Dynamics in piecewise linear and continuous models of complex switching networks

Shahrear, Pabel; Glass, Leon; Wilds, Roy; Edwards, Roderick

doi:10.1016/j.matcom.2013.12.002

Cited by 7 publications

(7 citation statements)

References 18 publications

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“…While a Boolean approximation to sigmoidal switching interactions seems robust in many cases [24][25][26], several oversimplifications of established models, such as synchronous update of node states, negligence of transmission delays, and homogeneous network elements, can restrict the dynamics or yield artifacts. For example, it is well-known that a large number of periodic cycles in Boolean networks become unstable for an autonomous updating scheme [23] and that even very simple Boolean networks can display complex dynamics [27,28].…”

Section: Introductionmentioning

confidence: 99%

Transient dynamics and their control in time-delay autonomous Boolean ring networks

et al. 2017

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Biochemical systems with switch-like interactions, such as gene regulatory networks, are well modeled by autonomous Boolean networks. Specifically, the topology and logic of gene interactions can be described by systems of continuous piecewise-linear differential equations, enabling analytical predictions of the dynamics of specific networks. However, most models do not account for time delays along links associated with spatial transport, mRNA transcription, and translation. To address this issue, we we have developed an experimental testbed to realize a time-delay autonomous Boolean network with three inhibitory nodes, known as a repressilator, and use it to study the dynamics that arise as time delays along the links vary. We observe various nearly-periodic oscillatory transient patterns with extremely long lifetime, which emerge in small network motifs due to the delay, and which are distinct from the eventual asymptotically stable periodic attractors. For repeated experiments with a given network, we find that stochastic processes give rise to a broad distribution of transient times with an exponential tail. In some cases, the transients are so long that it is doubtful the attractors will ever be approached in a biological system that has a finite lifetime. To counteract the long transients, we show experimentally that small, occasional perturbations applied to the time delays can force the trajectories to rapidly approach the attractors.

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Section: Introductionmentioning

confidence: 99%

Transient dynamics and their control in time-delay autonomous Boolean ring networks

et al. 2017

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“…Our experiments suggest that supertransients generally arise in ABNs, if link delays are taken into account. The Boolean approximation of a sigmoidal function in general holds well, 11 and thus we conjecture that similar transients occur in networks with switchlike interactions. Moreover, the same transient patterns, however of shorter duration, are observed in more biologically realistic simulations of a toggle switch and repressilator when accounting for the explicit delays arising from transcription and translation processes.…”

Section: Discussionmentioning

confidence: 67%

“…Without noise, we recover the maps Eqs. (11) and (21). The initial conditions are a 0 ¼ e Às 1 ; b 0 ¼ e Às 2 ; Dt i;0 ¼ s i þ n i;0 þ lnð2 À e Às i Þ.…”

Section: E Stochastic Delaysmentioning

confidence: 99%

Super-transient scaling in time-delay autonomous Boolean network motifs

D'Huys

Lohmann

Haynes

et al. 2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Autonomous Boolean networks are commonly used to model the dynamics of gene regulatory networks and allow for the prediction of stable dynamical attractors. However, most models do not account for time delays along the network links and noise, which are crucial features of real biological systems. Concentrating on two paradigmatic motifs, the toggle switch and the repressilator, we develop an experimental testbed that explicitly includes both inter-node time delays and noise using digital logic elements on field-programmable gate arrays. We observe transients that last millions to billions of characteristic time scales and scale exponentially with the amount of time delays between nodes, a phenomenon known as super-transient scaling. We develop a hybrid model that includes time delays along network links and allows for stochastic variation in the delays. Using this model, we explain the observed super-transient scaling of both motifs and recreate the experimentally measured transient distributions.

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“…The pioneer epidemiologist Kermack and McKendrick (1927) extensively used mathematical models to get features of these infectious diseases [11] . Following the footprints of Kermack and McKendrick, some of the eminent models have been proposed for the transmission dynamics of some illnesses which include: HIV model [12] ; Heathcote and Yorke (1984) model for the spread and control of gonorrhoea [13] ; Ronald Ross model for control of malaria [14] ; Glass network [15] etc.…”

Section: Introductionmentioning

confidence: 99%

“…We checked the numerical results utilizing simulation in MATLAB. Sometimes researchers used unique or a couple of software to validate numerical results [15] , [18] , [19] , [20] , but in this manuscript, we utilize fancy software MATLAB.…”

Section: Introductionmentioning

confidence: 99%

Prediction and mathematical analysis of the outbreak of coronavirus (COVID-19) in Bangladesh

Shahrear¹,

Rahman²,

Nahid³

2021

Results in Applied Mathematics

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In this study based on Bangladesh, a modified SIR model is produced and analysed for COVID-19. We have theoretically investigated the model along with numerical simulations. The reproduction number has been calculated by using the method of the next-generation matrix. Due to the basic reproduction number, we have analysed the local stability of the model for disease-free and endemic equilibria. We have investigated the sensitivity of the reproduction number to parameters and calculate the sensitivity indices to determine the dominance of the parameters. Furthermore, we simulate the system in MATLAB by using the fourth-order Runge–Kutta (RK4) method and validate the results using fourth order polynomial regression (John Hopkins Hospital (JHH), 2020). Finally, the numerical simulation depicts the clear picture of the upward, and the downward trend of the spread of this disease along with time in a particular place, and the parameters in the mathematical model indicate this change of intensity. This result represents, the effect of COVID-19 from Bangladesh’s perspective.

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Dynamics in piecewise linear and continuous models of complex switching networks

Cited by 7 publications

References 18 publications

Transient dynamics and their control in time-delay autonomous Boolean ring networks

Transient dynamics and their control in time-delay autonomous Boolean ring networks

Super-transient scaling in time-delay autonomous Boolean network motifs

Prediction and mathematical analysis of the outbreak of coronavirus (COVID-19) in Bangladesh

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