Mathematics and Bioterrorism: Graph-theoretical Models of Spread and Control of Disease

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Mathematics and Bioterrorism Graph-theoretical
Models of Spread and Control of Disease
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• Great concern about the deliberate introduction
  of diseases by bioterrorists has led to new
  challenges for mathematical scientists.
• 
  

smallpox
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• Bioterrorism issues are typical of many homeland
  security issues.
• This talk will emphasize bioterrorism, but many
  of the messages apply to homeland security in
  general.

Waiting on line to get smallpox vaccine during
New York City smallpox epidemic
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• Outline
• 1. The role of mathematical sciences in the fight
  against bioterrorism.
• 2. Methods of computational and mathematical
  epidemiology
• 2a. Other areas of mathematical sciences
• 2b. Discrete math and theoretical CS
• 3. Graph-theoretical models of spread and control
  of disease

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• Dealing with bioterrorism requires detailed
  planning of preventive measures and responses.
• Both require precise reasoning and extensive
  analysis.

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Understanding infectious systems requires being
able to reason about highly complex biological
systems, with hundreds of demographic and
epidemiological variables.
Intuition alone is insufficient to fully
understand the dynamics of such systems.
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• Experimentation or field trials are often
  prohibitively expensive or unethical and do not
  always lead to fundamental understanding.
• Therefore, mathematical modeling becomes an
  important experimental and analytical tool.

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• Mathematical models have become important tools
  in analyzing the spread and control of infectious
  diseases and plans for defense against
  bioterrorist attacks, especially when combined
  with powerful, modern computer methods for
  analyzing and/or simulating the models.

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What Can Math Models Do For Us?
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What Can Math Models Do For Us?

• Sharpen our understanding of fundamental
  processes
• Compare alternative policies and interventions
• Help make decisions.
• Prepare responses to bioterrorist attacks.
• Provide a guide for training exercises and
  scenario development.
• Guide risk assessment.
• Predict future trends.

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• What are the challenges for mathematical
  scientists in the defense against disease?
• This question led DIMACS, the Center for Discrete
  Mathematics and Theoretical Computer Science, to
  launch a special focus on this topic.
• Post-September 11 events soon led to an emphasis
  on bioterrorism.

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DIMACS Special Focus on Computational and
Mathematical Epidemiology 2002-2005
Anthrax
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Methods of Math. and Comp. Epi.

• Math. models of infectious diseases go back to
  Daniel Bernoullis mathematical analysis of
  smallpox in 1760.

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• Hundreds of math. models since have
• highlighted concepts like core population in
  STDs

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• Made explicit concepts such as herd immunity for
  vaccination policies

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• Led to insights about drug resistance, rate of
  spread of infection, epidemic trends, effects of
  different kinds of treatments.

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• The size and overwhelming complexity of modern
  epidemiological problems -- and in particular the
  defense against bioterrorism -- calls for new
  approaches and tools.

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The Methods of Mathematical and Computational
Epidemiology

• Statistical Methods
• long history in epidemiology
• changing due to large data sets involved
• Dynamical Systems
• model host-pathogen systems, disease spread
• difference and differential equations
• little systematic use of todays powerful
  computational methods

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The Methods of Mathematical and Computational
Epidemiology

• Probabilistic Methods
• stochastic processes, random walks, percolation,
  Markov chain Monte Carlo methods
• simulation
• need to bring in more powerful computational
  tools

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Discrete Math. and Theoretical Computer Science

• Many fields of science, in particular molecular
  biology, have made extensive use of DM broadly
  defined.

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Discrete Math. and Theoretical Computer Science
Contd

• Especially useful have been those tools that make
  use of the algorithms, models, and concepts of
  TCS.
• These tools remain largely unused and unknown in
  epidemiology and even mathematical epidemiology.

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What are DM and TCS?

• DM deals with
• arrangements
• designs
• codes
• patterns
• schedules
• assignments

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TCS deals with the theory of computer algorithms.

