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Conditional distributions and waiting times in multitype branching processes. (English) Zbl 1276.92081
Summary: We present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distributions of type-specific population sizes over time, conditioned on the non-appearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

92D15 Problems related to evolution
92D30 Epidemiology
60J85 Applications of branching processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
92-08 Computational methods for problems pertaining to biology
Full Text: DOI Euclid
[1] Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10 , 5-87. · Zbl 0749.60013
[2] Alexander, H. K. (2010). Modelling pathogen evolution with branching processes. Masters Thesis, Queen’s University Kingston, Canada. Available at http://hdl.handle.net/1974/5947.
[3] Alexander, H. K. and Day, T. (2010). Risk factors for the evolutionary emergence of pathogens. J. R. Soc. Interface 7 , 1455-1474.
[4] Alexander, H. K. and Wahl, L. M. (2008). Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model. Evolution 62 , 1600-1609.
[5] Arinaminpathy, N. and McLean, A. R. (2009). Evolution and emergence of novel human infections. Proc. R. Soc. B 276 , 3937-3943.
[6] Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002
[7] Bartlett, M. S. (1951). The dual recurrence relation for multiplicative processes. Math. Proc. Camb. Phil. Soc. 47 , 821-825. · Zbl 0045.07604
[8] Becker, N. (1977). Estimation for discrete time branching processes with application to epidemics. Biometrics 33 , 515-522. · Zbl 0371.92026
[9] Conway, J. M. and Coombs, D. (2011). A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Comput. Biol. 7 , e1002033.
[10] Dorman, K. S., Sinsheimer, J. S. and Lange, K. (2004). In the garden of branching processes. SIAM Rev. 46 , 202-229. · Zbl 1069.60072
[11] Ferguson, N. M. et al. (2004). Public health risk from the avian H5N1 influenza epidemic. Science 304 , 968-969.
[12] Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes , 2nd edn. Oxford Univeristy Press. · Zbl 0759.60002
[13] Guttorp, P. (1991). Statistical Inference for Branching Processes . John Wiley, New York. · Zbl 0778.62077
[14] Harris, T. E. (1963). The Theory of Branching Processes . Springer, Berlin. · Zbl 0117.13002
[15] Hubbarde, J. E., Wild, G. and Wahl, L. M. (2007). Fixation probabilities when generation times are variable: the burst-death model. Genetics 176 , 1703-1712.
[16] Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology . Springer, New York. · Zbl 0994.92001
[17] Lange, K. (1982). Calculation of the equilibrium distribution for a deleterious gene by the finite Fourier transform. Biometrics 38 , 79-86. · Zbl 0479.62086
[18] Lange, K. and Fan, R. Z. (1997). Branching process models for mutant genes in nonstationary populations. Theoret. Pop. Biol. 51 , 118-133. · Zbl 0889.92018
[19] Macken, C. A. and Perelson, A. S. (1988). Stem Cell Proliferation and Differentiation: A Multitype Branching Process Model (Lecture Notes Biomath. 76 ). Springer, Berlin. · Zbl 0681.92016
[20] Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications (Modern Anal. Comput. Methods Sci. Math. 34 ). American Elsevier Publishing, New York. · Zbl 0219.60061
[21] Saff, E. B. and Snider, A. D. (2003). Fundamentals of Complex Analysis: with Applications to Engineering and Science , 3rd edn. Pearson Education, Upper Saddle River, NJ. · Zbl 0331.30001
[22] Sagitov, S. and Serra, M. C. (2009). Multitype Bienaym√©-Galton-Watson processes escaping extinction. Adv. Appl. Prob. 41 , 225-246. · Zbl 1161.60031
[23] Serra, M. C. (2006). On the waiting time to escape. J. Appl. Prob. 43 , 296-302. · Zbl 1097.60069
[24] Serra, M. C. and Haccou, P. (2007). Dynamics of escape mutants. Theoret. Pop. Biol. 72 , 167-178. · Zbl 1123.92027
[25] Strikwerda, J. C. (2004). Finite Difference Schemes and Partial Differential Equations , 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA. · Zbl 1071.65118
[26] Sun, F. (1995). The polymerase chain reaction and branching processes. J. Comput. Biol. 2 , 63-86.
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