Models for managing the impact of an influenza epidemic

Transcription

1 Models for managing the impact of an influenza epidemic A Cecilia Zenteno Daniel Bienstock Columbia University February 24th, 2012

2 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

3 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

4 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

5 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

6 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

7 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

8 Motivation Mutations of influenza virus fears of a pandemic Research: Mortality and morbidity public health interventions Workforce shortfall Congestion in hospitals and clinics Figure: H5N1. Objective: Alleviate the impact of the epidemic on the operations continuity of critical infrastructure

9 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

10 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

11 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

12 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

13 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

14 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

15 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

16 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

17 Contingency planning Workers getting sick bring in surge staff Restrictions: Pool is finite Available for a fixed period of time Lag between request and availability Exposed to epidemic too Planning horizon - Full preplanned strategy When and how many to bring in?

18 Building blocks 1. Disease Modeling 2. Hiring Restrictions & Implementation 3. System Utilization Measure

19 Building blocks 1. Disease Modeling 2. HiringRestrictions RobustOptimization & Implementation Algorithm 3. System Utilization Measure

20 Disease Modeling S λβp E µ E I µ RR R S Susceptible E Exposed (Latent) I Infectious R Recovered λ contact rate; β P{meet I}; p P{infection}; µ E incubation rate; µ RR recovery rate.

21 Disease Modeling S λβp E µ E I µ RR R S Susceptible E Exposed (Latent) I Infectious R Recovered λ contact rate; β P{meet I}; p P{infection}; µ E incubation rate; µ RR recovery rate.

22 Disease Modeling S λβp E µ E I µ RR R S Susceptible E Exposed (Latent) I Infectious R Recovered Account for workforce separately: λ contact rate; β P{meet I}; p P{infection}; µ E incubation rate; µ RR recovery rate. λ 1βp S 1 E 1 λ 2βp S 2 E 2 µ E1 I 1 µ E2 I 2 µ RR1 µ RR2 R 1 General Population R 2 Workers

23 Disease Modeling S Susceptible S E Exposed (Latent) I Infectious R Recovered λβp E µ E I µ RR R λ contact rate; β P{meet I}; p P{infection}; µ E incubation rate; µ RR recovery rate. Discrete Time Model: for subgroup j at time t + 1: S j t+1 = St j e λ j β t p E j t+1 = Et j e µ E j + St j (1 e λ j β t p ) I j t+1 = It j e µ RR j + Et j (1 e µ E j ) R j t+1 = Rt j + It j (1 e µ RR j ).

24 A drawback Use of SEIR model rely on its parameters

25 A drawback Use of SEIR model rely on its parameters New epidemic - noisy estimations

26 A drawback Use of SEIR model rely on its parameters New epidemic - noisy estimations Incubation and recovery rates (µ E, µ RR ) are easy

27 A drawback Use of SEIR model rely on its parameters New epidemic - noisy estimations Incubation and recovery rates (µ E, µ RR ) are easy λβp?

28 A drawback Use of SEIR model rely on its parameters New epidemic - noisy estimations Incubation and recovery rates (µ E, µ RR ) are easy λβp? Embed uncertainty on p (prob of contagion)

29 Hedge against uncertainty 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time p =

30 Hedge against uncertainty 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time p = p = 0.013

31 Hedge against uncertainty 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time p = 0.01 p = p = 0.013

32 2. Implementing a procurement strategy When do we start bringing in volunteers?

33 2. Implementing a procurement strategy When do we start bringing in volunteers? An epidemic is declared when growth rate of infected> threshold

34 2. Implementing a procurement strategy When do we start bringing in volunteers? An epidemic is declared when growth rate of infected> threshold We bring in surge staff only after epidemic is declared.

35 Implementing a procurement strategy When do we start bringing in volunteers? An epidemic is declared when growth rate of infected> threshold We bring in surge staff only after epidemic is declared. 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time p = 0.01 p = p = 0.013

36 Implementing a procurement strategy When do we start bringing in volunteers? An epidemic is declared when growth rate of infected> threshold We bring in surge staff only after epidemic is declared. 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time p = 0.01 p = p = epidemic is declared

37 Implementing a procurement strategy When do we start bringing in volunteers? An epidemic is declared when growth rate of infected> threshold We bring in surge staff only after epidemic is declared. 100% 98% 96% % Available Workforce 94% 92% 90% 88% 86% 84% Time from declaration of epidemic p = 0.01 p = p = 0.013

38 3. System Utilization Measures Compute cost per day and add up. Consider 2 scenarios: Min WF level to operate, m - Threshold z t 1 m N 1 % av. workforce at t Cost = t z t.

