Analyzing HIV/AIDS dynamics with a novel Caputo-Fabrizio fractional order model and optimal control measures

In this manuscript, we present a novel mathematical model for understanding the dynamics of HIV/AIDS and analyzing optimal control strategies. To capture the disease dynamics, we propose a new Caputo-Fabrizio fractional-order mathematical model denoted as SEIEUPIATR, where the exposed class is subdivided into two categories: exposed-identified EI and exposed-unidentified EU individuals. Exposed-identified individuals become aware of the disease within three days, while exposed-unidentified individuals remain unaware for more than three days. Simultaneously, we introduce a treatment compartment with post-exposure prophylaxis (PEP), represented as P, designed for individuals of the exposed identified class. These individuals initiate treatment upon identification and continue for 28 days, resulting in full recovery from HIV. Additionally, we categorize infectious individuals into two groups: under-treatment individuals, denoted as T, and those with fully developed AIDS not receiving antiretroviral therapy (ART) treatment, denoted as A. We establish that the proposed model has a unique, bounded, and positive solution, along with other fundamental characteristics. Disease-free and endemic equilibrium points and their associated properties (such as the reproduction number R 0 and stability analysis) are determined. Sensitivity analysis is performed to assess the impact of parameters on R 0 and hence on the disease dynamics. Finally, we formulate a fractional optimal control problem to examine strategies for minimizing HIV/AIDS infection while keeping costs at a minimum. We adopt the use of condoms and changes in sexual habits as optimal controls. The numerical results are presented and discussed through graphs.