR-SEI model for Zika virus dynamics

human population

Sh

(βhI )v /Nv Ehαh Ih

γRh

vector population

Nδv Sv

δ

(βvI )h /Nh Ev

δ

αv Iv

δ

A. J. Kucharski et al. Transmission Dynamics of Zika Virus in Island Populations: A Modelling

Analysis of the 2013–14 French Polynesia Outbreak. PLOS Neglected Tropical Diseases, 2016.

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Associated dynamical system

dSh

dt= −βh

IvNv

Sh

dEh

dt= βh

IvNv

Sh − αh Eh

dIhdt

= αh Eh − γ Ih

dRh

dt= γ Ih

dSv

dt= δ − βv Sv

IhNh− δ Sv

dEv

dt= βv Sv

IhNh− (δ + αv )Ev

dIvdt

= αv Ev − δ Iv

dC

dt= αh Eh

+ initial conditions

S - Population of susceptibleV - Population of vaccinatedE - Population of exposedI - Population of infectiousR - Population of recovered

C - Infected humans cumulativeN - Total populationα - Incubation ratioβ - Transmission rateγ - Recovery rate

δ - Vector lifespan ratioσ - Infection rate of vaccinatedν - Fraction of vaccinatedh - Human-relatedv - Vector-related

A. J. Kucharski et al. Transmission Dynamics of Zika Virus in Island Populations: A Modelling

Analysis of the 2013–14 French Polynesia Outbreak. PLOS Neglected Tropical Diseases, 2016.

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Model parameters and outbreak data

open scientific literature

Brazilian health system

parameter value unit

αh 1/5.9 days−1

αv 1/9.1 days−1

γ 1/7.9 days−1

δ 1/11 days−1

βh 1/11.3 days−1

βv 1/8.6 days−1

N 206× 106 people

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Time series of susceptible humans

10 20 30 40 50time (weeks)

2057.0

2057.8

2058.5

2059.3

2060.0

num

ber

of p

eopl

e

105

SIS model

10 20 30 40 50time (weeks)

2057.0

2057.8

2058.5

2059.3

2060.0

num

ber

of p

eopl

e

105

SIR model

10 20 30 40 50time (weeks)

2057.0

2057.8

2058.5

2059.3

2060.0

num

ber

of p

eopl

e

105

SEIR model

10 20 30 40 50tempo (semanas)

2057.0

2057.8

2058.5

2059.3

2060.0

núm

ero

de p

esso

as

105

SEIR-SEI model

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Time series of infectious humans

10 20 30 40 50time (weeks)

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

SIS model

10 20 30 40 50time (weeks)

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

SIR model

10 20 30 40 50time (weeks)

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

SEIR model

10 20 30 40 50time (weeks)

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

SEIR-SEI model

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Quantities of interest (QoI)

mathematical

model

parameters modelresponse

observation

operator

observable

QoI 1: cumulative number of infectious

Ct =

∫ t

τ=0αh Eh(τ) dτ

QoI 2: new infectious cases

Nw = Cw − Cw−1, (w = 2, 3, · · · , 52)

N1 = C1

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Time series for QoI’s (SEIR-SEI model)

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

cumulative number of infectious

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

new infectious cases

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Time series for QoI’s (SEIR-SEI model)

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

cumulative number of infectious

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

new infectious cases

Mathematical model doesnot represent the reality

with this set of parameters

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Section 3

Inverse Problem

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Calibration of the model

Uncalibrated Model

Model

Observations

Calibrated Model

Model

Observations

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Forward and inverse problem

forward problem

inverse problem

mathematical

model

parameters

fittingparameters

observable

observations

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Inverse problem formulation

data space: F = RM

parameter space: C ={α ∈ R12 | αmin ≤ α ≤ αmax

}observation vector: y = (y1, y2, · · · , yM) ∈ F

prediction vector: φ (α) = (φ1, φ2, · · · , φM) ∈ F

misfit function:

J(α) = ||y − φ (α) ||2F =M∑

m=1

∣∣∣ym − φm (α)∣∣∣2

Find a vector of parameters such that

α∗ = arg minα∈C

J(α).

