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Deterministic and Stochastic Epidemic Models
Staněk, Jakub ; Štěpán, Josef (advisor) ; Hlubinka, Daniel (referee) ; Dohnal, Gejza (referee)
Kermack-McKendrick model and its version with vaccination are presented. First, we introduce a model with vaccination and then a numerical study that includes comparison of di erent vaccination strategies and searching for optimal vaccination strategy is presented. We proceed to introduce a stochastic model with migration and consequently we suggest its generalization and prove the existence and uniqueness of a solution to the stochastic di erential equation (henceforth SDE) describing this model. Three stochastic versions of Kermack-McKendrick model with vaccination are suggested and compared. A procedure of nding the optimal vaccination strategy is presented. We also prove the theorem on the existence and uniqueness of a solution to the SDE that drives a model with multiple pathogens. Finally, the stochastic di erential equation describing the general model is presented. We study properties of a solution to this SDE and present sufficient conditions for the existence of a solution that is absorbed by the natural barrier of the model.
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The arc sine laws
Mordinová, Katarína ; Lachout, Petr (referee) ; Hlubinka, Daniel (advisor)
Nazcv praee: /akony arku sinu Autor: Katarina Mordinova Katedra (iistuv): Kaledra pravdepodobnosti a matematicke statistiky MFF UK Vedouci bakalafske prace: RNDr. Daniel Hlubinka, PhD e-mail vedouciho: Daniel.IIlubinka(«}rnff.ciini.c/. Abstrakt: Tato bakalarska praca pojednava o Zakonc arku sinu pre roznc typy nahodnych velicin.Praca je clenena do troch casti.Prva kapitola su /aobera najjednoduchsou variantou altcrnativne rozdeleuyeh nezavislych scitancov.Zaoberame sa vyslovenim a dokazom Zakonu arku sinu.Druha kapitola formuje /akon arku sinu pro Wicncrov proccs.Tretia kapitola sa venuje vyuzitiu Wiencrovho proccsu v praxi pri occhovuni finanCnych dcrivatov pomocou Black-Scholesovcj tbrmule. Klicova slova: Zakon arku sinu. Nahodna prcchadzka, Wienerov proccs, Oceiiovunie akcii, Black-Scholesova formula Title: The arc sine laws Author: Katarina Mordinova Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Daniel Hlubinka, PhD Supervisor's e-mail address: Daniel.11lubinkafa>,mff.cuni.cz Abstract: This bachelor thesis threats of the Arc sine law for different types of random variables.The work is divided into three parts.First chapter talks about the simplest alternative distribution function for independent summands.We concerned about vote and proof of the Arc sine law.Second...
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