OPTIMAL CONTROL APPLIED TO RIFT VALLEY FEVER

Abstract Rift Valley Fever (RVF) virus is a mosquito‐born pathogen that infects livestock but it also has the capability to infect humans through direct or indirect contact with blood or organs of infected animals and by bites from infected mosquitos. The economic and social cost of the disease to rural populations can lead to a cascade of negative effects on the sustainability of animal and human populations. Vaccines exist to protect against this disease. Through a compartment model depicting the interactions leading to the spread of RVF in Aedes and Culex mosquitos and a livestock population, an optimal control problem is developed to minimize the number of vaccinated livestock at the final time while minimizing the negative effects of the infected Aedes and Culex mosquitos and the cost of the vaccination process. The unique optimal vaccination strategy is produced for given high transmission parameters and numerical results portray that vaccination depends on the level of effectiveness of the protocol.     

“Magic” Dispersion Maps with Distributed Raman Amplification

We analyze pulse propagation in an optical fiber with a periodic dispersion map and distributed amplification. Using an asymptotic theory and a momentum method, we identify a family of dispersion management schemes that are advantageous for massive multichannel soliton transmission. For the case of two-step dispersion maps with distributed Raman amplification to compensate for the fiber loss, we find special schemes that have optimal (chirp-free) launch point locations that are independent of the fiber dispersion. Despite the variation of dispersion with wavelength due to the fiber dispersion slope, the transmission in several different channels can be optimized simultaneously using the same optimal launch point. The theoretical predictions are verified by direct numerical simulations. The obtained results are applied to a practical multichannel transmission system.     

Diffraction in periodic structures and optimal design of binary gratings. Part I: direct problems and gradient formulas

The aim of the paper is to provide the mathematical foundation of effective numerical algorithms for the optimal design of periodic binary gratings. Special attention is paid to reliable methods for the computation of diffraction efficiencies and of the gradients of certain functionals with respect to the parameters of the non-smooth grating profile. The methods are based on a generalized finite element discretization of strongly elliptic variational formulations of quasi-periodic transmission problems for the Helmholtz equation in a bounded domain coupled with boundary integral representations in the exterior. We prove uniqueness and existence results for quite general situations and analyse the convergence of the numerical solutions. Furthermore, explicit formulas for the partial derivatives of the reflection and transmission coefficients with respect to the parameters of a binary grating profile are derived. Finally, we briefly discuss the implementation of the generalized finite element method for solving direct and adjoint diffraction problems and present some numerical results. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.     

Optimal control of the solidification process in metal casting

The optimal control of the solidification process in metal casting is considered. The mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem is given. The problem is solved numerically by direct optimization methods. The numerical results are described and analyzed. Some of the results are illustrated by plots.     

A dynamical model of echinococcosis with optimal control and cost-effectiveness

A dynamical model of echinococcosis transmission with optimal control strategies is first presented. The basic reproduction number of the model is determined and employed to study the global stability of the disease-free and endemic equilibrium points. The optimal control problem is formulated and solved analytically. Numerical simulations show that optimal control strategies could effectively reduce the transmission of echinococcosis. The cost-effectiveness analysis suggests that a combination of health education, anthelmintic treatment, and home slaughter inspection could provide the best cost-effective strategy to control the transmission of echinococcosis. Furthermore, it finds that anthelmintic treatment and environmental disinfection may shorten the time of eliminating the disease. The results may be helpful for prevention and control of echinococcosis in Ganzi Tibetan Autonomous Prefecture, China and other areas of echinococcosis.     

Domain Decomposition,Optimal Control of Systems Governed by Partial Differential Equations,and Synthesis of Feedback Laws

We present an iterative domain decomposition method for the optimal control of systems governed by linear partial differential equations. The equations can be of elliptic, parabolic, or hyperbolic type. The space region supporting the partial differential equations is decomposed and the original global optimal control problem is reduced to a sequence of similar local optimal control problems set on the subdomains. The local problems communicate through transmission conditions, which take the form of carefully chosen boundary conditions on the interfaces between the subdomains. This domain decomposition method can be combined with any suitable numerical procedure to solve the local optimal control problems. We remark that it offers a good potential for using feedback laws (synthesis) in the case of time-dependent partial differential equations. A test problem for the wave equation is solved using this combination of synthesis and domain decomposition methods. Numerical results are presented and discussed. Details on discretization and implementation can be found in Ref. 1.     

