Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology

Davydovych, Vasyl’; Dutka, Vasyl’; Cherniha, Roman

doi:10.3390/sym15112025

Symmetry

2023

DOI: 10.3390/sym15112025

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Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology

Vasyl’ Davydovych,

Vasyl’ Dutka,

Roman Cherniha

Abstract: The review is devoted to an analysis of mathematical models used for describing epidemic processes. Our main focus is on the models that are based on partial differential equations (PDEs), especially those that were developed and used for the COVID-19 pandemic modeling. Most of our attention is given to the studies in which not only results of numerical simulations are presented but analytical results as well. In particular, traveling fronts (waves), exact solutions, and the estimation of key epidemic paramete… Show more

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2024

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Cited by 4 publications

References 74 publications

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Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox

Shanmugam,

Byeon

2024

Medicine

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The challenge of developing comprehensive mathematical models for guiding public health initiatives in disease control is varied. Creating complex models is essential to understanding the mechanics of the spread of infectious diseases. We reviewed papers that synthesized various mathematical models and analytical methods applied in epidemiological studies with a focus on infectious diseases such as Severe Acute Respiratory Syndrome Coronavirus-2, Ebola, Dengue, and Monkeypox. We address past shortcomings, including difficulties in simulating population growth, treatment efficacy and data collection dependability. We recently came up with highly specific and cost-effective diagnostic techniques for early virus detection. This research includes stability analysis, geographical modeling, fractional calculus, new techniques, and validated solvers such as validating solver for parametric ordinary differential equation. The study examines the consequences of different models, equilibrium points, and stability through a thorough qualitative analysis, highlighting the reliability of fractional order derivatives in representing the dynamics of infectious diseases. Unlike standard integer-order approaches, fractional calculus captures the memory and hereditary aspects of disease processes, resulting in a more complex and realistic representation of disease dynamics. This study underlines the impact of public health measures and the critical importance of spatial modeling in detecting transmission zones and informing targeted interventions. The results highlight the need for ongoing financing for research, especially beyond the coronavirus, and address the difficulties in converting analytically complicated findings into practical public health recommendations. Overall, this review emphasizes that further research and innovation in these areas are crucial for addressing ongoing and future public health challenges.

show abstract

Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox

Shanmugam,

Byeon

2024

Medicine

View full text Add to dashboard Cite

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The Exact Solutions of a Diffusive SIR Model via Symmetry Groups

Naz,

Johnpillai,

Mahomed

2024

Journal of Mathematics

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The focus of this paper is to investigate the exact solutions of a diffusive susceptible‐infectious‐recovered (SIR) epidemic model, characterized by a nonlinear incidence. A four‐dimensional Lie point symmetry algebra is obtained for this model. We utilize the Lie symmetries to deduce the optimal system of one‐dimensional subalgebras. The reductions and group‐invariant solutions are obtained with the aid of these subalgebras. We also derive new group‐invariant solutions and reductions for the underlying model via subalgebras that are related to the optimal system by adjoint maps. We developed the diffusive susceptible‐infectious‐quarantined (SIQ) model with quarantine‐adjusted incidence function to understand the transmission dynamics of COVID‐19.

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A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine

Cherniha,

Dutka,

Davydovych

2024

Symmetry

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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.

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Reaction–Diffusion Equations in Mathematical Models Arising in Epidemiology

Cited by 4 publications

References 74 publications

Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox

Comprehending symmetry in epidemiology: A review of analytical methods and insights from models of COVID-19, Ebola, Dengue, and Monkeypox

The Exact Solutions of a Diffusive SIR Model via Symmetry Groups

A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine

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