A Fractional-Order Infectivity and Recovery SIR Model

Angstmann, Christopher N.; Henry, B. I.; McGann, Anna V.

doi:10.3390/fractalfract1010011

Cited by 33 publications

(38 citation statements)

References 20 publications

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“…Angstmann et al [34] use the master equation of a continuous-time random walk to derive an FDEM involving Riemann-Liouville fractional derivatives. Power laws are postulated to model time of infectiousness and recovery.…”

Section: Discussionmentioning

confidence: 99%

“…However, often these approaches lack mathematical basis or physical interpretation except for exchanging integer differentiation with fractional ones [26,33]. Angstmann et al [34] and Sardar et al [35] provided a valid variation by considering the memory of the non-Markovian infection process. The result is a mixed system of integer and fractional derivatives of the Riemann-Liouville type.…”

Section: Introductionmentioning

confidence: 99%

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Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam

Peace

Medina

et al. 2020

IJERPH

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In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms. Laplace transform of a function f (t) is defined as L( f )(s) =f (s) = ∞ 0 e −st f (t)dt.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam

Peace

Medina

et al. 2020

IJERPH

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“…Applications of ordinary and fractional calculus to competition models in biology to bio-medical models based on suitable generalization of classical fluid-dynamics and diffusion have recently attracted the interest of several authors. In [3], the authors derive a fractional-order infectivity and recovery susceptible infectious recovered (SIR) model from the stochastic process of a continuous time random walk that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence between infectivity and recovery, the authors show that fractional-order derivatives appear in generalized master equations that govern the evolution of SIR populations.…”

Section: Fractional Methods In Bio-medical Areasmentioning

confidence: 99%

Fractional Dynamics

Cattani

Spigler

2018

Fractal Fract

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Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].

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“…The existent literature has innumerous examples of applications of SIR models, namely in [21][22][23] where the population is divided into Susceptible (S), Infectious (I) and Recovery (R) . However, in SIR models, recovered individuals are assumed to develop lifelong immunity, which for some diseases such as seasonal flu, influenza or venereal diseases is not necessarily true.…”

Section: An Epidemic Model Of An Sirs System With Nonlinear Incidencementioning

confidence: 99%

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

Fialho

Minhós

2019

Axioms

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The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions.

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A Fractional-Order Infectivity and Recovery SIR Model

Cited by 33 publications

References 20 publications

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Fractional Dynamics

First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model

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