A Computational Framework for Modeling the Spread of Pathogens and Generating Effective Containment

1
A Computational Framework for Modeling the
Spread of Pathogens and Generating Effective
Containment Strategies in Weakly Connected Island
Models

• Lucas R. Shaw
• Masters Thesis Defense
• April 23, 2007
• Advisor Dr. William M. Spears
• University of Wyoming
• Department of Computer Science

2
Contributions

• Applied agent-based simulation, mathematical
  analysis, and Evolutionary Algorithm (EA)
  optimization tool to a simulation of virus spread
  between cities
• Implemented a parallelized simulation of virus
  spread between cities in Starlogo and also in C
• The general ideas used by our simulation could be
  used as a computational framework for similar
  simulation problems
• Analyzed the properties of the simulation model
  using EAs and mathematical analysis
• Developed vaccine allocation policies to minimize
  the impact of the virus on our simulated
  population and compared them to EA generated
  policies
• Developed more effective vaccine allocation
  policies based on mathematical analyses of the
  model

3
Overview

• Goal Create a simulation of viral spread in the
  United States.
• Focus on the 12 major airplane hub cities in the
  US, and the air travel between them.
• Create a macro-level model of the air travel
  between those cities.
• Create a micro-level model within the cities.
• Investigate vaccination policies using this
  simulation.

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Two-Level Architecture

• Macro-Level models approximately 2000 flights
  between the 12 busiest airports in the U.S.
• Airplane flights are governed by a probability
  matrix.
• Micro-Level models the spread of a virus inside
  each of the 12 cities
• Differential equations model the spread of the
  virus inside of each city.
• The number of people captured by the model is
  about 1/6th the population of the U.S.

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The 12 Cities
Minneapolis
Chicago
Detroit
San Francisco
Denver
Las Vegas
Los Angeles
Phoenix
Atlanta
Dallas
Houston
Miami
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The Air Travel Model (Macro-level)

• We found a list of the busiest airports in the
  world and picked (all) the top 12 US airports in
  the list.
• We also compiled data on each of the 12 cities
  (population, area, etc.)
• Using airline flight data, we compiled the
  frequencies of direct (non-stop) flights from one
  airport to another for all 12 cities.
• This data is compiled into a probability
  transition matrix Q, where each entry Qij in Q
  contains the probability a flight will leave from
  cityi for cityj.

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Population Sizes
The total number of people is approximately
55,000,000
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Probability Matrix
Atlanta Chicago LA Dallas Denver Phoenix
LV Houston Minn Detroit SanFran Miami
Atlanta 0.00 0.1061 0.0909 0.1162
0.1162 0.0707 0.0505 0.1111 0.0859 0.0808
0.0758 0.0960 Chicago 0.0909 0.00
0.0818 0.0100 0.0818 0.0909 0.0818 0.0864
0.100 0.100 0.100 0.0864 LA
0.0966 0.1023 0.00 0.1193 0.1023
0.1193 0.0909 0.0852 0.0682 0.0398 0.0966
0.0795 Dallas 0.1061 0.1111 0.101
0.00 0.0960 0.0909 0.096 0.0909 0.0859
0.0707 0.1061 0.0455 Denver 0.1128
0.1077 0.0872 0.1026 0.00 0.1026 0.1026
0.0923 0.1077 0.0513 0.0923 0.0410
Phoenix 0.0814 0.1163 0.1105 0.1163 0.1163
0.00 0.064 0.093 0.093 0.0698
0.1279 0.0116 LasVegas 0.0803 0.1241
0.1168 0.1387 0.1460 0.0803 0.00 0.0511
0.0584 0.0511 0.1314 0.0219 Houston
0.1438 0.1125 0.0938 0.100 0.1125 0.100
0.0938 0.00 0.0563 0.0563 0.0813 0.0500
Minn. 0.1224 0.1497 0.0748
0.1156 0.1361 0.1088 0.0544 0.0612 0.00
0.102 0.0544 0.0204 Detroit 0.1417
0.1750 0.0583 0.1167 0.0833 0.100 0.0583
0.0750 0.1167 0.00 0.0250 0.0500 San
Fran 0.0750 0.1375 0.1125 0.1250 0.1188
0.1375 0.1063 0.0813 0.050 0.0188 0.00
0.0375 Miami 0.2083 0.1771 0.1458
0.0938 0.0938 0.0208 0.0313 0.0833 0.0313
0.0625 0.0521 0.00 Total
1.2593 1.4192 1.0734 1.2441 1.2029 1.0219
0.8297 0.9108 0.8532 0.7030 0.9428 0.5398