=⇒ Q-wellposed: existence, uniqueness, unimodality and local stability

=⇒ Solution algorithm: bounded trust-region-reflective

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Inverse problem formulation

data space: F = RM

parameter space: C ={α ∈ R12 | αmin ≤ α ≤ αmax

}observation vector: y = (y1, y2, · · · , yM) ∈ F

prediction vector: φ (α) = (φ1, φ2, · · · , φM) ∈ F

misfit function:

J(α) = ||y − φ (α) ||2F =M∑

m=1

∣∣∣ym − φm (α)∣∣∣2

Find a vector of parameters such that

α∗ = arg minα∈C

J(α).

=⇒ Q-wellposed: existence, uniqueness, unimodality and local stability

=⇒ Solution algorithm: bounded trust-region-reflective

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Inverse problem formulation

data space: F = RM

parameter space: C ={α ∈ R12 | αmin ≤ α ≤ αmax

}observation vector: y = (y1, y2, · · · , yM) ∈ F

prediction vector: φ (α) = (φ1, φ2, · · · , φM) ∈ F

misfit function:

J(α) = ||y − φ (α) ||2F =M∑

m=1

∣∣∣ym − φm (α)∣∣∣2

Find a vector of parameters such that

α∗ = arg minα∈C

J(α).

=⇒ Q-wellposed: existence, uniqueness, unimodality and local stability

=⇒ Solution algorithm: bounded trust-region-reflective

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Calibrated model response

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

cumulative number of infectious

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

new infectious cases

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Calibrated model response

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

cumulative number of infectious

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

new infectious cases

Robust predictions demandssome kind of “certification”.

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Section 4

Sensitivity Analysis

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Parametric study for βh

10 20 30 40 50time (weeks)

0.0

100.0

200.0

300.0

400.0

num

ber

of p

eopl

e

104

85%90%95%nominal105%110%

cumulative number of infectious

0.0

25.0

50.0

75.0

100.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Parametric study for αh

10 20 30 40 50time (weeks)

0.0

50.0

100.0

150.0

200.0

num

ber

of p

eopl

e

103

85%90%95%nominal105%110%

cumulative number of infectious

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Parametric study for γ

10 20 30 40 50time (weeks)

0.0

50.0

100.0

150.0

200.0

num

ber

of p

eopl

e

103

85%90%95%nominal105%110%

cumulative number of infectious

0.0

6.3

12.5

18.8

25.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Parametric study for βv

10 20 30 40 50time (weeks)

0.0

50.0

100.0

150.0

200.0

num

ber

of p

eopl

e

104

85%90%95%nominal105%110%

cumulative number of infectious

0.0

12.5

25.0

37.5

50.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Parametric study for αv

10 20 30 40 50time (weeks)

0.0

75.0

150.0

225.0

300.0

num

ber

of p

eopl

e

103

85%90%95%nominal105%110%

cumulative number of infectious

0.0

7.5

15.0

22.5

30.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Parametric study for δ

10 20 30 40 50time (weeks)

0.0

75.0

150.0

225.0

300.0

num

ber

of p

eopl

e

103

85%90%95%nominal105%110%

cumulative number of infectious

0.0

6.3

12.5

18.8

25.0

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

85%90%95%nominal105%110%

new infectious cases

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Variance-based sensitivity analysis

Mathematical model:

Y =M(X), Xi ∼ U(0, 1), (i.i.d.)

