Birth rates definition

A birth process with birth rates ${\displaystyle (\lambda _{n},n\in \mathbb {N} )}$ and initial value ${\displaystyle k\in \mathbb {N} }$ is a minimal right-continuous process ${\displaystyle (X_{t},t\geq 0)}$ such that ${\displaystyle X_{0}=k}$ and the interarrival times ${\displaystyle T_{i}=\inf\{t\geq 0:X_{t}=i+1\}-\inf\{t\geq 0:X_{t}=i\}}$ are independent exponential random variables with parameter ${\displaystyle \lambda _{i}}$.^[2]

Infinitesimal definition

A birth process with rates ${\displaystyle (\lambda _{n},n\in \mathbb {N} )}$ and initial value ${\displaystyle k\in \mathbb {N} }$ is a process ${\displaystyle (X_{t},t\geq 0)}$ such that:

• ${\displaystyle X_{0}=k}$
• ${\displaystyle \forall s,t\geq 0:s<t\implies X_{s}\leq X_{t}}$
• ${\displaystyle \mathbb {P} (X_{t+h}=X_{t}+1)=\lambda _{X_{t}}h+o(h)}$
• ${\displaystyle \mathbb {P} (X_{t+h}=X_{t})=o(h)}$
• ${\displaystyle \forall s,t\geq 0:s<t\implies X_{t}-X_{s}}$ is independent of ${\displaystyle (X_{u},u<s)}$

(The third and fourth conditions use little o notation.)

These conditions ensure that the process starts at ${\displaystyle i}$, is non-decreasing and has independent single births continuously at rate ${\displaystyle \lambda _{n}}$, when the process has value ${\displaystyle n}$.^[3]

Continuous-time Markov chain definition

A birth process can be defined as a continuous-time Markov process (CTMC) ${\displaystyle (X_{t},t\geq 0)}$ with the non-zero Q-matrix entries ${\displaystyle q_{n,n+1}=\lambda _{n}=-q_{n,n}}$ and initial distribution ${\displaystyle i}$ (the random variable which takes value ${\displaystyle i}$ with probability 1).^[4]

${\displaystyle Q={\begin{pmatrix}-\lambda _{0}&\lambda _{0}&0&0&\cdots \\0&-\lambda _{1}&\lambda _{1}&0&\cdots \\0&0&-\lambda _{2}&\lambda _{2}&\cdots \\\vdots &\vdots &\vdots &&\vdots \ddots \end{pmatrix}}}$

Variations

Some authors require that a birth process start from 0 i.e. that ${\displaystyle X_{0}=0}$,^[3] while others allow the initial value to be given by a probability distribution on the natural numbers.^[2] The state space can include infinity, in the case of an explosive birth process.^[2] The birth rates are also called intensities.^[3]

Simple birth process

A simple birth process, where birth rates are equal to the size of the current population.

A simple birth process is a birth process with rates ${\displaystyle \lambda _{n}=n\lambda }$.^[10] It models a population in which each individual gives birth repeatedly and independently at rate ${\displaystyle \lambda }$. Udny Yule studied the processes, so they may be known as Yule processes.^[11]

The number of births in time ${\displaystyle t}$ from a simple birth process of population ${\displaystyle n}$ is given by:^[3]

${\displaystyle p_{n,n+m}(t)={\binom {n}{m}}(\lambda t)^{m}(1-\lambda t)^{n-m}+o(h)}$

In exact form, the number of births is the negative binomial distribution with parameters ${\displaystyle n}$ and ${\displaystyle e^{-\lambda t}}$. For the special case ${\displaystyle n=1}$, this is the geometric distribution with success rate ${\displaystyle e^{-\lambda t}}$.^[12]

The expectation of the process grows exponentially; specifically, if ${\displaystyle X_{0}=1}$ then ${\displaystyle \mathbb {E} (X_{t})=e^{\lambda t}}$.^[10]

A simple birth process with immigration is a modification of this process with rates ${\displaystyle \lambda _{n}=n\lambda +\nu }$. This models a population with births by each population member in addition to a constant rate of immigration into the system.^[3]