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Airplanes Flying
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The Virus Spread Model (Micro-level)

• People can be susceptible, asymptomatic,
  infected, or recovered.
• Susceptible people are healthy and can catch the
  disease.
• Asymptomatic and infected people are contagious.
  Asymptomatic people do not show symptoms, while
  infected people do. Asymptomatic people can fly.
• Recovered people will not catch the disease again
  for a long time.

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The Disease Cycle
P(Recovered Infected)
Infected
Recovered
P(Infected Asymptomatic)
P(Susceptible Recovered)
Susceptible
Asymptomatic
P(Asymptomatic Susceptible)
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Differential Equations
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Interpretation of Parameters

• Average infection rate
• Expected asymptomatic time 1/µ
• Expected infected time 1/d
• Expected time you are immune 1/d
• Everything is a constant, except To prove
  this, we can derive a power series that computes
  expected time ineach state and show it converges
  to 1/parameter for all states (1/ could be
  thought of as expected time before becoming
  asymptomatic, though thisvalue constantly
  changes as the population proportions change).

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What is the Average Infection Rate?

• Suppose some fraction f of the population is
  vaccinated (0 f 1).
• Let be the infection rate for the fraction of
  the population that is not vaccinated
• Let be the infection rate for the fraction
  of the population that is vaccinated
• Then the average infection rate is given by

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What are the Infection Rates a and a'?

• a and a' depend on the proportion of asymptomatic
  a and infected i people, and on the number of
  people n (neighbors) that you encounter daily.
• Let ß and ß' be the probability that a contagious
  neighbor will infect you
• ß is used if you are not vaccinated
• ß' is used if you are vaccinated
• ß'
• Then the infection rates areNote We assume
  n is the same for all cities

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Simulating City Population
People can be susceptible, contagious (sick or
asymptomatic), or recovered.
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Vaccine Allocation Policies

• Vaccines reduce the probability of infection.
• Given a fixed number V of vaccines, what is the
  optimal way to distribute them to the 12 cities?
• We compare hand-crafted benchmark policies versus
  policies found by an evolutionary algorithm (EA).

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Vaccine Allocation Policies

• Experiments on vaccine supplies of 5 to 55
  million vaccines (in increments of 5 million)
• The comparison will be done based on the number
  of sick days that occur better policies result
  in fewer sick days.
• Other possible measures are total number of
  contagious people or deaths due to the virus.

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Vaccine Allocation Policies

• Benchmark Policies
• Uniform distribution
• Uniformly distribute vaccine supply
• Proportional distribution
• Give each city a proportion of the vaccine supply
  equal to its proportion of the total population
• Thought experiment Which of these two policies
  is better?

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Results
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Discussion

• Why is a proportional distribution policy worse
  than a uniform distribution policy?
• Where are you more likely to run into one of 100
  contagious people a small city or a big city?
  (dilution factor)
• Why is the EA outperforming both hand-crafted
  policies? What is special about the EA policies?

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Example EA Policy
Note that the city of origin, Atlanta, receives a
lot of vaccine, as do the smaller cities!
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New Policy

• Give the city of origin a lot of vaccine and then
  use an inverse proportional distribution policy
  where smaller cities receive more vaccine.

