Notes, intro to modeling, Nimbios REU, day 2

S<-->I is introduced using matrix.

S(t+1) = 0.9 S + 0.2 I
I(t+1) = 0.1S + 0.8 I

[S(t+1), I(t+1)]^T = [(0.9, 0.1)^T, (0.2, 0.8)^T] [S, I]

X = A X for equilibrium, and eigen value will be found.


Leslie matrix model is introduced using locust with  3 stages: egg, nymph, adult. This simple case actually oscillates.

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Differential equations

dx/dt = r x ( 1 - x/K)    logistic growth

dx/dt = r x (K-x) ( x-b)     Allee effect


Two interacting populations (predator prey model)
 dx/dt = a x - bxy
 dy/dt = cxy - dy

More on disease models:
  S-I-R model
I can be infected or immune
R can be recovered (immue) or removed.

Discussion focused on SIR without demographics (no birth or death).

R0 basic reproduction ratio is introduced.


SIR with waning immunity
  dS/dT = - beta SI + gamma R
  dI/dt = beta SI- v I
  dR/dt = vI - gamma R

SEIR (E for exposed, latent, not able to transmit disease)


Student group exercise is to model 3 population:
 pop1: logistic growth
 pop2: gowth with Allee effect
 pop3: exponential growht
 pop1 and pop2 compete with each other. Pop 3 cooperate with the other two populations.