Sliding Bifurcations of Filippov Two Stage Pest Control Models with Economic Thresholds

Communications in Nonlinear Science and Numerical Simulation

2015

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Please cite this article as: Yang, J., Tang, S., Cheke, R.A., Modelling the regulatory system for diabetes mellitus with a threshold window, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns. 2014.08.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.Modelling the regulatory system for diabetes mellitus with a threshold window AbstractPiecewise (or non-smooth) glucose-insulin models with threshold windows for type 1 and type 2 diabetes mellitus are proposed and analyzed with a view to improving understanding of the glucose-insulin regulatory system. For glucose-insulin models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between regular equilibria and a pseudo-equilibrium are studied. Furthermore, the sufficient and necessary conditions for the global stability of regular equilibria and the pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Sliding bifurcations related to boundary node bifurcations were investigated with theoretical and numerical techniques, and insulin clinical therapies are discussed. For glucose-insulin models with a threshold window, the effects of glucose thresholds or the widths of threshold windows on the durations of insulin therapy and glucose infusion were addressed. The duration of the effects of an insulin injection is sensitive to the variation of thresholds. Our results indicate that blood glucose level can be maintained within a normal range using piecewise glucose-insulin models with a single threshold or a threshold window. Moreover, our findings suggest that it is critical to individualize insulin therapy for each patient separately, based on initial blood glucose levels.

Section: Resultsmentioning

confidence: 99%

Section: Ag(t)(c + Mi(t)/(n + I(t))mentioning

confidence: 99%

Modelling the regulatory system for diabetes mellitus with a threshold window

Tang

Communications in Nonlinear Science and Numerical Simulation

2015

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“…To achieve this we treated the insect populations as a Filippov system, variants of which have been widely used for modeling real world biological problems with switching surfaces [Tang et al, 2012;da Silveira Costa & Faria, 2010;da Silveira Costa, 2007;da Silveira Costa & Meza, 2006;Dercole et al, 2007]. The Filippov system is determined by different differential equations according to which the regions of phase space the solutions pass through [Filippov & Arscott, 1988].…”

Section: Discussion and Biological Implicationsmentioning

confidence: 99%

“…The classification and stability of all types of equilibria of these systems have been defined and investigated in [Glendinning, 2015;Dercole et al, 2011;Kuznetsov et al, 2003;Xiao et al, 2013;Tang et al, 2012]. In order to investigate the dynamics of locust populations and address how they change between the solitarious phase and the gregarious phase, we define the regular equilibrium and virtual equilibrium for the discrete switching system (4) based on the references [Glendinning, 2015;Dercole et al, 2011;Kuznetsov et al, 2003].…”

Section: Equilibria For the Switching Systemmentioning

confidence: 99%

A Locust Phase Change Model with Multiple Switching States and Random Perturbation

Xiang

Tang

Int. J. Bifurcation Chaos

et al. 2016

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Insects such as locusts and some moths can transform from a solitarious phase when they remain in loose populations and a gregarious phase, when they may swarm. Therefore, the key to effective management of outbreaks of species such as the desert locust Schistocercagregaria is early detection of when they are in the threshold state between the two phases, followed by timely control of their hopper stages before they fledge because the control of flying adult swarms is costly and often ineffective. Definitions of gregarization thresholds should assist preventive control measures and avoid treatment of areas that might not lead to gregarization. In order to better understand the effects of the threshold density which represents the gregarization threshold on the outbreak of a locust population, we developed a model of a discrete switching system. The proposed model allows us to address: (1) How frequently switching occurs from solitarious to gregarious phases and vice versa; (2) When do stable switching transients occur, the existence of which indicate that solutions with larger amplitudes can switch to a stable attractor with a value less than the switching threshold density?; and (3) How does random perturbation influence the switching pattern? Our results show that both subsystems have refuge equilibrium points, outbreak equilibrium points and bistable equilibria. Further, the outbreak equilibrium points and bistable equilibria can coexist for a wide range of parameters and can switch from one to another. This type of switching is sensitive to the intrinsic growth rate and the initial values of the locust population, and may result in locust population outbreaks and phase switching once a small perturbation occurs. Moreover, the simulation results indicate that the switching transient patterns become identical after some generations, suggesting that the * Author for correspondence evolving process of the perturbation system is not related to the initial value after some fixed number of generations for the same stochastic processes. However, the switching frequency and outbreak patterns can be significantly affected by the intensity of noise and the intrinsic growth rate of the locust population.

“…A common assumption in such models is that the human control activities occur instantaneously, but this is seldom the case with interventions or control strategies usually lasting for a given period. Recently, a threshold policy (TP) has been proposed to describe density-dependent and persistent interventions, which are implemented when the case numbers exceeds a certain value and are suspended when they fall below a critical level [23][24][25][26][27][28][29]. Therefore, our main purpose is to extend the existing models on WNV as a non-smooth system by considering density-dependent and non-instantaneous control measures, based on the threshold policy idea, to examine whether a threshold policy could be used to control the transmission dynamics of WNV more effectively than reliance on existing impulsive differential equations.…”

Section: A C C E P T E D Mmentioning

confidence: 99%

A threshold policy to interrupt transmission of West Nile Virus to birds

Zhou

Xiao

Applied Mathematical Modelling

2016

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Highlights• Propose a Filippov model of West Nile Virus with density-dependent culling strategy.• Solutions approach either endemic equilibrium for subsystems or a pseudo-equilibrium.• Results indicate that a previously chosen level of infected birds can be maintained.• Strengthening mosquito culling is beneficial to curbing the spread of WNV. AbstractThis paper proposes a model of West Nile Virus (WNV) with a Filippov-type control strategy of culling mosquitoes implemented once the number of infected birds exceeds a threshold level. The long-term dynamical behaviour of the proposed non-smooth system is investigated. It is shown that as the threshold value varies, model solutions ultimately approach either one of two endemic equilibria for two subsystems or a pseudo-equilibrium on the switching surface, which is a novel steady state. The results indicate that a previously chosen level of infected birds can be maintained when the threshold policy and other parameters are chosen properly. Numerical studies show that under the threshold policy, strengthening mosquito culling together with protecting bird population is beneficial to curbing the spread of WNV.