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2024 | Buch

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Mathematical Modeling and Control in Life and Environmental Sciences

Regional Control Problems

verfasst von: Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Verlag: Springer International Publishing

Buchreihe : Modeling and Simulation in Science, Engineering and Technology

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph explores the use of mathematical modeling and control theory in a variety of contemporary challenges in mathematical biology and environmental sciences. Emphasizing an approach of learning by doing, the authors focus on a set of significant case studies emerging from real-world problems and illustrate how mathematical techniques and computational experiments can be employed in the search for sustainable solutions.
The following topics are extensively discussed:Eradicability and control of a paradigmatic epidemic model, with a view to the existence of endemic states, their stability, and the existence of travelling wavesA spatially structured epidemic model concerning malaria as an example of vector-borne epidemicsOptimal harvesting problems for space-structured and age-structured population dynamicsControlling epidemics in agriculture due to pest insectsThe role of predators as a possible biocontrol agent of epidemics in agricultureControl by taxation of the environmental pollution produced by human activitiesThe originality of this text is in its leitmotif – regional control – along the principle of “Think Globally, Act Locally.” Indeed, for example, in many real spatially structured ecosystems, it is practically impossible to control the relevant system by global interventions in the whole habitat.

Proofs are given whenever they may serve as a guide to the introduction of new concepts. Each chapter includes a comprehensive description of the numerical methods used for the computational experiments, and MATLAB© codes for many of the numerical simulations are available for download. Several challenging open problems are also provided to stimulate future research.
This text is aimed at mathematicians, engineers, and other scientists working in areas such as biology, medicine, and economics. Graduate and advanced undergraduate students of a quantitative subject related to the analysis and applications of dynamical systems and their control will also find it to be a valuable resource.

Inhaltsverzeichnis

Frontmatter

Regional Control of Spatially Structured Epidemics

Frontmatter

Chapter 1. Regional Control for a Class of Spatially Structured Epidemics: Think Globally, Act Locally

Abstract

This chapter introduces the problem of controlling a paradigmatic spatially structured epidemic system modelled as an integro-differential reaction–diffusion system. The spread of the epidemic is due to the interaction of the relevant population (humans, animals, trees, etc.) with a polluted environment. The public health concern consists of reducing the disease in the relevant habitat, as fast as possible, at an optimal cost. The underlying issue of our presentation is that very often the entire domain of interest for the epidemic is either unknown or difficult to manage, for an affordable implementation of suitable environmental sanitation programs. This is the reason why regional control has been proposed; it has been shown that, under suitable modelling assumptions, it is indeed sufficient to implement such programs only in a given subregion, conveniently chosen, so to lead to an effective reduction, and eventual eradication, of the epidemic in the whole habitat; it is evident that this practice may have an enormous importance in real cases with respect to both financial and practical affordability.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Chapter 2. Controlling the Spread of a Vector Borne Epidemic: The Case of Malaria

Abstract

As an application of the man–environment model presented in Section 1.3, we shall consider a malaria epidemic system. In this case the environmental pollution is represented by the infective insect vector population, i.e., the infected mosquito population. We have considered a generalization of the classical Ross–Macdonald model, according to which malaria is an SIS system for which human infectives after recovery may go back to the susceptible state. We have extended the (linear) response of the Ross–Macdonald model, by using a possibly nonlinear functional response. This choice may allow possible saturation effects, \(\sigma -\)type responses, etc. For example behavioral changes can be taken into account; for a very large density of the infective population, the force of infection may tend to reduce itself because of the reduction of open exposure of the human population. We concentrate on a problem of diminishment of a spatially structured malaria epidemic, by acting on the segregation of the human and the mosquito populations via, e.g., treated bed nets. Optimal control problems have been analyzed with respect to both the parameters of the model and with respect to the region of intervention. All has been supported by a set of numerical simulations.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Optimal Harvesting

Frontmatter

Chapter 3. Optimal Harvesting: Space Dependence

Abstract

Here we investigate the regional control for some optimal harvesting problems related to population dynamics; namely we consider the problem of maximizing the profit for spatially structured harvesting problems with respect to both the harvesting effort and the selected subregion \(\omega \) (of the whole domain \(\varOmega \)) where the effort is localized. For a fixed subregion \(\omega \), we state the necessary optimality conditions and use them to get the structure of the optimal effort and to reformulate the maximization problem with respect to the subregion \(\omega \), where the harvesting effort is localized, in a more convenient way. We derive an iterative algorithm to increase at each iteration the profit by changing the subregion where the effort is localized. Some numerical tests are given to illustrate the effectiveness of the results for a particular optimal harvesting problem. Final comments are given as well concerning further directions to extend the results and methods presented here.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Chapter 4. Optimal Harvesting: Age Dependence

