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Math 4600 Homework 6 Please, note that even though this homework is due after the midterm, the problems here can and should be used for midterm practice. So, you are strongly encouraged to do them before the midterm. 1. Here is the HIV dynamics model that we considered in class: dT T ) − dT T − kV T = s + pT (1 − dt Tmax dT ∗ = kV T − δT ∗ dt dV = N δT ∗ − cV dt The RT-inhibitor drug blocks the reverse transcription, i.e. in the presence of this drug the virus is not able to infect T-cells. a) Which parameter is affected by the RT-inhibitor? Explain Imagine that the inhibitor is perfect, i.e. it sets the appropriate parameter to zero. In the resulting system T ∗ and V equations are uncoupled from the T equation. Use this system to answer questions b) and c). b) In the (T ∗ , V ) system find the steady states and their stability analytically (by computing the eigenvalues). Then, draw a phase plane with nullclines and representative direction field arrows. Finally, draw qualitative time courses for T ∗ (t) and V (t) starting with some high initial values. (Assume c = N kT0 if you need to). c) In the T equation with s = 0 find the steady states and stability analytically and using the phase line. Discuss what will be the long-term condition of T-cell population according to this equation. 1