Global stability for a 2n+1 dimensional HIV/AIDS epidemic model with treatments

In this work, we derive and analyze a 2n+1-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R_{0, n}, R_{t, n} and R_t,n as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R_{0, n}, R_{t, n} and R_t,n. We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if R_t,n=1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if R_{t, n} < 1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing R_t,n>1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.