Modeling Influenza Pandemic and Strategies for Food Distribution
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1 Modeling Influenza Pandemic and Strategies for Food Distribution Ali Ekici, Pinar Keskinocak, Pengyi Shi, Julie Swann H. Milton Stewart School of Industrial & Systems Engineering Georgia Institute of Technology 1
2 Pandemic Flu Historical Information Pandemic cases in history (Spanish flu) (A/H1N1) (Asian flu) (A/H2N2) (Hong Kong flu) (A/H3N2) Excess Mortality million ( %) 2 million (0.069%) 1 million (0.028%) Populations Affected Persons <65 years Infants, elderly Infants, elderly Middle-ground estimate: 25% of entire population will be infected (Gibbs and Soares 2005) 2
3 Potential Impact of Pandemic Flu $ billion economic impact in the U.S. (CDC) U.S. Department of Health & Human Services and U.S. Department of Commerce estimates 20% of working adults will become ill 40% workforce loss during peak outbreak periods 3
4 Response Plans Preparation efforts Local governments Federal government NGOs Companies Response plans focus on How to treat ill people Food, vaccine and medicine distribution 4
5 Focus of Analysis Modeling and understanding the disease spread geographically and over time Constructing a food distribution network Number of facilities and their locations (over time) Allocation of resources among the facilities In collaboration with 5
6 Disease Spread Model An individual-based stochastic model Natural history for pandemic flu (Wu et al. 2006): Sus. E P A S R H D Sus.: Susceptible E: Exposed P: Presymptomatic A: Asymptomatic S: Symptomatic R: Recovered H: Hospitalized D: Dead Similar to S-E-I-R models 5 age groups (0-5, 6-11, 12-18, 19-64, 65+) 6
7 Disease Contact Network Households Peer groups (work place and schools) Community: Census Tract-based Population ~ ( ) Test case: State of Georgia Household statistics, work flow data, classroom sizes, age statistics 7
8 Disease Spread in Georgia R0 Value Peak Value 2.48% 5.27% 8.01% Peak Time IAR 49.65% 67.49% 78.27% Percentage Infected for Different R0 Values Percentage 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Day
9 Day 10 Initial Seed Fulton 9
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42 Day 160 Initial Seed Atkinson 42
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44 Day 180 Initial Seed Atkinson 44
45 Infections over Time Fulton Seed Atkinson Seed Infectio ns Bibb Atkinson Cobb Fulton Catoosa Jackson In fe c tio n s Day Day Peak infection rate is similar regardless of starting location Time of peak depends on the starting locations Similar for densely populated areas Different for less dense areas 45
46 Infections over Time Fulton Seed Atkinson Seed In fectio n s Bibb Atkinson Catoosa Jackson In fectio n s Day Day 46
47 Impact of Seasonality Flu starting from March (non-flu season), decreasing, then increasing with flu season) Flu starting from Sept (flu season), increasing, then decreasing with two R0 values Infected Population Infected Population time time
48 Impact of Mutations Mutation begins after 3 (or 4) months 5% chance to be reinfected Combination of seasonality (starting in Sept-flu season) with mutations Infected Population Infected Population time Combine Mutation Seasonality time
49 Estimating the food need (Metropolitan Atlanta Area) Daily Number of Meals for HHs with an Infected Individual Daily Number of Meals for HHs with an Infected Adult 1,000, , , , , , , , , ,000 Below Poverty Income Below 25,000 All 400, ,000 Below Poverty Income Below 25,000 All 300, , , , , Day Day Daily Number of Meals for HHs with All Adults Infected 70,000 60,000 50,000 40,000 30,000 Below Poverty Income Below 25,000 All 20,000 10, Day 49
50 Estimating the food need (Metropolitan Atlanta Area) Cumulative Number of Meals for HHs with an Infected Individual Cumulative Number of Meals for HHs with an Infected Adult 40,000,000 25,000,000 35,000,000 30,000,000 20,000,000 25,000,000 20,000,000 15,000,000 Below Poverty Income Below 25,000 All 15,000,000 10,000,000 Below Poverty Income Below 25,000 All 10,000,000 5,000,000 5,000, Day Day Cumulative Number of Meals for HHs with All Adults Infected 3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 Below Poverty Income Below 25,000 All 500, Day 50
51 Estimated food need over time (for all HHs with an infected individual) (Day 20) 51
