# Mathematica - How to solve for the equilibrium?

So, I used Mathematica to create a basic SIR Disease model using the following equations:

Neqs=eqs/.{b->u,a->0}

eqs={

S'[t] == b N[t] - u S[t] - B S[t](I[t] / N[t]),

I'[t] == B S[t] (I[t] / N[t]) - u I[t] - a I[t] - g I[t],

R'[t] == g I[t] - u R[t],

N'[t] == b N[t] - u N[t] - a I[t],

S[0] == N[0] - I[0],

I[0] == 1,

R[0] == 0,

N[0] == n

}


I then used Manipulate to find the solution using NDSolve and then plot it. This code all worked and I have some neat sliders for $u$, $B$, $g$, and $n$ that are pretty fun to play around with.

Now, I am supposed to solve for the equilibrium, or when the disease spreads ($dI/dt > 0$), and plot the isoclines for $S$ and $I$ populations. I tried to write an if loop to do this, but I am really confused and it just gives me errors. Does anyone know a way to approach this? Any help is appreciated!

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## 1 Answer

Is that "BSt" supposed to be B*S[t] or is it B*S[t]*t? If the latter, this is a time-dependent system, not an autonomous system, and there is no such thing as an equilibrium. All the SIR models I've seen would have a term involving S[t]*I[t], to model susceptibles becoming infective by contact with infectives.

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