1 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 lineto get smallpoxvaccine during New York Citysmallpox epidemic

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Outline1. 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 Bernoulli’s mathematical analysis of smallpox in 1760.

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Hundreds of math. models since have:

•highlighted concepts like core population in STD’s;

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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 today’s 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 Cont’d

•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 ModelSocial Network = GraphVertices = PeopleEdges = 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 StatesOnce 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 t+1 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 t+1 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 (can’t 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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PeriodicityState 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(t+P) = s(t) for all t > T.

The smallest such P is called the period.

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Periodicity IIFirst 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 IIIWhen 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 IVInterpretation: 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 Setsk = 1.

What are the conversion sets in a basic 1-threshold process?

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1-Conversion Setsk = 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 Setsk = 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

C2p+1

2-threshold process.

What is a conversion set?

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Conversion Sets for Odd Cycles

C2p+1.

2-threshold process.

Place p+1 1’s in “alternating” positions.

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Conversion Sets for Odd Cycles

We have to be careful where we put the initial 1’s. p+1 1’s 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 p+1 vertices is an irreversible conversion set. Moreover, everyone gets infected in one step.

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Vaccination StrategiesIf you didn’t know whom a bioterrorist might infect, what people would you vaccinate to be sure that a disease doesn’t 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 adversary’s 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 Setsk-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-CompletenessProblem: 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 > 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 GraphsG 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 ceiling[n/2].

(b). The size of the smallest 2-conversion set in Cn is ceiling[(n+1)/2].

ceiling[x] = 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 floor[n/2], where floor[x] = largest integer no bigger than x. Thus, the smallest D so that V-D is independent has size ceiling[n/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 GridsThe 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

max{n(ceiling[m/2]), m(ceiling[n/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 ceiling[n/2] vertices in D and the ith column at least ceiling[m/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 max{n(ceiling[m/2]), m(ceiling[n/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 GridsConsider 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 < 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 ModelsOur 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 = 1/2Suppose 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 shouldn’t these approaches work in the defense against bioterrorism?