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.

Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper, we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals, in which only susceptible individuals can travel freely between the patches. The model has multiple equilibria. We determine conditions that lead to the appearance of a backward bifurcation. The results show that the TB model can have exogenous reinfection among the treated individuals and, at the same time, does not exhibit backward bifurcation. Also, conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained. In case without reinfection, the model has four equilibria. In this case, the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations (FDEs). Numerical simulations confirm the validity of the theoretical results.