COURSE OBJECTIVES:
Students will learn the basic theory of Branching Processes, with the strong emphasis on applications in biomedicine. Students will be able to recognize various biomedical examples that can be modelled via branching processes. They will learn how to set up the model and how to analyse it using its generating function, moments, branching mechanism, asymptotic behaviour, extinction probability. They will also learn how to interpret the elements of mathematical analysis of their models in biomedical environment.
COURSE CONTENT:
1.Introduction. Conditional Probability. Wald's Identity. Generating Function.
2.Galton-Watson Process. Extinction Probability. Examples (Last Names Extinction; DNA Haplogroups). Compound Poisson Process.
3.Asymptotic Behaviour. Exponential Growth: Stochastic vs. Deterministic Models. Strong Convergence.
4.Branching Processes and Markov Property. Limit Theorems.
5.Branching Processes with Immigration and Emigration. Branching Processes with Multiple Types.
6.Case Studies:
6.1.DNA and Chromosomes.
6.2.Cell Cycles.
6.3.Modelling of the Eye Lens Growth. Organ Growth Implications.
6.4.Basic Epidemiological Models.
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Branching Processes in Biology, 2nd ed., M.Kimmel, D.E.Axelrod, Springer, 2015.
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Branching Processes, K.B.Athreya, P.E.Ney, Springer, 1972.
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The Lens Growth Process, S.Bassnett, H.Šikić, Progress in Retinal and Eye Research, 60 (2017), 181 - 200.
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Branching Processes: Variation, Growth, and Extinction of Populations, P.Haccou, P.Jagers, V.A.Vatutin, Cambridge University Press, 2005.
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Branching Processes with Biological Applications, P.Jagers, Wiley, 1975.
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Stochastic Population and Epidemic Models, L.J.S.Allen, Springer, 2015.
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Branching Processes: Their Role in Epidemiology, Ch.Jacob, Int. J. Environ. Res. Public Health 2010, 7, 1186 - 1204. (open access).
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