DELAY DIFFERENTIAL EQUATION MODEL OF HIV-1 IN VIVO DYNAMICS IN THE PRESENCE OF ARV TREATMENT

Abstract

ABSTRACT Many mathematical models have been developed to describe the immunological response to infection with human immunodeficiency virus (HIV-1). The models have been used to predict the evolutions of HIV-1 in vivo and in- vitro dynamics. This study looked into an HIV-1 in-vivo dynamics in the presences of antiretroviral (ARVs) using delay differential equations. The delay is used to account for latent period of time that elapsed between exposure of a host cell to HIV-1 and the production of infectious virus from this host cell. This is the time needed to enable HIV-1 to reproduce within the host cell in sufficient number to become infectious. The model has four variables: healthy CD4+T-cells (T), infected CD4+T-cells (T*), infectious virus (virus not affected by treatment with ARV) (VI) and finally noninfectious virus (virus affected by treatment with ARV) (VNI). Stability analysis of disease free equilibrium (DFE) of the model and endemic equilibrium point (EEP) of the model were studied. The effects of time delay on the stability of equilibrium points were also considered. The study revealed that the stability of equilibrium points are affected by delay and efficacy of the drug. Analytical results showed that DFE is stable for all ð€€‚ > 0. Similarly, there is a critical value of delay ð€€‚ð€€… > 0, such that for all ð€€‚ > ð€€‚ð€€…, the EEP is stable and unstable for ð€€‚ < ð€€‚ð€€…. When the value of delay(ð€€‚) is equal to the critical valueð€€‚ð€€…, the HIV-1 in vivo dynamics undergoes a Hopf bifurcation and remains stable for all values ð€€‚ > ð€€‚ð€€… as confirmed by the transversality condition. Numerical simulations show that this stability is achieved at the drug efficacy of 0.79 and ð€€‚ð€€… = 0.65 days, or approximately 16 hours. This verifies the fact that if CD4+T cells remain inactive for long periods ð€€‚ > ð€€‚ð€€… the HIV-1 viral materials will not be reproduced, and the immune system together with treatment will have enough time to clear the viral materials in the blood and thus the EEP is maintained.