Optimal‐Control Techniques for Managing Dengue Outbreaks: An Advanced Mathematical Modeling

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols. In this paper we consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications (see e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro‐differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin′s maximum principle, e.g. (Gabasov and Kirilowa, 1971). The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang‐bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems (Duda 1994; 1997).

Systematic application of Pontryagin's maximum principle to a general class of nonlinear mixed boundary value control problems yields necessary conditions for optimal control. Also, the features of certain pathological situations including the abnormal case and singular control are clearly revealed in this treatment.

This paper presents a mathematical formalism with the Liouville–Caputo fractional derivative (LCFD) to investigate the transmission dynamics of Human papillomavirus (HPV) infection and related cervical cancer. The proposed fractional-order model (FOM) is designed through six fractional differential equations (FDEs), each equation capturing different compartments of individuals as well as the most important epidemiological features of HPV infection and related cervical cancer. We illustrate the solutions existence, positivity, uniqueness and boundedness to verify the well-posedness of this FOM. The basic reproduction number ([Formula: see text]) is evaluated to show whether the infection is growing in a population or not. Moreover, we demonstrate the stability analysis for disease-free equilibrium and endemic equilibrium points (EPs). A sensitivity analysis is conducted to identify key epidemiological parameters. The fractional optimal control problem (FOCP) is formulated using an objective function that incorporates three time-dependent control variables, each representing realistic interventions, namely [Formula: see text] represents measures promoting safe sexual practices, [Formula: see text] represents HPV prevention strategies, including vaccination and immunity enhancement, and [Formula: see text] represents cervical cancer prevention methods, such as regular screenings and medications to inhibit cancerous cell development. The necessary optimality conditions (NOCs) for this FOCP are derived using Pontryagins Maximum Principle (PMP). Four different control strategies resulting from a combination of the proposed controls are presented. The numerical simulations presented reveal that each strategy can reduce HPV infection and cervical cancer to some extent, however, the combination of the three controls gives the best scenario for controlling HPV infection and cervical cancer. It was also shown that the fractional operator plays a pivotal role in controlling the disease in addition to suggested controls.

Dengue virus is transmitted by Aedes mosquitoes, posing threat to people's health and leading to great economic cost in many tropical and subtropical regions. We develop an ordinary differential equation model taking into account multiple strains of dengue virus. Using the model, we assess the effectiveness of human vaccination considering its waning and failure. We derive the lower bound and upper bound for the final size of the epidemic. Sensitivity analysis quantifies the impact of parameters on the basic reproduction number. Different scenarios of vaccinating humans show that it is better to vaccinate humans at early stages. We find that the cumulative number of infected humans is small when the vaccination rate is high or the waning rate is low for previously infected humans. We analyze the necessary conditions for implementing optimal control and derive the corresponding optimal solutions for mitigation dengue virus transmission by applying Pontryagin's Maximum Principle. Our findings may provide guidance for the public health authorities to implement human vaccination and other mitigation strategies.

SummaryLimited bandwidth and limited saturation in actuators are practical concerns in control systems. Mathematically, these limitations manifest as constraints imposed on the control actions, their rates of change, and, more generally, the global behavior of their paths. While the problem of actuator saturation has been studied extensively, little attention has been devoted to the problem of actuators having limited bandwidth. While attempts have been made in the direction of incorporating frequency constraints on state‐action trajectories before, rate constraints on the control at the design stage have not been studied extensively in the discrete‐time regime. This article contributes toward filling this lacuna: We establish a new discrete‐time Pontryagin maximum principle with rate constraints imposed on the control trajectories and derive first‐order necessary conditions for optimality. A brief discussion on the existence of optimal control is also included.

Optimal control strategies and cost-effectiveness analysis of a malaria model

We study a model for the exploitation of renewable stocks developed in [5]. In this particular control problem, the control law contains a measurable and an impulsive control component. We formulate Pontryagin's maximum principle for this kind of control problems (see [3]), proving first order necessary conditions of optimality. Manipulating the correspondent Lagrange multipliers we manage to define two special switch functions, that allow the complete description the optimal trajectories and control policies, for all possible initial conditions in the phase plane.

