Dr Rachel NormanDr Rachel Norman

University of StirlingUniversity of Stirling

1010thth June 2010. June 2010.

Why Multi scale modelling of biological systems is important.

My background

•Rabies in ethiopian wolves

•Louping ill in red grouse

•Fish parasites

Why is changing scale Why is changing scale important?important?

Arises in a large number of systemsArises in a large number of systems

• Physical: From individuals to populations- Physical: From individuals to populations- cells, enzymes, people, animals…cells, enzymes, people, animals…

• Spatial: From local to global- bee hives, Spatial: From local to global- bee hives, villages, farms….villages, farms….

• Temporal: From short term to evolutionary Temporal: From short term to evolutionary time scales – transient dynamics vs time scales – transient dynamics vs equilibrium, present time vs evolutionary equilibrium, present time vs evolutionary time…time…

Individuals to populations: Examples

Population growth

Epidemiology

Immunology

Population growth

Exponential growth

rXXbabXaXdt

dX )(

Exponential growth

0

200

400

600

800

1000

1200

1400

1600

0 0.2 0.4 0.6 0.8 1 1.2

time

X

r=-0.2 r=0 r=0.15 r=0.25

Saturated growth

Assume birth decreases or death increases linearly with density

XsXrdt

dX)(

Logistic Growth

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7

Time

X(t

)

Other possibilities

))1(1( KXrX tt

KXrX tt 1exp

trXK exp1

tt

XKXK

2

11

ctt

XKXK

2

1

1

ctt

XKXK

2

1

1

•Name (ref)

•Quadratic [6]

•Ricker [7]

•Skellam [8]

•Beverton Holt [9]

•Hassell [10]

•Maynard-Smith-Slatkin [11]

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?

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?

Epidemiology

Example :1 simple SIR model

Susceptible Infected Immune

births

deaths

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 =

SI

N

SI

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.

baIS

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.

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?

Immunology

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

IIPefdt

dI

PIfrPdt

dP

)(

)(

Questions

Are these the right assumptions to make about interaction terms?

Can we derive better functions for this?

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?