Question

A) HIV functions 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 manner. Normally, T cells are produced at a rate s and die at a rate d. The virus, when present in the bloodstream as free virus, infect health T cells at a rate β. Also, the viruses reproduce through the T cell multiplication process at a rate k. Free viruses die at a rate μ. This description yields the following differential equations: dT dt のーの dt dv dt where T = number of healthy cells, T·= number of infected cells, and number of free viruses. (i) Indicate which equations, and specifically which terms within the equation, are nonlinear. [4 pts] (ii) Find the system's equilibrium points. There are two. One is readily achieved by inspection. The other requires solving simultaneous equations, as we did in the neuromuscular reflex lab. [6 pts] B) Although there is no cure for the disease, AIDS is treated with retroviral drugs, reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). Incorporating administration of these drugs changes the system equations to: dT の. 、の dv where 0 S u S1,0S u 1 represent the effectiveness of RTI and Pl, respectively (ii) Obtain a complete state-space representation of the HIV/AIDS model by linearizing about an equilibrium point defined by (To.To,o). Let the output equal the number of free viruses. [10 pts] Hint: Matrices A and B should be -(d + β%) 0-87,

Answer 1

Based on provided information, we have (a) partialdifferential/ partialdifferential phi (s sin theta cos phi) = s sin theta partialdifferential/ partialdifferential phi (cos phi) = -s sin theta cos (sin phi) = -s sin theta sin phi (i) we have three system of non linear differential equation dT/dt = s - dT - beta T nu -----(i) dT/dt = beta T nu - mu T*-----(ii) d nu/dt = kT* - c nu----(iii)-----(1) In (1) the term RT nu is non - linear In (ii) the term PT nu is non linear equation (ii) is linear in variable (ii) Equilibrium points of system (1) can be find out by putting dT = 0: dT* = 0: d nu/dt = 0 Rightarrow S - dT - beta T nu = 0---(2) beta T nu - mu T* = 0----(3) KT* - c nu = 0 ----(4)}-----(*) adding (2) and (3) we get s - dT - mu T* = 0 Rightarrow dT + mu T* = s-----(5) adding c times (equation (3)) and beta T times(equation (4)) we get c beta T nu - c mu T* + k beta TT* - C beta T nu = 0 T* (k beta T - C mu) = 0 Rightarrow T* = 0 and T = c mu/kbeta From (2), T = S - mu T*/d