Why Model?

Fred S. Roberts

Department of Mathematics andDIMACS (Center for Discrete Mathematics and

Theoretical Computer Science)

Rutgers UniversityPiscataway, New Jersey

There are All Kinds of Models

•Maps

•Scale Models

•Computer Models

•Mathematical Models

Mathematical Models: *Make use of the Most Precise Language ever Invented By Man.*Make use of the Power of this Language to Enable us to Reason and Analyze

Why Modeling and Bioterrorism?

Why Modeling and Bioterrorism?•Components of host-pathogen systems are sufficiently numerous and their interactions sufficiently complex that intuition alone is insufficient to fully understand the dynamics of such interactions.

•Experimentation or field trials are often prohibitively expensive or unethical or impossible.

•We don’t have real data to go on.Mathematical Modeling becomes an important experimental and analytical tool.

What Can Math Models Do For Us?•Sharpen our understanding of fundamental processes

•Compare alternative policies and interventions

•Help make decisions.

•Provide a guide for training exercises and scenario development.

•Guide risk assessment.

•Aid forensic analysis.

•Predict future trends.

Math Models are Widely Used with Great Success for Such Purposes

•By government and industry.

•For economic policy, transportation planning, logistics, scheduling, resource allocation, …

•By such federal agencies as Transportation, Commerce, Defense, Energy, ...

•In military planning.

•In the private sector in such industries as: Airlines, Oils, Biotechnology, Financial, ...

A Simplified Example:Air Pollution

Meeting Air Pollution StandardsOne pollutant.

n locations in a region.

Environmental Standards give maximum pollutant concentration gi at each location i.

g = (g1,g2,…,gn)

Problem: What policies will achieve the standards?

ci(t) = concentration of pollutant at location i at time t.

c(t) = (c1(t), c2(t), …, cn(t))

Building the Model

Time: continuous or discrete? Observe every hour, day, week, … : discrete.

Observe: Some of the pollutant at location i moves to location j each time period.

qij = fraction of pollutant at i that moves to j each time period. Or: probability that molecule of pollutant moves from i to j each time period.

(Deterministic model vs. probabilistic model.)

Fleshing out the ModelA learning process.

Forces us to identify vital assumptions.

Simplifying Assumption: qij is same every time period.

Simplifying Assumption: No pollutant comes to j from locations not in the set of locations considered.

Observe : . In fact: < 1.

Data: How do we get data to fit our model?

1 j ijq j ij

q

Using the Model to Formalize the Problem

Goal: For t sufficiently large: ci(t) gi for all i:

c(t) g.

Question: How do we achieve this?

Mathematical Analysis:

Q = (qij)

c(t) = c(0)Qt

Under our assumptions (or weaker ones):

Qt 0

The Prediction from the Model

c(t) 0.

There is no air pollution!

Model predictions need to be checked against data if possible.

Checking Against the Data

So What if a Model Fails?

Model failures are good learning experiences.

They help in problem formulation.

They help in forensics.

What is missing? No pollutant is added.

A Modified ModelAssumption: A certain amount of pollutant fi is emitted from location i each time period.

f = (f1,f2,…,fn)

Simplification: fi is the same each time period.

fi is something we can control; it gives us a way to achieve our goal.

Mathematical Analysis:

c(t) = c(0)Qt + f Qk

t

k 0

Under our assumptions, Qk (I-Q)-1

Thus: c(t) f(I-Q)-1

Our goal: c(t) g

This is achieved after awhile for all practical purposes if

f(I-Q)-1 g

We can now find f satisfying this condition and this gives us a policy for achieving the pollution standards.

1/3 0 1/3

Q = 1/3 1/3 1/3 g = (25, 25, 25)

0 2/3 1/3

3 3 3

(I-Q)-1 = 3 6 4.5

3 6 6

3f1 + 3f2 +3f3 25

f(I-Q)-1 g 3f1 + 6f2 + 6f3 25

3f1 + 4.5f2 + 6f3 25

Finding Policy Options

A “policy” to achieve the goal corresponds to a vector f = (f1, f2, f3) satisfying these 3 inequalities.

