# 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

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

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)