Bistability and State Transition of a Delay Differential Equation Model of Neutrophil Dynamics

This paper studies the existence of bistable states and control strategies to induce state transitions of a delay differential equation model of neutrophil dynamics. We seek the conditions that a stable steady state and an oscillatory state coexist in the neutrophil dynamical system. Physiologically, stable steady state represents the healthy state, while oscillatory state is usually associated with diseases such as cyclical neutropenia. We study the control strategies to induce the transitions from the disease state to the healthy state by introducing temporal perturbations to system parameters. This study is valuable in designing clinical protocols for the treatment of cyclical neutropenia.

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Linear Analysis of an Integro-Differential Delay Equation Model

International Journal of Differential Equations ◽

10.1155/2018/5035402 ◽

2018 ◽

Vol 2018 ◽

pp. 1-6

Author(s):

Anael Verdugo

Keyword(s):

Differential Equation ◽

Steady State ◽

Differential Equations ◽

Gene Networks ◽

Computational Study ◽

Linear Analysis ◽

Equation Model ◽

Stable Steady State ◽

Differential Delay Equation ◽

Delay Differential

This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of N delay differential equations (DDEs) coupled to N ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as N becomes large.

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Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽

10.3390/math8112069 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2069

Author(s):

Anton V. Kovalev ◽

Evgeny A. Viktorov ◽

Thomas Erneux

Keyword(s):

Differential Equation ◽

Delay Differential Equation ◽

Optical Feedback ◽

Laser Pulses ◽

Equation Model ◽

Rate Equations ◽

Differential Equation Model ◽

The Stability ◽

Delay Differential ◽

Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

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