52 Estimated the food need over time (for all HHs with an infected individual) (Day 30) 52
53 Estimated the food need over time (for all HHs with an infected individual) (Day 40) 53
54 Estimated the food need over time (for all HHs with an infected individual) (Day 50) 54
55 Estimated the food need over time (for all HHs with an infected individual) (Day 60) 55
56 Estimated the food need over time (for all HHs with an infected individual) (Day 70) 56
57 Estimated the food need over time (for all HHs with an infected individual) (Day 80) 57
58 Estimated the food need over time (for all HHs with an infected individual) (Day 90) 58
59 Estimated the food need over time (for all HHs with an infected individual) (Day 100) 59
60 Estimated the food need over time (for all HHs with an infected individual) (Day 110) 60
61 Estimated the food need over time (for all HHs with an infected individual) (Day 120) 61
62 Estimated the food need over time (for all HHs with an infected individual) (Day 130) 62
63 Intervention strategies: Quarantine Quarantine leads to two peaks Comparison of 2 Week Quarantine Starting at Different Times Comparison of 2 Week Quarantine Starting at Different Times 3.00% 3.00% 2.50% 2.50% Percentage Infected 2.00% 1.50% 1.00% Week 3 Week 4 Week 5 Week 6 Percentage Infected 2.00% 1.50% 1.00% Week 6 Week 7 Week 8 Week % 0.50% 0.00% 0.00% Day Day Comparison of 2 Week Quarantine Starting at Different Times 3.00% 2.50% Percentage Infected 2.00% 1.50% 1.00% 0.50% 0.00% Day Week 9 Week 10 Week 11 Week 12 63
64 When to start the quarantine? To minimize the maximum peak, or maximum food distribution capacity needed Start sometime during weeks 7-9 If shorter quarantine start later Percentage Infected at the Peak 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 2 Week Q - Peak 3 Week Q - Peak 4 Week Q - Peak 5 Week Q - Peak 6 Week Q - Peak 7 Week Q - Peak 8 Week Q - Peak 0.00% Start Week of Quarantine 64
65 When to start the quarantine? To minimize the total % of infected population, i.e, the total food distribution Start quarantine between weeks 8-11 Infection Attack Rate 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 0.00% Start Week of Quarantine 65
66 How long to impose quarantine? A longer quarantine delays the timing and reduces the magnitude of the second peak reduces the percentage affected at the maximum peak % Day of the Peak P e ecen tag e In fected at th e P eak 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Week Q - 1. Peak 2 Week Q - 2. Peak 4 Week Q - 1. Peak 4 Week Q - 2. Peak 6 Week Q - 1. Peak 6 Week Q - 2. Peak 8 Week Q - 1. Peak 8 Week Q - 2. Peak Start Week of Quarantine Start Week of Quarantine 66
67 Effect of Compliance Rate on the Attack Rate Base case : 50% Infection Attack Rate 55.00% 50.00% 45.00% 40.00% 4 Week Q Week Q Week Q % 30.00% Start Week of Quarantine 67
68 Effect of Compliance Rate on the Peak Value Day Start Week of Quarantine 4 Week Q Peak 4 Week Q Peak 4 Week Q Peak 4 Week Q Peak 4 Week Q Peak 4 Week Q Peak P e a k V a lu e 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% Start Week of Quarantine 3.00% 2.50% Peak Value 2.00% 1.50% 1.00% 0.50% 0.00% Start Week of Quarantine 68
69 Intervention strategies: School Closure Effectiveness of the school closure plans proposed for the state of Georgia 6 weeks for R 0 = weeks for R 0 = 1.8 Infection Attack Rate for R0 = 1.5 Infection Attack Rate for R0 = % 68.00% 49.00% 48.00% 66.00% 47.00% 46.00% 45.00% 44.00% 6 Week School Closure 64.00% 62.00% 60.00% 12 Week School Closure 43.00% 42.00% 58.00% 41.00% 56.00% Start Week of Closure Start Week of Closure 69
70 Intervention strategies: School Closure Peak infectivity Percentage Infected at the Peak for R0 = 1.5 Percentage Infected at the Peak for R0 = % 6.00% 2.50% 5.00% 2.00% 1.50% 1.00% 6 Week School Closure 4.00% 3.00% 2.00% 12 Week School Closure 0.50% 1.00% 0.00% % Start Week of Closure Start Week of Closure 70
71 No Intervention vs. School Closure Percentage Infected for Different for R0=1.5 Percentage Infected for Different for R0= % 6.00% 2.50% 5.00% P e rc e n ta g e 2.00% 1.50% 1.00% No intervention 6 Week Closure Starting at Week 7 6 Week Closure Starting at Week 8 P e r c e n ta g e 4.00% 3.00% 2.00% No intervention 12 Week Closure Starting at Week % 1.00% 0.00% Day 0.00% Day 71