Hoeffding-Sobol’ decomposition:

Y =M0 +∑

1≤i≤nMi (Xi ) +

∑1≤i<j≤n

Mij(Xi ,Xj) + · · ·+M1···n(X1 · · ·Xn)

An orthogonal decomposition in terms of conditional expectations:

M0 = E {Y }Mi (Xi ) = E

{Y |Xi

}−M0

Mij(Xi ,Xj) = E{Y |Xi ,Xj

}−M0 −Mi −Mj

etc

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Sobol’ indices

Total variance:

D = Var[M(X )

]=∑

u⊂{1,··· ,k}

Var[Mu(Xu)

]First order Sobol’ indices:

Si = Var[Mi (Xi )

]/D

(quantify the additive effect of each input separately)

Second order Sobol’ indices:

Sij = Var[Mij(Xi ,Xj)

]/D

(quantify interaction effect of inputs Xi and Xj)

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Metamodelling via Polynomial Chaos

Assuming Y =M(X) has finite variance, then it admits aPolynomial Chaos expansion

Y =∑α∈Nk

yα Φα(X)

where

Φα(X): multivariate orthonormal polynomials

yα: real-valued coeficients to be determined

D. Xiu, and G. Karniadakis, The Wiener-Askey Polynomial Chaos for Stochastic Differential Equa-

tions. SIAM Journal on Scientific Computing, 24: 619-644, 2002.

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PC-based Sobol’ indices

For computational purposes, a truncated PCE is employed

Y ≈∑α∈A

yα Φα(X)

Thus, Sobol’ indices are given by

Su = Du/D =∑α∈Au

y2α

/∑α∈A\0

y2α

Au ={α ∈ A : i ∈ u⇐⇒ αi 6= 0}

Sobol’ indices of any order can be obtained, analytically, from thecoefficients of the PC expansion!

B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering &

System Safety, 2016, 93(7): 964–979, 2008.

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Global sensitivity analysis: first order

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Global sensitivity analysis: second order

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Global sensitivity analysis: total order

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Global sensitivity analysis: general overview

→ Two most relevant: δ and βH (75% variance around 7th EW )

→ Third most, γ, mainly by nonlinear interactions with δ and βH

→ Parameters limited to nonlinear interactions have, in general,delayed effects (significant for EW > 15)

→ (sparsity-of-effects principle) Higher order interactions haveminor effect: 1st and 2nd are 99.8–96.7% variance on 5–10th EW

Around 7th EW → uncertainty propagation of {βh,δ}

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Section 5

Uncertainty Quantification

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Uncertainty Quantification (UQ) framework

Mathematical model:Y =M(X)

General steps for UQ:

1 Stochastic modeling−→ characterization of inputs uncertainties

(MaxEnt Principle)

2 Uncertainty propagation−→characterization of output uncertainties

(Monte Carlo Method)

3 Response certification−→ specification of reliability levels for predictions

(Nonparametric Statistical Inference)

C. Soize, Uncertainty Quantification: An Accelerated Course with Advanced Applications in Com-

putational Engineering, Springer, 2017.

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Maximum Entropy Principle (MaxEnt)

Among all the probability distributions, consistent with the knowninformation about a random parameter, choose the one whichcorresponds to the maximum of entropy (MaxEnt).

MaxEnt distribution = most unbiased distribution

Entropy of the random variable X is defined as

S (pX ) = −∫RpX (x) ln pX (x) dx ,

“measure for the level of uncertainty”

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MaxEnt optimization problem

Maximize

S (pX ) = −∫RpX (x) ln pX (x) dx ,

respecting N + 1 constraints (known information) given by∫Rgk (X ) pX (x) dx = mk , k = 0, · · · ,N,

where the gk are known real functions, with g0(x) = 1.

MaxEnt general solution

pX (x) = 1K(x) exp (−λ0) exp

− N∑k=1

λk gk(x)

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MaxEnt optimization problem

Maximize

S (pX ) = −∫RpX (x) ln pX (x) dx ,

respecting N + 1 constraints (known information) given by∫Rgk (X ) pX (x) dx = mk , k = 0, · · · ,N,

where the gk are known real functions, with g0(x) = 1.