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Results
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Discussion

• So, vaccinating smaller cities is important.
• However, further analysis of the EA-generated
  policy made us realize that one should also
  vaccinate cities that are flown to more often!
• This information can be obtained from the
  probability matrix.

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New Policy

• Give the city of origin as much vaccine as
  possible.
• Then order the cities important cities are
  those that are flown to often and have small
  population sizes.
• Give as much vaccine to the important cities as
  possible, until you run out of vaccine.

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Results
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Discussion

• Clearly, this new policy works very well we are
  now outperforming the EA policy!
• Question, do we really have to give as much
  vaccine as possible to the important cities?

Not at 1.0! Why?
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Phase Transition

• An initially small infection will die off quickly
  if the following inequality holds
• For this simulation, f is approximately 0.9!

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New Policy

• Give the city of origin as much vaccine as
  possible.
• Then order the cities important cities are
  those that are flown to often and have small
  population sizes.
• Give f 0.9 vaccine to the important cities,
  until you run out of vaccine.
• This leaves more vaccine for other cities.

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Results
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R and the Phase Transition

• We can relate our phase transition to R the
  replacement number, which is used in mathematical
  epidemiology
• R estimates how fast a virus will spread
• R is the average number of secondary infections
  generated by a typical infective over the
  infectious period
• When R 1, the virus will spread
• When R 1, the virus will not spread
• When R

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R and Phase Transition
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Vaccine Efficacy

• We can use the phase transition to define a
  feasibility region in terms of f and ß' when h
  1, letting f and ß' range over 0..1recall
  d, µ and ß are constants defined by the virus
  characteristics, and n is assumed constant as
  well
• We could also use the ratio c of ß' to ß to
  define the feasibility region if we relate ß' and
  ß by c ß' /ß.
• We used a c 1/3 in our experiments.

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Vaccine Efficacy
This shows the possible vaccine efficacies in
terms of ß'. If there is no f value between
(0..1, then the virus cannot be controlled with
any amount of vaccine.
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Vaccine Efficacy
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Vaccine Efficacy

• The region of feasible vaccines in the experiment
  corresponds directly with the theoretical limit.

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The 2005 Flu Season

• Given the initial flu outbreak, we can
  potentially use our probability matrix to figure
  out where the flu will go next!
• During the 2005 Christmas week, the flu became
  widespread in California and Arizona. Of the 10
  states (12 cities) we model, our simulation
  indicated that Nevada, Texas, and Colorado would
  be next.

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Week Ending December 24
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Week Ending December 31
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Week Ending January 7
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2006 Flu Season

• The model predicted spread from Florida to
  Georgia.

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2006 Flu Season

• Then the model predicted Minnesota, Colorado,
  Texas, and Nevada. Only Nevada avoided the worst.

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Summary

• Lessons learned
• Vaccinating populations that are visited more
  often is important.
• Vaccinating smaller populations may be much
  better than vaccinating larger populations!
• If conditions change, the Evolutionary Algorithm
  can automate the search for good policies.
• This method can be applied to a variety of
  similar problems (animal disease, computer
  viruses, etc)!

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Loosely Connected Island Model
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Acknowledgements

• Thanks go to
• My Committee Members Dr. William Spears, Dr.
  Diana Spears, Dr. Steven Barrett, and Dr. Lora
  Billings
• BRIN for funding development of the simulation
• INBRE for funding development of the Parallel
  Evolutionary Algorithm toolkit
• Dr. William M. Spears and Dr. Diana F. Spears for
  choosing me to do this work and their guidance
• Paul Maxim for his help with gathering data and
  other work with the first version of this
  simulation
• Stormy Knight the UW Beowulf Cluster
  administrator for his input regarding the
  parallelization of the virus simulation

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Discussion

• Clearly, this is much closer to the EA-generated
  policy. But why does it work?
• Because the rate at which an infection takes off
  depends on the proportion of contagious people
• In other words,100 contagious people flying into
  a small city have a greater impact than the same
  people flying into a large city.