New Applications of Perov’s Fixed Point Theorem

Muresan, Sorin; Iambor, Loredana Florentina; Bazighifan, Omar

doi:10.3390/math10234597

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…In the limit of small cumulative fractions J ≪ 1, which very often is fulfilled, this approximation provides accurate analytical expressions for all epidemic quantities of interest such as the rate of new infections J(t) and the fraction I(t) of infected persons. One of the referees has kindly informed us that our analytical approach is related to the Perov's fixed point theorem [18]. As an aside, when determining τ d in Appendix C we provided accurate approximative solutions to Wright's transcendental Equation (A16), or equivalently, Equation (186).…”

Section: Discussionmentioning

confidence: 99%

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Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models

Schlickeiser,

Kröger

2024

Mathematics

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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.

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Section: Discussionmentioning

confidence: 99%

“…Before proceeding to the approximation, we note that in terms of the real time, the Equations ( 21)-(26) read, with the help of Equations ( 9), ( 10), ( 12), ( 14), ( 15), ( 17), (18), and (31),…”

Section: Solution Of the Sirvd Equationsmentioning

confidence: 99%

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models

Schlickeiser,

Kröger

2024

Mathematics

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“…One of the extensions of the principle by replacing the contractive constant with a family of functions was established by Rakotch [2]. For some recent refinements of the Banach fixed point theorem, one can consult [3][4][5] and the references therein. A few of these earlier improvements that are of interest to us in this current project include the work of Ciric [6], Geraghty [7], Jaggi [8], and Dass-Gupta [9].…”

Section: Introductionmentioning

confidence: 99%

On Two-Point Boundary Value Problems and Fractional Differential Equations via New Quasi-Contractions

Noorwali

Shagari

2023

Mathematics

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The aim of this paper is to introduce new forms of quasi-contractions in metric-like spaces and initiate more general conditions for the existence of invariant points for such operators. The proposed notions are then applied to study novel existence criteria for the existence of solutions to two-point boundary value problems in the domains of integer and fractional orders. To attract further research in this direction, important consequences are deduced and discussed to indicate the novelty and generality of our proposed concepts.

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Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD and SIR Compartment Models

Schlickeiser,

Kröger

2024

Preprint

View full text Add to dashboard Cite

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 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. \Revised{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.

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New Applications of Perov’s Fixed Point Theorem

Cited by 4 publications

References 21 publications

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models

On Two-Point Boundary Value Problems and Fractional Differential Equations via New Quasi-Contractions

Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD and SIR Compartment Models

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