Mathematical modeling of R5 and X4 HIV : from within host dynamics to the epidemiology of HIV infection.
Most existing models have considered the immunological processes occurring within the host and the epidemiological processes occurring at population level as decoupled systems. We present a new model using continuous systems of non linear ordinary differential equations by directly linking the within host dynamics capturing the interactions between Langerhans cells, CD4+ T-Cells, R5 HIV and X4 HIV and the without host dynamics of a basic compartmental HIV/AIDS, susceptible, infected, AIDS model. The model captures the biological theories of the cells that take part in HIV transmission. The study incorporates in its analysis the differences in time scales of the fast within host dynamics and the slow without host dynamics. In the mathematical analysis, important thresholds, the reproduction numbers, were computed which are useful in predicting the progression of the infection both within the host and without the host. The study results showed that the model exhibits four within host equilibrium points inclusive of three endemic equilibria whose effects translate into different scenarios at the population level. All the endemic equilibria were shown to be globally stable using Lyapunov functions and this is an important result in linking the within host dynamics to the population dynamics, because the disease free equilibrium point ceases to exist. The linked models had no effect on the basic reproduction numbers of the within host dynamics but on the basic reproduction number of the population dynamics. The effects of linking were observed on the endemic equilibrium points of both the within host and population dynamics. Therefore, linking the two dynamics leads to the increase in the viral load within the host and increase in the epidemic levels in the population dynamics.