Parameter estimation in Matlab

This problem describes the immune response triggered by encountering foreign antigens. This is called the cytotoxic T cell or CTL response. Suppose that z eliminates infected cells. There are 4 variables: uninfected cells, x, infected cells, y, free virus particles v, and CTL cells z. The equations are:

Uninfected cells are produced at a constant rateλ , die at a rate dx and are lost to infection at a rate ofβxv . Infected cells are produced from uninfected cells and free virus at the rate,βxv and die at a rate ay and are lost to CTL cells at a

rate of pyz. Free virus is produced from infected cells at a rate ky and die

at a rate uv. The CTL cells are produced at a rate of c and die at a rate of b. Parameter values are

λ= 1;d= 0.01;a= 0.05; β = ?; p=?;k= 50;u= 5;b= ?

Initial values of x,y,v,z are as follows

x(0) = 10 y(0) = 1; v(0) = 1; z(0) = 0

b, p and beta are unknown, but are between 0 and 1.5 in value. The file prob_1_data.mat has the data from a simulation where the first column is time, the second is x, the third is y, the fourth is v and the fifth is z. In the data, the initial condition for z is 0.5 not 0. Write the Matlab® code to estimate the unknown parameters, b p and beta.

Use a minimum square error criterion between the data and the model output to generate the parameter estimates.