Control of HIV/AIDS. in Chapter 2 introduced a model for

Chapter 3, Problem 31

(choose chapter or problem)

Control of HIV/AIDS. Problem 67 in Chapter 2 introduced a model for HIV infection. If retroviral drugs, RTIs and PIs as discussed in Problem 22inChapter 1, are used, the model is modified as follows (Craig, 2004):

\(\begin{aligned} & \frac{d T}{d t}=s-d T-\left(1-u_1\right) \beta T v \\ & \frac{d T^*}{d t}=\left(1-u_1\right) \beta T v-\mu T^* \\ & \frac{d v}{d t}=\left(1-u_2\right) k T^*-c v \end{aligned}\)

where \(0 \leq u_1 \leq 1,0 \leq u_2 \leq 1\) represent the effectiveness of the RTI and PI medication, respectively.

(a) Obtain a state-space representation of the HIV/AIDS model by linearizing the equations about the

\(\left(T_0, T_0^*, \quad v_0\right)=\left(\frac{c \mu}{\beta k}, \frac{s}{\mu}-\frac{c d}{\beta k}, \frac{s k}{c \mu}-\frac{d}{\beta}\right)\)

equilibrium with \(u_{10}=u_{20}=0\). This equilibrium represents the asymptomatic HIV-infected patient. Note that each one of the above equations is of the form \(\dot{x}_i=f_i\left(x_i, u_1, u_2\right), i=1,2,3\).

(b) If Matrices A and B are given by

\(\mathbf{A}=\left[\begin{array}{lll}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3}\end{array}\right]_{T_0, T_0^*, v_0} ; \quad \mathbf{B}=\left[\begin{array}{ll}\frac{\partial f_1}{\partial u_1} & \frac{\partial f_1}{\partial u_2} \\ \frac{\partial f_2}{\partial u_1} & \frac{\partial f_2}{\partial u_2} \\ \frac{\partial f_3}{\partial u_1} & \frac{\partial f_3}{\partial u_2}\end{array}\right]_{T_0, T_0^*, v_0}\)

and we are interested in the number of free HIV viruses as the system's output,

\(\mathbf{C}=\left[\begin{array}{lll} 0 & 0 & 1 \end{array}\right]\)

show that

\(\mathbf{A}=\left[\begin{array}{crc} -\left(d+\beta v_0\right) & 0 & -\beta T_0 \\ \beta v_0 & -\mu & \beta T_0 \\ 0 & k & -c \end{array}\right] ; \quad \mathbf{B}=\left[\begin{array}{cc} \beta T_0 v_0 & 0 \\ -\beta T_0 v_0 & 0 \\ 0 & -k T_0^* \end{array}\right]\)

(c) Typical parameter values and descriptions for the HIV/AIDS model are shown in the following table. Substitute the values from the table into your model and write as

\(\begin{aligned} & \dot{\mathbf{x}}=\mathbf{A} \mathbf{x}+\mathbf{B u} \\ & \mathbf{y}=\mathbf{C x} \end{aligned}\)