Abstract
This paper explores an alternative approach starting from first principles, to the derivation of probability generating functions (pgfs) of death, birth-death and immigration processes in continuous time, subject to random catastrophes. A more elementary version of the general method proposed by Economou and Fakinos (2003) is presented. We examine the simple death process, the survival of susceptibles in a carrier-borne epidemic, the birth-death and immigration process, the unbiased random walk and the barber shop queue, all of them subject to random catastrophes occurring as a Poisson process. The stationary pgfs and the expected values of the processes are derived.
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References
Bailey, N.T.J., 1975. The Mathematical Theory of Infectious Diseases and its Applications. Charles Griffin, London.
Brockwell, P.J., Gani, J., Resnick, S.I., 1982. Birth, immigration and catastrophe processes. Adv. Appl. Prob. 14, 709–731.
Daley, D.J., Gani, J., 1999. Epidemic Modelling. Cambridge University Press.
Economou, A., Fakinos, D., 2003. A continuous time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes. Europ. J. Operat. Res. 149, 625–640.
Gani, J., 2004. A versatile birth-death model applicable to four distinct problems. Aust. N.Z. J. Stat. 46(1), 13–21.
Gani J., Swift, R.J., 2006. A simple approach to birth processes with random catastrophes. J. Combin. Inform. System Sci. 31, 1–7.
Stirzaker, D., 2006. Processes with catastrophes. Math. Scientist 31, 107–118.
Swift, R.J., 2000. A simple immigration-catastrophe process. Math. Scientist 25, 32–36.
Switkes, J., 2004. An unbiased random walk with catastrophe. Math. Scientist 29, 115–121.
Weiss, G.H., 1965. On the spread of epidemics by carriers. Biometrics 21, 481–490.
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Gani, J., Swift, R.J. Death and Birth-Death and Immigration Processes with Catastrophes. J Stat Theory Pract 1, 39–48 (2007). https://doi.org/10.1080/15598608.2007.10411823
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DOI: https://doi.org/10.1080/15598608.2007.10411823