Transcript Medical Mathematics - Engineering and Information Sciences @ UOW

University of Wollongong
Mathematics Teachers Day
June 2011
Where’s the Mathematics
In
Medicine?
School of Mathematics and Applied Statistics
Faculty of Informatics
University of Wollongong
Presenter: Dr Annette L. Worthy
What’s On?
• General Talk (30mins)
• An Exercise in Spreading a Disease
• The Pitch:
Where`s the Maths in Medicine?
• A Solution!
• Questions??
2
Traditions
• Mathematics has played a huge role in
physics and engineering!
• Biomedical Research has relied on verbal
reasoning and statistical experiments!
3
Recent Developments!
• Dynamical Systems have become of much
interest to Mathematicians
• Powerful tool to describe mathematically
complex interactions
• Mathematics is bridging the gap between
medicine and science
Examples of studies:
– How drug diffuses through a body,
– Growth of tumor cells,
– Spread of diseases and viruses,
– Behaviour of nerves, etc
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So Where’s the Maths In
Medicine??
A Classroom Interactive Activity
School of Maths & Applied Statistics
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Modelling Scenario:
• The whimpy kid (from Summer High) is a
disaster/disease waiting to happen!
(One student from the class is chosen at random here.)
• Within the Summer High grounds lays that
mouldy cheese that was never ever touched
for ions. (Ewwh!) It was grossly blue black
with lots of sickening bacteria that causes a
very, very dabiliatating disease called
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Modelling Scenario:
(students give a name for the disease here eg Cafaterium Hyotis.)
• Cafaterium Hyotis is contagious. The kid accidentally falls
on that cheese and becomes infected with Cafaterium
Hyotis!
• Each day, the disease either re-infects the same infectious
student or infects someone else that has come into contact
with the infected student (called probability with replacement).
• One student (chosen at random using a random number generator)
comes to the aid of the Whimpy kid.
• Thus starts the spread of the infection: Cafaterium Hyotis.
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Rules of Infection Encounter
• Count the number of infected students per day.
• The infected student(s) infects only one
student per day.
• An infected student can re-infect themselves
(probability with replacement).
• Class size 12+ students.
• No recovery from Infection.
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How is it done?
• Using a random number generator (Excel)
• Each infected student member picks a number
at random by the generator to choose those
students who will be next to become infected (or
re-infected).
• Number of infected each day are counted
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Random Number Generator
(Excel)
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Graph of Time vs Infection
In Days
I
n
f
e
c
t
i
o
n
What is happening??
In the long term,
will the Infection
continue to increase
Or
will it stop??
Days
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Graph of Time vs Infection
In Days
I
n
f
e
c
t
i
o
n
Exponential
Function
Experimental
Data
Days
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Graph of Time vs Infection
In Days
I
n
f
e
c
t
i
o
n
Corrected
Exponential
Function
Exponential
Function
Experimental
Data
Days
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Are the rules sufficient to
model a contagious disease?
• What if we had more students in the
class? Will the result/graph change?
• What else can we consider?
Births/Deaths/Immigration?
Recovery?
Immunity?
Immunisation?
Delay in Infection and Recovery
????
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Mathematical Analysis
• Word Equation
{Rate of Change in Infection I}
= {Reasons for Change}
= {Increase in Infection}
• That is,
dI
dt
proportional I
=
aI
But need correction term!
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Mathematical Analysis
dI = a I – b I 2
dt
Here Max I = N (Population Size)
Correction term!
and Min I = 0 (No Infection)
Don’t know how to solve this differential equation!!!!!
No! But we can analysis what happens using
a Phase Line
We can find out the long term behaviour of Infection!
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Using Derivatives
Given I=I(t) then
dI = 0
dt
Physically means no
change in rate of
Infection
Generates the stationary values for I.
Here
I(a–bI)= 0
Ie I = 0
and
I = a/b .
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Examination of Stationary
Values
dI = I ( a – b I )
dt
Note:
I=0
dl > 0
dt
I is an increasing fn
I=a/b
0 < a/b <= N .
dl > 0
dt
I=N
I is an decreasing fn
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What have we found out about
the Spread of Infection
• Long term infection
• Infection increases to the stationary
value of
I= a/b.
for this model.
• Improved mathematical models can
give us further details.
eg: What parameters, will disease spread? The effects
of recovery, immunisation etc
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Examples of Mathematical
Modelling in Medicine
• Called 1D models
– Infection Model (previous example)
– Populations (harvesting, births/deaths etc)
– Patients in a Hospital
• Dynamical Systems
– 2 D + compartmental models
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Dynamical Systems
• Rate of Flow of Toxins in a Body
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Dynamical Systems
• Simple SIR Model
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Dynamical Systems
• Drug Release Into A Body
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Where’s the Maths in
Medicine?
• Modelling in many areas
–
–
–
–
–
–
–
–
Epidemiology
Cancer Research
Pharmokinetics
Neurology
Forensic Entomology
Enzyme Kinetics
Population Dynamics
Others!!
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Where’s the Maths in
Medicine?
“Mathematical Modelling is a key to the future
breakthroughs in the treatment of diseases.”
Millennium Maths Project 2008
“The frontiers of biological and medical research
are becoming increasingly dependent on
sophisticated modeling, analysis, and
computational techniques. Operations
research and mathematics are rapidly
emerging as vital tools..”
Institute for Operations Research and the Management Sciences Workshop Pittsburgh, 2006 27
Mathematics in Medicine??
Undertake:
Bachelor of Medical Mathematics Degree!
Code Number: 756530
ATAR: 80-85
Background Recommendation:
Math Extension 1
and
2 Unit Chem
or
Enjoy Science.
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Content Of BMedMath Degree?
Overview
• 3 years full-time
• Indicative ATAR: 85
• UAC code: 756530
• Entry requirement : HSC Mathematics (not General Maths)
• Recommended: HSC Mathematics Ext 1 is highly recommended but
not essential; HSC Chemistry is recommended but not essential.
• The program is comprised of subjects from Mathematics, Biology,
Chemistry, Psychology, Physics and Statistics.
First year
Introduction to Anatomy and Physiology I
Structure and Reactivity of Molecules for Life
Molecules, Cells and Organisms
Introductory Physical and General Chemistry
Introduction to Behavioural Science
Algebra and Differential Calculus
Series and Integral Calculus
Introduction to Biomedical Physics
Second year
Multivariate and Vector Calculus
Linear Algebra
Estimation & Hypothesis Testing
Probability & Random Variables
Differential Equations 2
Applied Mathematical Modelling 2
Introductory Genetics
Principles of Biochemistry
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Careers In Medical
Mathematics?
• Health Industry
• Pharmaceutical Industry
• Government
–
–
–
–
ABS
CSIRO
Universities
Forensic
• Research Organisations
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Other Maths Degrees?
• Bmath
• BMath(Adv)
Graduate
• BMathFin
• BMathEd
• Other joint or double degrees
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Questions????
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References
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•
•
•
•
•
http://www.childrenspanadol.com.au/Health/InfectiousDiseases.aspx
http://justdamnstupid.wordpress.com/2011/03/04/wanna-bet/
http://www.sfjohnson.com/acad/studying/studying.htm
Millennium Maths Project, University of Cambridge, 2008
http://meetings2.informs.org/Pittsburgh06/nsfworkshop.html
http://www.google.com.au/search?q=cartoon+school+students
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