1 Birth-Death Processes
1.1 What is a Birth-Death (BD) Process?
Birth-death processes are continuous-time models for population growth where \(\{ X_{t};t\geq 0 \}\) records the number of individuals alive at time \(t\).
Assume that \(X_{t}=i\) that is, at time \(t\), the population numbers \(i\) individuals. We define the following:
- \(B(i)\sim\text{Exponential}(\lambda_{i})\) be the time until the next birth.
- \(D(i)\sim\text{Exponential}(\mu_{i})\) be the time until the next death.
Here \(\lambda_{i}\) is the birth rate and \(\mu_{i}\) is the death rate. From standard results for exponential random variables we have that:
- Mean birth time. \(\frac{1}{\lambda_{i}}\), \(i \geq 0\).
The underlying assumption here is that the birth and death rates depend on the size of the population. We also make the following further simplifying assumptions:
- Only one event happens at a time e.g. two births do not happen at the same time.
- \(\{ B(i) \}\)’s and \(\{ D(i) \}\)’s are independent sequences that are independent of each other.
- We have rates \(\lambda_{i}\geq 0\), \(\mu_{i}\geq 0\), \(\mu_{0}=0\) and \(\mu_{i}+\lambda_{i}>0\) for all \(i\geq 1\).
1.2 BD Process Transition Probabilities
From this construction we have a a continuous process \((X_{t})\) bounded below by zero that has jumps of size \(\pm 1\). We are naturally interested in the transition probabilities \(p_{i,i+1}\) and \(p_{i,i-1}\).
If the population size increases by 1 then \(B(i)\) occurred before \(D(i)\) hence \[ p_{i,i+1}=\mathbb{P}(B(i)<D(i))= \frac{{\lambda_{i}}}{\lambda_{i}+\mu_{i}}. \] by properties of the Exponential distribution.
Proof: We can compute \[ \begin{align} \mathbb{P}(B(i)<D(i)) & =\int_{0}^\infty \int_{0}^s f_{D(i)}(s)f_{B(i)}(t)~dtds \\ & = \int_{0}^\infty \mu_{i}\exp(-\mu_{i}s)(1-\lambda_{i}\exp(-\lambda_{i}t))~dtds \end{align} \] \(\square\)
Similarly, if the population decreases by 1, that is \(X_{t}\) makes a transition \(i \to i-1\), if the next event is death rather than birth, so that \(D(i)<B(i)\) with probability \[ p_{i,i-1}=\mathbb{P}(D(i)<B(i))= \frac{{\mu_{i}}}{\lambda_{i}+\mu_{i}}. \]
1.3 BD Process Holding Times
Since the process is continuous with jumps we are also interested in how long the process stays in each state known as the holding times.
Intuitively, since \(B(i), D(i)\) are times until either birth or death (i.e. a jump) the holding time of the process in state \(i\) is which ever happens first.
Therefore, the holding time for state \(X_{t}=i\) is then \[ T_{i}=\min(B(i), D(i))\sim\mathcal{E}(\lambda_{i}+\mu_{i}). \] Thus, the holding rate is \(q(i)=\lambda(i)+\mu(i)\). Note \(q_{0}=\lambda_{0}\) and furthermore, the mean time until a transition is \(\frac{1}{\lambda_{i}+\mu_{i}}\).
1.4 BD Process Examples
- Pure Birth Process
- Linear Birth-Death Process
- Yule Process
- [[poisson-processes]]
- Thinned Poisson Process