Transcript ppt - Virginia Tech

Optimal Interventions in Infectious
Disease Epidemics: A Simulation
Methodology
Jiangzhuo Chen
INFORMS at Virginia Tech
November 30th, 2011
Network Dynamics & Simulation Science Laboratory
Talk Outline
• Background:
– Propagation of infectious disease on social contact
networks
– Intervention strategies
• Vaccine Assignment Problem
– Mathematical formulation
– Simulations
Network Dynamics & Simulation Science Laboratory
Background: SEIR Disease Model
• Influenza like illness
• Each person in one of four states: S, E, I ,R
• Disease can only transmit from infectious person
to susceptible person
• r(u,v): prob. that u transmits disease to v per unit
time
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Background: Social Contact Network
• Daily activities move people between locations
• People staying at the same location at the same
time may have contacts with each other (physical
proximity)
• Contact network G(V,E)
– V: people
– E: (u,v) if u and v have contacts
– w(u,v): edge weight for contact duration
• Disease may spread from node to node along the
edge (contact)
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Disease Spread in Contact Network
• Within-host disease model: SEIR
– State transitions are probabilistic and timed.
• Between-host disease model: transmission
occurs along edges of a social contact
network
– People + Locations => Contacts.
– Transmissions are probabilistic.
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Disease Spread in Contact Network
• Transmission depends on
–
–
–
–
Duration of contact
Type of contact
Characteristics of the infectious person
Characteristics of the susceptible person
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Synthetic Social Contact Network
• Synthetic population based on census data
– Individual demographics: age, gender…
– Household characteristics: size, income…
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Synthetic Social Contact Network
• Locations: Dun&Bradstreet data
• Synthetic activities based on activity surveys.
– Matched to individuals by demographics
– Matched to locations by activity type
• Synthetic social contact network
– People follow activity schedules
– Activities take them to locations
– At locations they interact with each other
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Synthetic Social Contact Network
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Synthetic Social Contact Network
• We have generated
networks for major urban
regions of US: Miami,
Seattle, Chicago, NYC, etc.
• We have generated
network for regions
outside US: Beijing, Delhi.
• These networks are of
large-scale and very
complex
– E.g. NYC synthetic contact
network has 18 million
people and about 1 billion
contacts
Network Dynamics & Simulation Science Laboratory
Background: Interventions
• Pharmaceutical interventions: vaccination or
antiviral changes an individual’s role in the
transmission chain
– Lower susceptibility or infectiousness
• Non-pharmaceutical interventions: social
distancing measures change people activities and
hence the social network
– Sick leave, school closure, isolation, etc.
Network Dynamics & Simulation Science Laboratory
Complications in Interventions
• Supply: vaccines may not be ready; antiviral
stockpile; production capacity; available leave
days
• Compliance: not all individuals will be able or
willing to comply with an intervention policy
• Cost: drug cost; productivity loss
• Delay: vaccine takes a few days to become
effective
Network Dynamics & Simulation Science Laboratory
Optimal Interventions
Effectiveness of an intervention depends on when and
to whom it is applied
• When is it applied?
– Too early: unnecessary cost; too late: outbreak out of
control
• Who are targeted?
– Supply constraints may require prioritization of groups for
different interventions
Objective varies
• Mitigating epidemic: minimize number of cases;
reduce mortality; delay outbreak
• Cost-benefit analysis
Network Dynamics & Simulation Science Laboratory
Talk Outline
• Background:
– Propagation of infectious disease on social contact
networks
– Intervention strategies
• Vaccine Assignment Problem
– Mathematical formulation
– Simulations
Network Dynamics & Simulation Science Laboratory
Vaccine Assignment: A Mathematical Formulation
• VA(G, r, τE , τI , x, k)
– G(V,E) contact network
– SEIR disease model (r,τE ,τI)
• r(u,v): prob. that u infects v per unit time
• τE : incubation duration (time in state E)
• τI : infectious duration (time in state I)
– x in [0,1]n: prob. that each node is infected initially
– k: vaccine supply
• Choose subset of nodes S with |S| at most k, so that
expected number of infected nodes is minimized
– Nodes in S are removed (assuming 100% vaccine efficacy)
• Stochastic combinatorial optimization
Network Dynamics & Simulation Science Laboratory
Vaccine Assignment Problem is Hard
Theorem
VA(G, r, τE , τI , x, k) is NP-complete if r(u,v)=1
and there is a node s such that x(s)=1 and x(v)=0 for
any other node v.
•Difficult to solve analytically for
– realistic settings
– large scale, unstructured network
– Complicated intervention strategies
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Simulation Methodology
• Synthetic contact network
• Fast simulation tool: EpiFast (MPI code)
– A few seconds for simulating a flu season in a multi-million
population (e.g. Seattle)
– Can handle sophisticated intervention strategies
• Find optimal from a set of feasible intervention
strategies by comparing simulation results
• Factorial experiment design + replicates = many
runs!