Sample solutions:

f = (4,1,1)

f = (4,2,0)

How to choose between policies?

We have not built this into the model.

The model has provided options.

Building a More Realistic ModelSimplified models are useful in formulating ideas, thinking about relevant factors, forcing us to define terms and goals precisely.

Making the model more realistic: qij and fi can change each time period.

Now, no closed form solution. Computer simulation is necessary. Simulations allow us to do “what if” experiments.

A Second Example: Money

Achieving the Desired Distribution of Currency

We have n cities with central banks.

We have an idealized amount gi of currency in

city i.

Problem: What policies will help us achieve and maintain the idealized distribution g = (g1,g2,…,gn) of currency in central bank cities?

ci(t) = amount of currency in city i at time t.

Fleshing out the ModelDiscrete times.

Let qij be the fraction of currency in city i that goes to city j each time period.

Simplifying Assumption: qij is same every time period.

Simplifying Assumption: No currency comes to j from locations not in the set of locations considered.

Observe: . In fact: < 1.

Data: How do we get data to fit our model?

1 j ijq j ij

q

Learning from Analogous Models

Note the analogy to the pollution model.

A model in one area can teach us something about another area. (And besides, we only have to do the mathematics once!)

So what is fi?

fi = amount of currency the federal reserve adds to city i each time period.

Modifying fi gives us possible plans to achieve the desired distribution of currency.

Mathematical AnalysisAs before:

c(t) = c(0)Qt + f Qk

Thus:

c(t) f(I-Q)-1

Our goal: c(t) = g (for t sufficiently large).

Thus:

f(I-Q)-1 = g, f = g(I-Q).

t

k 0

We can now find f satisfying this condition and this gives us a policy for achieving the pollution limits. Note: In contrast to pollution example, f is unique.

1/3 0 1/3

Q = 1/3 1/3 1/3 g = (12, 6, 3)

0 2/3 1/3

f = g(I-Q) = (6, 2, -4)

This is the only possibility.

What does -4 mean?

Examining the Policy FurtherSuppose c(0) = (1, 1, 4).

Using the policy f = (6, 2, -4), we find that

c(1) = (28/3, 3, -2/3).

What does -2/3 mean?

City 3 has negative currency.

If we wait long enough, we will achieve the desired currency distribution.

But, we go through an impossible intermediate phase. The policy is a failure!

So What Have We Learned?

Since f is the only possible policy (under our restrictions), no policy could work.

The goal is infeasible.

Models can help us discover that our goals are unrealistic and help us to modify them.

Taking the Analysis FurtherDifferent initial conditions c(0) can make the goal feasible.

Example: Same g, c(0) = (6, 5, 4)

Now, one can prove that c(t) never has negative components before converging to g.

The goal is feasible under a different initial distribution of currency.

Checking Whether a Goal is Feasible

How can we tell whether a goal g is feasible in the sense that the unique policy f that leads c(t) to converge to g never leads to negative components in c(t)?

Mathematicans have developed an efficient computer algorithm for checking feasibility.

What Have We Learned from the Modeling?

We have a precise notion of policy -- even if a simplified one. The next step would be to look at more complex policies that allow fi to vary over time.

We have been led to understand that our initial analysis left out an important criterion: no negativity in components of c(t).

We are now ready to do “what-if” experiments and make the model increasingly more realistic.

So What About Models for Defense Against Bioterrorism?

Are these simplified models convincing?Would modeling help with a deliberate outbreak of Anthrax?

What about a deliberate release of smallpox?

Similar approaches have proven useful in many other fields, to:

•make policy

•plan operations

•analyze risk

•compare interventions

•identify the cause of observed events

Why shouldn’t these approaches work in the defense against bioterrorism?