Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability

1. COVID-19 pandemic control:
balancing detection policy and lockdown
intervention under ICU sustainability
Arthur Charpentier, UQAM
(with Romuald Elie, Mathieu Laurière & Viet Chi Tran)
https://www.medrxiv.org/content/10.1101/2020.05.13.20100842v2
Septembre 2Q20 - modcov19
Analyse coût-efficacité de stratégies de contrôle épidémique
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2. Valeur de la vie (introduction)
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3. SIR model with controls & constraints
S I R
β γ
dSt
dt
= −βStIt,
dIt
dt
= βStIt − γIt, and
dRt
dt
= γIt.
Important quantity: R0 =
β
γ
(reproductive ratio).
lockdown: S → (1 − δ)S
asymptomatic: I → (I+, I−) and R → (R+, R−)
more categories: H, ICU and D
testing/detection: I−→I+ and R−→R+
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4. The SIDUHR+/−
model
S I−
R−
I+
H
R+
U D
(1 − δ)β
λ1
γHR
γUR
γHU γUD
γIR
γIH
γIR
γIH
λ2
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5. The SIDUHR+/−
model



dSt = −(1 − δt)βtI−Stdt, Susceptible
dI−
t = (1 − δt)βI−
t Stdt − λ1
t I−
t dt − (γIR + γIH)I−
t dt, Infected –
dI+
t = λ1
t I−
t dt − (γIR + γIH)I+
t dt, Infected +
dR−
t = γIRI−
t dt − λ2
t R−
t dt, Recovered –
dR+
t = γIRI+
t dt + λ2
t R−
t dt + γHRHtdt + γUR(Ut)Utdt, Recovered +
dHt = γIH I−
t + I+
t dt − (γHR + γHU)Htdt, Hospitalized
dUt = γHUHtdt − (γUR(Ut) + γUD(Ut))Utdt, ICU
dDt = γUD(Ut)Utdt, Dead
R0 =
(1 − δ0)β
λ1
0 + γIR + γIH
and Rt =
(1 − δt)βSt
λ1
t + (γIR + γIH)
.
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6. Controls and constraints
Controls
δ: lower social contacts
lockdown / quarantine / masks
λ1: virologic tests, type-1 (short term)
identify I− (→ I+)
λ2: antibody tests, type-2 (long term)
identify R− (→ R+)
Constraints
"flatten the curve" : ICU sustainability, Ut ≤ u
Objective
min
(δt ),(λt )
wC Csanitary + wE Cecon + wT Ctest
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7. The objective function



Qt = R−
t + I−
t + St (suceptible to be) quarantined/lockdowned
Wt = (1 − δt)Qt + R+
t work force
N1
t = λ1
t Qt + γIHI−
t virologic tests, type-1 (short term)
N2
t = λ2
t Qt antibody tests, type-2 (long term)



