# Answered Essay: The system of problem l has the same parameters except that b, p and beta are different, but are between 0 and 1.5 in problem l. The file p

The system of problem l has the same parameters except that b, p and beta are different, but are between 0 and 1.5 in problem l. 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. 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: x = lambda – dx – beta xv y = beta xv – ay – pyz v = ky – uv z = c – bz Uninfected cells are produced at a constant rate lambda, die at a rate dx and are lost to infection at a rate of beta xv. Infected cells are produced from uninfected cells and free virus at the rate, beta xv 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 lambda = 1: d = 0.01: alpha = 0.05: beta = 0.005: p = l: k = 50: u = 5: b = 0.05. x(0) = 10 y(0) = 1: v(0) = l: z(0) = 0

d = diff(BP_max);
> HR = [d(1)*ones(1,d(1)) d(2)*ones(1,d(2)) d(3)*ones(1,d(3))
> d(4)*ones(1,d(4))];
>
> Of course, this would get cumbersome if you had to use it for a
> much
> longer BP_max, but as long as you preallocate HR (with HR = ones(1,
> max(BP_max);), a for loop shouldn’t slow the code down too much.

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