#### Posts

2017-08-22: A rebuttal on the beauty in applying math

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-20: Random dispersal speeds invasions

2014-04-14: More on fairer markets

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

## Dependent Non-Commuting Random Variable Systems

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.

### Sex Structure

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.

### Reed--Frost

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.

### Wright--Fisher

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}

### Beverton--Holt

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}

### More

• 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: $N_{t+1} \cong \mathcal{L}_t ( N_t F_1(t), N_t F_2(t), \ldots)$