Birth and death models

See also: Time series introduction, Time series modeling in finance, Geometric Brownian Motion models, Autoregressive models, Markov Chain models, Time series in ModelRisk

There are two strongly related probabilistic time series models called the Yule (or pure birth) and Pure Death models. We have certainly found them very useful in modelling numbers in a bacterial population but they could be helpful in modelling other variables modelling numbers of individuals that increase or decrease according to their population size.

Yule growth model

This is a pure birth growth model, and is a stochastic analogue to the deterministic exponential growth models one often see in, for example, microbial risk analysis. In exponential growth models, the rate of growth of a population of n individuals is proportional to the size of the population:

where b is the mean rate of growth per unit time t. This gives the number of individuals n_t in the population after time t as:

where n₀ is the initial population size. The model is limited because it takes no account of any randomness in the growth. It also takes no account of the discrete nature of the population, which is important at low values of n. Moreover, there are no defensible statistical tests to apply to fit an exponential growth curve to observations (regression is often used as a surrogate) because an Exponential growth model is not probabilistic, so no probabilistic (i.e. statistical) interpretation of data is possible.

The Yule model starts with the premise that individuals have offspring on their own (e.g. by division), that they procreate independently, that procreating is a Poisson process in time, and that all individuals in the population are the same. The expected number of offspring from an individual per unit time (over some infinitesimal time increment) is defined as b. This leads to the results that an individual will have, after time t, Geometric(Exp(-bt)) offspring, giving a new total population of Geometric(Exp(-bt))+1. Thus, if we start with n₀ individuals, then by some later time t we will have:

                 

from the relationship:

                

with mean , corresponding to the Exponential growth model.

A possible problem in implementing this type of model is that n₀ and n_t can be very large and simulation programs tend to produce errors for discrete distributions like the Negative Binomial for large input parameters and output values. The optional boolean Log10 parameter in ModelRisk 's VoseTimeYule function eliminates this problem:

{=VoseTimeYule(n₀,Beta,t,FALSE)}

which generates values for n_t, and

{=VoseTimeYule(n₀,Beta,t,TRUE)}

which generates values for Log₁₀(n_t), and interprets the first and second parameters as Log₁₀n₀  and LogIncrease. LogIncrease is the number of logs (in base 10) that one expects the population to increase by per time unit. The parameters b and LogIncrease are related by:

.

Pure Death model

The pure death model is a stochastic analogue to the deterministic exponential death models one often sees in, for example, microbial risk analysis. Individuals are assumed to die independently and randomly in time following a Poisson process. Thus, the time until death can be described by an Exponential distribution, which has a cdf:

where l is the expected instantaneous death of an individual. The probability that an individual is still alive at time t is therefore:

Thus, if n₀ is the initial population, the number n_t surviving until time t follows a Binomial distribution:

which has a mean of

i.e. the same as the exponential death model. The cdf for the time until extinction t_E of the population is given by:

                

The binomial Death model offered here is an improvement over the exponential death model for several reasons:

• The exponential death model takes no account of any randomness in the growth, so cannot interpret variations from an exponential line fit;

• The exponential death model takes no account of the discrete nature of the population, which is important at low values of n;

• There are no defensible statistical tests to apply to fit an exponential growth curve to observations (regression is often used as a surrogate) because an Exponential model is not probabilistic, so there can be no probabilistic interpretation of data. A likelihood function is possible, however, for the Death model described here.

A possible difficulty in implementing this Death model is that n₀ and n_t can be very large and simulation programs tend to produce errors for discrete distributions like the Binomial for large input parameters and output values. The optional boolean Log10 parameter in ModelRisk 's VoseTimeDeath function eliminates this problem:

{=VoseTimeDeath(n₀,l,t,FALSE)}

which generates values for n_t, and

{=VoseTimeDeath(n₀,l,t,TRUE)}

which generates values for Log₁₀(n_t), and interprets the first and second parameters as Log₁₀n₀  and LogDecrease. LogDecrease is the number of logs (in base 10) that one expects the population to decrease by per time unit. The parameters l and LogDecrease are related by

 .

Read on: Wilkie models