Stability and Dynamic of HIV-1 Mathematical Model with Logistic Target Cell Growth, Treatment Rate, Cure Rate and Cell-to-cell Spread

Abstract

One way of HIV infection spreading is through the cell division of infected cells by mitosis‎ ‎expressed in mathematical models as a logistic process‎. ‎Cell-to-cell transmission is another factor in the spread and speed of disease‎. ‎In this work‎, ‎we present a five-dimensional Ordinary Differential Equation model (ODE) with the logistic form for proliferation of uninfected cells‎, ‎cell-to-cell and virus-to-cell transmission rate‎, ‎two types of cellular and humoral immune responses‎, ‎the cure rate for returning infected cells to non-infectious cells‎, ‎and two treatment rates‎, ‎one for reducing infectious cells and the other for blocking free viruses‎. ‎We discuss the positivity and boundedness of solutions‎, ‎free-equilibrium points‎, ‎steady-state equilibrium points‎, ‎and stability by the Routh Hurwitz criterion‎. ‎The rate of reproduction is analyzed, ‎and the useful parameters for increasing or decreasing it are identified‎. ‎Numerical simulations are performed to investigate the dynamic behavior of model responses to treatment effects on disease.