An Introduction to Delay Differential Equations with Applications to the Life Sciences

Smith, Hal L.

doi:10.1007/978-1-4419-7646-8

Cited by 749 publications

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“…Modeling of such systems by fractional-order (or arbitrary order) models provides the systems with long-time memory and gains them extra degrees of freedom [27]. A large number of mathematical models, based on ordinary and delay differential equations with integer-orders, have been proposed in modeling the dynamics of epidemiological diseases [18,20,28,29]. In recent years, it has turned out that many phenomena in different fields can be described very successfully by models using fractionalorder differential equations (FODEs) [13,6,27].…”

Section: Fractional-order Sirc Epidemic Modelmentioning

confidence: 99%

Dynamics of Salmonella Infection

Rihan¹

2017

Current Topics in Salmonella and Salmonellosis

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In this chapter, we propose a mathematical epidemic model, with integer and fractional order to describe the dynamics of Salmonella infection in animal herds. We investigate the qualitative behaviors of such model and find the conditions that guarantee the asymptotic stability of disease-free and endemic steady states. To assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control, we estimate basic reproduction number R 0 . This threshold parameter specifies the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak. We also provide an unconditionally stable implicit scheme for the fractional-order epidemic model. The theoretical and computational results give insight into the modelers and infectious disease specialists.

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Section: Fractional-order Sirc Epidemic Modelmentioning

confidence: 99%

Dynamics of Salmonella Infection

Rihan¹

2017

Current Topics in Salmonella and Salmonellosis

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“…By way of contradiction, we assume that 0 < l ≤ κ. Again from the fluctuation lemma [12,Lemma A.1. ], there exists a sequence {t k } k≥1 such that…”

Section: Lemma 21 (See [[11] Theorem 21]) Assume Thatmentioning

confidence: 99%

Global Exponential Stability of Periodic Solutions for a Nicholson’s Blowflies Model with a Linear Harvesting Term

Jiang¹

2014

Appl. Math. Inf. Sci.

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Abstract:In this paper, we study a generalized Nicholson's blowflies model with a linear harvesting term, which is defined on the positive function space. Under proper conditions, we employ a novel proof to establish some criteria for the existence and global exponential stability of positive periodic solutions for this model. Moreover, we give an example and its numerical simulations to illustrate our main results.

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“…In the following we apply the Hopf bifurcation Theorem as from [30] (see Theor.4.8. p.55, and references therein) (the solution λ(r) of (3.33) is parameterized in terms of r ≥ 0).…”

Section: Let Us Denote By X(t) = (U(t) S(t)) T and By X T := (U(t mentioning

confidence: 99%

Mathematical Modelling of Cancer Stem Cells Population Behavior

Beretta

Capasso

Morozova

2012

Math. Model. Nat. Phenom.

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Abstract. Recent discovery of cancer stem cells in tumorigenic tissues has raised many questions about their nature, origin, function and their behavior in cell culture. Most of current experiments reporting a dynamics of cancer stem cell populations in culture show the eventual stability of the percentages of these cell populations in the whole population of cancer cells, independently of the starting conditions. In this paper we propose a mathematical model of cancer stem cell population behavior, based on specific features of cancer stem cell divisions and including, as a mathematical formalization of cell-cell communications, an underlying field concept. We compare the qualitative behavior of mathematical models of stem cells evolution, without and with an underlying signal. In absence of an underlying field, we propose a mathematical model described by a system of ordinary differential equations, while in presence of an underlying field it is described by a system of delay differential equations, by admitting a delayed signal originated by existing cells. Under realistic assumptions on the parameters, in both cases (ODE without underlying field, and DDE with underlying field) we show in particular the stability of percentages, provided that the delay is sufficiently small. Further, for the DDE case (in presence of an underlying field) we show the possible existence of, either damped or standing, oscillations in the cell populations, in agreement with some existing mathematical literature. The outcomes of the analysis may offer to experimentalists a tool for addressing the issue regarding the possible non-stem to stem cells transition, by determining conditions under which the stability of cancer stem cells population can be obtained only in the case in which such transition can occur. Further, the provided description of the variable corresponding to an underlying field may stimulate further experiments for elucidating the nature

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An Introduction to Delay Differential Equations with Applications to the Life Sciences

Cited by 749 publications

References 0 publications

Dynamics of Salmonella Infection

Dynamics of Salmonella Infection

Global Exponential Stability of Periodic Solutions for a Nicholson’s Blowflies Model with a Linear Harvesting Term

Mathematical Modelling of Cancer Stem Cells Population Behavior

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