Why Model?
Fred S. Roberts
Department of Mathematics and
DIMACS (Center for Discrete Mathematics and
Theoretical Computer Science)
Rutgers University
Piscataway, 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 Standards
One 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 Model
A 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 :  j qij 1. In fact:  j qij < 1.
Data: How do we get data to fit our model?
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 Model
Assumption: 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:
t
c(t) = c(0)Qt + f  Qk
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
0 2/3 1/3
(I-Q)-1 =
f(I-Q)-1  g
g = (25, 25, 25)
3 3 3
3 6 4.5
3 6 6
3f1 + 3f2 +3f3  25
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 Model
Simplified 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 Model
Discrete 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:  j q 1 . In fact:  j qij < 1.
ij
Data: How do we get data to fit our model?
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 Analysis
As before:
t
c(t) = c(0)Qt + f Qk
Thus:
k 0
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).
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 Further
Suppose 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 Further
Different 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?