how to predict an epidemic of zika virus?coloquiomea/apresentacoes/cunha_2019.pdf · c. manore and...
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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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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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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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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(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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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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Introduction Dynamic Model Inverse Problem Sensitivity Analysis Uncertainty Quantification Ongoing Final Remarks
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!
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.