Mathematical Modeling of HIV-T Co-Infection With Optimal Control: A Caputo Fractional Approach

Abstract

Human immunodeficiency virus (HIV) and tuberculosis (TB) remain among the most devastating infectious diseases worldwide, particularly in sub-Saharan Africa. The interaction between these diseases through co-infection produce significant challenges for effective disease management. HIV compromises the immune system which in creases risk of developing active TB, while TB exacerbates the progression of HIV to AIDS. Understanding the dynamics of these diseases and evaluating the impact of in tervention strategies is therefore essential for effective control. In this dissertation, we develop and analyze a series of Caputo fractional order mathematical models to study the transmission dynamics of HIV, TB, and their co-infections. First, an HIV/AIDS model incorporating PrEP and drug-resistant strain is proposed. Analysis shows that reducing transmission rate, PrEP discontinuation rate, and drug resistance develop ment rate, while increasing PrEP uptake and detection rates of symptomatic cases, significantly decreases HIV prevalence. Next, a TB model is formulated to investigate the impact of vaccination and treatment interventions. The finding indicates that in creased vaccination and treatment rates, coupled with reduced effective contact, substan tially decrease TB prevalence. Extending this model into a Caputo fractional optimal control framework, we proposed and analyzed seven different intervention strategies: prevention alone, vaccination alone, treatment alone, prevention combined with vac cination, prevention combined with treatment, vaccination combined with treatment, and all three combined together, among which vaccination alone emerges as the most cost-effective intervention strategy. Finally, two HIV-TB co-infection models are de veloped, with parameters estimated using real-data obtained from Ethiopia. An optimal control analysis demonstrates that simultaneous HIV and TB treatment strategy yields the most effective reduction in disease burden at minimal cost. Numerical simulations across models support the analytical findings, and stability analyses confirm that the disease-free equilibrium is locally asymptotically stable when all the eigenvalues of the linearized matrix satisfy the Matignon’s fractional stability criterion and globally stable whenever the constructed Lyapunov functions monotonically decay to zero. Overall, the dissertation illustrates the importance of Caputo fractional-order models and frac tional optimal control theory in understanding the complex dynamics of HIV, TB, and their co-infection and provides valuable insights for designing cost-effective public health strategies with long-term policy effects in Ethiopia and similar burden regions.