System Of Impulsive Differential Equations Research Articles

Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.

Modulo periodic Poisson stable solutions of dynamic equations on a time scale

Existence, uniqueness, and asymptotic stability of modulo periodic Poisson stable solutions of dynamic equations on a periodic time scale are investigated. The model under investigation involves a term which is constructed via a Poisson stable sequence. Novel definitions for Poisson stable as well as modulo periodic Poisson stable functions on time scales are given, and the reduction technique to systems of impulsive differential equations is utilized to achieve the main result. An example which confirms the theoretical results is provided.

A Mathematical Study of Metal Biosorption on Algal–Bacterial Granular Biofilms

A multiscale mathematical model describing the metals biosorption on algal–bacterial photogranules within a sequencing batch reactor (SBR) is presented. The model is based on systems of partial differential equations (PDEs) derived from mass conservation principles on a spherical free boundary domain with radial symmetry. Hyperbolic PDEs account for the dynamics of sessile species and their free sorption sites, where metals are adsorbed. Parabolic PDEs govern the diffusion, conversion and adsorption of nutrients and metals. The dual effect of metals on photogranule ecology is also modelled: metal stimulates the production of EPS by sessile species and negatively affects the metabolic activities of microbial species. Accordingly, a stimulation term for EPS production and an inhibition term for metal are included in all microbial kinetics. The formation and evolution of the granule domain are governed by an ordinary differential equation with a vanishing initial value, accounting for microbial growth, attachment and detachment phenomena. The model is completed with systems of impulsive differential equations describing the evolution of dissolved substrates, metals, and planktonic and detached biomasses within the granular-based SBR. The model is integrated numerically to examine the role of the microbial species and EPS in the adsorption process, and the effect of metal concentration and adsorption properties of biofilm components on the metal removal. Numerical results show an accurate description of the photogranules evolution and ecology and confirm the applicability of algal–bacterial photogranule technology for metal-rich wastewater treatment.

Stability and controllability of nonautonomous neutral system with two delays

This article investigates the exact and approximate controllability, finite time stability, and –Hyers–Ulam–Rassias stability of a neutral impulsive system of differential equations with two delays. Fixed point theory is associated in the analysis of controllability of the system. The stability analysis is carried out with the help of Grönwall type inequality.

A generalization of practical stability of nonlinear impulsive systems

Abstract In this paper, we introduce a new type of stability for nonlinear impulsive systems of differential equations, namely practical h-stability. By using the Lyapunov stability theory, some sufficient conditions which guarantee practical h-stability are established. Our original results generalize well-known fundamental stability results, practical stability, practical exponential stability and practical asymptotic stability for nonlinear time-varying impulsive systems. Then two classes of nonlinear impulsive systems, namely perturbed and cascaded impulsive systems, are discussed. Furthermore, the problem of practical h-stabilization for certain classes of nonlinear impulsive systems is considered. Finally, two numerical examples are given to show the effectiveness of our theoretical results.

Variational approach to instantaneous and noninstantaneous impulsive system of differential equations

In this paper, the existence and multiplicity of solutions for a coupled system of differential equations with instantaneous and noninstantaneous impulses are studied. By the virtue of variational methods, some new existence theorems of solutions are obtained. In addition, two examples are given to demonstrate our main results.

Integral Condition with Nonlinear Kernel for an Impulsive System of Differential Equations with Maxima and Redefinition Vector

Integral Condition with Nonlinear Kernel for an Impulsive System of Differential Equations with Maxima and Redefinition Vector

Modeling the Impact of Climate Change on Cervid Chronic Wasting Disease in Semi-Arid South Texas.

Chronic wasting disease (CWD) is a spongiform encephalopathy disease caused by the transmission of infectious prion agents. CWD is a fatal disease that affects wild and farmed cervids in North America with few cases reported overseas. Social interaction of cervids, feeding practices by wildlife keepers and climate effects on the environmental carrying capacity all can affect CWD transmission in deer. Wildlife deer game hunting is economically important to the semi-arid South Texas region and is affected by climate change. In this paper, we model and investigate the effect of climate change on the spread of CWD using typical climate scenarios. We use a system of impulsive differential equations to depict the transmission of CWD between different age groups and gender of cervids. The carrying capacity and contact rates are assumed to depend on climate. Due to the polygamy of bucks, we use mating rates that depend on the number of bucks and does. We analyze the stability of the model and use simulations to study the effect of harvesting (culling) on eradicating the disease, given the climate of South Texas. We use typical climate change scenarios based on published data and our assumptions. For the climate indicator, we calculated and utilized the Standard Precipitation Evapotranspiration Index (SPEI). We found that climate change might hinder the efforts to reduce and effectively manage CWD as it becomes endemic to South Texas. The model shows the extinction of the deer population from this region is a likely outcome.

