Simulating birth-death processes

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

\[\begin{eqnarray}\frac{{d}p_0(t)}{{d}t} &=& \mu p_1(t),\\\frac{{d}p_n(t)}{{d}t} &=& \lambda(n-1)p_{n-1}(t) - (\lambda + \mu)np_n(t) + \mu(n+1)p_{n+1}(t),~~(n \geq 1).\tag{Eq:BDn}\end{eqnarray}\]

with the initial condition

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

See Combining birth and death processes for the detail of how to analytically solve the above differential-difference equations. Here, we show how to simulate this process numerically as we did for the birth process and death process; See also:

First, we discretize the 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-death process: Every birth or death is the first birth or death since the last one. That is, given \(N(\tau) = n_\tau\) at time \(t = \tau\), we may regard this as the "initial" condition for the next step: \(N(\tau+\delta t) = n_{\tau}\) or \(n_{\tau} + 1\) or \(n_{\tau}-1\). 

Unlike the simple birth or death process, we have three mutually disjoint events. That is, we assume that during the short period \(\delta t\), one of the following occurs: one birth \(N(\tau + \delta t) = n_{\tau} + 1\), one death \(N(\tau+\delta t) = n_{\tau} -1\), or nothing \(N(\tau + \delta t) = n_{\tau}\). Thus, we have the following differential-difference equations with \(p_{n}(\tau) = \delta_{n,n_\tau}\)
\[\begin{eqnarray} \frac{dp_{n_{\tau}-1}(t)}{dt}&=& \mu n_{\tau} p_{n_{\tau}}(t),\\ \frac{dp_{n_\tau}(t)}{dt}&=& -(\lambda + \mu)n_{\tau} p_{n_{\tau}}(t),\\ \frac{dp_{n_\tau+1}(t)}{dt}&=& \lambda n_{\tau} p_{n_\tau}(t). \end{eqnarray}\]
Solving these, we have
\[\begin{eqnarray} p_{n_\tau-1}(\tau + \delta t) &=& \frac{\mu}{\lambda + \mu}(1 - e^{-(\lambda + \mu)n_{\tau}\delta t})\\ p_{n_\tau}(\tau + \delta t) &=& e^{-(\lambda + \mu)n_{\tau}\delta t},\\ p_{n_\tau+1}(\tau + \delta t) &=& \frac{\lambda}{\lambda + \mu}(1 - e^{-(\lambda + \mu)n_{\tau}\delta t}). \end{eqnarray}\]
Note that \(p_{n_\tau-1}(\tau + \delta t) + p_{n_\tau}(\tau + \delta t) + p_{n_\tau+1}(\tau + \delta t) = 1.\) 

Based on this, we have the following algorithm for simulating the birth-death process:
  1. Set \(N(0) = n_0\); set \(k = 0\).
  2. Generate a uniformly distributed random number \(r \in [0, 1)\).
    1. If \(r < \frac{\mu}{\lambda + \mu}(1 - e^{-(\lambda + \mu)N(k\delta t)\delta t})\), then \(N((k+1)\delta t) = N(k\delta t) - 1\);
    2. Otherwise, if \(r < 1 - e^{-(\lambda + \mu)N(k\delta t)\delta t}\), then \(N((k+1)\delta t) = N(k\delta t) + 1\);
    3. Otherwise, \(N((k+1)\delta t) = N(k\delta t)\).
  3. Update \(k := k + 1\), go to Step 2, and repeat.
A sample path obtained from such simulations is shown below:
A sample path of a birth-death process with birth rate \(\lambda = 0.1\) and death rate \(\mu = 0.09\).


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