MaxEnt general solution

pX (x) = 1K(x) exp (−λ0) exp

− N∑k=1

λk gk(x)

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Philosophy of MaxEnt Principle

The parameter of interesthas a unknown distribution

Distributions arbitrarilychosen can be coarse andbiased

A conservative strategy isto use the most unbiased(MaxEnt) distribution

real

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Philosophy of MaxEnt Principle

The parameter of interesthas a unknown distribution

Distributions arbitrarilychosen can be coarse andbiased

A conservative strategy isto use the most unbiased(MaxEnt) distribution

realbiased

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Philosophy of MaxEnt Principle

The parameter of interesthas a unknown distribution

Distributions arbitrarilychosen can be coarse andbiased

A conservative strategy isto use the most unbiased(MaxEnt) distribution

realbiased

unbiased

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Uncertainty propagation through the model

Monte Carlo Method

pre-processing

generationof scenarios

X1

...

XM

known FX

generator ofrandom vector X

processing

solution ofmodel equations

U = h(X)

computationalmodel

deterministic solverof u = h(x)

post-processing

computationof statistics

U1 = h(X1)

...

UM = h(XM )

estimated FU

statistical inferenceto estimate convergenceand distribution of U

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Probabilistic model 1

Random variables: βh and δ

Available information: support and mean (nominal) value

MaxEnt distribution

pX (x) = 1[a,b](x) exp (−λ0 − λ1 x)

“truncated exponential (2 parameters)”

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Confidence band for the QoIs

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

cumulative number of infectious

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

new infectious cases

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Probabilistic model 2

Random variables: βh and δ

Available information: support, mean (nominal) value and dispersion

MaxEnt distribution

pX (x) = 1[a,b](x) exp(−λ0 − λ1 x − λ2 x

2)

“truncated exponential (3 parameters)”

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Confidence band for the QoIs

βh dispersion = 5% , δ dispersion = 5%

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

cumulative number of infectious

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

new infectious cases

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Confidence band for the QoIs

βh dispersion = 10% , δ dispersion = 5%

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

cumulative number of infectious

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

new infectious cases

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Confidence band for the QoIs

βh dispersion = 10% , δ dispersion = 10%

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

cumulative number of infectious

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

new infectious cases

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Probabilistic model 3

Random variables: βh, δ and σ

Available information for βh and δ: support, mean (nominal) value

Distribution for βh and βv

pX (x) = 1[a,b](x) exp(−λ0 − λ1 x − λ2 x

2)

Available information for σ: support

MaxEnt distribution for σ

pX (x) = 1[a,b](x)1

b − a

“uniform”

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Confidence band for the QoIs

random dispersion ∼ U(5%, 10%)

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

cumulative number of infectious

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

new infectious cases

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Confidence band for the QoIs

no dispersion

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

σ = {5%,5%}

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

σ = {10%,5%}

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0nu

mbe

r of

peo

ple

104

95% prob. mean nominal data

σ = {10%,10%}

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

σ ∼ U(5%, 10%)

10 20 30 40 50

time (weeks)

0.0

1.5

3.0

4.5

6.0

num

ber

of p

eopl

e

105

10 20 30 40 50

time (weeks)

0.0

0.8

1.5

2.3

3.0

num

ber

of p

eopl

e

104

95% prob. mean nominal data

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Evolution of QoIs PDFs

random dispersion ∼ U(5%, 10%)

0.0 1.5 3.0 4.5 6.0

number of people

0.0

0.4

0.7

1.1

1.5

prob

abili

ty

10-4

105

57101520

2530354552

cumulative number of infectious

0.0 0.8 1.5 2.3 3.0

number of people

0.0

0.8

1.5

2.3

3.0

prob

abili

ty

10-3

104

57101520

2530354552

new infectious cases

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Time-averaged 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

pro

babi

lity

dens

ity fu

nctio

n

105

PDFmeanmean std 95% prob.

