The HIV-1 virus attacks so-called helper T-cells ofthe immune system; causing infected cells to produce more copies of the virus that can further spread the infection. A simple model of HW infection can be written as the following set of UDEs: ':_:=p+qq*(1%)kVTdr. (1a) 3: = {1 rr}li:VTei; {lb} :1 {11: = {1 aw fv. {In} Here T{t} is the number of uninfected helper T cells circulating in an individual, t} is the number of infected T cells and FE} represents the viral load {the number 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 fl; I 2 fl and V 2 fl. 3]; q; r; it; K; d; e and f are positive constants: p is the rate of production of new T cells; 43 is the growth rate of the population of uninfected T cells; -.'r' measures the rate of production of vi111s by infected T cells; fi.' measures the rate of infection of T cells by the virus; It" is the carrying capacity of the miinfected T cell population in the body; of 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 class of drug {a so-called reverse transcriptase inhibitor} is represented by the parameter or, which satises fl 5 o: 5 1; so that the drug is absent if o: = fl and maao'nlally e'ective if d = 1. The action of a. second class of drug {a so-called protease inhibitor} is represented by the parameter fl; satisfying fl 5 ,3 E 1; again; the drug is maximally eective if f! = 1 and absent if fl = fl. {a}. Taking lfq as a timescale, show how {El} can be written in dimensionless form as E=H+X{1X}EXX; {Ea} il: =(1_ cuss HY. (2b) :1 f ={1_ mpv as. (20} Here X {1'} Y[1'} and 3(1'} represent the uninfected T cell population; the infected T cell population and viral load at {dimensionless} time 1'. Express the dimensionless parameters 11; p; :5; d and qb in terms of the original dimensional parameters; and show that flaw}: is independent of q. [ marks] [cont inued over leaf] (b). Consider the case o = 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 ear (that is, write (X, Y, Z) (t) = (X, Y, Z) + (X, Y, Z)ed, 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. " [7 marks] (c). When Z = 0 (again with a = = 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 eat demonstrate that two of the three eigenvalues satisfy A + x(0to) + 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]