72 Quarantine vs. School Closure For R 0 = 1.5 A voluntary 4-week quarantine with 50% percent compliance rate is at least as effective as a 6-week school closure both in terms of IAR and peak activity School closure is more disruptive Quarantine is voluntary 72
73 Food Distribution Network S S S S S MF MF MF MF MF POD POD POD POD POD POD POD S MF POD D Supply point Major facility Point of Distribution Demand point D D D D D D D D 73
74 Decisions (in each time period) Which facilities to open/close How much food to send: from the suppliers to each of the major facilities from the major facilities to each of the PODs Subject to Capacity constraints Potential locations To minimize Opening/closing, operating, and distribution costs 74
75 Computational Results: Georgia Case 1615 census tracts in Georgia (for disease spread model) 603 census tracts (15 counties) in Metropolitan Atlanta Area Distances between census tracts Estimated data Cost figures Supply amount Potential facility locations and capacities 75
76 Models Dynamic model Decisions made and updated in each time period 5 scenarios are generated in each time period for the remaining time horizon and the average is used as an estimate Static model Decisions about opening the facilities is done at the beginning of the planning horizon Resource allocation decisions are made in each time period 76
77 Dynamic Model Computational Results Number of Major Facilities Number of Distribution Points Number of Major Facilities Number of Distribution Points Number of Open Major Facilities Nu m b er o f O p en Distrib u tio n P o in ts Week Week 77
78 Opening Closing PODs over Time (Week 4) 78
79 Opening Closing PODs over Time (Week 6) 79
80 Opening Closing PODs over Time (Week 8) 80
81 Opening Closing PODs over Time (Week 10) 81
82 Opening Closing PODs over Time (Week 12) 82
83 Opening Closing PODs over Time (Week 14) 83
84 Opening Closing PODs over Time (Week 16) 84
85 Preliminary Computational Results: Georgia Case Static: ~10% higher cost than dynamic Cost under dynamic updates vs. the perfect hindsight solution 32.71% (compared to the lower bound) 7.87% (compared to best integer solution) Service level Under dynamic updates 63.21% of the demand served from a POD within 10 miles Best integer solution 69.95% of the demand served from a POD within 10 miles 85
86 Conclusions & Food Distribution Dynamic facility location and resource allocation decisions for food distribution during a pandemic Next steps Optimization with a fixed budget Maximize number of people served subject to a budget Heuristics for obtaining solutions efficiently Effect of other intervention strategies for the disease spread and optimizing their decisions Effect of temperature on the disease spread Sensitivity analysis on R 0 Acknowledgement: Special thanks to Joseph T. Wu for sharing his C++ code for the disease spread model 86
87 Other Next Steps Impact of school closures Allocation of limited medicines (anti-virals or vaccines) Supply chain impact Others? 87
88 Center for Humanitarian Logistics MISSION: Positive humanitarian impact worldwide To improve humanitarian logistics and ultimately the human condition by system transformation and organization effectiveness through education, outreach, and solutions. Education Outreach Research/Applications Co-directors: Ozlem Ergun, Pinar Keskinocak, Julie Swann 88 88
89 Questions/Discussions? Ali Ekici: Pinar Keskinocak: Pengyi Shi: Julie Swann: Humanitarian Logistics: 89
90 Literature Review-Disease Spread Models Natural History Spatial Structure Age Specific Night-Day Wu et al. (2006) Detailed S-E-I-R No No No Germann et al. (2006) S-E-I-R Yes Yes Yes Ferguson et al. (2006) S-E-I-R Yes Yes No Longini et al. (2005) S-E-I-R Yes Yes No Our model Detailed S-E-I-R Yes Yes Yes 90
91 Natural History Parameters State durations are as follows (Wu et al. 2006): Exposed + Presymptomatic ~ Weibull with mean 1.48 days (including 0.5 days offset) and std. dev days Presymptomatic = 0.5 days Asymptomatic ~ Exponential with mean days Symptomatic ~ Exponential with mean days Hospitalized ~ Exponential with mean 14 days Transition probabilities (Wu et al. 2006, Carrat et al. 2006): Probability of developing symptoms 0.75 for children and elderly and 0.6 for adults Probability of hospitalization 0.18 for children between for elderly 0.06 for others Probability of death for children between 0-5 and elderly for others 91
