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 4

So Where’s the Maths In Medicine??

6 A Classroom Interactive Activity School of Maths & Applied Statistics

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 7

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. 8

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. 9

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 10

Random Number Generator ( Excel) 11

Graph of Time vs Infection In Days InfectionInfection Days What is happening?? In the long term, will the Infection continue to increase Or will it stop?? 12

Graph of Time vs Infection In Days InfectionInfection Days Exponential Function Experimental Data 13

Graph of Time vs Infection In Days InfectionInfection Exponential Function Experimental Data Days Corrected Exponential Function 14

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 ???? 15

Mathematical Analysis Word Equation {Rate of Change in Infection I} = {Reasons for Change} 16 = {Increase in Infection} That is, dI dt proportional I = a I But need correction term!

17 Mathematical Analysis dI dt = a I – b I 2 Correction term! Don’t know how to solve this differential equation!!!!! No! But we can analysis what happens using a Phase Line Here Max I = N (Population Size) and Min I = 0 (No Infection) We can find out the long term behaviour of Infection!

Using Derivatives Given I=I(t) then dI dt = 0 Generates the stationary values for I. Here Physically means no change in rate of Infection I ( a – b I ) = 0 Ie I = 0 and I = a/b. 18

Examination of Stationary Values I=0I=NI=a/b dI dt = I ( a – b I ) I is an increasing fn dl dt > 0 I is an decreasing fn Note: 0 < a/b <= N. 19 dl dt > 0

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 20

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 21

Dynamical Systems Rate of Flow of Toxins in a Body 22

Dynamical Systems Simple SIR Model 23

Dynamical Systems Drug Release Into A Body 24

Dynamical Systems Movement of Impulses Through the Nervous System 25

Where’s the Maths in Medicine? Modelling in many areas –Epidemiology –Cancer Research –Pharmokinetics –Neurology –Forensic Entomology –Enzyme Kinetics –Population Dynamics –Others!! 26

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,

Mathematics in Medicine?? Undertake : Bachelor of Medical Mathematics Degree! Code Number: ATAR: Background Recommendation: Math Extension 1 and 2 Unit Chem or Enjoy Science. 28

Content Of BMedMath Degree? 29 Overview 3 years full-time Indicative ATAR: 85 UAC code: 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 yearSecond year Introduction to Anatomy and Physiology I Multivariate and Vector Calculus Structure and Reactivity of Molecules for Life Linear Algebra Molecules, Cells and Organisms Estimation & Hypothesis Testing Introductory Physical and General Chemistry Probability & Random Variables Introduction to Behavioural Science Differential Equations 2 Algebra and Differential Calculus Applied Mathematical Modelling 2 Series and Integral Calculus Introductory Genetics Introduction to Biomedical PhysicsPrinciples of Biochemistry

Careers In Medical Mathematics? Health Industry Pharmaceutical Industry Government –ABS –CSIRO –Universities –Forensic Research Organisations 30

Other Maths Degrees? Bmath BMath(Adv) BMathFin BMathEd Other joint or double degrees Graduate 31

Questions???? 32

References Millennium Maths Project, University of Cambridge,