Yakubovich's theorem for impulsive differential equations with piecewise constant argument of generalized type

The main goal of our paper is to obtain sufficient conditions for the asymptotic equivalence of a linear differential system and a quasilinear system of impulsive differential equations with piecewise constant argument of generalized type, in short IDEPCAG. A deviating argument is of the advanced and delayed type. As an auxiliary result, the structure of the set of solutions of the quasilinear system is specified. Several examples are also given to show the feasibility of results.

Impulsive simulation model for the dynamics of allergen immunotherapy

This study aims to analyze the effects of allergen immunotherapy, used to treat allergic symptoms such as pollen allergy. Mathematical models are used as a methodological approach to simulate from a system of impulsive differential equations the dynamics of the model. Immunotherapy is based of supplying small amounts of pollen to the patient, which leads to minimizing severe allergic symptoms when patients are subsequently exposed to higher amounts of pollen in the environment. Lymphocyte concentrations are considered state variables, allowing the behavior and efficacy of allergen immunotherapy to be identified. The manuscript proposes a method that allows to model mixed systems. Phenomena that present continuous times in some instants and discrete times in others, these are phenomena that are frequently found in the field of physics. Allergen immunotherapy is most effective when a treatment is created with pollen dose increments in a linear form.

Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.

Dynamical Modelling of Bone Formation and Resorption under Impulsive Estrogen Supplement: Effects of Parathyroid Hormone and Prolactin

Osteoporosis, a bone metabolic disease, is one of the major diseases occurring in aging population especially in postmenopausal women. A system of impulsive differential equations is developed in this paper in order to investigate the effects of parathyroid hormone and prolactin on bone-forming cells, namely, osteoblasts, and bone-resorbing cells, namely, osteoclasts, under the impulsive estrogen supplement. The theoretical analysis of the developed model is carried out so that we obtain the conditions on the system parameters in which the stability and permanence of the model can occur. Computer simulations are also provided to illustrate the theoretical predictions.

Longitudinal exposure to pyrethroids (3-PBA and trans-DCCA) and 2,4-D herbicide in rural schoolchildren of Maule region, Chile

BackgroundSeveral studies showed that early exposure to pesticides affects the development and health of children. In Maule, there is previous evidence of the high exposure to organophosphate pesticides (OP) of schoolchildren. However, to date, there are no studies assessing exposure to pyrethroids and the herbicide 2,4-D.Objetive.To evaluate children's exposure to pyrethroids insecticides 3-phenoxybenzoic acid (3-PBA), trans 3-(2,2-dichlorovinyl)-2,2-dimethylcyclopropane carboxylic acid (trans-DCCA) and 2,4 dichlorophenoxyacetic acid (2,4-D) herbicides. MethodLongitudinal study with 48 schoolchildren from two rural schools in the Maule region, Chile. Urinary metabolites of pyrethroids 3-PBA, Trans-DCCA and 2,4-D herbicides were evaluated in 2016 and 2017. Mann-Whitney U for repeated measurements and Spearman's rho correlation tests were used for data analysis. Also, we used a system of impulsive differential equations for mathematical modeling. ResultsAll the schoolchildren assessed had more than two pesticide urinary metabolites in both years, with the 3-PBA metabolite being the most frequent. There was an increase in concentrations of urinary 3-PBA in November 2017, compared to 2016 (from 0.69 μg/L to 1.90 μg/L). In 2016, the specific metabolites of 3-PBA were correlated with Trans-DCCA, 2,4-D, chlorpyrifos, diazinon, and para-nitrophenol. In 2017, 3-PBA was correlated with 2,4-D, and Trans-DCCA. The concentrations of 3-PBA of Chilean children were higher than studies conducted in the USA that found an association of prenatal exposure to these metabolites with cognitive difficulties. ConclusionsWe found high concentrations of pyrethroid metabolites among all the schoolchildren assessed, which may impact on their health and development. These insecticides had received no attention from the scientific community in Chile, and neither from the government agencies, despite the increased use of these chemicals in recent years. This is the first study in South America that confirms the exposure to pyrethroids and herbicides through biomarkers in human population living near farm fields.

Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely without human intervention and derive conditions for this to occur. In this case, the system of impulsive differential equations has a periodic solution. We show that success is equivalent to the convergence of solutions to this periodic solution. We provide conditions that ensure that a periodic solution exists. When it exists, it is unique and attracting. However, we also show that whether a solution converges to this periodic solution, and hence whether the model predicts that the reactor operates successfully, is initial condition dependent. The analysis is illustrated with numerical examples.

Modeling the effect of a combinatorial anti-CTLA antibody and radiation therapy on metastatic lung cancer.

Cytotoxic T-lymphocyte antigen 4 (CTLA-4) is an inhibitory receptor that is upregulated after T-cell activation, and transmits signals that suppress T-cell activation and proliferation through the depletion of CD8+ T cells. Antibodies targeting CTLA-4 can selectively deplete regulatory T cells by antibody-mediated cellular cytotoxicity mechanisms. However, anti-CTLA-4 antibody therapy as a single treatment is limited on effectiveness to fight against immunogenic diseases like metastatic cancers. Radiation therapy (RT) on the other hand is a longstanding pillar of cancer treatment, historically utilized to treat a discrete target and provide local tumor control. Recent preclinical studies demonstrated that RT is synergistic with anti-CTLA antibody and induces systemic anti-tumor responses. In order to better understand the effect of the combination therapy on metastatic lung cancer, we developed a mathematical model using a system of impulsive differential equations. The objective of this study was to develop a predictive simulation platform to improve cancer management by manipulating dose and fractionation schedule of RT and dose and timing for the anti-CTLA-4 antibody therapy. This study is supported by NIH Grant P20GM103434 to the West Virginia IDeA Network for Biomedical Research Excellence.

On the solvability of the modified Cauchy problem for linear systems of impulsive differential equations with singularities

On the solvability of the modified Cauchy problem for linear systems of impulsive differential equations with singularities

Medidas de conservación ex situ: Un enfoque metapoblacional a partir del modelo clásico de Levins

Given a metapopulation, which as such, is distributed in fragments or patches, we can think that this is at risk of extinction if the number of occupied patches is decreasing. In this sense, a measure of conservation to consider could be the incorporation of individuals to unoccupied patches. This process is known as ex situ conservation. In the present work we will suppose a metapopulation that is becoming extinct, whose dynamics will be modeled mathematically through the model proposed by Richard Levins in his work “Some demographic and genetic consequences of environmental heterogeneity for biological control”. The effect of an ex situ conservation measure, consisting of the artificial colonization of unoccupied patches, will be incorporated into the model, with the purpose of preventing the total population from becoming extinct. The incorporation of this conservation measure will be expressed through a system of impulsive differential equations, from which a series of simulations will be generated, in order to understand the long-term dynamics and determine what factors influence the success of the measure of conservation.

Modeling the effect of a combinatorial anti-CTLA antibody and radiation therapy on metastatic melanoma

Cytotoxic T-lymphocyte antigen 4 (CTLA-4) is an inhibitory receptor that is upregulated after T-cell activation, and transmits signals that suppress T-cell activation and proliferation. Antibodies targeting CTLA-4 can selectively deplete regulatory T cells by antibody-mediated cellular cytotoxicity mechanisms. Radiation therapy (RT) on the other hand is a longstanding pillar of cancer treatment, historically utilized to treat a discrete target and provide local tumor control. Recent preclinical studies demonstrated that RT is synergistic with anti-CTLA antibody and induces systemic anti-tumor responses. In order to better understand the effect of the combination therapy on metastatic melanoma, we developed a multi-scale mechanistic model using a system of impulsive differential equations. The objective of this study was to develop a predictive simulation platform to improve cancer management by manipulating dose and fractionation schedule of RT in the combination therapy. The project was supported by NIH Grant P20GM103434 to the West Virginia IDeA Network for Biomedical Research Excellence.

Growth on two limiting essential resources in a self-cycling fermentor.

A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.

Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.