Analysis of a Delayed Multiscale AIDS/HIV-1 Model Coupling Between-Host and Within-Host Dynamics

Abstract

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h<1, the disease-free equilibrium is globally asymptotically stable; if R˜0h>1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.