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1. Advertisement:Possible independent work or project EOS: Economics via Object-oriented Simulation An open-source project devoted to the highly structured simulation of complete economies, making strong use of inheritance, and a very few high-level primitives.

2. Simulation Generally speaking, this means there is one program variable for each element in the system being simulated, … as opposed to • analytical solution • formulation of algebraic or differential eqs.

3. Example: Epidemics • [Dur95] R. Durrett, "Spatial Epidemic Models," in Epidemic Models: Their Structure and Relation to Data, D. Mollison (ed.), Cambridge University Press, Cambridge, U.K., 1995. • Discrete-time, discrete-space, discrete-state

4. Durrett’s epidemic model • Time, t = 0, 1, 2, … • Space: orthogonal (square) grid • State: {susceptible, infected, removed} Rules tell us how to get from t to t+1 for each spatial location Each site has 4 neighbors, contains 0 or 1 individual

5. Durrett’s Rules (“SIR” Model) • Susceptible individuals become infected at rate proportional to the number of infected neighbors • Infected individuals become healthy (removed) at a fixed rate δ • Removed individuals become susceptible at a fixed rate α

6. Time, t = 0, 1, 2, … Space: orthogonal (square) grid State: {susceptible, infected, removed}

7. Simulation results α = 0 : No return from removed; immunity is permanent. If δ, recovery rate, is large, epidemic dies out. If δ is less than some critical number, the epidemic spreads linearlyand approaches afixed shape.  Can be formulated and proven as a theorem! α > 0 : behavior is more complicated

8. Empirical verification • measles in Glasgow, 1929: 440 ft/week • Muskrats escape in Bohemia, 1905: square-root of area grows linearly • Other models: ODEs, PDEs with spatial diffusion. For example, rabies: NSF Mathematical Sciences Institutes SARS: http://www.scielosp.org/img/revistas/bwho/v84n12/a12fig01.jpg

9. More recent work: "Epidemic Thresholds and Vaccination in a Lattice Model of Disease Spread“, C.J. Rhodes and R.M. Anderson, Theoretical Population Biology52, 101118 (1997) Article No. TP971323. Note ring of vaccinated individuals.

10. Some questions: • How do you choose the language? • Can you parallelize? • How do you display? • Why are random numbers needed? • How do you debug with random numbers when every run is different? • How do you test?

11. Simulating population genetics(assignment 1) • review of very basic genetics genes alleles If there are two possible alleles at one site, say A and a, there are in a diploid organism three possible genotypes: AA, aa, Aa, the first two homozygotes, the last heterozygote Question: How are these distributed in a population as functions of time?

12. Why study this? • Understanding history of evolution, human migration, human diversity • Understanding relationship between species • Understanding propagation of genetic diseases • Agriculture

13. Approaches, pros and cons • Field experiment + realistic - hard work for one particular situation • Mathematical model + can yields lots of insight, intuition - usually uses very simplified models - not always tractable • Simulation + very flexible + works when math doesn’t - not easy to make predictions

14. 19th Century: Darwin et al. didn’t know about genes, etc., and used the idea of blendedinheritance • But this requires an unreasonably large mutation rate to explain variation, evolution Enter Mendel…

15. Gregor Mendel (1822 - 1884)

16. http://bio.winona.edu/berg/241f00/Lec-note/Mendel.htm, Steven Berg,Winona State

17. Simplest model • A little history, Mendelian laws • Hardy-Weinberg equilibrium • A little probability/statistics • Wahlund effect in segregated population • Example: Da Cunha’s data on Drosophila polymorpha; abdomen color [Smi89] • Assignment 1: goal, limitations of theoretical model

18. www.nd.edu/~hholloch/pi.html, Hope Holloche, U. Chicago