Free terminal time optimal control of a fractional-order model for the HIV/AIDS epidemic

In this paper, we present a fractional model for the HIV/AIDS epidemic and incorporate into the model control parameters of pre-exposure prophylaxis (PrEP), behavioral change and antiretroviral therapy (ART) aimed at controlling the spread of diseases. We prove the local and global asymptotic stability of disease-free and endemic equilibria of the model. We present a general fractional optimal control problem (FOCP) with free terminal time and develop the Adapted Forward-Backward Sweep method for numerical solving of the FOCP. Necessary conditions for a state/control/terminal time triplet to be optimal are obtained. The results show that the use of all controls increases the life expectancy of HIV-treated patients with ART and remarkably increases the number of people undergoing PrEP and changing their sexual habits. Also, when the derivative order [Formula: see text] ([Formula: see text]) limits to 1, the value of optimal terminal time increases while the value of objective functional decreases.

A Resource Extraction Model with Technology Adoption under Time Inconsistent Preferences

A two-stage non-standard optimal control problem with time inconsistent preferences is studied. In an infinite horizon setting, a time consistent (sophisticated) decision maker chooses the time of switching between two consecutive regimes. The second regime corresponds to the implementation of a new technology, and a cost must be paid at the switching time. Although the problem is formulated for a general discount function, special attention is devoted to models with nonconstant discounting and heterogeneous discounting. The problem is solved by transforming it into a problem in a finite horizon and free terminal time. The corresponding dynamic programming equations are presented, and conditions for the derivation of the switching time by decision makers with different degrees of sophistication are studied. A resource extraction model with technology adoption is solved in detail. Effects of the adoption of different discount functions are illustrated numerically.

Free terminal time optimal control problem for the treatment of HIV infection

In this work, an optimal control approach is presented in order to propose an optimal therapy for the treatment HIV infection using a combination of two appropriate treatment strategies. The optimal treatment duration and the optimal medications amount are considered. The main objective of this study is to be able to maximize the benet based on number of healthy CD4+ T-cells and CTL immune cells and to minimize the infection level and the overall treatment cost while optimizing the duration of therapy. The free terminal time optimal control problem is formulated and the Pontryagin's maximum principle is employedto provide the explicit formulations of the optimal controls. The corresponding optimality system with the additional transversality condition for the terminal time is derived and solved numerically using an adapted iterative method with a Runge-Kutta fourth order scheme and a gradient method routine.