Mathematical modeling of the HIV/Kaposi's sarcoma coinfection dynamics in areas of high HIV prevalence

Optimal Control and Bifurcation Analysis of HIV Model

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R ₀ is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R ₀ ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R ₀ > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R ₀ = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

A novel mathematical model of AIDS-associated Kaposi's sarcoma: Analysis and optimal control

Kaposi's sarcoma (KS) has been the most common HHV-8 virus-induced neoplasm associated with HIV-1 infection. Although the standard KS therapy has not changed in 20 years, not all cases of KS will respond to the same therapy. The goal of current AIDS-KS treatment modalities is to reconstitute the immune system and suppress HIV-1 replication, but newer treatment modalities are on horizon. There are very few mathematical models that have included HIV-1 viral load (VL) measures, despite VL being a key determinant of treatment outcome. Here we introduce a mathematical model that consolidates the effect of both HIV-1 and HHV-8 VL on KS tumor progression by incorporating low or high VLs into the proliferation terms of the immune cell populations. Regulation of HIV-1/HHV-8 VL and viral reservoir cells is crucial for restoring a patient to an asymptomatic stage. Therefore, an optimal control strategy given by a combined antiretroviral therapy (cART) is derived. The results indicate that the drug treatment strategies are capable of removing the viral reservoirs faster and consequently, the HIV-1 and KS tumor burden is reduced. The predictions of the mathematical model have the potential to offer more effective therapeutic interventions based on viral and virus-infected cell load and support new studies addressing the superiority of VL over CD4⁺ T-cell count in HIV-1 pathogenesis.