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The next meeting of the National Seminar on Probability and Statistics will be held on .June 19, 2019 (Wednesday), at 2.00 p.m., in Eoom 403 of the Institute of Mathematics and Informatics, BAS. A talk on

A decomposable multi-type Bellman-Harris branching process
with mutations. Application in modeling of cancer evolution

will be delivered by Kaloyan Vitanov (PhD student at Sofia University).

Everybody is invited.

Abstract: A prominent feature of biological systems is their capacity to adapt and mutation is a key mechanism for achieving adaptation. As a continuation of [3] we propose a decomposable multi-type Bellman-Harris branching process, which incorporates a possibility for mutation.
Metastasis, the spread of cancer cells from a primary tumour to secondary location(s) in the human organism, is the ultimate cause of death for the majority of cancer patients. We consider a metastasized cancer cell population after some medical treatment (e.g. chemotherapy). Arriving in a di erent environment the cancer cells may change their characteristics concerning lifespan and reproduction, thus they may di erentiate into di erent types. Even if the treatment is e ective (resulting in subcritical reproduction of all cancer cell types), however, it is possible during cell division for mutations to occur. These mutations can produce a new cancer cell type that is adapted to the treatment (having supercritical reproduction). Cancer cells from this new type may lead to a non-extinction process.
We model the above scenario with the proposed model. Expanding [1] and [2] we investigate relevant quantities such as the probability of extinction of the process until time t and as t approaches in nity, the number of occurred supercritical mutants until time t and as t approaches in nity, the time until the rst occurrence of a mutant starting a non-extinction process and the immediate risk for the process to escape extinction. We also propose numerical schemes for performing calculations.
Keywords: Mutations, Decomposable multi-type branching process, Probability of extinction, Waiting time to escape mutant, Immediate risk of escaping extinction
Acknowledgements: The research of these authors was supported, in part, by the Bulgarian National Science Fund at the Ministry of Science and Education, Grant No. KP-06-H22/3.
[1] K. Vitanov, M. Slavtchova-Bojkova, Multitype branching processes in continuous time as models of cancer, Annuaire de l’Universite de So a St. Kl. Ohridski”, Fac. Math and Inf., 104, 2017, 193-200.
[2] M. Slavtchova-Bojkova, K. Vitanov, Modelling cancer evolution by multi-type age-dependent branching processes, Compt. rend. de l’acad. bulgare des Sci. 71, 2018, 1297{1305.
[3] M. Slavtchova-Bojkova, P. Trayanov, S. Dimitrov, Branching processes in continuous time as models of mutations: Computational approaches and algorithms, Computational Statistics and Data Analysis, 113, 2017, 111{124.

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