A mathematical and numerical study of an age-dependent transactivation model for dynamics of HIV infection

We analyze a mathematical model for the dynamics of HIV infection that incorporates an age-dependent transactivation rate and reversion into latency. The basic reproduction number $ R_0 $ for viral infection has been determined, and it completely characterizes the global behavior of model solutions: If $ R_0 > 1 $, the infection is chronic, and if $ R_0 \leq 1 $, the infection is cleared. By constructing suitable Volterra-type Lyapunov functionals, it has been demonstrated that the corresponding equilibrium is globally stable in each of the two settings for $ R_0 $. We used the gamma distribution to define the age-dependent transactivation rate, which is the ratio of the probability density function to the complement of the cumulative distribution function. We present some numerical simulations in each of the two settings for the $ R_0 $ of the age-structured model, where the subsystem of ODE was solved by the fourth-order Runge–Kutta method and the hyperbolic partial differential equation (PDE) is solved by the Crank–Nicolson method, which is second-order in time and age. In conclusion, we have provided the complete classification for the global dynamics of the age-structured model. Possible strategies to control a viral infection have been identified through sensitivity analysis.