On a fractional alternating Poisson process

Crescenzo, Antonio Di; Meoli, Alessandra

doi:10.3934/math.2016.3.212

“…which is simply the case when α = 1. But that leads to another interesting fact that defines the relationship between the fractional and integer stochastic SEIR model; that is, the former process is a random-time subordination of the latter one, as established for other fractional processes like the fractional Poisson process [37,45,52], and the fractional birth and/or death processes [39,40,42,43,53].…”

Section: Fractional Stochastic Processmentioning

confidence: 98%

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam¹,

Peace²,

Medina³

et al. 2020

Preprint

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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 as approximations of some type of fractional nonlinear birth--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 FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

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“…3 (t)) which is simply the case when α = 1. Interestingly, the fractional stochastic SEIR process is a random-time subordination of the integer stochastic SEIR model, as established for other fractional processes like the fractional Poisson process [37,45,52], and the fractional birth and/or death processes [39,40,42,43,53]. In Mandelbrot and Taylor [54], the stochastic time process T 2α is called the operational time, and t is the physical time.…”

Section: Transitionmentioning

confidence: 99%

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam

¹

,

Peace

²

,

Medina

³

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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“…It is worth pointing out that under assumptions (2.3) and (2.4), the process {N t , t ≥ 0} constitutes a fractional alternating Poisson process previously investigated in [10].…”

Section: )mentioning

confidence: 99%

“…In the light of the previous investigations, and aiming to construct a more general model that takes into account both the occurrence of jumps and the fractional nature, in this paper we propose and study a one-dimensional jump-telegraph process with deterministic jumps occurring at velocity changes, and with intertimes governed by a fractional alternating counting process studied in detail in [10]. We obtain the probability law of the new process, which is given in a series form involving the generalized Mittag-Leffler function, also known as the Prabhakar function, and defined as…”

Section: Introductionmentioning

confidence: 99%

On a jump-telegraph process driven by an alternating fractional Poisson process

Crescenzo

¹

,

Meoli

²

2018

Self Cite

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The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

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On a fractional alternating Poisson process

Cited by 6 publications

References 11 publications

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

On a jump-telegraph process driven by an alternating fractional Poisson process

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