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

pro

babi

lity

dens

ity fu

nctio

n

105

PDFmeanmean std 95% prob.

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

pro

babi

lity

dens

ity fu

nctio

n

105

PDFmeanmean std 95% prob.

no dispersion σ = {10%,10%} σ ∼ U(5%, 10%)

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(mean) Cumulative infectious CDF until EW 20

σ ∼ U(5%, 10%)

1.0 1.3 1.5 1.8 2.0

number of people

0.0

0.3

0.5

0.8

1.0

prob

abili

ty

100

105

Statistics of C

mean = 1,47× 105

std. dev. = 1,53× 104

skewness = 0.084kurtosis = 2.605

P(C ≥ c∗) = 87.10%

c∗ = 130,000

Half the maximum C (data)

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

(mean) New cases CDF until 10th EW

σ ∼ U(5%, 10%)

1.0 1.3 1.5 1.8 2.0

number of people

0.0

0.3

0.5

0.8

1.0

prob

abili

ty

100

104

Statistics of Nw

mean = 1,57× 104

std. dev. = 1,35× 103

skewness = −0.032kurtosis = 2.656

P(Nw ≥ NC ∗) = 83.40%

NC∗ = 14,440

average NC (data) until EW 10

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Section 6

Ongoing

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Investigation of control strategies

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

SVEIR-SEI model for Zika virus dynamics

human population

Sh

ν

Vh

(βhI )v /Nv

(σβhI )v /Nv

Ehαh Ih

γRh

vector population

Nδv Sv

δ

(βvI )h /Nh Ev

δ

αv Iv

δ

H. S. Rodrigues et al. Vaccination models and optimal control strategies to dengue. Mathematical

Biosciences, 247 (2014) 1–12.

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Associated dynamical system

dSh

dt= −

(βh

IvNv

+ ν

)Sh

dVh

dt= νSh − σβh

IvNv

Vh

dEh

dt= βh (Sh + σ Vh)

IvNv−αh Eh

dIhdt

= αh Eh − γ Ih

dRh

dt= γ Ih

dSv

dt= δ − βv Sv

IhNh− δ Sv

dEv

dt= βv Sv

IhNh− (δ + αv )Ev

dIvdt

= αv Ev − δ Iv

dC

dt= αh Eh

+ initial conditionsS - Population of susceptibleV - Population of vaccinatedE - Population of exposedI - Population of infectiousR - Population of recovered

C - Infected humans cumulativeN - Total populationα - Incubation ratioβ - Transmission rateγ - Recovery rate

δ - Vector lifespan ratioσ - Infection rate of vaccinatedν - Fraction of vaccinatedh - Human-relatedv - Vector-related

H. S. Rodrigues et al. Vaccination models and optimal control strategies to dengue. Mathematical

Biosciences, 247 (2014) 1–12.

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Time series for QoI’s (SVEIR-SEI model)

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

cumulative number of infectious

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

new infectious cases

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Quantification of model discrepancy

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Calculation of model discrepancy

Conventional statistical calibration:

y︸︷︷︸truth

= f (x ,p)︸ ︷︷ ︸model

+ ε︸︷︷︸error

Novel approach:

y︸︷︷︸truth

≈ f (x ,pε)︸ ︷︷ ︸model

, pε =∑k

αk Φk(ξ)

Bayesian inversion to identify α

π(model | data)︸ ︷︷ ︸posterior

∝ π(data | model)︸ ︷︷ ︸likelihood

×π(model)︸ ︷︷ ︸prior

K. Sargsyan, H. N. Najm and R. Ghanem, On the statistical calibration of physical models.

International Journal of Chemical Kinetics, 47 (2015) 246-276.

