MATLAB-SIR流行病模型的优化

Question

I'm trying to model a flu epidemic using the SIR model: http://en.wikipedia.org/wiki/Epidemic_model#The_SIR_Model .

It basically requires me to solve 3 differential equations simultaneously.

My task is to fit a graph to a set of data and I was just wondering how I should go about doing this.

I've managed to setup a system to solve the ODEs like this:

function dydt = sir_ode(t,y,p)
B = p(1);
r = p(2);
S = y(1);
I = y(2);
R = y(3);
dydt = [-B*I*S; B*I*S - r*I; r*I];

and the m file to call it:

time = 1:24;
y0 = [400 1 0];
tspan = time;
p0 = [.01 .5];
[t,y] = ode45(@sir_ode,tspan,y0,[],p0);

so I have a tspan, I have a y0 matrix for S, I and R. For my function, the output contains 3 of the differential equations so I can solve them together with the ode45 function.

Now all I need to do now is find the appropriate p0 matrix so I can fit the curves to my data.

Firstly, can anyone see any flaws in my solving method and secondly, who would have any suggestions on how I would go about finding the best p0 matrix?

Thanks!!

Answer 1

The solving method seems correct to me.

For the second part of your question, in every optimisation problem there is a function that you want to maximize or minimize (using various algorithms, more on that later). Here you could minimize an error function . One possible design for the error function is the L2 distance between your functions (or in your case the sum of the squared distances of the points between your original data curve and the curve given by your model at each timestep). This should be fairly simple given the code you already have.

Now you have a function whose argument is p0 and output is the error you have. The minimum is clearly 0, and the function is deterministic. My guess is that it should be rather smooth but I have no proof of that. Now to minimize it, there are a lot of ways (the simplest ones are the gradient descent and Newton's method). But the fminsearch function in Matlab should do the trick. Make sure to test several starting points, as you might find several local minimas.