• During the first 30-40 years of the computer age,
  TCS, aided by powerful mathematical methods,
  especially DM, probability, and logic, had a
  direct impact on technology, by developing
  models, data structures, algorithms, and lower
  bounds that are now at the core of computing.

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DM and TCS Continued

• These tools are made especially relevant to
  epidemiology because of
• Geographic Information Systems

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DM and TCS Continued

• Availability of large and disparate computerized
  databases on subjects relating to disease and the
  relevance of modern methods of data mining.

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DM and TCS Continued

• Availability of large and disparate computerized
  databases on subjects relating to disease and the
  relevance of modern methods of data mining
• Issues involve
• detection
• surveillance (monitoring)
• streaming data analysis
• clustering
• visualization of data

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DM and TCS Continued

• The increasing importance of an evolutionary
  point of view in epidemiology and the relevance
  of DM/TCS methods of phylogenetic tree
  reconstruction.

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DM and TCS Continued

• The increasing importance of an evolutionary
  point of view in epidemiology and the relevance
  of DM/TCS methods of phylogenetic tree
  reconstruction.
• Heavy use of DM in phylogenetic tree
  reconstruction
• Might help in identification of source of an
  infectious agent

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Models of the Spread and Control of Disease
through Social Networks

• Diseases are spread through social networks.
• This is especially relevant to sexually
  transmitted diseases such as AIDS.
• Contact tracing is an important part of any
  strategy to combat outbreaks of diseases such as
  smallpox, whether naturally occurring or
  resulting from bioterrorist attacks.

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The Basic Model
Social Network Graph Vertices People Edges
contact State of a Vertex simplest model 1
if infected, 0 if not infected (SI Model) More
complex models SI, SEI, SEIR, etc. S
susceptible, E exposed, I infected, R
recovered (or removed)
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More About States
Once you are infected, can you be cured? If you
are cured, do you become immune or can you
re-enter the infected state? We can build a
digraph reflecting the possible ways to move from
state to state in the model.
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The State Diagram for a Smallpox Model
The following diagram is from a Kaplan-Craft-Wein
(2002) model for comparing alternative responses
to a smallpox attack. This is being considered by
the CDC and Office of Homeland Security.
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The Stages

• Row 1 Untraced and in various stages of
  susceptibility or infectiousness.
• Row 2 Traced and in various stages of the queue
  for vaccination.
• Row 3 Unsuccessfully vaccinated and in various
  stages of infectiousness.
• Row 4 Successfully vaccinated dead

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Moving From State to State
Let si(t) give the state of vertex i at time
t. Two states 0 and 1. Times are discrete t
0, 1, 2,
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Threshold Processes
Basic k-Threshold Process You change your state
at time t1 if at least k of your neighbors
have the opposite state at time t. Disease
interpretation? Cure if sufficiently many of your
neighbors are uninfected. Does this make sense?
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Threshold Processes II
Irreversible k-Threshold Process You change
your state from 0 to 1 at time t1 if at
least k of your neighbors have state 1 at
time t. You never leave state 1. Disease
interpretation? Infected if sufficiently many of
your neighbors are infected. Special Case k
1 Infected if any of your neighbors is
infected.
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Basic 2-Threshold Process
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Irreversible 2-Threshold Process
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Complications to Add to Model

• k 1, but you only get infected with a certain
  probability.
• You are automatically cured after you are in the
  infected state for d time periods.
• You become immune from infection (cant re-enter
  state 1) once you enter and leave state 1.
• A public health authority has the ability to
  vaccinate a certain number of vertices, making
  them immune from infection.

