We study a birth and death process (BDP) with immigrations and catastrophes. Births occur in the population at linear rate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$i\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> when the population size is i. The death process remains in OFF mode until the population size becomes N. However, once started, the death process remains in the ON mode until the population size again becomes 0. We assume that the cumulative death rate is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> when the population size is i. Catastrophes strike the population with inter-catastrophe time following the exponential distribution with mean <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mi>γ</mi> </mfrac> </math></EquationSource> </InlineEquation>. The control policy for the catastrophe is such that it may occur when and only when the population size is greater than or equal to N. Immigration of one member is allowed in the model when the population size is 0. The immigration time is exponentially distributed with mean <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{1}{\alpha _1}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <msub> <mi>α</mi> <mn>1</mn> </msub> </mfrac> </math></EquationSource> </InlineEquation>. For the transient analysis of the model, we use the generating function approach, leading to a Volterra integral equation of the first kind, which can be solved numerically to obtain performance measures such as the average population size. The steady state probabilities are also obtained using the generating function method. Our numerical study indicates that the present model could be apt for modelling certain biological phenomena such as tumor metastasis and HIV blips.

Transient Analysis of the Effect of Control Policies for Catastrophes and Death Process on a Birth and Death Process with Immigration

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发表日期：

2025年11月7日

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Shimna A.,
Narayanan C. Viswanath,
A Krishnamoorthy

摘要

We study a birth and death process (BDP) with immigrations and catastrophes. Births occur in the population at linear rate $i\alpha $ $i α$ when the population size is i. The death process remains in OFF mode until the population size becomes N. However, once started, the death process remains in the ON mode until the population size again becomes 0. We assume that the cumulative death rate is $i\beta $ $i β$ when the population size is i. Catastrophes strike the population with inter-catastrophe time following the exponential distribution with mean $\frac{1}{\gamma }$ $\frac{1}{γ}$ . The control policy for the catastrophe is such that it may occur when and only when the population size is greater than or equal to N. Immigration of one member is allowed in the model when the population size is 0. The immigration time is exponentially distributed with mean $\frac{1}{\alpha _1}$ $\frac{1}{α_{1}}$ . For the transient analysis of the model, we use the generating function approach, leading to a Volterra integral equation of the first kind, which can be solved numerically to obtain performance measures such as the average population size. The steady state probabilities are also obtained using the generating function method. Our numerical study indicates that the present model could be apt for modelling certain biological phenomena such as tumor metastasis and HIV blips.