003. Факультет інформатики

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Browsing 003. Факультет інформатики by Author "Cherniha, Roman"

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    An Age-Structured Diffusive Model for Epidemic Modelling: Lie Symmetries and Exact Solutions
    (2025) Cherniha, Roman; Davydovych, Vasyl’
    A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for similar purposes. The Lie symmetry classification of the model is derived. It is shown that themodel admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. Examples of application of exact solutions with the correctly-specified parameters for calculation of the total number of infected individuals during an epidemic are presented.
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    Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows
    (2025) Cherniha, Roman; King, John
    A wide class of two-component evolution systems is constructed admitting an infinite-dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross-diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above-mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two-component evolution systems with an infinite-dimensional Lie symmetry that possess essentially different structures is presented.
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    A Reaction-Diffusion System with Nonconstant Diffusion Coefficients: Exact and Numerical Solutions
    (2025) Cherniha, Roman; Kriukova, Galyna
    A Lotka–Volterra-type system with porous diffusion, which can be used as an alternative model to the classical Lotka–Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly specified. In particular, this occurs when the system models the prey–predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.
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    A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine
    (2024) Cherniha, Roman; Dutka, Vasyl; Davydovych, Vasyl
    A space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model.