Advances in Dynamical Systems, Differential Equations, and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 984

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Faculdade de Economia, Universidade do Porto, 4200-464 Porto, Portugal
Interests: dynamical systems; manifolds; lie algebras; holomorphic flows; geometry

Special Issue Information

Dear Colleagues,

We cordially invite you to submit articles for a Special Issue of Mathematics on dynamical systems and differential equations. The title of this Special Issue not only reflects the topicality of the Special Issue itself but also alludes to the upcoming International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMASC’24). ICMASC’24 is a peer-reviewed conference that highlights various aspects of mathematical analysis and its applications in science and engineering.

The focus of this Special Issue will revolve around the broad spectrum of dynamical systems, encompassing iterative dynamics, ordinary differential equations, and (evolutionary) partial differential equations. We encourage submissions addressing these topics either from a theoretical perspective or exploring their diverse applications in physics (such as in cosmology, quantum physics, matter theory, thermodynamics) or in conventional fields like control theory, artificial intelligence, diagnosis algorithms, among others. We want to stress that this Special Issue is also open to submissions from authors who are interested in the topic but did not attend the conference.

We kindly invite the submission of both original research works and exceptional review articles for consideration in this Special Issue. 

Dr. Carla Pinto
Dr. Helena Reis
Guest Editors

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Keywords

  • iterative dynamics
  • ordinary differential equations
  • (evolutionary) partial differential equations
  • applications to physics (cosmology, quantum physics, matter theory, thermodynamics) and other sciences

Published Papers (1 paper)

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Research

45 pages, 4025 KiB  
Article
Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models
by Reinhard Schlickeiser and Martin Kröger
Mathematics 2024, 12(7), 941; https://doi.org/10.3390/math12070941 - 22 Mar 2024
Viewed by 416
Abstract
The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between [...] Read more.
The susceptible–infected–recovered–vaccinated–deceased (SIRVD) epidemic compartment model extends the SIR model to include the effects of vaccination campaigns and time-dependent fatality rates on epidemic outbreaks. It encompasses the SIR, SIRV, SIRD, and SI models as special cases, with individual time-dependent rates governing transitions between different fractions. We investigate a special class of exact solutions and accurate analytical approximations for the SIRVD and SIRD compartment models. While the SIRVD and SIRD equations pose complex integro-differential equations for the rate of new infections and the fractions as a function of time, a simpler approach considers determining equations for the sum of ratios for given variations. This approach enables us to derive fully exact analytical solutions for the SIRVD and SIRD models. For nonlinear models with a high-dimensional parameter space, such as the SIRVD and SIRD models, analytical solutions, exact or accurately approximative, are of high importance and interest, not only as suitable benchmarks for numerical codes, but especially as they allow us to understand the critical behavior of epidemic outbursts as well as the decisive role of certain parameters. In the second part of our study, we apply a recently developed analytical approximation for the SIR and SIRV models to the more general SIRVD model. This approximation offers accurate analytical expressions for epidemic quantities, such as the rate of new infections and the fraction of infected persons, particularly when the cumulative fraction of infections is small. The distinction between recovered and deceased individuals in the SIRVD model affects the calculation of the death rate, which is proportional to the infected fraction in the SIRVD/SIRD cases but often proportional to the rate of new infections in many SIR models using an a posteriori approach. We demonstrate that the temporal dependence of the infected fraction and the rate of new infections differs when considering the effects of vaccinations and when the real-time dependence of fatality and recovery rates diverge. These differences are highlighted for stationary ratios and gradually decreasing fatality rates. The case of stationary ratios allows one to construct a new powerful diagnostics method to extract analytically all SIRVD model parameters from measured COVID-19 data of a completed pandemic wave. Full article
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