Searcharxiv⌕ Search

arXiv subjects

Cristiana J. Silva

Publications and source records attributed to Cristiana J. Silva.

46 records · Page 3Linked to original sources
arXiv2014

Multiobjective approach to optimal control for a tuberculosis model

Mathematical modelling can help to explain the nature and dynamics of infection transmissions, as well as support a policy for implementing those strategies that are most likely to bring public health and economic benefits. The paper addresses the application of optimal control strategies in a tuberculosis model. The model consists of a system of ordinary differential equations, which considers reinfection and post-exposure interventions. We propose a multiobjective optimization approach to find optimal control strategies for the minimization of active infectious and persistent latent individuals, as well as the cost associated to the implementation of the control strategies. Optimal control strategies are investigated for different values of the model parameters. The obtained numerical results cover a whole range of the optimal control strategies, providing valuable information about the tuberculosis dynamics and showing the usefulness of the proposed approach.

math.OC↗
arXiv2014

Cost effectiveness analysis of optimal control measures for tuberculosis

We propose and analyse an optimal control problem where the control system is a mathematical model for tuberculosis that considers reinfection. The control functions represent the fraction of early latent and persistent latent individuals that are treated. Our aim is to study how these control measures should be implemented, for a certain time period, in order to reduce the number of active infected individuals, while minimizing the interventions implementation costs. The optimal intervention is compared along different epidemiological scenarios, by varying the transmission coefficient. The impact of variation of the risk of reinfection, as a result of acquired immunity to a previous infection for treated individuals on the optimal controls and associated solutions, is analysed. A cost-effectiveness analysis is done, to compare the application of each one of the control measures, separately or in combination.

math.OC↗
arXiv2014

Optimal Control of Tuberculosis: A Review

We review the optimal control of systems modeling the dynamics of tuberculosis. Time dependent control functions are introduced in the mathematical models, representing strategies for the improvement of the treatment and cure of active infectious and/or latent individuals. Optimal control theory allows then to find the optimal way to implement the strategies, minimizing the number of infectious and/or latent individuals and keeping the cost of implementation as low as possible. An optimal control problem is proposed and solved, illustrating the procedure. Simulations show an effective reduction in the number of infectious individuals.

math.OC↗
arXiv2014

Modeling TB-HIV syndemic and treatment

Tuberculosis (TB) and human immunodeficiency virus (HIV) can be considered a deadly human syndemic. In this article, we formulate a model for TB and HIV transmission dynamics. The model considers both TB and acquired immune deficiency syndrome (AIDS) treatment for individuals with only one of the infectious diseases or both. The basic reproduction number and equilibrium points are determined and stability is analyzed. Through simulations, we show that TB treatment for individuals with only TB infection reduces the number of individuals that become co-infected with TB and HIV/AIDS, and reduces the diseases (TB and AIDS) induced deaths. Analogously, the treatment of individuals with only AIDS also reduces the number of co-infected individuals. Further, TB-treatment for co-infected individuals in the active and latent stage of TB disease, implies a decrease of the number of individuals that passes from HIV-positive to AIDS.

math.DS↗
arXiv2013

An optimal control approach to malaria prevention via insecticide-treated nets

Malaria is a life threatening disease, entirely preventable and treatable, provided the currently recommended interventions are properly implemented. These interventions include vector control through the use of insecticide-treated nets (ITNs). However, ITN possession does not necessarily translate into use. Human behavior change interventions, including information, education, communication (IEC) campaigns and post-distribution hang-up campaigns are strongly recommended. In this paper we consider a recent mathematical model for the effects of ITNs on the transmission dynamics of malaria infection, which takes into account the human behavior. We introduce in this model a supervision control, representing IEC campaigns for improving the ITN usage. We propose and solve an optimal control problem where the aim is to minimize the number of infectious humans while keeping the cost low. Numerical results are provided, which show the effectiveness of the optimal control interventions.

math.OC↗
arXiv2013

Optimal control for a tuberculosis model with reinfection and post-exposure interventions

We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions, considering reinfection and post-exposure interventions. They depend on the parameters of the model and reduce effectively the number of active infectious and persistent latent individuals. The time that the optimal controls are at the upper bound increase with the transmission coefficient. A general explicit expression for the basic reproduction number is obtained and its sensitivity with respect to the model parameters is discussed. Numerical results show the usefulness of the optimization strategies.

math.OC↗
arXiv2009

The optimal control and its multiple applications

In this work we refer to motivations, applications, and relations of control theory with other areas of mathematics. We present a brief historical review of optimal control theory, from its roots in the calculus of variations and the classical theory of control to the present time, giving particular emphasis to the Pontryagin maximum principle.

math.OC↗
arXiv2006

Absolute Extrema of Invariant Optimal Control Problems

Optimal control problems are usually addressed with the help of the famous Pontryagin Maximum Principle (PMP) which gives a generalization of the classical Euler-Lagrange and Weierstrass necessary optimality conditions of the calculus of variations. Success in applying the PMP permits to obtain candidates for a local minimum. In 1967 a direct method, which permits to obtain global minimizers directly, without using necessary conditions, was introduced by Leitmann. Leitmann's approach is connected, as showed by Carlson in 2002, with "Caratheodory's royal road of the Calculus of variations". Here we propose a related but different direct approach to problems of the calculus of variations and optimal control, which permit to obtain global minima directly, without recourse to needle variations and necessary conditions. Our method is inspired by the classical Noether's theorem and its recent extensions to optimal control. We make use of the variational symmetries of the problem, considering parameter-invariance transformations and substituting the original problem by a parameter-family of optimal control problems. Parameters are then fixed in order to make the problem trivial, in some sense. Finally, by applying the inverse of the chosen invariance-transformation, we get the global minimizer for the original problem. The proposed method is illustrated, by solving concrete problems, and compared with Leitmann's approach.

math.OC↗
arXiv2006

Two-dimensional Newton's Problem of Minimal Resistance

Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and ask at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two-dimensional problem is more rich than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem doesn't admit a local minimum), we prove that in dimension two the unrestricted problem is also well-posed when the ratio height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.

math.OC↗