BRANCHING PROCESS MODELLING: A TOOL FOR DECIPHERING COMPLEX REAL-WORLD PHENOMENA USING R PROGRAMMING
DOI:
https://doi.org/10.33003/fjs-2023-0703-1964Keywords:
Branching process, Mathematical Modelling Population-dependence, Extinction time, R ProgrammingAbstract
The reproduction of organisms such as human beings, cells or neutrons can be modeled via branching processes.The theory of branching processes offers suitable mathematical models to depict the chance-based progression of systems, where elements like cells, particles, or general individuals replicate and perish over a certain lifespan. These models are employed to illustrate stochastic systems like chain reactions, lineage continuation of surnames, elimination of pests, population growth, and gene spread. This study clearly showcases the likelihood of extinction, its timing, and the odds of complete offspring through vivid examples. The decline of prominent families from history has often been observed and has sparked numerous speculations. The branching process is a very useful tool in many real life problems (non-deterministic problems) and random phenomenon. Research indicates that when the average number of descendants for each entity exceeds 1 (meaning individuals reproduce at a rate slightly higher than self-replacement), the branching process doesn't necessarily cease. On the other hand, if this average number is 1 or lower, the process will inevitably face extinction.
References
Abdulazeez, S.A (2022). An Analysis of the Variation in Human Height based on Phenotypic traits in heredity. FUDMA, Journal of Sciences (FJS) VOL. 6(2), 243 – 250
Albertsen, K. (1995). Extinction of family,. International Statistical Institute (ISI),2,234-239
Alsmeyer, G., R & Osler, U.(2002). Asexual versus promiscuous bisexual Galton-Watson processes: the extinction probability ratio, Annals of Applied Probability12,125 – 142.
Athreya, K.B. & Ney, P.E. (1972). Branching Processes; Springer-Verlag: Berlin, Germany.
Bagley, J. H.(1986). On the asymptotic properties of a supercritical branching process, Journal of Applied Probability. 23,820 – 826.
Becker, N.G (1989). Analysis of Infectious Disease Data; Chapman and Hall/CRC Press: Boca Raton, FL,USA.
Bruss,F.T(1984). A note on extinction criteria for bisexual Galton-Watson processes,Journal of Applied Probability21, , 915 – 919.
Diekmann.O&Heesterbeek, J.A.P.(2000). Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation; Wiley Series in Mathematical and Computational Biology.John Wiley, Ltd.: Chichester, UK.
Farrington, C.P. & Grant, A.D.(1999). The distribution of time to extinction in subcritical branching processes:Applications to outbreaks of infectious disease. Journal of Applied Probability., 36, 771-779.
Galton, F .(1891). Hereditary genius Second American edition. New York: Appleton and Co,
Galton, F., & Watson, H.W.(1874). On the probability of the extinction of families. The Journal of the Royal Anthropological Institute of Great Britain and Ireland, 4, 138 - 144
Gonz_alez, M. & Molina, M.(1998). On the partial and total progeny of a bisexual Galton-Watson branching process, Applied Stochastic Models in Data Analysis13,225 – 232.
Gonz_alez, M., Molina, M.& Mota, M.(2000). Limit behavior for a subcritical bisexual Galton-Watson branching process with immigration, Statistics and Probability Letters49, 19 – 24.
Haccou, P., Jagers, P.&Vatutin, V.A. (2005) .Branching Processes Variation, Growth, and Extinction of Populations; Cambridge Studies in Adaptative Dynamics: Cambridge, UK.
Hefferman; J.M., Smith, R.J. .& Wahl, L M. (2005) . Perspectives on the basic reproductive ratio.Journal ofRoyal Statistical Society, , 2, ,281-293.
Hull, D.M.(1982). A necessary condition for extinction in those bisexual Galton-Watson branching processesgoverned by superadditive mating functions, Journal of Applied Probability. 19, 847 – 850.
Iglehart, D. L. (1976). Limiting diffusion approximations for the adjusted Galton-Watson process. Operations Research, 24(3), 509-536.
Karlin, S. & Kaplan, N.(1993) . Criteria for extinction of certain population growth processes with interacting types, Advanced Applied Probability 5, 183 – 199.
Klebaner, F.C.(1985). A limit theorem for population-size-dependent branching processes, Journal of Applied Probability22, 48 – 57.
Linda J.S. (2011) :An Introduction to Stochastic Processes with Applications to Biology. 2nd Edition. CRC Press. New York
Lotka, A. J. (1931). The extinction of families-I. Journal of Washington Academy. Science21, 377 – 380.
Lyons R. & Peres Y. (2016). Probability on trees and networks. Cambridge Series in Statistical and Probabilistic Mathematics. 4, 22 - 37
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