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Section 7

Final Remarks

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Concluding remarks

Contributions:

Development of an epidemic model to describe Brazilianoutbreak of Zika virus

Calibration of this model with real epidemic data

Construction of a parametric probabilistic model ofuncertainties

Ongoing research:

Investigate the effectiveness of different control strategies

Quantify model discrepancy in a nonparametric way

Future directions:

Scenarios exploration with active subspace method

Data-driven identification of epidemiological models

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Acknowledgments

Invitation for the talk:

Profa. Maria Eulalia Vares

Prof. Leandro Pimentel

Financial support:

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Thank you for your attention!

[email protected]

www.americocunha.org

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.https://doi.org/10.1016/j.amc.2018.06.024

E. Dantas, M. Tosin and A. Cunha Jr,

Uncertainty quantification in the nonlinear dynamics of Zika virus, 2019 (in preparation).https://hal.archives-ouvertes.fr/hal-02005320

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nominal parameters

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Nominal parameters and initial conditions

α value unit

αh 1/5.9 days−1

αv 1/9.1 days−1

γ 1/7.9 days−1

δ 1/11 days−1

βh 1/11.3 days−1

βv 1/8.6 days−1

N 206× 106 people

S ih 205,953,959 people

E ih 8,201 people

I ih 8,201 people

R ih 29,639 people

S iv 0.99956 ——

E iv 2.2× 10−4 ——

I iv 2.2× 10−4 ——

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Model response with nominal parameters

10 20 30 40 50time (weeks)

205.8

205.9

206.0

num

ber

of p

eopl

e

106

Susceptible humans

10 20 30 40 50time (weeks)

20

80

140

200

num

ber

of p

eopl

e

103

Recovered humans

5 10 15 20 25 30 35 40 45 50time (weeks)

0.99955

0.99970

0.99985

1.00000

prop

ortio

n of

vec

tors

100

Susceptible vectors

10 20 30 40 50time (weeks)

0

7

13

20

num

ber

of p

eopl

e

103

Exposed humans

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

Cumulative infectious

10 20 30 40 50time (weeks)

0.0

0.8

1.7

2.5

prop

ortio

n of

vec

tors

10-4

Exposed vectors

10 20 30 40 50time (weeks)

0

1

2

3

num

ber

of p

eopl

e

104

Infectious humans

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

New cases

10 20 30 40 50time (weeks)

0.0

0.8

1.7

2.5

prop

ortio

n of

vec

tors

10-4

Infectious vectors

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Inverse Problem

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Well-posedness

Let the forward map φ : E → F associates to each parametervector x , restricted to be on the set of admissible values C in theparameter space E , an observable vector in the data space F . TheNLS problem is Quadratically (Q-) wellposed if, and only if, φ(C )possesses an open neighborhood ϑ such that

1 Existence and uniqueness: for every z ∈ ϑ, the inverseproblem has a unique solution x

2 Unimodality: for every z ∈ ϑ, the objective function x J(x)has no parasitic stationary point

3 Local stability: the mapping z x is locally Lipschitzcontinuous from (ϑ, || · ||F ) to (C , | · ||E ).

G. Chavent. Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and

Step-by-Step Guide for Applications. Springer, 2010.

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Well-posedness

Theorem

Let the follow finite dimension minimum set of hypothesis hold:

E = finite dimensional vector space, with norm || · ||E ,C = closed, convex subset of E ,

Cη = convex open neighborhood of C in E ,

F = Hilbert space, with norm || · ||F ,z ∈ F ,

φ : Cη F is twice differentiable along segments of Cη,

and: V =∂

∂tφ((1− t)x0 + tx1),A =

∂2

∂t2φ((1− t)x0 + tx1)

are continuous functions of x0, x1 ∈ Cη and t ∈ [0, 1].

Then, if moreover C is small enough for the deflection condition θ ≤ π/2 to

hold, x is OLS-identifiable on C , or equivalently: the NLS problem is

Q-wellposed on C .

G. Chavent. Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and

Step-by-Step Guide for Applications. Springer, 2010.

