Admistrivia

• Homework due tomarrow
• Read sections 6.1,2,3.

Continuous markov models: Pure death processes

Pure death process

1. P(X(t+h) - X(t) = -1|X(t) = k) = mu_k h + o(h)
2. P(X(t+h) - X(t) = 0|X(t) = k) = 1 - mu_k h + o(h)
3. P(X(t+h) - X(t) > 0|X(t) = k) = 0
4. X(0) = N.

Cool new way of representing "Yule"/linear death process

1. Each person has chance alpha h of dieing in time periods h.
2. So expected number of deaths is alpha X(t) h.
3. But each person is easy to solve individually
4. Each treated independently: call death time eta_i
5. P(X_t = n) = P(exactly N-n people die up to t)
6. but this is just a Bernulli sequence!

Birth / death processes

1. P_i,i+1(h) = lambda_k h + o(h)
2. P_i,i-1(h) = mu_k h + o(h)
3. P_i,i(h) = 1 - mu_k h - lambda_k h + o(h)
4. P_i,j(0) = 0 unless i = 1
5. mu₀ = 0. All others, positive

• P_i,j(h) basically looks like the identity matrix for small h
• So we can write P_i,j(h) = I + h A + o(h)
• What is A?
• Exponential asside:
  • P_i,j(t) = P_i,j(h)^t/h
  • Taking limit: P_i,j(t) = exp(tA).
  • Make it computable later (at least now we have an exact definition)

Chapman-Kolmogorov equations

• Write also scalar form
• Continuous time analog of discrete case: P^t+s =P^tP^s

Sojourn times

• Exponential holding times
• up with probablity lambda/(mu+lambda)
• Enough to simulate it all

Slight worry

Unique if sum (1/lambda)'s is infinite.

Otherwise, there is a positive probability of visiting +infinity and maybe coming back down. Very cool, but not very realistic.