Simulating birth processes

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The birth process \(\{N(t)\}\) with birth rate \(\lambda\) and initial population size \(N(0) = n_0\) is characterized by the following differential-difference equations for the probability mass functions \(p_n(t) = \Pr(N(t) = n)\):

\[\begin{eqnarray} \frac{{d}p_{n_0}(t)}{{d}t} &=& -\lambda n_0 p_{n_0}(t),\tag{Eq:Birth0}\\ \frac{{d}p_{n}(t)}{{d}t} &=& \lambda(n-1)p_{n-1}(t)-\lambda n p_{n}(t), ~~ (n \geq n_0 + 1)\tag{Eq:Birthn} \end{eqnarray}\]

with the initial condition

\[p_{n}(0) = \delta_{n, n_0}.\]

See also: Birth process

Here, we want to simulate this process numerically to obtain some concrete sample paths like this one:

To do so, we first discretize the time variable so that we consider time steps with some small interval \(\delta t\): \(N(0), N(\delta t), N(2\delta t), N(3\delta t), \cdots, N(k\delta t), \cdots\). 

Next, we exploit the Markov property of the birth process: Every birth is the first birth since the last one. That is, given \(N(t) = n_t\) at time \(t\), we may regard this as the "initial" condition for the next step: \(N(t+\delta t) = n_t\) or \(n_t+1\). We can solve (Eq:Birth0) (with \(n_0\) replaced with \(n_t\)) to find \(p_{n_t}(t + \delta t)\) with the "initial" condition \(p_{n}(t) = \delta_{n,n_t}\). A bit of exercise gives
\[p_{n_t}(t + \delta t) = e^{-\lambda n_t \delta t},\]
which is the probability that \(N(t + \delta t) = n_t\) (i.e., no birth during \(\delta t\)). Accordingly, we have the probability that \(N(t+\delta t) = n_t + 1\) (i.e., one birth during \(\delta t\)) as
\[p_{n_t+1}(t + \delta t) = 1 - e^{-\lambda n_t \delta t}.\]

Thus, the algorithm for simulating the birth process is the following:
  1. Set \(N(0) = n_0\); set k = 0.
  2. Generate a uniformly distributed random number \(r \in [0, 1)\).
  3. If \(r < 1 - e^{-\lambda \delta t N(k\delta t)}\), then set \(N((k+1)\delta t) = N(k\delta t) + 1\); Otherwise, set \(N((k+1)\delta t) = N(k\delta t)\).
  4. Update \(k := k + 1\), go to Step 2, and repeat.


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