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Periodicity
State vector s(t) (s1(t), s2(t), ,
sn(t)). First example, s(1) s(3) s(5)
, s(0) s(2) s(4) s(6) Second example
s(1) s(2) s(3) ... In all of these
processes, because there is a finite set of
vertices, for any initial state vector s(0),
the state vector will eventually become periodic,
i.e., for some P and T, s(tP) s(t) for
all t gt T. The smallest such P is called the
period.
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Periodicity II
First example the period is 2. Second example
the period is 1. Both basic and irreversible
threshold processes are special cases of
symmetric synchronous neural networks. Theorem
(Goles and Olivos, Poljak and Sura) For
symmetric, synchronous neural networks, the
period is either 1 or 2.
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Periodicity III
When period is 1, we call the ultimate state
vector a fixed point. When the fixed point is
the vector s(t) (1,1,,1) or (0,0,,0), we
talk about a final common state. One problem of
interest Given a graph, what subsets S of the
vertices can force one of our processes to a
final common state with entries equal to the
state shared by all the vertices in S in the
initial state?
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Periodicity IV
Interpretation Given a graph, what subsets S
of the vertices should we plant a disease with so
that ultimately everyone will get it? (s(t) ?
(1,1,,1)) Economic interpretation What set of
people do we place a new product with to
guarantee saturation of the product in the
population? Interpretation Given a graph, what
subsets S of the vertices should we vaccinate
to guarantee that ultimately everyone will end up
without the disease? (s(t) ? 0,0,,0))
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Conversion Sets
Conversion set Subset S of the vertices that
can force a k-threshold process to a final common
state with entries equal to the state shared by
all the vertices in S in the initial state. (In
other words, if all vertices of S start in same
state x 1 or 0, then the process goes to a
state where all vertices are in state
x.) Irreversible k-conversion set if
irreversible process.
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1-Conversion Sets
k 1. What are the conversion sets in a basic
1-threshold process?
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1-Conversion Sets
k 1. The only conversion set in a basic
1-threshold process is the set of all vertices.
For, if any two adjacent vertices have 0 and 1 in
the initial state, then they keep switching
between 0 and 1 forever. What are the
irreversible 1-conversion sets?
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Irreversible 1-Conversion Sets
k 1. Every single vertex x is an
irreversible 1-conversion set if the graph is
connected. We make it 1 and eventually all
vertices become 1 by following paths from x.
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Conversion Sets for Odd Cycles
C2p1 2-threshold process. What is a conversion
set?
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Conversion Sets for Odd Cycles
C2p1. 2-threshold process. Place p1 1s in
alternating positions.
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Conversion Sets for Odd Cycles
We have to be careful where we put the initial
1s. p1 1s do not suffice if they are next to
each other.
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Irreversible Conversion Sets for Odd Cycles
What if we want an irreversible conversion set
under an irreversible 2-threshold process? Same
set of p1 vertices is an irreversible
conversion set. Moreover, everyone gets infected
in one step.
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Vaccination Strategies
If you didnt know whom a bioterrorist might
infect, what people would you vaccinate to be
sure that a disease doesnt spread very much?
(Vaccinated vertices stay at state 0 regardless
of the state of their neighbors.) Try odd cycles
again. Consider an irreversible 2-threshold
process. Suppose your adversary has enough supply
to infect two individuals. Strategy 1 Mass
vaccination make everyone 0 and immune in
initial state.
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Vaccination Strategies
In C5, mass vaccination means vaccinate 5
vertices. This obviously works. In practice,
vaccination is only effective with a certain
probability, so results could be different. Can
we do better than mass vaccination? What does
better mean? If vaccine has no cost and is
unlimited and has no side effects, of course we
use mass vaccination.
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Vaccination Strategies
What if vaccine is in limited supply? Suppose we
only have enough vaccine to vaccinate 2
vertices. Consider two different vaccination
strategies
Vaccination Strategy I
Vaccination Strategy II
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Vaccination Strategy I Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy Ia
Adversary Strategy Ib
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The alternation between your choice of a
defensive strategy and your adversarys choice of
an offensive strategy suggests we consider the
problem from the point of view of game
theory.The Food and Drug Administration is
studying the use of game-theoretic models in the
defense against bioterrorism.
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Vaccination Strategy I Adversary Strategy Ia
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Vaccination Strategy I Adversary Strategy Ib
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Vaccination Strategy II Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy IIa
Adversary Strategy IIb
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Vaccination Strategy II Adversary Strategy IIa
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Vaccination Strategy II Adversary Strategy IIb
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Conclusions about Strategies I and II

