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Autonomous systems of differential equations - classical vs fractional ones
Glozigová, Anna ; Zatočilová, Jitka (referee) ; Nechvátal, Luděk (advisor)
Hlavním zaměřením této práce je hlubší studium a porovnání dvou oblastí diferenciálních rovnic, kde důraz je kladen na neceločíselné řády, neboť během posledních desítek let se tato oblast nejenže stala populární, ale dokonce bylo zjištěno, že standardní přístupy řešení nenaplňují očekávání, tudíž jsou vyžadovány speciální postupy. Práce také obsahuje příklady, experimenty a simulaci pro ověření, případné vyvrácení teoretických výsledků.
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Advanced epidemic models and their analysis
Skácelová, Radka ; Šremr, Jiří (referee) ; Čermák, Jan (advisor)
This diploma thesis analyzes several SIR epidemiological models which are described by a system of non-linear differental equations; it is mainly focused on SIR models with biths and deaths describing long-term epidemics. The goal of this thesis is to develop and analyze models with a time delay, and to extend some of the studied models using the graph theory, find their stationary states and analyze their stability. The thesis is particularly focused on spatially heterogenous stationary states for special types of graphs - complete graphs, stars and cycles.
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Epidemiological modeling of Covid-19
Motlíčková, Klára ; Ředina, Richard (referee) ; Mézl, Martin (advisor)
This bachelor thesis deals with matematical compartment modeling of the COVID-19 pandemic. The basic epidemiological models are presented and six studies that deal with the behavior of the SARS-CoV-2 virus under different conditions are presented. On the SIR model are estimated parameters for first and fourth wave of the epidemic, using algorithm in MATLAB is made optimalization for all the waves of epidemic in the Czech Republic and also for the fourth wave on the SEIR model. The impact of antiepidemic measures is modelled on the SIR model.
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Spatial-temporal epidemiologic models of Covid-19
Schubert, Richard ; Ředina, Richard (referee) ; Mézl, Martin (advisor)
This work aims to establish a fundamental framework for studying spatially diffusive models that describe the dynamics of infectious disease spread with constant parameters in a homogeneous domain. Initially, compartmental models and their extension to spatial domains are examined, followed by the theory of metapopulation models, where the degree of coupling between populations and the overall reproductive number R0 is discussed. Furthermore, the relationship between R0 and the shape of the spatial distribution of infected individuals in a simple diffusive SIR model is modeled. The influence of Neumann boundary conditions versus Dirichlet boundary conditions on R0 is demonstrated. In the second part of the work, selected findings and conclusions of studies that applied models in the spatiotemporal domain to analyze and predict the COVID-19 pandemic are summarized. In the third part of the work, a model with diffusive and metapopulation elements is fitted to epidemiological data from Lombardy in 2020, and the suitability of this approach is discussed.
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Simulation of Spread of Infectious Diseases in Human Population
Vaňo, Michal ; Peringer, Petr (referee) ; Strnadel, Josef (advisor)
Epidémie predstavujú trvalú hrozbu pre život a ekonomiku krajín. Napriek ich ničivému potenciálu, ich dopad je možné zjemniť študovaním chorôb, ktoré tieto epidémie spôsobujúu, a predpoveďou ich dosahu. Modelovanie epidémií je kľúčové pre pochopenie šírenia infekčných chorôb a hodnotenie účinnosti zdravotných zásahov. Táto bakalárska práca najprv predstavuje a porovnáva kompartmentálne a multiagentné modely, dva primárne prístupy v oblasti epidemiologického modelovania. Zvoleným prístupom pre túto prácu je multiagentné modelovanie, ktoré bolo dôkladne analyzované a následne transformované na abstraktný model šírenia chorôb. Model je navrhnutý tak, aby bol spustiteľný na akomkoľvek slovenskom meste a poskytol všestranný nástroj na pochopenie dynamiky chorôb v rôznych lokalitách. Abstraktný model bol implementovaný v Pythone. Záverečná časť práce pozostáva z vykonania experimentov na analýzu rôznych scenárov a rôznych opatrení. Tieto experimenty prispievajú k hlbšiemu pochopeniu dynamiky šírenia chorôb a môžu byť zdrojom informácií pri tvorbe politiky v oblasti verejného zdravia. Práca ponúka prehľadnú štúdiu rôznych prístupov k modelovaniu a vyvíja všestranný model založený na agentoch, ktorý je možné rozšíriť podľa dostupných údajov.
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Spatial-temporal epidemiologic models of Covid-19
Schubert, Richard ; Ředina, Richard (referee) ; Mézl, Martin (advisor)
This work aims to establish a fundamental framework for studying spatially diffusive models that describe the dynamics of infectious disease spread with constant parameters in a homogeneous domain. Initially, compartmental models and their extension to spatial domains are examined, followed by the theory of metapopulation models, where the degree of coupling between populations and the overall reproductive number R0 is discussed. Furthermore, the relationship between R0 and the shape of the spatial distribution of infected individuals in a simple diffusive SIR model is modeled. The influence of Neumann boundary conditions versus Dirichlet boundary conditions on R0 is demonstrated. In the second part of the work, selected findings and conclusions of studies that applied models in the spatiotemporal domain to analyze and predict the COVID-19 pandemic are summarized. In the third part of the work, a model with diffusive and metapopulation elements is fitted to epidemiological data from Lombardy in 2020, and the suitability of this approach is discussed.
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Epidemické modely s vakcinací a behaviorálními změnami
LUNGA, Petr
Compartmental models used in mathematical epidemiology were studied in order to learn about basic principles of those models and to create new ones. First two chapters are about basic SIR model, it's history and analysis. Next chapters then contain models with behavioral changes and vaccination. Models were illustrated by simulations created in the software AnyLogic.
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Advanced epidemic models and their analysis
Skácelová, Radka ; Šremr, Jiří (referee) ; Čermák, Jan (advisor)
This diploma thesis analyzes several SIR epidemiological models which are described by a system of non-linear differental equations; it is mainly focused on SIR models with biths and deaths describing long-term epidemics. The goal of this thesis is to develop and analyze models with a time delay, and to extend some of the studied models using the graph theory, find their stationary states and analyze their stability. The thesis is particularly focused on spatially heterogenous stationary states for special types of graphs - complete graphs, stars and cycles.
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Epidemiological modeling of Covid-19
Motlíčková, Klára ; Ředina, Richard (referee) ; Mézl, Martin (advisor)
This bachelor thesis deals with matematical compartment modeling of the COVID-19 pandemic. The basic epidemiological models are presented and six studies that deal with the behavior of the SARS-CoV-2 virus under different conditions are presented. On the SIR model are estimated parameters for first and fourth wave of the epidemic, using algorithm in MATLAB is made optimalization for all the waves of epidemic in the Czech Republic and also for the fourth wave on the SEIR model. The impact of antiepidemic measures is modelled on the SIR model.
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