39 3. System Utilization Measures Compute cost per day and add up. Consider 2 scenarios: Min WF level to operate, m - Threshold Queue z t exp(ρ t) System utilization at time t: 1 ρ t = λt s t µ m N 1 % av. workforce at t av. workforce at t Cost = t z t. Cost = t e K(ρt ρ ) + 1

40 Our problem V (h p) : total cost of a deployment strategy h given p

41 Our problem V (h p) : total cost of a deployment strategy h given p Robust optimization problem V = min h H max p P V (h p) H set of feasible deployment vectors P uncertainty set

42 Our problem V (h p) : total cost of a deployment strategy h given p Robust optimization problem V = min h H max p P V (h p) H set of feasible deployment vectors P uncertainty set Generalized Benders decomposition

43 More general uncertainty sets Very flexible and fast algorithm - handle more general uncertainty sets

44 More general uncertainty sets Very flexible and fast algorithm - handle more general uncertainty sets Analyze impact of multiple values of p during one epidemic. 100% Daily cumulative incidence Flu epidemic San Francisco 90% 80% Cu mulative Inci idence 70% 60% 50% 40% 30% 20% 10% 0% Time (days) SanFran pcont = 0.18 pcont = 0.2 pcont = 0.194

45 More general uncertainty sets Very flexible and fast algorithm - handle more general uncertainty sets Analyze impact of multiple values of p during one epidemic. 100% Daily cumulative incidence Flu epidemic San Francisco 90% 80% Cu mulative Inci idence 70% 60% 50% 40% 30% 20% 10% 0% Time (days) SanFran pcont = 0.18 pcont = 0.2 pcont = p (p 1, p 2, d)

46 Example - Profile Demographics General Population High Risk Population Size 900,000 20,000 Initial infected 5 0 Contact rates (per day) Incubation rate (µe) 10/19 10/19 Removal rate (µr) 10/41 10/41 Survival prob (f) 1 1 Uncertainty Set P = [0.01, 0.012] [0.0125, ] p can change on day {140,..., 160} Procurement Considerations Can bring up to 3,000 volunteers Stay up to 1 week Social Contact Model Nonhomogeneous-mixing Damp contact rates by 30% when epidemic is declared Queueing Cost

47 Example - Scenario 1 Worst case: (p 1, p 2, d) = (0.0109, , 140) Cost: ρ Time from declaration of epidemic ρ (No Interv); Cost = 4.58 ρ*

48 Example - Scenario 1 Worst case: (p 1, p 2, d) = (0.0109, , 140) Cost: 4.58

49 ρ 1 7 Example - Scenario 1 Worst case: (p 1, p 2, d) = (0.0109, , 140) Cost: 4.58 Robust Cost: Time from declaration of epidemic Procurement Strategy ρ (Robust); Cost = ρ (No Interv); Cost = 4.58 ρ* Robust Strategy

50 ρ 1 7 ρ 1 7 Example - Scenario 1 Worst case: (p 1, p 2, d) = (0.0109, , 140) Cost: 4.58 Robust Cost: Naive worst-case Cost: Procurement Strategy Procurement Strategy Time from declaration of epidemic Time from declaration of epidemic ρ (Robust); Cost = ρ (No Interv); Cost = 4.58 ρ (Worst Scen); Cost = 0 ρ (No Interv); Cost = 4.58 ρ* Robust Strategy ρ* Worst case Strategy

51 Example - Scenario 2 Given Robust strategy is implemented: Worst case:(p 1, p 2, d) = ( , , 140) Cost: 1.43

52 ρ 1 7 Example - Scenario 2 Given Robust strategy is implemented: Worst case:(p 1, p 2, d) = ( , , 140) Cost: 1.43 Robust Cost: Time from declaration of epidemic Procurement Strategy ρ (Robust); Cost = 0.05 ρ (No Interv); Cost = 1.43 ρ* Robust Strategy

53 ρ 1 7 ρ 1 7 Example - Scenario 2 Given Robust strategy is implemented: Worst case:(p 1, p 2, d) = ( , , 140) Cost: 1.43 Robust Cost: 0.05 Naive worst-case Cost: Procurement Strategy Procurement Strategy Time from declaration of epidemic Time from declaration of epidemic ρ (Robust); Cost = 0.05 ρ (No Interv); Cost = 1.43 ρ (Worst Scen); Cost = 0.69 ρ (No Interv); Cost = 1.43 ρ* Robust Strategy ρ* Worst case Strategy

54 Example - Comparing strategies No Intervention Robust Worst Case Strategy Strategy No intervention: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1) Robust Strategy: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1) Worst case Strategy: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1)

55 Example - Comparing strategies Takeaway: Planning against worst-case scenario may not be enough! No Intervention Robust Worst Case Strategy Strategy No intervention: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1) Robust Strategy: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1) Worst case Strategy: Cost worst tuple Maximum ρ ( , , 140) Critical days ( ρ > 1)

56 Final Remarks Consider robust models of surge capacity planning in view of a flu pandemic.

57 Final Remarks Consider robust models of surge capacity planning in view of a flu pandemic. Focus on critical staff levels.

58 Final Remarks Consider robust models of surge capacity planning in view of a flu pandemic. Focus on critical staff levels. SEIR model + adversarial models (contagion rate).

59 Final Remarks Consider robust models of surge capacity planning in view of a flu pandemic. Focus on critical staff levels. SEIR model + adversarial models (contagion rate). Present efficient and accurate algorithms procurement strategies which optimally hedge against uncertainty.

60 Final Remarks Consider robust models of surge capacity planning in view of a flu pandemic. Focus on critical staff levels. SEIR model + adversarial models (contagion rate). Present efficient and accurate algorithms procurement strategies which optimally hedge against uncertainty. Need to prepare for more than just the worst case scenario.

61 Thank you! Cecilia Zenteno acz2103