3 Birth Process, Death Process and Birth-Death Process.

A birth-death process is a continuous-time Markov chain used to describe the evolution of the system by counting the number of individuals in the system over time.
In this dynamic system, each individual has the potential to either give birth to a new individual or undergo a death event.
The rates of these birth and death events at any given time depends upon the current number of existing individuals in the system.

3.1 Applications of Birth-Death Process

Genetics, epidemiology: Study of infectious diseases, birth-death processes can be employed to model the spread of infections within a population
Ecology: model the birth and death rates of different species in a habitat
Epidemiology: model number of infected individuals

3.2 Definition: Transition Probability

Let \(N(t)\) be the number of individuals at arbitrary time \(t\) for \(t \geq 0\) and \(\{N(t): t \geq 0\}\) be the sequence of random variables which define a birth-death Markov chain with a birth rate \(\lambda_i\) and death rate \(\mu_i\) for state \(i\). Assume that an arbitrary community with \(r\) individuals, thus \(N(0) = r\) for some \(r > 0\).

The stationary transition probability is defined as follows:

\[P_{ij}(t) = P\{N(s+t) = j|N(s) = i\} \text{ for all } s \text{and } t>0\]

3.3 Instantaneous birth rate and instantaneous death rate

Consider a population of \(N(t)\) individuals. Suppose in next time interval \((t, t+h)\) probability of population increase of 1 (called a birth) is \(\lambda_ih + o(h)\) and probability of decrease of 1 (death) is \(\mu_ih + o(h)\). Here \(\lambda_i\) is called the instantaneous birth rate and \(\mu_i\) is called instantaneous death rate.

Further, we have

\[ q_{ij}= \begin{cases} 0,& \text{if } |i-j|\geq 1\\ \lambda_i, & \text{if } j=i+1\\ \mu_{i}, & \text{if } j=i-1 \end{cases} \]

3.4 Postulates

Let \[P_{ij}(h) = P(N(t+h) = j|N(t) = i)\text{ for all }t \geq 0,\]

Further, \(P_{ij}(t)\) satisfy,

\(P_{i, i+1}(h) = P(N(t+h) = i+1|N(t)=i) = \lambda_ih + o(h), i \geq 0\)
\(P_{i, i-1}(h)=P(N(t+h) = i-1|N(t)=i) = \mu_ih + o(h), i \geq 1\)
\(P_{i, i}(h)= P(N(t+h) = i|N(t)=i) = 1- (\lambda_i + \mu_i)h + o(h), i \geq 0\)
\(P(N(t+h) = i+m|N(t)=i) = o(h), |m| > 1\)
\(\mu_0 = 0, \lambda_0 > 0, \mu_i, \lambda_i > 0, i=1, 2, 3,,,,\)

3.5 Exercise

Show that the birth process, death process and birth-death process are Markov processes

3.6 Special cases

When all \(\mu_i = 0\), called a pure birth process.
When all \(\lambda_i = 0\) called a pure death process
When \(\lambda_n = n\lambda\) and \(\mu_n = n\lambda\) is a linear birth and death process.
When \(\lambda_n = \lambda\) and \(\mu_n = 0\) is called a Poisson process.

3.7 Exercise

Derive the distribution of length of stay (elapsed time between two consecutive occurrences ) for a birth, death and birth-death process.

Help 1

Important facts about the exponential distribution

Fact 1:…………………………….

Fact 2:…………………………….

Fact 3:…………………………….

Fact 4:…………………………….

Help 2

Distribution of length of stay - Continuous parameter Markov chain processes

Suppose that a continuous-time time-homogeneous Markov chain \(\{N(t): t \geq 0\}\) enters state \(i\) at some time, (say at time \(s \geq 0\))) and let \(T_i\) denote the amount of time that the process stays in state \(i\). Then, for any \(u > 0\)

\[P[T_i \geq u] = P[N(t)=i; s < t < s+u|N(s)=i)]\] for any \(s \geq 0\).

3.8 Instanttaneous Probability, Transition Probability, Probability Mass Function

In-class discussion