How to solve nonlinear system of equations?

How to solve nonlinear system of equations in mathematical epidemiology

S'(t)= A- bS(t)- (cS(t)I(t))/(1+kI(t))
I'(t)= (cS(t)I(t))/1+kI(t)-(d+e+f)I(t)-rI(t)/(1+pI(t))
R'(t)= fI(t)+(rI(t)/1+pI(t))-bR(t)

where A, b, c, d, e, f, k and p are positive constants.

A= 15, b=0.1, c=0.01, e=0.01, f=0.1, r=0.8, p=0.1, k=0.1 another set of values A=12, b=0.1, c=0.01, e=0.01, f=0.1, r=2, p=1, k=0.1 send the mathlab code. thank you

I need code for this request. Please send me a sample code for nonlinear system of equations

[Walter Roberson]

Could you confirm that bS(t) is b*S(t), not a separate function?

[Rajasekar S P]

bS(t) is b*S(t),cS(t)I(t)is c*S(t)*I(t) and rI(t) is r*I(t)

[Walter Roberson]

I think you might not have written I' and R' correctly. You have divisions by 1, then add something. I suspect those should be divisions by (1 plus something)

Answer 1

[Rajasekar S P]

bS(t) is b*S(t),cS(t)I(t)is c*S(t)*I(t) and rI(t) is r*I(t)

Answer 2

[Walter Roberson]

Provided that I was correct that the division should not be by 1, that the actual equations are:

[diff(S(t), t) = A-b*S(t)-c*S(t)*I_(t)/(1+k*I_(t)),
 diff(I_(t), t) = c*S(t)*I_(t)/(1+k*I_(t))-(d+e+f)*I_(t)-r*I_(t)/(1+p*I_(t)),
 diff(R(t), t) = f*I_(t)+r*I_(t)/(1+I*p)-b*R(t)]

Then, one of the solutions is

I(t) = 0
R(t) = R(0)*exp(-b*t)
S(t) = a/b + exp(-b*t)*(S(0)-a/b)

In the above, R(0) and S(0) are the boundary conditions at 0 for R and S, which is something you did not mention.

If I(t) might not be identical to 0, then even with specific numeric values, about the best you can get symbolically is a complicated system of relationships about what the various functions would have to look like. When I look at that system of relationships, I think it might even constrain it down to a single time that the relationship is valid.

I think you are probably going to need to supply boundary conditions and do a numeric solution using ode45 or similar.

[Rajasekar S P]

Let us take the boundary values are S(0)=0.6,R(0)=0.2,I(0)=0.2. Please show me or attached file for detailed mathlab code. It is very helpful for my research. thank you.