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Mathematics and Bioterrorism: Graphtheoretical Models of Spread and Control of Disease

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Title: Mathematics and Bioterrorism: Graphtheoretical Models of Spread and Control of Disease


1
Mathematics and Bioterrorism Graph-theoretical
Models of Spread and Control of Disease
2
  • Great concern about the deliberate introduction
    of diseases by bioterrorists has led to new
    challenges for mathematical scientists.


smallpox
3
  • 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
4
  • 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

5
  • Dealing with bioterrorism requires detailed
    planning of preventive measures and responses.
  • Both require precise reasoning and extensive
    analysis.

6
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.
7
  • 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.

8
  • 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.

9
What Can Math Models Do For Us?
10
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.

11
  • 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.

12
DIMACS Special Focus on Computational and
Mathematical Epidemiology 2002-2005
Anthrax
13
Methods of Math. and Comp. Epi.
  • Math. models of infectious diseases go back to
    Daniel Bernoullis mathematical analysis of
    smallpox in 1760.

14
  • Hundreds of math. models since have
  • highlighted concepts like core population in
    STDs

15
  • Made explicit concepts such as herd immunity for
    vaccination policies

16
  • Led to insights about drug resistance, rate of
    spread of infection, epidemic trends, effects of
    different kinds of treatments.

17
  • The size and overwhelming complexity of modern
    epidemiological problems -- and in particular the
    defense against bioterrorism -- calls for new
    approaches and tools.

18
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

19
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

20
Discrete Math. and Theoretical Computer Science
  • Many fields of science, in particular molecular
    biology, have made extensive use of DM broadly
    defined.

21
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.

22
What are DM and TCS?
  • DM deals with
  • arrangements
  • designs
  • codes
  • patterns
  • schedules
  • assignments

23
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.

24
DM and TCS Continued
  • These tools are made especially relevant to
    epidemiology because of
  • Geographic Information Systems

25
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.

26
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

27
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.

28
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

29
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.

30
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)
31
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 dig
raph reflecting the possible ways to move from
state to state in the model.
32
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.
33
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34
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

35
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,
36
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 m
any of your neighbors are uninfected. Does this
make sense?
37
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 sufficientl
y many of your neighbors are infected.
Special Case k 1 Infected if any of your ne
ighbors is infected.
38
Basic 2-Threshold Process
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Irreversible 2-Threshold Process
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44
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.

45
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, becau
se 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 T.
The smallest such P is called the period.
46
Periodicity II
First example the period is 2.
Second example the period is 1. Both basic a
nd 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.
47
Periodicity III
When period is 1, we call the ultimate state
vector a fixed point. When the fixed point is t
he 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 sub
sets 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?
48
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))
49
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 pr
ocess.
50
1-Conversion Sets
k 1. What are the conversion sets in a basic
1-threshold process?

51
1-Conversion Sets
k 1. The only conversion set in a basic 1-thr
eshold 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-conv
ersion sets?
52
Irreversible 1-Conversion Sets
k 1. Every single vertex x is an irrevers
ible 1-conversion set if the graph is connected.
We make it 1 and eventually all vertices become 1
by following paths from x.
53
Conversion Sets for Odd Cycles
C2p1 2-threshold process. What is a convers
ion set?

54
Conversion Sets for Odd Cycles
C2p1. 2-threshold process. Place p1 1s
in alternating positions.

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57
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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60
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 co
nversion set. Moreover, everyone gets infected in
one step.
61
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.
62
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 vaccinati
on? What does better mean? If vaccine has no
cost and is unlimited and has no side effects, of
course we use mass vaccination.
63
Vaccination Strategies
What if vaccine is in limited supply? Suppose we
only have enough vaccine to vaccinate 2
vertices. Consider two different vaccination stra
tegies
Vaccination Strategy I
Vaccination Strategy II
64
Vaccination Strategy I Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy Ia
Adversary Strategy Ib
65
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.
66
Vaccination Strategy I Adversary Strategy Ia
67
Vaccination Strategy I Adversary Strategy Ib
68
Vaccination Strategy II Worst Case (Adversary
Infects Two)Two Strategies for Adversary
Adversary Strategy IIa
Adversary Strategy IIb
69
Vaccination Strategy II Adversary Strategy IIa
70
Vaccination Strategy II Adversary Strategy IIb
71
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.

72
k-Conversion Sets
k-conversion sets are complex.

Consider the graph K4 x K2.
73
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 on
e 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.
74
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 i
s NP-complete for fixed k 2.
(Whether or not it is NP-complete for k 2 re
mains open.) Same conclusions for irreversible
k-conversion set.
75
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-con
version 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.
76
k-Conversion Sets in Regular Graphs II
Corollary (Dreyer 2000) (a). The size of the s
mallest irreversible 2- conversion set in Cn is
ceilingn/2. (b). The size of the smallest 2
-conversion set in Cn is ceiling(n1)/2.
ceilingx smallest integer at least as big as
x. This result agrees with our observation.
77
k-Conversion Sets in Regular Graphs III
Proof (a). Cn is 2-regular. The largest indep
endent set has size floorn/2, where floorx
largest integer no bigger than x. Thus, the
smallest D so that V-D is independent has siz
e ceilingn/2. (b). If n is odd, taking the f
irst, 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.
78
k-Conversion Sets in Grids
Let G(m,n) be the rectangular grid graph with
m rows and n columns.
G(3,4)
79
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 rect
angular grids because they form regular graphs
Every vertex has degree 4. Thus, we can make use
of the results about regular graphs.
80
T(3,4)
81
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.

82
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 n
eed at least maxn(ceilingm/2),
m(ceilingn/2) vertices in an irreversible
4-conversion set.
83
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
84
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-conver
sion set and why 8?
85
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 degr
ee 4-conversion set. They give a conversion set.

86
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?

87
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-thre
shold 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 ...
88
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?

89
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 en
d with no one infected?
90
The Case d 2, p 1/2
Consider the following initial state
91
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.
92
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, ve
rtices c and d have been infected for two
time periods and thus enter the uninfected
state. Thus, with probability 1/4, we get to th
e following state at time 2
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94
The Case d 2, p 1/2
Thus, with probability 1/4 x 1/4 1/16, we ente
r this state with no one infected at time 2.
However, we might enter this state at a later ti
me. It is not hard to show (using the theory o
f 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.
95
The Case d 2, p 1/2
Is this realistic? What might we do to modify t
he model to make it more realistic?
96
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.

97
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.

98
Would Graph Theory help with a deliberate
outbreak of Anthrax?

99
  • What about a deliberate release of smallpox?

100
  • 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

101
  • Why shouldnt these approaches work in the
    defense against bioterrorism?
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