15 birth and death processes

Question 1

Formally, a birth and death process is defined by the transition probabilities from state n
to state n + 1 (15.1a) and to state n − 1 (15.1b) in a time increment ∆t → 0:

PROBABILITIES for X_t+∆t for transition to other states

Answer 1

P {X_t+∆t = n + 1|Xt = n} = λ_n∆t
P {Xt+∆t = n − 1|Xt = n} = µ_n∆t
P {Xt+∆t = n|Xt = n} = 1 − (λ_n + µ_n)∆t

Last one: no transitions from state n occur in ∆t. Here, the birth rate λ_n ≥ 0 and death rate µn ≥ 0 are assumed to depend on n = 0, 1, 2, . . ..

Question 2

birth an death process
probabilities for a state

Answer 2

p_n(t+∆t) =
λ_{n−1}∆t *p_{n−1}(t)
+
µ_{n+1}∆t *pn+1(t)
+
(1 − (λ_n + µ_n)∆t) *p_n(t)

birth: n − 1 → n death: n + 1 → n nothing happens

{z }

Question 3

birth an death process
rearranging
probabilities for a state
to obtain dp_n(t)/dt

MASTER EQUATION

e.g.

p_n(t+∆t) =
λ_{n−1}∆t *p_{n−1}(t)
+
µ_{n+1}∆t *pn+1(t)
+
(1 − (λ_n + µ_n)∆t) *p_n(t)

Answer 3

dp_n/dt=
λ_{n−1} *p_{n−1}(t)
+
µ_{n+1} *pn+1(t)
-
(λ_n + µ_n)) *p_n(t)

gain terms loss terms

dp_0(t)/dt=µ_1p_1(t) − λ_0p_0(t).

Question 4

The Gillespie algorithm:

If X_t = n, after how long does the process leave the state n? That is, for how long do
we need to wait until a birth or a death occurs in this population of n individuals?

Answer 4

we define the probability of no jump before time τ :
s(τ ) = P {X_{t+τ} = n|X_t = n}.

Considering a time increment ∆t → 0:
P {X_{t+∆t} = n|X_t = n} = 1 − (λ_n + µ_n)∆t

s(t + ∆t) = s(t)(1 − (λn + µn)∆t).
rearranging gives ds/dτ=
−(λ_n + µ_n)s, with s(0) = 1

solved
s(τ ) = exp(−(λn+µn)τ)

The time of leaving n: INTER EVENT TIME is a RV with exponential distribution and mean
1/(λn + µn):

Probability that T_n, separating two successive events into and from state n
(e.g. a death, n + 1 → n, followed by another death, n → n − 1), exceeds τ is exponentially small:
P {T_n > τ} = P {X_{t+τ} = n|X_t = n} =exp(−(λ_n+µ_n)τ)
.
.

Question 5

The Gillespie algorithm:
If X_t = n, which state does the process jump to, n − 1 (death) or n + 1 (birth)?

That is, when an event occurs, what’s the probability that it is a death or a birth?
MENTIONED

Answer 5

when a transition occurs after a mean time 1/(λn +µn),

P {jump is up to n + 1} =λ_n/(λ_n + µ_n)
and

P {jump is down to n − 1} = µ_n/(λ_n + µ_n)

Question 6

The Gillespie algorithm mentioned

Answer 6

Sample paths of birth and death process aew found by each time step sampling two uniformly distributed Random numbers in (0,1) U_1 and U_2
X_t=n>0

Next jump occurs after
∆t = −1/(λ_n+µ_n)
U_1 ∼ Exp(λ_n + µ_n).
U_1 is the simulation clock

if U_2 ≤µ_n/(λ_n + µ_n)
X_{t+∆t} = n − 1,
otherwise X_{t+∆t} = n + 1.

Question 7

Extinction in birth and death processes
u_k

Answer 7

u_k

λ_k/(λ_k + µ_k)*u_{k+1}
+
µ_k/(λ_k + µ_k} *u_{k−1}

for k ≥ 1, with u_0 = 1.

Question 8

Extinction in birth and death processes
In summary, we have found two possible types of extinction/absorption results:

Answer 8

(i) Either u_m = 1 for all m = 1, 2, . . . and absorption is certain for any m = 1, 2, . . . .
This is true if
lim_{m→∞}
SUM of k=1 to m of ρk =
limm→∞
SUM of k=1 to m of
(µ_1µ_2 . . . µ_k)/(λ_1λ_2 . . . λ_k)
=
(µ_1/λ_1) + (µ_1µ_2)/(λ_1λ_2_
+· · ·+ lim_(m→∞) (µ_1µ_2 . . . µ_m)/(λ_1λ_2 . . . λ_m)
= ∞.

(ii) Or u_1 < 1 and u_m is a decreasing function of m.
This means that the extinction/absorption probability is finite for all m = 1, 2, . . . and is a decreasing function of m.
This is true if limm→∞ sum k=1 to m ρk is finite.

In this case extinction/absorption probability can be explicitly written as
u_m = (SUM of k=mto infinity of ρk)/(1 + SUM of k=1 to infinity of ρk)

Hence, when limm→∞
Xm
k=1
ρk < ∞ the population either goes extinct with a finite probability u_m OR GROWS WITHOUT BOUND with the complementary probability
1 − u_m

Question 9

Simple (linear) birth and death process
When we have
λ_n = nλ and µ_n = nµ

Answer 9

Master equation

p_n(t): nth partial derivative of G(z,t) wrt z and set z=0, dividing by factorial