1. Dr Rachel Norman University of Stirling 10 th June 2010. Why Multi scale modelling of biological systems is important.

2. My background Rabies in ethiopian wolves Louping ill in red grouse Fish parasites

3. Why is changing scale important? Arises in a large number of systems Physical: From individuals to populations- cells, enzymes, people, animals… Physical: From individuals to populations- cells, enzymes, people, animals… Spatial: From local to global- bee hives, villages, farms…. Spatial: From local to global- bee hives, villages, farms…. Temporal: From short term to evolutionary time scales – transient dynamics vs equilibrium, present time vs evolutionary time… Temporal: From short term to evolutionary time scales – transient dynamics vs equilibrium, present time vs evolutionary time…

4. Individuals to populations: Examples Population growth Epidemiology Immunology

5. Population growth Exponential growth

6. Saturated growth Assume birth decreases or death increases linearly with density

7. Other possibilities Name (ref) Quadratic [6] Ricker [7] Skellam [8] Beverton Holt [9] Hassell [10] Maynard-Smith-Slatkin [11]

8. Brannstrom and Sumpter (Proc Roy soc, 272, 2065-2072. 2005) Built model based on distribution of individuals amongst discrete resource sites. Changed rules about competition. Some models, for example the quadratic model cannot be derived this way. Why not?

9. Questions Why can’t you get the quadratic model? What is the best form of a population growth model under different assumptions about the way individuals interact?

10. Epidemiology

11. Example :1 simple SIR model

13. Transmission Term f(S,I) Transmission rate per susceptible – –contact*prob(infection)*prop infected = c*p*I/N Density dependent transmission – –Contact rate = constant * N – –Transmission = Frequency dependent transmission – –Contact rate = constant – –Transmission =

14. Other forms Hochberg (JTB, 153, 301-321, 1991) Fenton, Fairbairn, Norman and Hudson (JAE, 71, 893-905, 2002) Fitted experimental data on insect parasites to this transmission rate and compared with others in the literature.

15. Turner, Begon and Bowers (Proc Roy soc, 270, 105-112, 2003) Cellular Automata Defined contacts locally for density and frequency dependent transmission. Look at what happens globally. They found that they got frequency dependent transmission globally in both cases.

16. Questions: Does that mean that you cannot get density dependent transmission from an IBM? What is the “correct” form of the transmission form under different assumptions about interaction?

17. Immunology Fenton and Perkins (Parasitology, 137, 1027-1038, 2010)

19. Questions Are these the right assumptions to make about interaction terms? Can we derive better functions for this?

20. Conclusion There are many systems where we make population level assumptions about interaction terms. How do we write down rules about how be observe that individuals behave and derive the population level terms?