Csanitary = E Dτ =
∞
0
e−αt
dDt
Cecon = E
τ
0
(1 − Wt)2
dt =
∞
0
e−αt
(1 − Wt)2
dt
Cprevalence = E
τ
0
|N1
t |2
dt =
∞
0
e−αt
|N1
t |2
dt
Cimmunity = E
τ
0
|N2
t |2
dt =
∞
0
e−αt
|N2
t |2
dt
Computational issue: ∞ = 700 days
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8. With optimal (δ∗
t ) (Fig. 3)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
0.8
1.0
S S
S (no control)
0 200 400 600
time (in days)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
I
I
I (no control)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
0.8
R
R
R (no control)
0 200 400 600
time (in days)
0.0000
0.0001
0.0002
0.0003
0.0004
U
U
U (no control)
Umax
0 200 400 600
time (in days)
0.000
0.002
0.004
0.006
0.008
0.010
D D
D (no control)
0 200 400 600
time (in days)
0.0
0.5
1.0
1.5
2.0
2.5
3.0 (no control)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W
W (nocontrol)
0 200 400 600
time (in days)
0
100
200
300
400
500
cumulatedeconomiccost
econ. cost
econ. cost (no control)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
0.8
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9. Optimal (δ∗
t ) with increase of ICU (Fig. 4)
0 200 400 600
time (in days)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
S S (benchmark)
S (increased capacity)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I (benchmark)
I (increased capacity)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R (benchmark)
R (increased capacity)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
0.00030
U
U (benchmark)
U (increased capacity)
Umax
Umax (increased capacity)
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D
D (benchmark)
D (increased capacity)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
(benchmark)
(increased capacity)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W (benchmark)
W (increased capacity)
0 200 400 600
time (in days)
0
100
200
300
400
500
cumulatedeconomiccost
econ. cost (benchmark)
econ. cost (increased capacity)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 (benchmark)
(increased capacity)
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10. Impact of (λ1
t ) (constant, Fig. 10, B5)
0 200 400 600
time (in days)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
S S ( 1 =0)
S ( 1 =1.10 2)
S ( 1 =5.10 2)
S ( 1 =1.10 1)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I ( 1 =0)
I ( 1 =1.10 2)
I ( 1 =5.10 2)
I ( 1 =1.10 1)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R ( 1 =0)
R ( 1 =1.10 2)
R ( 1 =5.10 2)
R ( 1 =1.10 1)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U ( 1 =0)
U ( 1 =1.10 2)
U ( 1 =5.10 2)
U ( 1 =1.10 1)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D
D ( 1 =0)
D ( 1 =1.10 2)
D ( 1 =5.10 2)
D ( 1 =1.10 1)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
( 1 =0)
( 1 =1.10 2)
( 1 =5.10 2)
( 1 =1.10 1)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W ( 1 =0)
W ( 1 =1.10 2)
W ( 1 =5.10 2)
W ( 1 =1.10 1)
0 200 400 600
time (in days)
0
100
200
300
400
500
cumulatedeconomiccost
econ. cost ( 1 =0)
econ. cost ( 1 =1.10 2)
econ. cost ( 1 =5.10 2)
econ. cost ( 1 =1.10 1)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 ( 1 =0)
( 1 =1.10 2)
( 1 =5.10 2)
( 1 =1.10 1)
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11. Impact of (λ1∗
t ) (Fig. 12, B7)
0 200 400 600
time (in days)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
S
S ( opt)
S ( and 1 opt)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I ( opt)
I ( and 1 opt)
0 200 400 600
time (in days)
0.0
0.1
0.2
0.3
0.4
0.5
1
1 ( opt)
1 ( and 1 opt)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U ( opt)
U ( and 1 opt)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D D ( opt)
D ( and 1 opt)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
3.0 ( opt)
( and 1 opt)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W ( opt)
W ( and 1 opt)
0 200 400 600
time (in days)
0
100
200
300
400
500
cumulatedeconomiccost
econ. cost ( opt)
econ. cost ( and 1 opt)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 ( opt)
( and 1 opt)
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12. Impact of (λ2∗
t ) (Fig. 16, B11)
0 200 400 600
time (in days)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
S
S ( opt)
S ( , 1 and 2 opt)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I ( opt)
I ( , 1 and 2 opt)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R ( opt)
R ( , 1 and 2 opt)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U ( opt)
U ( , 1 and 2 opt)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D D ( opt)
D ( , 1 and 2 opt)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
3.0 ( opt)
( , 1 and 2 opt)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 ( opt)
( , 1 and 2 opt)
0 200 400 600
time (in days)
0.0
0.1
0.2
0.3
0.4
1
1 ( opt)
1 ( , 1 and 2 opt)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
2
2 ( opt)
2 ( , 1 and 2 opt)
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13. Taking in account ages
Susceptibles S = (Sc, Sa, Ss), with children, adults and seniors,
d
dt


