Simulating death processes

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Let \(\{N(t)\}\) be a death process with the death rate \(\mu\) and initial population size \(N(0) = n_0\). It is characterized by the following differential-difference equations for the probability mass function \(p_n(t) = \Pr(N(t)=n)\):

\[\begin{eqnarray} \frac{{d}p_n(t)}{{d}t} &=& \mu(n+1) p_{n+1}(t) - \mu np_n(t),~~(0 \leq n \leq n_0 - 1)\tag{Eq:DeathN}\\ \frac{{d}p_{n_0}(t)}{{d}t} &=& - \mu n_0p_{n_0}(t)\tag{Eq:DeathN0} \end{eqnarray}\]

with the initial condition

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

See also: Death process

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

We can do so just like we did for the birth process.

That is, 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 death is the first death 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:Death0) (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
\[p_{n_t}(t + \delta t) = e^{-\mu n_t \delta t},\]
which is the probability that \(N(t + \delta t) = n_t\) (i.e., no death during \(\delta t\)). Accordingly, we have the probability that \(N(t+\delta t) = n_t + 1\) (i.e., one death during \(\delta t\)) as
\[p_{n_t-1}(t + \delta t) = 1 - e^{-\mu n_t \delta t}.\]

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 \(r < 1 - e^{-\mu \delta t N(k\delta t)}\), 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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