Solving non-linear differential system

I want to solve this non-linear differential system using matlab where :

sigma=2/3
mu=1/20
alpha=1/3
beta=2/3
gamma=1/3
b1=0
b2=0
b3=1/3
(S0,T0,I0,M0,R0)=(100000,50000,2753,2000,1645) 

[Walter Roberson]

If you have the Symbolic Toolbox, then that can make it easier to prepare the equations; after that you would follow the workflow in the first example in odeFunction() in order to construct a function handle to pass to a numeric solver.

Answer 1

[Sulaymon Eshkabilov]

You can use: syms to introduce the variables and then employ dsolve to get symbolic solutions, e.g.:

syms S(t) M(t) I(t) T(t) R(t)
dS = diff(S, t);
dM = diff(M, t);
...
sigma=2/3
mu=1/20
...
    EQN = [dS == -sigma*S*T+mu*T-b1*S, dT == sigma*S*T+mu*T-(mu+alpha)*TS, ..];
    SOL = dsolve(EQN, S(0)==100000, T(0)==50000,..);
    SOL.S    % Solution: S(t)
    SOL.T    % SOlution: T(t) 
    ...
     fplot(SOL.S, [0, 15]) 
    ...

[Walter Roberson]

dsolve() works well... until it doesn't.

There are a lot of systems that dsolve() cannot resolve symbolically. In those cases it may be necessary to use odeFunction() to convert the symbolic system to a function for numeric solution.