These are some notes I made back in April, 2012, building off the basic ideas in Daley 1986, as suggested by Hal Caswell at the MBI invasion meeting in February. When I was messing with the idea of non-commuting random variables was initially developed in the context of density-independent branching processes, but these random variables can also be applied to model density-dependent processes. Hereare some simple examples that I find interesting. One of the interesting aspects of these models is that we have to pull a few tricks with variable-orderings that are not usually needed in continuous-time models.

Let the number of offspring produced by a fertilized female be Poisson distributed with expectation \(R\). Represent this by noncommuting random variable (NCRV), See Reluga 2009. In non-commuting random variable notation, \(x\) and \(y\) represent positive integer-valued random variables, and \(x y\) represents the sum of \(x\) realization of the variable \(y\). \(\mathcal{P}(R)\).

Let \(\mathcal{B}(x)\) represent a binomial NCRV equal to \(1\) with probability \(x\) and \(0\) with probability \(1-x\). Let \(p\) be the probability that a single male fertilizes a single female in a given generation, and assume that a female only needs to fertilized by just one male while males can fertilize all the females they find. Let \(s\) be the probability that an offspring is female while \(1-s\) is the probability that an offspring is male.

Let state variables \(F_t\), \(M_t\), and \(N_t=F_t+M_t\) represent the number of females, the number of males, and the total number of individuals in a given generation. Then the governing rules for this process can represented with the equations \[\begin{align} N_{t+1} &\cong \left[ F_t \mathcal{B}(1-(1-p)^{M_t}) \right] \mathcal{P}(R), \\ F_{t+1} &\cong N_{t+1} \mathcal{B}(s), \\ M_{t+1} &\cong N_{t+1} - F_{t+1}. \end{align}\] Density-dependent NCRV models seem to be a bit tricky to construct. But they seem much easier to specify than the corresponding birth-death process for this model. The females-only model without a sex-induced Allee effect, and corresponding to a regular branching process, would be \[\begin{align} F_{t+1} &\cong F_t \mathcal{P}(R). \end{align}\]

The probability of extinction for System 1 has to be calculated numerically right now, as far as I can tell. Figure 1 shows a simulation-based calculation of the probability of extinction of various initial conditions.

Let \(I_t\) represent the number of cases of infection at time \(t\). Assume all cases recover or are removed in exactly one time-step. Let \(S_t\) represent the number of susceptible individuals at time \(t\). Let \(\mathcal{B}(x)\) be a Bernoulli random variable that returns \(1\) with probability \(x\) and \(0\) with probability \(1-x\). Making use of the random-variable multiplication convention from my paper on non-commuting random variables, we can write the Reed-Frost model as \[\begin{align} I_{t+1} &\cong S_t \mathcal{B}(1-(1-p)^{I_t}), \\ S_{t+1} &\cong \eta + (S_t - I_{t+1}) \mathcal{B}(1-m) \end{align}\] with initial condition \((S_0,I_0)\) given. Here, \(p\) is the probability that a person comes in contact with another person in one time-step and that that contact results in disease transmission.

In the classic Reed--Frost model, we take \(\eta = m = 0\), so that the dynamics are those of a simple epidemic without any effects from population turnover. \[\begin{align}S_{t+1} = S_t \mathcal{B}(q^{I_t}) \\ I_{t+1} = S_t - S_{t+1} \end{align}\]

The deterministic limit is found by replacing the random variables with their expectations. \[\begin{align} I_{t+1} &= S_t (1-(1-p)^{I_t}), \\ S_{t+1} &= \eta + S_t (1-p)^{I_{t}} (1-m) \end{align}\]

A major question with this model is how \(p\) might change depending on population size and activity patterns.

The Wright--Fisher model is a classic stochastic model from population genetics. It describes random changes in the prevalence of an allele in a haploid population. If \(P_t\) represents the number of copies of the allele in a population of \(N\) individuals, then the Wright--Fisher model in non-commuting random variable form is \[\begin{gather} P_{t+1} \cong N \mathcal{B}\left( \frac{P_t}{N} \right) \end{gather}\] (this is a discrete, non-overlapping generation model)

The **Moran** model, while simpler in some senses, is not as easy to write. So far, I think the best representation requires a temporary variable. \[\begin{align}
X_{t+1} &\cong P_{t} - \mathcal{B}\left( \frac{P_t}{N} \right)
\\
P_{t+1} &\cong X_{t+1} + \mathcal{B}\left( \frac{X_{t+1}}{N-1} \right)
\end{align}\]

A simple NCRV version of the Beverton--Holt model is \[\begin{gather} N_{t+1} \cong \left[ N_t \mathcal{P}(R) \right] \mathcal{B} \left( \frac{A}{A+N_t} \right) . \end{gather}\] Age structure can be included by using Bernoulli variables for mortality. \[\begin{align} N_{t+1,1} &\cong \left[ N_{t,2} \mathcal{P}(R) \right] \mathcal{B} \left( \frac{A}{A+N_{t,1}} \right), \\ N_{t+1,2} &\cong N_{t,1} \mathcal{B} \left( m \right), \end{align}\]

Symmetrized random product:

\[( X_t Y_t + Y_t X_t ) \mathcal{B}(1/2)\]

But note that this involves many more realizations than a single product.

Discrete Logistic: \[N_{t+1} \cong N_t \mathcal{P}(R_0 - k N_t)\] \[N_{t+1} \cong N_t \mathcal{P}(R_0 - k N_t) + N_t \mathcal{B}(m)\]

Random environments: \[N_{t+1} \cong \mathcal{L}_t ( N_t F_1(t), N_t F_2(t), \ldots)\]