HIV inflicts its damage by infecting healthy CD4 + T cells (a type of white blood cell) that are necessary to fight infection. As the virus embeds in a T cell and the immune system produces more of these cells to fight the infection, the virus propagates in an opportunistic fashion. As we now develop a simple HIV model, refer to Figure P2.39. Normally T cells are produced at a rate s and die at a rate d. The HIV virus is present in the bloodstream in the infected individual. These viruses in the bloodstream, called free viruses, infect healthy T cells at a rate β. Also, the viruses reproduce through the T cell multiplication process or otherwise at a rate k. Free viruses die at a rate c. Infected T cells die at a rate μ.
simple mathematical model that illustrates these interactions is given by the following equations (Craig, 2004):
where
T = number of healthy T cells
T* = number of infected T cells
v = number of free viruses
a. The system is nonlinear; thus linearization is necessary to find transfer functions as you will do in subsequent chapters. The nonlinear nature of this model can be seen from the above equations. Determine which of these equations are linear, which are nonlinear, and explain why.
b. The system has two equilibrium points. Show that these are given by
