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Control epidemics
The document provides a mathematical model for controlling the COVID-19 pandemic by balancing detection policies and lockdown interventions to ensure the sustainability of ICU capacity. It discusses various strategies, including virological and antibody testing, and their impact on disease spread and healthcare resources. The objective is to minimize a composite cost function that considers public health and economic factors over time.
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Control epidemics
- 1. COVID-19 pandemic control: balancingdetection 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 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 1 / 18
- 2. Valeur de lavie (introduction) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 2 / 18
- 3. SIR model withcontrols & 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+ @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 3 / 18
- 4. The SIDUHR+/− model S I− R− I+ H R+ UD (1 − δ)β λ1 γHR γUR γHU γUD γIR γIH γIR γIH λ2 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 4 / 18
- 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) . @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 5 / 18
- 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 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 6 / 18
- 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 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 7 / 18
- 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 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 8 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 9 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 10 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 11 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 12 / 18
- 13. Taking in accountages 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 13 / 18
- 14. Taking in accountages 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 @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 14 / 18
- 15. To go further... Evolutionof 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 15 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 16 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 17 / 18
- 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) @freakonometrics freakonometrics https://doi.org/10.1101/2020.05.13.20100842 18 / 18