Abstract

Age is an important parameter in population dynamics. This chapter concerns two regional control problems related to continuous models for age-dependent populations. The first problem is devoted to an optimal harvesting problem of a linear model when the harvesting effort acts only on the age interval \((a_1,a_2)\). The second one is an optimal harvesting problem as well, related this time to a nonlinear model in a time-periodic environment. The attention is focused on the determination of the position and length of the time intervals when the harvesting is allowed/prohibited.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Controlling Epidemics in Agriculture

Frontmatter

Chapter 5. Controlling the Spread of an Epidemic in Agriculture: The Case of Xylella fastidiosa

Abstract

The etiological agent of olive trees disease known as olive quick decline syndrome (OQDS) is the plant pathogenic bacterium Xylella fastidiosa, which is a vector borne bacterium.

The main vector of Xylella fastidiosa in Southern Italy has been identified in the so-called meadow spittlebug, i.e., the Philaenus spumarius, a xylem sap-feeding specialist. Their juvenile form (nymphs) develops on weeds or ornamental plants, confined in a foam produced by themselves for protection from predators and temperature, while their adult forms move to olive tree canopies.

Once a plant is infected, bacteria multiply within the xylem vessels inducing the production of a gel in the plant xylem, which occludes the xylem vessels, thus inhibiting the flux of water through the lymph vessels eventually blocking the nutrition of the plant. Typical symptoms are leaf scorch, dieback of twigs, branches, and even of the whole plant (Carlucci et al., Phytopathol Mediterr., 52, 541–544).

Sanitation of infected olive trees is unfeasible.

The aim of this chapter is to present results obtained by the authors of this monograph in a series of recent papers. The three steps are (i) mathematical modelling of the population dynamics of the ecosystem in the presence of the infection, (ii) possible strategies for the eventual eradication of the disease, (iii) computational experiments supporting the theoretical outcomes.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Chapter 6. The Role of Predators in Controlling the Spread of an Epidemic in Agriculture: The Case of Xylella fastidiosa

Abstract

In this chapter, the role of a predator is analyzed as a possible biocontrol agent of epidemics in agriculture. As a concrete application, the case of Zelus renardii has been considered as a predator of Philaenus spumarius, for a possible control of a Xylella epidemic, which has been the subject of the previous chapter. By taking into account previous investigations, we show that the choice of the functional response to predation is crucial. Possible choices adopted in literature are the Holling type II predator (specialist) and the Holling type III predator (generalist). The analysis presented here shows that a H. II predator, whenever identified, would indeed lead to the eventual eradication of a Xylella epidemic. Unfortunately, Z. renardii has been reported to be a generalist predator, which implies an Holling type III functional response to predation; we show that it is not an efficient control strategy to eradicate a Xylella epidemic.

All of that has been illustrated by a set of computational experiments, within a variety of different possible parameter scenarios.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Controlling Environmental Pollution in Geographical Economics

Frontmatter

Chapter 7. Economic Growth and Pollution Diffusion

Abstract

As an additional application of the methods presented in this volume, in this chapter, we introduce an environmental issue concerning the control of the pollution produced by human activities. The proposed model consists of a spatially structured dynamic economic growth model that takes into account the level of pollution induced by production and a possible taxation based on the amount of produced pollution. Then we analyze an optimal harvesting control problem with an objective function composed of three terms, namely the intertemporal utility of the decision maker, the space-time average of the level of pollution in the habitat, and the disutility due to the imposition of taxation. The system is subject to two controls, namely the level of consumption and, in addition, the taxation on physical capital rate. The optimal controls are determined by means of a sweep-type numerical algorithm that allows to solve a system of four backward–forward reaction–diffusion equations. Open problems are proposed at the end of the chapter.

Sebastian Aniţa, Vincenzo Capasso, Simone Scacchi

Backmatter

Titel: Mathematical Modeling and Control in Life and Environmental Sciences
verfasst von: Sebastian Aniţa
Vincenzo Capasso
Simone Scacchi
Verlag: Springer International Publishing
Electronic ISBN: 978-3-031-49971-5
Print ISBN: 978-3-031-49970-8
DOI: https://doi.org/10.1007/978-3-031-49971-5

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