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Calibrated parameters and initial conditions

α TRR input lb ub TRR output

αh 1/5.9 1/12 1/3 1/12αv 1/9.1 1/10 1/5 1/10γ 1/7.9 1/8.8 1/3 1/3δ 1/11 1/21 1/11 1/21βh 1/11.3 1/16.3 1/8 1/10.40βv 1/8.6 1/11.6 1/6.2 1/7.77

S ih 205,953,959 0.9× N N 205,953,534

E ih 8,201 0 10,000 6,827

I ih 8,201 0 10,000 10,000

S iv 0.9996 0.99 0.999 0.999

E iv 2.2× 10−4 0 1 4.14× 10−4

I iv 2.2× 10−4 0 1 0

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Remarks on the calibration

Reasonable parameters

cumulative number of infectious overshoots data by only 5.74%

Initial infectious humans is approximately 10,000 individuals

Peak value of new infectious cases differs from the data maximumby 10.57%

Peak of new infectious cases occurs two weeks before the peakof the data

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Comparison of infectious humans curves

10 20 30 40 50time (weeks)

0

1

2

3

num

ber

of p

eopl

e

105

First calibration

10 20 30 40 50time (weeks)

0

3

7

10

num

ber

of p

eopl

e

103

Second calibration

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Comparison of infectious humans curves

10 20 30 40 50time (weeks)

0

20

40

60

num

ber

of p

eopl

e

103

10005000100002000030000400005000060000

Curves for various initial infectioushumans values

10 20 30 40 50time (weeks)

0

5

10

15

num

ber

of p

eopl

e

103

10005000100002000030000400005000060000

Zoom in the local peak regionof the image to the left

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Comparison of cumulative and new infectious curves

10 20 30 40 50time (weeks)

0

167

333

500

num

ber

of p

eopl

e

103

Data10005000

100002000030000

400005000060000

cumulative number of infectious

10 20 30 40 50time (weeks)

0

10

20

30

num

ber

of p

eopl

e

103

Data10005000100002000030000400005000060000

new infectious cases

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Calibrated model response

10 20 30 40 50time (weeks)

205.7

205.8

206.0

num

ber

of p

eopl

e

106

Susceptible humans

10 20 30 40 50time (weeks)

0

117

233

350

num

ber

of p

eopl

e

103

Recovered humans

5 10 15 20 25 30 35 40 45 50time (weeks)

0.99900

0.99933

0.99967

1.00000

prop

ortio

n of

vec

tors

100

Susceptible vectors

10 20 30 40 50time (weeks)

0

12

23

35

num

ber

of p

eopl

e

103

Exposed humans

10 20 30 40 50time (weeks)

0

100

200

300

num

ber

of p

eopl

e

103

datamodel

Cumulative infectious

10 20 30 40 50time (weeks)

0.0

1.5

3.0

4.5

prop

ortio

n of

vec

tors

10-4

Exposed vectors

10 20 30 40 50time (weeks)

0

3

7

10

num

ber

of p

eopl

e

103

Infectious humans

0

6

13

19

25

num

ber

of p

eopl

e

103

10 20 30 40 50time (weeks)

datamodel

New cases

10 20 30 40 50time (weeks)

0.0

0.6

1.2

1.8

prop

ortio

n of

vec

tors

10-4

Infectious vectors

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Monte Carlo convergence

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Study of convergence for MC simulation

Stochastic dynamic model:

U(t, ω) = f (U(ω, t))

Convergence metric for of Monte Carlo simulation:

conv(ns) =

1

ns

ns∑n=1

∫ tf

t0

‖ U(t, ωn) ‖2 dt

1/2

C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties

for predictive models in structural dynamics. Journal of Sound and Vibration, 288: 623–652, 2005.

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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks

Study of convergence for MC simulation

0.00 0.34 0.68 1.02 number of MC realizations

38.910

38.915

38.921

38.926 c

onve

rgen

ce m

etric

108

103

Figure: MC convergence metric as function of the number of realizations.