Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays.

*(English)* Zbl 1260.92088
Summary: We consider a mathematical model that describes the interactions of the HIV virus, CD4 cells and cytotoxic T lymphocytes (CTLs) within the host, which is a modification of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function $f$ for virus infections, and (iii) a nonlinear removal rate function $h$ for infected cells. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we show that the model has a threshold dynamics: if the basic reproduction number (BRN) is less than or equal to one, then the infection free equilibrium is globally asymptotically stable, meaning that the HIV virus will be cleared; whereas if the BRN is larger than one, there exists an infected equilibrium which is globally asymptotically stable, implying that the HIV-1 infection will persist in the host and the viral concentration will approach a positive constant level. This together with the dependence/independence of the basic reproduction number (BRN) on $f$ and $h$ reveals the effect of the adoption of these nonlinear functions.

##### MSC:

92C60 | Medical epidemiology |

34D23 | Global stability of ODE |

34D05 | Asymptotic stability of ODE |