Sc,t
Sa,t
Ss,t

 = −


Sc,t
Sa,t
Sa,t

 ·


βc,c βc,a βc,s
βa,c βa,a βa,s
βs,c βs,a βs,s




Ic,t
Ia,t
Ia,t

 ,
i.e.
d
dt
St = −St · BIt
for some 3 × 3 WAIFW (Who Acquires Infection From Whom)
matrix,
d
dt
It = St · BIt − γIt and
d
dt
Rt = γIt
(to be extended in our larger model)
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14. Taking in account ages
Sc I−
c R−
c
Sa I−
a R−
a
Ss I−
s R−
s
I+
c
I+
a
I+
s
γc,R
γa,R
γs,R
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15. To go further...
Evolution of dI+
t (dI+∗
t ?) over time, in France
Can we use publicly available data to calibrate models ?
"when a measure becomes a target, it ceases to be a good measure"
(Goodhart’s law)
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16. Sensitivity in I0 (Fig. 6, B1)
0 200 400 600
time (in days)
0.4
0.6
0.8
1.0
S S (I0 =1.e 3)
S (I0 =5.e 3)
S (I0 =25.e 3)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
0.025
I
I (I0 =1.e 3)
I (I0 =5.e 3)
I (I0 =25.e 3)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R (I0 =1.e 3)
R (I0 =5.e 3)
R (I0 =25.e 3)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U (I0 =1.e 3)
U (I0 =5.e 3)
U (I0 =25.e 3)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D
D (I0 =1.e 3)
D (I0 =5.e 3)
D (I0 =25.e 3)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
3.0
(I0 =1.e 3)
(I0 =5.e 3)
(I0 =25.e 3)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W (I0 =1.e 3)
W (I0 =5.e 3)
W (I0 =25.e 3)
0 200 400 600
time (in days)
0
100
200
300
400
500
cumulatedeconomiccost
econ. cost (I0 =1.e 3)
econ. cost (I0 =5.e 3)
econ. cost (I0 =25.e 3)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8
(I0 =1.e 3)
(I0 =5.e 3)
(I0 =25.e 3)
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17. Sensitivity in R0 (Fig. 7, B2)
0 200 400 600
time (in days)
0.4
0.6
0.8
1.0
S S ( 0 =3.0)
S ( 0 =3.3)
S ( 0 =3.6)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I ( 0 =3.0)
I ( 0 =3.3)
I ( 0 =3.6)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R ( 0 =3.0)
R ( 0 =3.3)
R ( 0 =3.6)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U ( 0 =3.0)
U ( 0 =3.3)
U ( 0 =3.6)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D
D ( 0 =3.0)
D ( 0 =3.3)
D ( 0 =3.6)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
3.0 ( 0 =3.0)
( 0 =3.3)
( 0 =3.6)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W ( 0 =3.0)
W ( 0 =3.3)
W ( 0 =3.6)
0 200 400 600
time (in days)
0
100
200
300
400
500
600
cumulatedeconomiccost
econ. cost ( 0 =3.0)
econ. cost ( 0 =3.3)
econ. cost ( 0 =3.6)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 ( 0 =3.0)
( 0 =3.3)
( 0 =3.6)
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18. Sensitivity in α (Fig. 9, B4)
0 200 400 600
time (in days)
0.4
0.6
0.8
1.0
S S ( =0)
S ( =1/500)
S ( =1/250)
S ( =1/100)
0 200 400 600
time (in days)
0.000
0.005
0.010
0.015
0.020
I
I ( =0)
I ( =1/500)
I ( =1/250)
I ( =1/100)
0 100 200 300 400 500 600 700
time (in days)
0.0
0.2
0.4
0.6
R
R ( =0)
R ( =1/500)
R ( =1/250)
R ( =1/100)
0 200 400 600
time (in days)
0.00000
0.00005
0.00010
0.00015
0.00020
U
U ( =0)
U ( =1/500)
U ( =1/250)
U ( =1/100)
Umax
0 200 400 600
time (in days)
0.00000
0.00025
0.00050
0.00075
0.00100
0.00125
0.00150
0.00175
D
D ( =0)
D ( =1/500)
D ( =1/250)
D ( =1/100)
0 200 400 600
time (in days)
0.5
1.0
1.5
2.0
2.5
( =0)
( =1/500)
( =1/250)
( =1/100)
0 100 200 300 400 500 600 700
time (in days)
0.2
0.4
0.6
0.8
1.0
W
W ( =0)
W ( =1/500)
W ( =1/250)
W ( =1/100)
0 200 400 600
time (in days)
0
100
200
300
400
500
600
cumulatedeconomiccost
econ. cost ( =0)
econ. cost ( =1/500)
econ. cost ( =1/250)
econ. cost ( =1/100)
0 200 400 600
time (in days)
0.0
0.2
0.4
0.6
0.8 ( =0)
( =1/500)
( =1/250)
( =1/100)
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