• If you can only vaccinate two individuals
• Vaccination Strategy II never leads to more than
  two infected individuals, while Vaccination
  Strategy I sometimes leads to three infected
  individuals (depending upon strategy used by
  adversary).
• Thus, Vaccination Strategy II is better.

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k-Conversion Sets
k-conversion sets are complex. Consider the
graph K4 x K2.
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k-Conversion Sets II
Exercise (a). The vertices a, b, c, d, e form a
2-conversion set. (b). However, the vertices
a,b,c,d,e,f do not. Interpretation Immunizing
one more person can be worse! (Planting a
disease with one more person can be worse if you
want to infect everyone.) Note the same does
not hold true for irreversible k-conversion
sets.
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NP-Completeness
Problem Given a positive integer d and a graph
G, does G have a k-conversion set of size at
most d? Theorem (Dreyer 2000) This problem is
NP-complete for fixed k gt 2. (Whether or not
it is NP-complete for k 2 remains
open.) Same conclusions for irreversible
k-conversion set.
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k-Conversion Sets in Regular Graphs
G is r-regular if every vertex has degree
r. Set of vertices is independent if there are no
edges. Theorem (Dreyer 2000) Let G (V,E)
be a connected r-regular graph and D be a set
of vertices. (a). D is an irreversible
r-conversion set iff V-D is an independent
set. (b). D is an r-conversion set iff V-D
is an independent set and D is not an
independent set.
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k-Conversion Sets in Regular Graphs II
Corollary (Dreyer 2000) (a). The size of the
smallest irreversible 2- conversion set in Cn
is ceilingn/2. (b). The size of the smallest
2-conversion set in Cn is ceiling(n1)/2. ce
ilingx smallest integer at least as big as
x. This result agrees with our observation.
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k-Conversion Sets in Regular Graphs III
Proof (a). Cn is 2-regular. The largest
independent set has size floorn/2, where
floorx largest integer no bigger than x.
Thus, the smallest D so that V-D is
independent has size ceilingn/2. (b). If n
is odd, taking the first, third, , nth vertices
around the cycle gives a set that is not
independent and whose complement is independent.
If n is even, every vertex set of size n/2
with an independent complement is itself
independent, so an additional vertex is needed.
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k-Conversion Sets in Grids
Let G(m,n) be the rectangular grid graph with
m rows and n columns.
G(3,4)
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Toroidal Grids
The toroidal grid T(m,n) is obtained from the
rectangular grid G(m,n) by adding edges from
the first vertex in each row to the last and from
the first vertex in each column to the
last. Toroidal grids are easier to deal with
than rectangular grids because they form regular
graphs Every vertex has degree 4. Thus, we can
make use of the results about regular graphs.
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T(3,4)
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4-Conversion Sets in Toroidal Grids
Theorem (Dreyer 2000) In a toroidal grid T(m,n)
(a). The size of the smallest 4-conversion set
is maxn(ceilingm/2), m(ceilingn/2) m or n
odd mn/2 1 m, n even (b). The size of
the smallest irreversible 4-conversion set is as
above when m or n is odd, and it is mn/2
when m and n are even.

