Title: Optimal control treatment through quarantine of a mathematical model for cholera
Abstract: We propose a mathematical model for cholera with treatment through quarantine. The model is shown to be both epidemiologically and mathematically well posed. In particular, we prove that all solutions of the model are positive and bounded; and that every solution with initial conditions in a certain meaningful set remains in that set for all time. The existence of unique disease-free and endemic equilibrium points is proved and the basic reproduction number is computed. Then, we study the local asymptotic stability of these equilibrium points. An optimal control problem is proposed and analyzed, whose goal is to obtain a successful treatment through quarantine. We provide the optimal quarantine strategy for the minimization of the number of infectious individuals and bacteria concentration, as well as the costs associated with the quarantine. Finally, a numerical simulation of the cholera outbreak in the Department of Artibonite (Haiti), in 2010, is carried out, illustrating the usefulness of the model and its analysis.
Title: Vaccination games and evolutionary dynamics
Abstract: For diseases in which vaccination is not compulsory, individuals take into account different aspects when deciding between to vaccinate or not, such as the probability of becoming infected and also the morbidity risks from infection and vaccination. Using the basic the reinfection SIRI epidemiological model, we analyze the impact of education programs and vaccine scares on individuals decisions between to vaccinate or not. The presence of the reinfection provokes the existence of three Nash equilibria for the same level of the morbidity relative risk instead of a single Nash equilibrium as occurs in the SIR model. The existence of three Nash equilibria, with two of them being evolutionary stable, introduces two scenarios with relevant and opposite features for the same level of the morbidity relative risk: the low-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a low probability; and the high-vaccination scenario corresponding to the evolutionary stable vaccination strategy, where individuals will vaccinate with a high probability. We introduce the evolutionary vaccination dynamics for the SIRI model and we prove that it is bistable. The bistability of the evolutionary dynamics indicates that the damage provoked by false scares on the vaccination perceived morbidity risks can be much higher and much more persistent than in the SIR model. Furthermore, the vaccination education programs to be efficient need to implement a mechanism to suddenly increase the vaccination coverage level.
This is a joint work with Alberto Pinto. The authors thank the financial support of LIAAD-INESC TEC and FCT-Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) within project “Dynamics, optimization and modelling”, with reference PTDC/MAT-NAN/6890/2014.
Title: Chaos in a delay mathematical model for AIDS-related cancer
Abstract: We study a delay mathematical model for the dynamics of AIDS-related cancer. Cancer is a major burden in HIV infected patients, and as such, is extremely important to understand the epidemiology and the mechanisms behind it. Our model consists of four classes, the cancer cells, the healthy cells, the infected cells and the virus. We show numerically the existence of periodic orbits arising from a Hopf bifurcation from an endemic state. Moreover, we observe the appearance of chaos, due to a cascade of period doubling bifurcations.
Title: Within-host and synaptic transmissions: contributions to the spread of HIV infection
Abstract: We study the contributions of within-host (virus-to-cell) and synaptic (cell-to-cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease-free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell-to-cell transmission rate on the value of the reproduction number, R0. Moreover, we show evidence of a qualitative change in the models’ dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn.
Title: A stochastic SICA epidemic model for HIV transmission
Abstract: We propose a stochastic SICA epidemic model for HIV transmission, described by stochastic ordinary differential equations, and discuss its perturbation by environmental white noise. Existence and uniqueness of the global positive solution to the stochastic HIV system is proven, and conditions under which extinction and persistence in mean hold, are given. The theoretical results are illustrated via numerical simulations.
Title: Modeling dengue disease
Abstract: In this work the experience on modeling Dengue disease is shown. Optimal control problem, direct and indirect methods, software packages are presented. Two Dengue outbreaks are presented - Cape Verde and Madeira Island.
Title: Necessary optimality conditions for average cost minimization problems
Abstract: We provide necessary optimality conditions for an optimal control problem where unknown parameters intervene in the data, and the cost function is presented as an 'expected value' or what we call 'average cost'. The result intervenes in applications such as epidemiology and aerospace engineering.
Title: Challenges in Optimal Control with Mixed Constraints
Abstract: Focusing on necessary conditions for mixed constrained optimal control problems, we discuss the role of regularity conditions and give a short overview of the state of the art.
We also show how necessary conditions for mixed constraints problems are of help to provide results for implicit control problems, problems involving differential algebraic problems and problems with order one state constraints. We then discuss new challenges for such problem emphasizing the importance of mixed constraints for open and interesting problems that can be reformulated as problems with mixed constraints not satisfying the regularity conditions, including some special cases of sweeping problems.
Title: The dynamic programming approach in optimal control
Abstract: In this talk, we briefly discuss the main features of the dynamic programming approach to optimal control problems and the Hamilton-Jacobi equation.
Title: Optimal control for normalized SEIR models
Abstract: In this work we use optimal control to establish vaccination plans to control the spread of an infectious disease, within a population, described by a normalized model. We compare the obtained results with the ones obtained when classical SEIR models are used. We partially validate our numerical solutions using the Maximum Principle.