We propose a population model for TB-HIV/AIDS coinfection transmission dynamics, which considers antiretroviral therapy for HIV infection and treatments for latent and active tuberculosis. The HIV-only and TB-only sub-models are analyzed separately, as well as the TB-HIV/AIDS full model. The respective basic reproduction numbers are computed, equilibria and stability are studied. Optimal control theory is applied to the TB-HIV/AIDS model and optimal treatment strategies for co-infected individuals with HIV and TB are derived. Numerical simulations to the optimal control problem show that non intuitive measures can lead to the reduction of the number of individuals with active TB and AIDS.

Consider the minimal time control problem for a single-input control-affine system ẋ = X(x) + u₁Y₁(x) in Rⁿ, where the scalar control u₁(·) satisfies the constraint lu₁(·)l ≤. When applying a shooting method for solving this kind of optimal control problem, one may encounter numerical problems due to the fact that the shooting function is not smooth whenever the control is bang-bang. In this article we propose the following smoothing procedure. For ε > 0 small, we consider the minimal time problem for the control system ẋ = X(x) + u₁^εY₁(x) + εΣ_i=2^m u_i^εY_i(x), where the scalar controls u_i^ε(·), i = 1,... , m, with m ≥ 2, satisfy the constraint Σ_i=1^m u_i^ε(u_i^ε(t))² ≤ 1. We prove, under appropriate assumptions, a strong convergence result of the solution of the regularized problem to the solution of the initial problem.

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to “normal life” and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.