• Realistic suggestions for public health policy makers
Network Dynamics & Simulation Science Laboratory
Simulation Design: Populations
• Two US cities
City
Miami
Seattle
population
2092076
3206897
average age
36.05
34.94
average household size
3.84
3.32
average household income
56335
72740
average degree
49
54
Network Dynamics & Simulation Science Laboratory
Simulation Design: H1N1 Flu
•
•
•
•
•
Catastrophic flu: very high infectivity
Average incubation duration = 1.2 days
Average infectious duration = 4.1 days
20 random seeds at beginning of epidemic
25 replicates for every configuration
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Simulation Design: Vaccines
• Limited supply: number of doses equal 10% of
city population size
• Vaccines are applied one month after epidemic
starts
• Vaccine reduces transmission probability by 80%
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Simulation Design: Vaccine Assignment
• To minimize attack rate, it is intuitive to give
vaccines to most vulnerable people.
• Vulnerability of each person is his probability of
getting infected.
• We use EpiFast simulations to compute the
vulnerability measure: 1000 replicates.
• How good is this strategy?
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Assign Vaccines to Most Vul People
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Optimal Implementable Policies
• Unfortunately it is not implementable to directly
assign vaccines to most vulnerable people.
• We can identify them in our synthetic population
through EpiFast simulations.
• But in real population, it is difficult to find them.
• Can we make use of vulnerability measure and
assign vaccines based on it?
Network Dynamics & Simulation Science Laboratory
Optimal Vaccine Assignment: idea 1
• Partition population into groups.
• Allocate vaccines to groups based on their
average vulnerability.
• Various ways for grouping; most naturally by age.
• 5 age groups: [0,19], [20,39], [40,59],[60,79],
[80,∞).
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Idea 1: allocation matters
• Fair allocation: give same amount of vaccines to each
group.
• Weighted allocation: fraction of vaccinated people in
each group is proportional to “group vulnerability”.
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Idea 1: In-Group Assignment also matters
• With same between-group allocation (weighted), it
matters how to assign vaccines within each group:
randomly, to most vulnerable, or to least vulnerable.
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Idea 1: Grouping by Multi-Dimension
• There are other natural variables: household size,
household income, degree in social network.
• Does it help to further partition the groups by
using more and more dimensions?
Dimension
Number of
groups
How
A (age)
5
see previous slides
S (household size)
5
1, 2, 3, 4, 5 and above
I (household income)
3
low, medium, high; evenly
D (degree in social network) 3
low, medium, high; evenly
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Idea 1: Grouping by Multi-Dimension
• Further grouping does not help much.
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Idea 1: Limited Effectiveness
• “Lower bound”: assigning to most vulnerable people.
• Weighted allocation performs much less effectively
than “lower bound”.
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Idea 2: Winner-Takes-All Allocation
• Assign all vaccines to most vulnerable age group.
• Which age group is most vulnerable? age group 1
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Idea 2: Winner-Takes-All Allocation
• Assign all vaccines to age group 1: outperforms
weighted allocation.
• Can we do better?
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Idea 2: Better Proxy for Vulnerability
• Contact of each person is sum of durations of all
his contacts. (weighted degree)
• Contact has strong correlation with vulnerability.
• Divide people into 3 contact groups (C): low,
medium, high.
• Or combine contact and age for grouping.
• Assign all vaccines to most vulnerable contact
group, or most vulnerable (age+contact) group.
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Idea 2: All Vaccines to Most Vul Group
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Why Winner-Takes-All Works Better?
• Same grouping by age; different allocation
schemes: fair, weighted, winner-takes-all.
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Why Winner-Takes-All Works Better: Age Groups
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Why Winner-Takes-All Works Better: Age Groups
• Large variance of vulnerability within each age
group: under random assignment vaccines often
do not go to most vulnerable people in each
group.
• Age group 1 is much more vulnerable than other
groups: in both Miami and Seattle, about 80% of
people in age group 1 has vulnerability larger
than average of any other age group.
• Giving all vaccines randomly to age group 1:
vaccines very likely go to highly vulnerable
people.
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Why Winner-Takes-All Works Better: Contact Groups
Network Dynamics & Simulation Science Laboratory
Why Winner-Takes-All Works Better: Contact Groups
• Even more obvious for contact grouping: when all
vaccines are given randomly to contact group 3,
more than 99% of the recipients have
vulnerability larger than average vulnerability of
any other contact group.
• Coefficient of correlation between vulnerability
and contact is more than 0.95 for either Miami or
Seattle!!
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CDC Recommendations
• Pre-2008 vaccination recommendation for
seasonal flu: age 6mo to 5yr and 50yr and above.
• For seasonal flu (after 2008): age 6mo to 18yr
and 50yr and above.
• Vaccination guideline for H1N1 flu (July 2009):
age 6mo to 5yr then 5yr to 18yr.
• Subsequent guideline for H1N1 flu vaccination
(Oct. 2009): age 6mo to 24yr.
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Compare CDC and our vaccination schemes
• CDC is improving.
• All its strategies are outperformed by ours.
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Compare CDC and our vaccination schemes
• The best CDC vaccination strategy decreases
epidemic peak by 52%, and delays the peak by 25
days.
• Our optimal vaccination strategy decreases
epidemic peak by 70%, and delays the peak by
46 days!
• The lower the peak, the better our logistics can
handle the worst case scenario. The more we
delay the outbreak, the better we can get
prepared and come up with other measures.
Network Dynamics & Simulation Science Laboratory
Optimal Vaccine Assignment: A Solution
• Group people by their total contact time with
others, or by age, or by both.
• Social contact network + EpiFast: tells which
group is most vulnerable.
• Assign all vaccines randomly to people in most
vulnerable group.
Network Dynamics & Simulation Science Laboratory