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# How to solve a system of three ODE

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James Beard on 14 Jun 2012
I was looking into epidemiology, and I came to a system of three ODEs to solve the spread. For the life of me I can't remember how to do this in Matlab. Here is my system, with the initial conditions.
S(0) = 19,465,197
E(0) = 0
I(0) = 0
dS/dt=-147199.4991*S+10*S*I
dE/dt=10*S*I-147203.5121*E
dI/dt=4*E-147199.5197*I
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### Answers (2)

Benjamin Schwabe on 15 Jun 2012
Hi James,
start writing it in a vectorform with x=(S,E,I); Then you have x(0)=x0 as defined and you can write dx/dt=f(x); create a function doin this. the you might use ode45 oder another ode solver for you problem. The help is quite detailed here.
Hope that helps, Benjamin
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Walter Roberson on 15 Jun 2012
WIth the symbolic toolbox, it would look somewhat like
syms S(t) E(t) I(t)
dsolve(S(0) == 19465197, E(0) == 0, I(0) == 0, diff(S(t), t) == -1.471994991*10^5*S(t)+10*S(t)*I(t), diff(E(t), t) == 10*S(t)*I(t)-1.472035121*10^5*E(t), diff(I(t), t) = 4*E(t)-1.471995197*10^5*I(t))
with solution
E(t) = 0, I(t) = 0, S(t) = 19465197*exp(-147199.4991*t)
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