The HIV-1 virus attacks so-called helper T-cells of the immune system; causing infected cells to produce more copies of the virus that can further spread the infection. A simple model of HIV infection can be written as the following set of C'DEs: g=p+q(1_%)kvr1 {13} '31: = (1 o-HHVTef; {\"3} av E = (1 s9}?! fv. (1c) Here TE} is the number of uninfected helper T cells circulating in an individual; {{t} is the nmnber of infected T cells and V{t} represents the viral load [the nmnber of virus particles circulating in the body}. While the ODE system allows the variables to take positive and negative values; we restrict attention to the physically admissible range in which T 2 II]; I 3: fl and V 3: fl. 3]; q; -r; it; It"; d; e and f are positive constants: p is the rate of production of new T cells; 9* is the growth rate of the population of uninfected T cells; -r measures the rate of production of virus by infected T cells; ii: measures the rate of infection of T cells by the virus; If is the carrying capacity of the munfected T cell population in the body; d and e are death rates of healthy and infected T cells respectively; and f is the rate of removal of virus particles. The action of one clam of drug {a so-called reverse transcriptase inhibitor} is represented by the parameter it; which satises II] 5 c: 5 1; so that the drug is absent if c: = II] and maximally eective if c: = 1. The action of a second class of drug {a so-called protease inhibitor} is represented by the parameter (3, satisfying 1] 5 fl 5 1; again; the drug is maximally eective if fl = 1 and absent if {3 = II]. {a}. Taking lfq as a timescale; show how [Ell can be written in dimensionless form as %=n+t(1x}EXM. {3'1} dY E ={1 max HY. {2b} as E = {1 _ mpv as. (2-?) Here X {1'} Y{r} and 3(1'} represent the mtinfected T cell population; the infectedT cell population and viral load at {dimensionless} time 1'. Express the dimensionless parameters 11; c; if; H and qt} in terms of the original dimensional parameters; and show that qufp is independent of q. ['5 marks] (b). Consider the case a = 1 and 8 = 1. Show that (2) has a unique equilibrium point, for which X = X > 0 (say). Linearise (2) about this point, considering perturbations that are proportional to eat (that is, write (X, Y, Z)(t) = (X, Y, Z) + (X, Y, Zjet, and assume that hatted variables are small.) By finding three possible values of A (using a 3 x 3 determinant, or otherwise), demonstrate that the equilibrium point is stable to small perturbations. I [7 marks] (c). When Z = 0 (again with o = =1), demonstrate that the equilibrium state X = X is stable to disturbances of arbitrary amplitude. [2 marks] (d). Now suppose that o = = 0, the case in which an individual is not treated with drugs and is vulnerable to infection. Show that (2) admits a second equilibrium point, for which X = X* (say). Again considering small disturbances around this equilibrium point that are proportional to ed demonstrate that two of the three eigenvalues satisfy 13 + x(0 + 0) + p (X * - X ) = 0. (3) Deduce that the equilibrium for which X = X becomes unstable (indicating the onset of infection) when X > X*. [5 marks]