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Mathematics and Bioterrorism Graph-theoretical

Models of Spread and Control of Disease

- Great concern about the deliberate introduction

of diseases by bioterrorists has led to new

challenges for mathematical scientists. -

smallpox

- 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

- 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

- Dealing with bioterrorism requires detailed

planning of preventive measures and responses. - Both require precise reasoning and extensive

analysis.

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.

- 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.

- 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.

What Can Math Models Do For Us?

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.

- 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.

DIMACS Special Focus on Computational and

Mathematical Epidemiology 2002-2005

Anthrax

Methods of Math. and Comp. Epi.

- Math. models of infectious diseases go back to

Daniel Bernoullis mathematical analysis of

smallpox in 1760.

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

STDs

- Made explicit concepts such as herd immunity for

vaccination policies

- Led to insights about drug resistance, rate of

spread of infection, epidemic trends, effects of

different kinds of treatments.

- The size and overwhelming complexity of modern

epidemiological problems -- and in particular the

defense against bioterrorism -- calls for new

approaches and tools.

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

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

Discrete Math. and Theoretical Computer Science

- Many fields of science, in particular molecular

biology, have made extensive use of DM broadly

defined.

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.

What are DM and TCS?

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

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.

DM and TCS Continued

- These tools are made especially relevant to

epidemiology because of - Geographic Information Systems

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.

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

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.

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

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.

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)

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.

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

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,

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?

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.

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.

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.

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.

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?

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))

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.

1-Conversion Sets

k 1. What are the conversion sets in a basic

1-threshold process?

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?

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.

Conversion Sets for Odd Cycles

C2p1 2-threshold process. What is a convers

ion set?

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 co

nversion set. Moreover, everyone gets infected in

one step.

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.

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.

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

Vaccination Strategy I Worst Case (Adversary

Infects Two)Two Strategies for Adversary

Adversary Strategy Ia

Adversary Strategy Ib

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.

Vaccination Strategy I Adversary Strategy Ia

Vaccination Strategy I Adversary Strategy Ib

Vaccination Strategy II Worst Case (Adversary

Infects Two)Two Strategies for Adversary

Adversary Strategy IIa

Adversary Strategy IIb

Vaccination Strategy II Adversary Strategy IIa

Vaccination Strategy II Adversary Strategy IIb

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.

k-Conversion Sets

k-conversion sets are complex.

Consider the graph K4 x K2.

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.

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.

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.

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.

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.

k-Conversion Sets in Grids

Let G(m,n) be the rectangular grid graph with

m rows and n columns.

G(3,4)

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.

T(3,4)

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.

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.

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

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?

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.

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?

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 ...

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?

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?

The Case d 2, p 1/2

Consider the following initial state

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.

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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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.

The Case d 2, p 1/2

Is this realistic? What might we do to modify t

he model to make it more realistic?

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.

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.

Would Graph Theory help with a deliberate

outbreak of Anthrax?

- What about a deliberate release of smallpox?

- 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

- Why shouldnt these approaches work in the

defense against bioterrorism?