An introduction to mathematical optimal control theory. On an optimal control strategy in a kinetic social dynamics model. In this paper an optimal control approach with a state dependent switching cost index is proposed to determine the best strategy to control via a vaccination the epidemic disease by using a sir model. However, the functions being optimized are in genera. Overview optimal control applications and methods wiley. Suzanne marie lenhart born november 19, 1954 is an american mathematician who works in partial differential equations, optimal control and mathematical biology. The aim is to encourage new developments in optimal control theory and design methodologies that may lead to advances in real control applications. Deterministic as well as stochastic approaches have been proposed by scholars and a variety of mathematical models given by nonlinear equations have been introduced, analyzed, and applied to real cases.
Pdf optimal control applied to a model for species augmentation. Optimal control applied to biological models crc press book. The 14 labs with matlab codes with userfriendly interface provided on the web offer a good resource to manipulate and learn the materials. Workman, optimal control applied to biological models. Optimal control of advective direction in reactiondiffusion population. Optimal control applications and methods will provide a forum for papers on the full range of optimal control and related control design methods. An introduction to optimal control applied to disease models. This is an excellent introductory book on optimal control applied to biological models. Applying optimal control theory to complex epidemiological. Develops the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models. Timeoptimal control strategies in sir epidemic models. Lenhart s, workman jt, optimal control applied to biological models, 1st edn. Optimal control applied to biological models thoroughly develops the mathematical aspects of. Optimal control applied to biological models thoroughly develops the mathematical aspects of optimal control theory and provides insight into the application of this theory to biological models.
Optimal control applied to biological models request pdf. Modelling optimal control of inhost hiv dynamics using. Optimality principles have been used to explain many biological processes and systems. Optimal control applied to biological models 1st edition suzanne. In particular, we are interested in the variant of maximum principle applicable to optimal control problems with free terminal time, concisely described by lenhart. Derive necessary conditions for the optimal control.
Dear colleagues, the literature on mathematical models for the spread of epidemics is now huge and has been developed over many decades. The wide range in projected caseload generated attention as a source of uncertainty, but debate did not address whether uncertainty affected choice of action. The goal is to characterize the optimal control in terms of the solution of the opti mality system. With the advances of computer technique, optimal control is now widely used in multidisciplinary applications such as biological systems, communi.
Mathematical model, optimal control and transmission dynamics of avian. In particular, we establish the optimal strategies for controlling the infection using three. University of tennessee, knoxville, tennessee, usa. An optimal strategy to solve the prisoners dilemma. The 2014 ebola outbreak illustrates the complexities of decision making in the face of explosive epidemics.
In this dissertation, we investigate optimal control of hybrid differential equations and elliptic partial differential equations with two biological applications. It introduces the optimal control of discrete systems and of partial differential equations pdes. Optimality and identification of dynamic models in systems. In this paper, we present a general formulation for a fractional optimal control problem focp, in which the state and costate equations are given in terms of the left fractional derivatives. Buy optimal control applied to biological models chapman. We prove the existence of an optimal control for which the objective functional is maximized. We outline the steps in solving such optimal control problems and discuss the.
It is further shown that the model with constant contact rates can also exhibit this phenomenon if the wnvinduced mortality in the avian population is high enough. This book examines the basic problem for continuous time ordinary differential equations odes. Optimal control applied to biological models taylor. Optimal control applied to biological models by suzanne lenhart. Therefore, it is possible to change control strategy whenever new different conditions in the epidemic spread are present. Introduction to optimal control for discrete time models with an. Lenhart introduce the theory of optimal control applied to systems of ordinary differential equations with an application on seir model while in 33 s. From economics and business to the biological sciences to physics and engineering, professionals successfully use the powerful mathematical tool of optimal control to make management and strategy decisions. Introduction to optimal control for discrete time models. Cooperation is so important that it has been suggested as a fundamental principle of evolution, besides reproduction, mutation and selection 1,2,3,4,5,6,7,8,9,10,11,12. As a guided tour to methods in optimal control and related computational methods for ode and pde models, an introduction to optimal control problems in life sciences and economics serves as an excellent textbook for graduate and advanced undergraduate courses in mathematics, physics, engineering, computer science, biology, biotechnology, and. Optimal control of a fractionalorder model for the hivaids.
The problem comprises an objective or cost functional, which is a function of the state and control variables, and a set of constraints. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile. More formally, an optimal control problem means endogenously controlling a parameter in a mathematical model to produce an optimal output, using some optimization technique. May 20, 2019 use oct in an approximate model to find optimal levels of chlamydia screening and contact tracing which are then mapped onto a network simulation. May 20, 2019 we highlight two frameworks for bridging the gap between detailed epidemic simulations and optimal control theory. Both these frameworks approximate a simulation model with a simpler model more amenable to mathematical analysis. In this paper, we develop a differential equations model and optimal control formulation for a.
In this paper, an in vivo deterministic model of differential equations is presented and analyzed for hiv dynamics. Optimal control and sensitivity analysis of an influenza model with treatment and vaccination. Optimal control of a fractionalorder model for the hiv. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view.
Optimal control applied to biological models thoroughly develops the mathematical aspects of optimal control theory and provides insight into the application of. Requiring only basic knowledge of multivariable calculus, simple odes, and mathematical models, this text shows how to adjust controls in biological. Two biological applications of optimal control to hybrid. Download it once and read it on your kindle device, pc, phones or tablets.
May 07, 2007 from economics and business to the biological sciences to physics and engineering, professionals successfully use the powerful mathematical tool of optimal control to make management and strategy decisions. Optimal control applied to biological models chapman. Pontragin ls, gamkrelize rv, boltyanskii vg, mishchenko ef, the mathematical theory of optimal processes, crc press, 1987. Solving optimal control problems with matlab indirect methods xuezhong wang 1 introduction the theory of optimal control has been well developed for over forty years. Solving optimal control problems with matlab indirect methods.
Hiv is one of the major causes of deaths, especially in subsaharan africa. It covers a variety of topics including optimal control on odes, pdes and discrete system. Optimal control model of biological movement provides a natural starting point for describing. Evans department of mathematics university of california, berkeley. An introduction to optimal control problems in life sciences. Optimal control theory is applied to investigate the key roles played by the various hiv treatment strategies. We study optimal control problem for an sir model with vaccination and treatment as controls. Optimal control applied to biological models lenhart. An introduction to mathematical optimal control theory version 0. Workman jt 2007 optimal control applied to biological models.
She is a chancellors professor of mathematics at the university of tennessee, an associate director for education and outreach at the national institute for mathematical and biological synthesis, and a parttime researcher at the oak. The model is extended to assess the impact of some antiwnv control measures, by reformulating the model as an optimal control problem with densitydependent demographic parameters. The model hierarchy is shown, with optimized controls from the approximate model directly lifted to the simulation model. Optimal control of sir epidemic model with state dependent. National institute for mathematical and biological synthesis and itr04274710. Optimal control applied to biological models thoroughly develops the mathematical aspects of optimal control theory and provides insight into t. In this paper, an sir epidemic model with variable size of population is considered. Professor emanuel todorov, chair professor robert e. Optimal control applied to biological models suzanne. Jul 21, 2014 modeling optimal control of rotavirus disease with different control strategies. This is a free offprint provided to the author by the publisher. Our analyses led to the nontrivial result that, even in the unconstrained optimal control problem. Optimal control applied to biological models 1st edition. Specifically, we characterized analytically the time optimal control strategies for prophylactic vaccination, isolation, nonselective culling, and reduction of transmission by using a family of simple sir models in an optimal control framework.
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