DOKYI, M. Diferenciální rovnice se zpožděním v dynamických systémech [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2021.

Posudek vedoucího

Opluštil, Zdeněk

The master’s thesis deals with Delay Differential Equations in Dynamical Systems. The text is divided into three main parts. Firstly, a mathematical apparatus relevant to the dynamic systems and delay differential equations are introduced. Then SIR epidemic models (with and without delay) are analyzed. Finally, the author performs numerical simulations of specific models. The author fulfilled the requirements and goals of the master´s thesis. The text is clear and logically structured. There are some mathematical inaccuracies and inconsistencies in the thesis, e.g., when the local stability of equilibria is derived in sections 5.0.1-5.0.4. The author chose current and interesting (epidemic) models for the demonstration and comparison of systems of differential equations with and without delay. Unfortunately, some analyses have not been completed and more detailed interpretations of numerical simulations are missing. On the other hand, I appreciate the following in the thesis. Mathematical modeling of epidemics is applicable in practice and is now a current topic. The author compared SIR models with and without delay using numerical calculations and many illustrative graphs in MATLAB software. She also presented different behavior of the models for different delays representing various incubation times. For the above reasons, I recommend the thesis for defence with classification satisfactory.

Posudek oponenta

Šremr, Jiří

The present thesis is focused to the study of a stability of dynamical systems and delay differential equations. I have, in particular, the following objections: * The text contains a lot of misprints and inaccuracies, some notions are not defined at all, e.g., topologically conjugate system, N_\delta(x^*), equilibrium of delay differential equations, system with a single delay, etc. * There are many uncited references. * On p. 28, you claim: "To find the local stability of E_2 we need to show that equation (5.2) has no pure imaginary roots". A reference to a theoretical part of your thesis or to a literature is missing. * In the case of generalized SIR model, a meaning of the parameters \gamma_1, \gamma_2 is not explained. Moreover, you assume that the total population N is constant, however, if the condition \mu=\gamma_1=\gamma_2 does not hold, then $N(t)\not\equiv Const.$. In such a case, the constant term \mu_0 is very often considered in the first equation of a system instead of \mu N(t) which allows us to elaborate a stability analysis similarly to the case of classical SIR model. On the other hand, I would like to appreciate: * SIR model and its generalizations are described very well. * I like how a comparison of the classical SIR models with and without delay is presented in Chapters 5.1.5 and 5.1.7. Conclusion: In my opinion, the main goals of the thesis have been achieved. In some parts, the thesis somehow lacks mathematical accuracy (which should be a standard for such type of work). Moreover, the parameters \gamma_1, \gamma_2, \mu are not chosen properly in the case of generalized model, which causes incorrectness of graphical results (outputs) in Chapters 5.1.3 and 5.1.4. In view of the above-said, I propose the evaluation D.

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