This thesis aims at developing efficient mathematical methods for solving hierarchical dynamic optimization problems. The main motivation is to model processes in nature, for which there is evidence to assume that they run optimally. We describe models of such processes by optimal control problems (called optimal control models (OCMs)). However, an OCM typically includes unknown parameters that cannot be derived entirely on a theoretical basis, which is in particular the case for the cost function. Therefore, we develop parameter estimation techniques to estimate the unknowns in an OCM from observation data of the process. Mathematically, this leads to a hierarchical dynamic optimization problem with a parameter estimation problem on the upper level and an optimal control problem on the lower level. We focus on multi-stage equality and inequality constrained optimal control problems based on nonlinear ordinary differential equations. The main goal of this thesis is to derive numerically efficient mathematical methods for solving hierarchical dynamic optimization problems, and to use these methods to estimate parameters in high-dimensional OCMs from real-world measurement data. We develop parameter-dependent OCMs for the gait of cerebral palsy patients and able-bodied subjects. The unknown parameters in the OCMs are then estimated from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg by using the mathematical methods developed within this work. The main novelties and contributions of this thesis to the field of hierarchical dynamic optimization are summarized herein. - We establish a novel mathematical method, a so-called direct all-at-once approach, for solving hierarchical dynamic optimization problems based on the direct multiple shooting method and first-order optimality conditions. - Furthermore, we propose an efficient numerical algorithm for large-scale hierarchical dynamic optimization problems, which fully exploits the structures inherited from both the hierarchical setting and the discretization. - Pontryagin's maximum principle is used to analyze solution properties of hierarchical dynamic optimization problems like second-order optimality conditions of the lower-level problem. - In addition, we propose and discuss alternative methods for hierarchical dynamic optimization that are based on derivative-free optimization and a bundle approach. These methods keep the hierarchical problem setting and do not reformulate the lower-level problem using first-order optimality conditions. - We establish a novel lifting method for regularizing mathematical programs with complementarity constraints, which is discussed and numerically investigated by means of a well-known collection of benchmark problems. - Proofs of regularity and convergence results for sequential quadratic programming methods applied to lifted mathematical programs with complementarity constraints are provided. - Efficient state-of-the-art implementations of all mathematical methods derived in this thesis, as well as a benchmark collection of hierarchical dynamic optimization problems are presented. - High-dimensional optimal control gait models for cerebral palsy patients and able-bodied subjects are developed. The mathematical methods derived in this thesis are used to estimate the unknown model parameters from real-world motion capture data provided by the Heidelberg MotionLab of the Orthopedic University Clinic Heidelberg. The theoretical and practical results presented in this thesis can be considered an initial motivating step towards answering open questions in current medical research in fields like treatment planning, classification of gaits, or the evaluation of surgeries by means of hierarchical dynamic optimization.

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

In this paper a class of non-linear hereditary control systems is considered and necessary conditions for optimality of the initial data {u(t),φ(t)/t∊[∝,t0]} and the control {u(t)/t∝[t0,t1]} are derived. The results are direct extensions of Pontryagin's Maximum Principle. An example demonstrates the effect of the optimal selection of the initial data.

A co-infection model between HIV and COVID-19 that takes into account COVID-19 vaccination and public awareness is discussed in this article. Rigorous analysis of the model is conducted to establish the existence and local stability conditions of the single-infection models. We discover that when the corresponding reproduction number for COVID-19 and HIV exceeds one, the disease continues to exist in both single-infection models. Furthermore, HIV will always be eradicated if its reproduction number is less than one. Nevertheless, this does not apply to the single-infection COVID-19 model. Even when the fundamental reproduction number is less than one, an endemic equilibrium point may exist due to the potential for a backward bifurcation phenomenon. Consequently, in the single-infection COVID-19 model, bistability between the endemic and disease-free equilibrium may arise when the basic reproduction number is less than one. From the co-infection model, we find that the reproduction number of the co-infection model is the maximum value between the reproduction number of HIV and COVID-19. Our numerical continuation experiments on the co-infection model reveal a threshold indicating that both HIV and COVID-19 may coexist within the population. The disease-free equilibrium for both HIV and COVID-19 is stable only if the reproduction numbers are less than one. Additionally, our two-parameter continuation analysis of the bifurcation diagram shows that the condition where both reproduction numbers equal one serves as an organizing center for the dynamic behavior of the co-infection model. An extended version of our model incorporates four different interventions: face mask usage, vaccination, and public awareness for COVID-19, as well as condom use for HIV, formulated as an optimal control problem. The Pontryagin's Maximum Principle is employed to characterize the optimal control problem, which is solved using a forward-backward iterative method. Numerical investigations of the optimal control model highlight the critical role of a well-designed combination of interventions to achieve optimal reductions in the spread of both HIV and COVID-19.

In this paper, we study optimal control problems whose behavior is described by second-order differential equations with nonlocal Bitsadze–Samarski boundary conditions. Necessary conditions of optimality are obtained in terms of the maximum principle; adjoint equations are constructed in the differential and integral form. Necessary and sufficient optimality conditions are obtained for a linear problem, a difference scheme is constructed and examined, and a numerical algorithm is proposed.

First order necessary conditions for optimality are the Pontryagin Maximal Principle (PMP) or the Euler-Lagrange and Legendre-Clebsch conditions. In many problems of practical importance, particularly those when the control enters linearly, these conditions are inconclusive for determining the optimal control. In 1964 Kelley [1] discovered a generalization of the Legendre-Clebsch condition and this was extended by Robbins [2,3], Tait [4], Goh [5,6], Kopp and Moyer [7], and Kelley, Kopp and Moyer [8]. An excellent survey of these papers and related results is Gabasov and Kirillova [9].

In this paper, the Linear Quadratic Regulator Problem with a positivity constraint on the admissible control set is addressed. Necessary and sufficient conditions for optimality are presented in terms of inner products, projections on closed convex sets, Pontryagin's maximum principle and dynamic programming. The main results are concerned with smoothness of the optimal control and the value function. The maximum principle will be extended to the infinite horizon case. Based on these analytical methods, we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem. The numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other.