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Part of the Proof Recall that D is an
irreversible 4-conversion set in a 4-regular
graph iff V-D is independent. V-D
independent means that every edge u,v in G
has u or v in D. In particular, the ith row
must contain at least ceilingn/2 vertices in D
and the ith column at least ceilingm/2 vertices
in D (alternating starting with the end vertex of
the row or column). We must cover all rows and
all columns, and so need at least
maxn(ceilingm/2), m(ceilingn/2) vertices
in an irreversible 4-conversion set.
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4-Conversion Sets for Rectangular Grids
More complicated methods give Theorem (Dreyer
2000) The size of the smallest 4-conversion set
and smallest irreversible 4-conversion set in a
grid graph G(m,n) is 2m 2n - 4
floor(m-2)(n-2)/2
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4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8?
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4-Conversion Sets for Rectangular Grids
Consider G(3,3) 2m 2n - 4
floor(m-2)(n-2)/2 8. What is a smallest
4-conversion set and why 8? All boundary
vertices have degree lt 4 and so must be included
in any 4-conversion set. They give a conversion
set.
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More Realistic Models

• Many oversimplifications. For instance
• What if you stay infected only a certain number
  of days?
• What if you are not necessarily infective for the
  first few days you are sick?
• What if your threshold k for changes from 0 to 1
  changes depending upon how long you have been
  uninfected?

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Alternative Models to Explore
Consider an irreversible process in which you
stay in the infected state (state 1) for d time
periods after entering it and then go back to the
uninfected state (state 0). Consider a
k-threshold process in which we vaccinate a
person in state 0 once k-1 neighbors are infected
(in state 1). Etc. -- let your imagination roam
free ...
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More Realistic Models

• Our models are deterministic. How do
  probabilities enter?
• What if you only get infected with a certain
  probability if you meet an infected person?
• What if vaccines only work with a certain
  probability?
• What if the amount of time you remain infective
  exhibits a probability distribution?

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Alternative Model to Explore
Consider an irreversible 1-threshold process in
which you stay infected for d time periods and
then enter the uninfected state. Assume that you
get infected with probability p if at least one
of your neighbors is infected. What is the
probability that an epidemic will end with no one
infected?
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The Case d 2, p 1/2
Consider the following initial state
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The Case d 2, p 1/2
With probability 1/2, vertex a does not get
infected at time 1. Similarly for vertex
b. Thus, with probability 1/4, we stay in the
same states at time 1.
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The Case d 2, p 1/2
Suppose vertices are still in same states at time
1 as they were at time 0. With probability 1/2,
vertex a does not get infected at time 2.
Similarly for vertex b. Also after time 1,
vertices c and d have been infected for two
time periods and thus enter the uninfected
state. Thus, with probability 1/4, we get to the
following state at time 2
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The Case d 2, p 1/2
Thus, with probability 1/4 x 1/4 1/16, we
enter this state with no one infected at time
2. However, we might enter this state at a later
time. It is not hard to show (using the theory
of finite Markov chains) that we will end in
state (0,0,0,0). (This is the only absorbing
state in an absorbing Markov chain.). Thus with
probability 1 we will eventually kill the disease
off entirely.
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The Case d 2, p 1/2
Is this realistic? What might we do to modify
the model to make it more realistic?
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How do we Analyze this or More Complex Models for
Graphs?

• Computer simulation is an important tool.
• Example At the Johns Hopkins University and the
  Brookings Institution, Donald Burke and Joshua
  Epstein have developed a simple model for a
  region with two towns totalling 800 people. It
  involves a few more probabilistic assumptions
  than ours. They use single simulations as a
  learning device. They also run large numbers of
  simulations and look at averages of outcomes.

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How do we Analyze this or More Complex Models for
Graphs?

• Burke and Epstein are using the model to do what
  if experiments
• What if we adopt a particular vaccination
  strategy?
• What happens if we try different plans for
  quarantining infectious individuals?
• There is much more analysis of a similar nature
  that can be done with graph-theoretical models.

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Would Graph Theory help with a deliberate
outbreak of Anthrax?
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• What about a deliberate release of smallpox?

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• Similar approaches, using mathematical models
  based in DM/TCS, have proven useful in many other
  fields, to
• make policy
• plan operations
• analyze risk
• compare interventions
• identify the cause of observed events

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• Why shouldnt these approaches work in the
  defense against bioterrorism?