Direct and indirect methods for trajectory optimization

This paper gives a brief list of commonly used direct and indirect efficient methods for the numerical solution of optimal control problems. To improve the low accuracy of the direct methods and to increase the convergence areas of the indirect methods we suggest a hybrid approach. For this a special direct collocation method is presented. In a hybrid approach this direct method can be used in combination with multiple shooting. Numerical examples illustrate the direct method and the hybrid approach.     

封闭空间中新型冠状病毒肺炎传播模型:以日本“钻石公主号”邮轮为例

该文以新型冠状病毒(SARS-Cov-2)在日本钻石公主号邮轮上传播为例,通过建立简单的易感者-感染者传染病模型,研究在封闭空间中新冠病毒肺炎(COVID-19)的传播机制.动力学分析和数值拟合预测了疾病传播过程和最终结果,讨论了不同隔离措施对疾病传播进程的影响,并给出防控策略建议.     

Direct Numerical Algorithm for Constrained Variational Problems

We propose a direct treatment for the numerical simulation of optimal solutions for vector, one-dimensional variational problems under pointwise constraints in the form of several inequalities. It is an iterative procedure to approximate the optimal solutions of such variational problems that rely on our ability to e?ciently approximate the optimal solutions of variational problems without restrictions, except possibly for end point constraints. One main advantage is that there is no need to control the free boundary, or the contact set, during the iterative process where constraints are active. In addition to proving some convergence results, the scheme is illustrated through several typical situations.     

Computational Control of an HIV Model

We consider a computational approach to solving an optimal control formulation of optimal drug scheduling in HIV infected individuals. The optimal control problem is transformed using the control parameterisation enhancing technique (CPET), which enables efficient computation of an optimal control using a relatively coarse discretisation. A number of numerical difficulties with the model are discussed, and for illustration, numerical examples are solved.     

一类具有病毒变异的Logistic死亡率SEIR传染病模型研究分析

本文建立了一类具有病毒变异的Logistic死亡率SEIR传染病模型,借助Lyapunov函数和LaSalle''s不变原理,证明了无病平衡点全局稳定性.利用代数方法构造Lyapunov函数,证明了地方病平衡点全局稳定性.另外,通过数值模拟分析了参数对疾病传播的影响.     

Study of the Optimal Integrated Control of a Dengue Transmission Model北大核心CSCD

建立了一个登革热在蚊子和人之间传播的模型,引入了Wolbachia、自我保护和杀虫剂三种控制措施,分别从常数控制和时变控制两个方面进行探讨。首先,分析了常数控制对模型基本再生数的影响,研究发现:Wolbachia有助于减小基本再生数,且基本再生数与自我保护和杀虫剂呈负相关。其次,以使得感染数最少且实施成本最低为目标,使用Pontryagin极值原理讨论最优控制。最后,通过数值模拟展示了最优控制的效果。     

Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner.The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction-diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.     

The effect of time delays on transmission dynamics of schistosomiasis

A 6-dimension dynamical schistosomiasis model incorporating five time delays is established in this paper. Two equilibrium points: a disease free equilibrium and an endemic equilibrium, are calculated respectively. The stability behaviors at the disease free equilibrium are analysed. Both analytical and numerical results are presented that prepatent periods in infection can affect the schistosomiasis transmission significantly. Thus, two effective measures on schistosomiasis prevention and control are obtained: lengthening the prepatent period in susceptible snails, and prolonging the incubation periods in miracidia and cercaria by temperature control or drug restraint. And then, numerical simulations are given to illustrate the validity and effectiveness of the model. At last a discussion is provided about our results and further work.     

Direct Shooting Method for the Numerical Solution of Higher-Index DAE Optimal Control Problems

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.     

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.     

Optimal control problem for a general reaction-diffusion eco-epidemiological model with disease in prey

This paper deals with an optimal control problem for a general reaction-diffusion predator-prey model with disease in prey population. Infected prey will recover from a medication considered as a control strategy. Our primary goal is to characterize an optimal control which minimizes the total density of infected prey and the costs of treatment. Firstly, we obtain the existence and some estimates of the unique strong solution for the controlled system by applying semigroup theory. Subsequently, the existence of optimal pair is proved by means of the technique of minimizing sequence. Furthermore, by proving the differentiability of the control-to-state mapping, we derive the first-order necessary optimality condition, and point out that the optimal is a Bang-Bang control in a special case. Finally, several numerical simulations are performed to illustrate the concrete realization and practical application of the theoretical results obtained in this contribution.