5.3 Simple Birth and Death Process

Every individual in the population is subject to giving birth or dying

\(\lambda \, \Delta t\):

chance of an individual giving birth in time \(\Delta t\).

Hence if \(X(t)\) represents population size at time \(t\), then chance of a birth in interval \(\Delta t\) is \(\lambda \, \Delta \, X(t)\).

\(\mu \, \Delta t\)

: chance of an individual dying in time \(\Delta t\).

Assume independence of events birthing and dying and that individual give birth or die independent of each other.

Let \(X(t)\) be the size of the population at time \(t\). Assume \(X(0) = a\).

Let \(P_n(t) = P(X(t) = n)\) \begin {align*} P_n(t + \Delta t) = & P_n(t)(1 - n\lambda \Delta t)(1 - n\mu \Delta t) + P_{n+1}(t)\left (1 - \overline {n+1}\, \lambda \Delta t\right )\left (\mu (n +1)\Delta t\right )\\ & + P_{n - 1}(t)\left (\overline {n - 1}\, \lambda \Delta t\right )\left (1 - \overline {n - 1}\, \mu \Delta t\right )\\ = & P_n(t)(1 - n\lambda \Delta t - n\mu \Delta t) + P_{n + 1}(t)\mu (n + 1)\Delta t + P_{n - 1}(t)(n - 1) \lambda \Delta t. \end {align*}

As \(\Delta t \longrightarrow 0\), \[P'_n(t) = -n(\lambda + \mu )\, P_n(t) + \mu (n + 1)\, P_{n + 1}(t) + \lambda (n - 1) \, P_{n - 1}(t)\, , \hspace {0.3cm} n\leq 1.\]

Let \[M(t,\theta ) = \sum ^{\infty }_{n = 0} e^{n\theta }\, P_n(t).\]

\[\frac {\partial M}{\partial t} = -(\lambda + \mu )\, \frac {\partial M}{\partial \theta } + \mu \, e^{-\theta }\, \frac {\partial M}{\partial \theta } + \lambda \, e^{\theta }\, \frac {\partial M}{\partial \theta }.\]

Assume \(X(0) = a\). Then it can be deduced that \[M(t,\theta ) = \left (\frac {1 + \left (e^{\theta } - 1\right )\left (1 - \lambda t\right )}{1 - \lambda t \, \left (e^{\theta } - 1\right )}\right )^a\] where \(\lambda = \mu \) which further gives \[E(X(t)) = \frac {\partial u}{\partial \theta }\Bigg |_{\theta = 0} = a.\]

When \(a = 1\), \[P(t, \theta ) = \frac {1 + (\theta - 1)(1 - \lambda t)}{1 - \lambda t\, (\theta - 1)}.\]

\[P_n(t) = \frac {\left (\lambda t\right )^{n - 1}}{\left (1 + \lambda t\right )^{n + 1}}\hspace {0.2cm} , \hspace {0.5cm} a = 1, \hspace {0.2cm} \lambda = \mu .\]

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