92 Simulation Model Details-1 Events 3 main type of events Disease progress within an individual Close type of infection (in a household or peer group) Nonclose type of infection (in the community or as an import) Compare time of next infection with the disease progress type of event Start the simulation with 30 initial seeds 92
93 Simulation Model Details-2 Assumed parameters Probability of developing symptoms, hospitalization and death (age based) Duration of the disease stages R 0 : the average number of secondary cases generated by an infectious individual θ : the proportion of transmission that occurs at either presymptomatic or asymptomatic stage ω : the proportion of infections generated by individuals who are never symptomatic γ : the proportion of transmission which occurs outside the households δ : the proportion of transmission outside the home that occurs in the community 93
94 Simulation Model Details-2 Parameters calculated based on the assumed parameters N: size of the community n i HA : active householdsize (excluding dead) β: coefficient of transmission h PS : relative hazard of presymptomatic with respect to symptomatic h AS : relative hazard of asymptomatic with respect to symptomatic h PH : relative hazard of peer group with respect to household h CH : relative hazard of community with respect to household ε A : probability that a symptomatic adult withdraws from peer group ε C : probability that a symptomatic child withdraws from peer group δ ijh : indicator function which is 1 if individuals i and j share the same household or 0 otherwise δ ij PG : indicator function which is 1 if individuals i and j share the same peer group or 0 otherwise Other parameters Probability of compliance to a quarantine (Base case = 0.5) 94
95 Calculation of Parameters-1 95
96 Calculation of Parameters-2 96
97 Instantaneous Force of Infection-1 The force of infection for individual i is calculated as follows: m j where j= N m H jβ m h PG λ i = δ ij + δ HA ij mjε jhph β + j= 1 ni N 1 if j was symptomatic hps if j was presymptomatic = has if j was asymptomatic 0 otherwise j CH β 0 with probability ε A if j was an adult and was symptomatic ε j = 0 with probability εc if j was a child and was symptomatic 1 otherwise 97
98 Our model vs. Wu et al Night-day differentiation During the day, infections occur in the peer groups During the night, infections occur in the households Age specific model Children are more susceptible and infective Infection attack rates are similar Number of children infected in our model is approximately 10% higher Spatial component More than one communities connected via schools and work places Real work flow data used while constructing the work places Spatial component decreases the infection attack rate and peak infectivity Infection attack rate decreases by 9% (for R 0 = 1.8 ) Peak value decreases by 26% (for R 0 = 1.8 ) 98
99 Models in the Literature Multi-period facility location problems are solved using Dynamic programming integrated with MIP Lagrangean relaxation Hormozi and Khumawala (1996) No capacity on the facilities No hierarchical structure Integrates dynamic programming and MIP methods Canel et al. (2001) Hierarchical structure but no supply points (Factory Facility Customer) Multi-commodity Integrates dynamic programming and MIP methods Hinosoja et al. (2000) Hierarchical structure but no supply points (Factory Facility Customer) Multi-commodity Lagrangean relaxation Main difference of our problem: Hierarchical structure 99
100 Generated Instances 6 major facilities to more populated census tracts 80 PODs assigned to census tracts according to the population size 30 supply points randomly assigned to census tracts For a POD or major facility Fixed operating cost / Closing cost 6 Fixed operating cost / Opening cost 3 Fixed, opening and closing costs are proportional to the capacity (Cost of a major facility / Cost of a POD) ϵ {10, 20, 50, 100} Unit transportation cost per mile between facilities / Unit transportation cost per mile to a demand point = 2 100
101 Opening Closing PODs over Time (Week 3) 101
102 Opening Closing PODs over Time (Week 5) 102
103 Opening Closing PODs over Time (Week 7) 103
104 Opening Closing PODs over Time (Week 9) 104
105 Opening Closing PODs over Time (Week 11) 105
106 Opening Closing PODs over Time (Week 13) 106
107 Opening Closing PODs over Time (Week 15) 107
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