1. Computational Methods for Design Lecture 2 – Some “ Simple ” Applications John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C enter for A pplied M athematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531 A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N∞M∞T Series Two Course Canisius College, Buffalo, NY

2. Today’s Topics Lecture 2 – Some “ Simple ” Applications  A Falling Object: Does F=ma ?  Population Dynamics  System Biology  A Smallpox Inoculation Problem  Predator - Prey Models  A Return to Epidemic Models

3. A Falling Object “Newton’s Second Law” WARNING !! THIS IS A SPECIAL CASE !! IF m(t) = m is constant, then ASSUME the only force acting on the body is due to gravity …. y(t)

4. A Falling Object (constant mass). y(t) ODE INITIAL VALUES GENERAL SOLUTION

5. A Falling Object: Problems?

7. Terminal Velocity AIR RESISTANCE FOR A FALLING OBJECT

8. Terminal Velocity

9. Comments About Modeling Newton’s Second Law IS Fundamental TWO PROBLEMS 1. FINDING ALL THE FORCES (OF IMPORTANCE) 2. KNOWING HOW MASS DEPENDS ON VELOCITY ASSUMING CONSTANT MASS “ CORRECTION ” FOR AIR RESISTANCE THE “ MODEL ” FOR AIR RESISTANCE IS AN APPROXIMATION TO REALITY

10. More Fundamental Physics ? HOW DOES THE MASS DEPENDS ON VELOCITY ? FOLLOWS FORM EINSTEIN’S FAMOUS ASSUMPTION

11. More Fundamental Physics EINSTEIN’S CORRECTED FORMULA

12. Comments About Mathematics  ONLY IMPORTANT WHEN  STILL DOESN’T HELP WITH MODELING FORCES  SCIENTISTS AND ENGINEERS MUST FIND THE “ IMPORTANT ” RELATIONSHIPS MATHEMATICIANS MUST DEVELOP NEW MATHEMATICS TO DEAL WITH THE MORE COMPLEX PROBLEMS AND MODELS

13. Comments About Modeling MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT 4465

14. Population Dynamics  Use growth of protozoa as example  A “population” could be … l Bacteria, Viruses … l Cells (Cancer, T-cells …) l People, Fish, Cows …Fish  “ Things that live and die ” ASSUME PLENTY OF FOOD AND SPACE sec, hrs, days, years …. Number of cells at time t Probability that a cell divides in unit time at time t Probability that a cell dies in unit time at time t

15. Population Dynamics Number of new cells on Number of cell deaths on Change in cell population TAKE LIMIT AS Malthusian LAW of population growth Thomas R. Malthus (1766-1834)

16. Population Dynamics BIRTH RATE DEATH RATE REPRODUCTIVE RATE ASSUME CONSTANT DO AN EXPERIMENT

17. Population After 5 Days

18. Population After 7.5 Days

19. Population After 10 days NOT WHAT REALLY HAPPENS

20. Improved Model COMPETITION FOR FOOD AND SPACE Malthus ASSUMED PLENTY OF FOOD AND SPACE Pierre-Fancois Verhulst (1804-1849)

21. Logistics Equation CARRYING CAPACITY NATURAL REPRODUCTIVE RATE LE IS THIS A BETTER (MORE ACCURATE) MODEL ? ?

22. A Comparison: First 5 Days Malthusian LAW of population growth Logistic LAW of population growth

23. A Comparison: First 7.5 Days Malthusian LAW of population growth Logistic LAW of population growth 10,0005,000

24. A Comparison: First 10 Days Logistic LAW of population growth Malthusian LAW of population growth 9,000100,000

25. Logistic Equation: 15 Days K

26. Modeling in Biology Malthusian LAW of population growth Logistic LAW of population growth MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT ? WHAT HAVE WE LEFT OUT ? ? WHAT IS THE CORRECT LAW ? NEED A NEWTON OR EINSTEIN FOR SYSTEM BIOLOGY

27. A Smallpox Inoculation Problem  Basic issue: Compute the gain in life expectancy if smallpox eliminated as a cause if death?  Very timely problem  What if smallpox is “injected” into a large city?  How does age impact the problem? a = AGE Fraction of susceptibles who survive & become immune Death rate at age a due to all causes Rate at which susceptibles become infected Fraction that dies due to the infection Probability that a newborn is alive and susceptible at age a Probability that a newborn is alive and immune at age a

28. A Smallpox Inoculation Model  Typical epidemiological model  Contains age dependent coefficients  Model applied to Paris  Not funded by Dept. of Homeland Defense  Work was done in 1760 and published in 1766 by … Daniel Bernoulli, “Essai d’une nouvelle analyse de la mortalité causée par la petite vérole”, Mém. Math. Phys. Acad. Roy. Sci., Paris, (1766),1.

29. Predator - Prey Models  Vito Volterra Model (1925)  Alfred Lotka Model (1926) THINK OF SHARKS AND SHARK FOOD Numbers of predators Numbers of prey Parameters

30. Numerical Issues: LV Model  Numerical Schemes? l Explicit Euler? l Implicit Euler? l ODE23? l ODE45? l ?????? a/b c/d o > > > y x o

31. Symplectic Methods Explicit Euler Symplectic Implicit Euler > >

32. Epidemic Models SusceptibleInfected Removed

33. Epidemic Models  SIR Models (Kermak – McKendrick, 1927) l S usceptible – I nfected – R ecovered/Removed

34. SIR Models NOT ISOLATED Equilibrium

35. SIR Model I(t) S(t) S(t) + I(t) = N = 1

36. Epidemic Models (SARS)  SEIJR: S usceptibles – E xposed - I nfected - Re moved Model of SARS Outbreak in Canada by Chowell, Fenimore, Castillo-Garsow & Castillo-Chavez (J. Theo. Bio.)  Multiple cities (patch models) l Hyman  Mass transportation l Castillo-Chavez, Song, Zhang  Delays l Banks, Cushing, May, Levin …  Migration – Spatial effects l Aronson, Diekmann, Hadeler, Kendall, Murray, Wu …

37. Spatial Model I  O’Callaghan and Murray (J. Mathematical Biology 2002) Spatial Epidemic Model Partial-Integro-Differential Equations NON-NORMAL DELAY = latent

38. Spatial Model II WHERE A generates a delay semi-group HERE

39. Remarks  GOOD COMPUTATIONAL MATHEMATICS WILL BE THE KEY TO FUTURE BREAKTHROUGHS APPROXIMATIONS MUST BE DONE RIGHT  LOTS OF SIMPLE APPLICATIONS  OPPORTUNITIES FOR MATHEMATICIANS TO GET INVOLVED WITH MODELING = JOB SECURITY FOR APPLIED MATHEMATICIANS  NEW MODELS NEED TO BE DEVELOPED … TOGETHER … MATHEMATICIANS WITH l PHYSICS, CHEMISTRY, BIOLOGY … l FLUID DYNAMICS, STRUCTURAL DYNAMICS …