Threshold conditions for integrated pest management models with pesticides that have residual effects

Communications in Nonlinear Science and Numerical Simulation

Cheke

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%

Modelling the regulatory system for diabetes mellitus with a threshold window

Communications in Nonlinear Science and Numerical Simulation

Cheke

2015

Self Cite

“…Recently, many mathematical models with impulsive chemical control tactics and releases of natural enemies have been proposed to model an IPM strategy such as spraying of pesticides [25,[28][29][30][31][32][33] or releases of natural enemies at critical times [27,[34][35][36][37][38][39][40][41]. Those studies mainly focused on the effects of chemical control and biological control on the permanence or extinction of pest populations, and did not consider the effects of pesticide resistance.…”

Section: Discussionmentioning

confidence: 99%

Analytical methods for detecting pesticide switches with evolution of pesticide resistance

Liang

Mathematical Biosciences

Nieto

et al. 2013

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Citation for this version held on GALA:Liang, Juhua, Tang, Sanyi, Nieto, Juan J. and Cheke, Robert A. (2013) After a pest develops resistance to a pesticide, switching between different unrelated pesticides is a common management option, but this raises the following questions: (1) What is the optimal frequency of pesticide use? (2) How do the frequencies of pesticide applications affect the evolution of pesticide resistance? (3) How can the time when the pest population reaches the economic injury level (EIL) be estimated and (4) how can the most efficient frequency of pesticide applications be determined? To address these questions, we have developed a novel pest population growth model incorporating the evolution of pesticide resistance and pulse spraying of pesticides. Moreover, three pesticide switching methods, threshold condition-guided, density-guided and EIL-guided, are modelled, to determine the best choice under different conditions with the overall aim of eradicating the pest or maintaining its population density below the EIL. Furthermore, the pest control outcomes based on those three pesticide switching methods are discussed. Our results suggest that either the density-guided or EIL-guided method is the optimal pesticide switching strategy, depending on the frequency (or period) of pesticide applications.

“…From the model (2), we can see that once the tumour size reaches the critical value V L , the comprehensive therapy is carried out, which leads to the effector cells increasing to y(t + ) from y(t) immediately and the tumour cells decreasing to x(t + ) from x(t) right away. Quantitative theory for such impulsive systems has been extensively developed [Bonotto, 2009;Bonotto & Federson, 2008;Liang et al, 2013;Qian et al, 2010;Zeng et al, 2006] which has applications in many domains of applied science, such as in pest management programmes [Tang & Cheke, 2005Tang et al, 2013], virus dynamical systems [Lou et al, 2012;Yang & Xiao, 2012;Yang et al, 2013], vaccination strategies and epidemiology [Nie et al, 2013;Shulgin et al, 1998], diabetes mellitus [Tang & Xiao, 2007], and neuron systems [Touboul & Brette, 2009]. Our paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

A Feedback Control Model of Comprehensive Therapy for Treating Immunogenic Tumours

Xiao

Int. J. Bifurcation Chaos

et al. 2016

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In the traditional therapy of cancers, surgery is a main method but it often fails to cure patients for complex reasons. Thus, a new therapeutic approach including both surgery and immunotherapy has been proposed and shown to be effective clinically in inhibiting cancer cells while retaining immunologic memory. This comprehensive strategy guided by a threshold of tumour cells in an immune tumour system is modelled and conditions for successful control of tumours and related dynamics are addressed. A mathematical model with state-dependent impulsive interventions is formulated to describe surgery combined with immunotherapy. By analyzing the properties of the Poincaré map we examine the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semi-trivial order-1 periodic solution and the positive order-k periodic solution. The main results showed that surgery alone can only control the tumour size below a certain level while there is no immunologic memory. If comprehensive therapy involving combining surgery with immunotherapy is considered, then not only can the cancers be controlled below a certain level, but the immune system can also retain its activity. The existence of positive order-k periodic solutions implies that periodical therapy is needed to control the cancers, however choosing the treatment frequency and the strength of the therapy remains challenging, and hence a strategy of individual-based therapy is suggested.