Effect of time delays in an HIV virotherapy model with nonlinear incidence

Abstract

In this paper, we propose a mathematical model for HIV infection with delays in cell infection and virus production. The model examines a viral therapy for controlling infections through recombining HIV with a genetically modified virus. For this model, we derive two biologically insightful quantities (reproduction numbers) [Formula: see text] and [Formula: see text], and their threshold properties are discussed. When [Formula: see text], the infection-free equilibrium E₀ is globally asymptotically stable. If [Formula: see text] and [Formula: see text], the single-infection equilibrium E_s is globally asymptotically stable. When [Formula: see text], there occurs the double-infection equilibrium E_d, and there exists a constant R_b such that E_d is asymptotically stable if [Formula: see text]. Some simulations are performed to support and complement the theoretical results.

Keywords: HIV-1 model; Hopf bifurcation; Lyapunov function; global stability; nonlinear incidence; recombinant virus.

Figure 1.

The curves defined by equations (2.4) and (2.5). (Online version in colour.)

Figure 2.

Simulation of (1.2) for τ₁=0.5,0.9,1.2, taken from the interval τ₁∈ (τ_h,τ_d), showing convergence to the equilibrium E_d. (Online version in colour.)

Figure 3.

Simulation of (1.2) for τ₁=0.15<τ_h, showing bifurcation to a limit cycle. (Online version in colour.)

Figure 4.

Bifurcation diagram projected on y−τ₁ plane of model (1.2). The dotted and solid lines indicate unstable and stable equilibria, respectively. (Online version in colour.)