# Equilibrium of the following system of non linear ODE's

How would find the equilibriums of the following system:

$$\frac{d I}{dt}=-\frac{\beta S I}{N} \qquad (1)$$

$$\frac{d S}{dt}=\frac{\beta S I}{N} -\gamma I \qquad (2)$$

$$\frac{d R}{dt}=\gamma I \qquad (3)$$

where $$S+I+R=N$$ and $$\beta,\gamma >0$$.

I know one such equilibrium is \begin{align} e_1 : \left( S_1^*, I_1^*, R_1^*\right)&= \left(N, 0, 0\right), \\[2ex] \end{align}

How do I find the other(if it exists)? I believe it should be similar to the problem below unless I'm mistaken.

I have solved the more complicated case:

$$\frac{d S}{dt}=\mu N -\frac{\beta S I}{N} - \nu S \qquad (1)$$

$$\frac{d I}{dt}=\frac{\beta S I}{N} -\gamma I - \nu I \qquad (2)$$

$$\frac{d R}{dt}=\gamma I - \nu R \qquad (3)$$

where $$S+I+R=N$$, $$\quad$$ $$\mu =\nu$$, $$\quad$$ $$\mu, \beta, \nu,\gamma >0$$.

Solutions are:

\begin{align} e_1 : \left( S_1^*, I_1^*, R_1^*\right)&= \left(N, 0, 0\right), \\[2ex] e_2 : \left( S_2^*, I_2^*, R_2^*\right)&= \left(\frac{N\left(\gamma+\nu\right)}{\beta}, N\nu\left(\frac{1}{\gamma +\nu}-\frac{1}{\beta}\right), N\gamma\left(\frac{1}{\gamma +\nu}-\frac{1}{\beta}\right)\right) \end{align}

Notice that a necessary condition for an equilibrium is given by: $$\frac{dR}{dt} = 0 \implies I = 0.$$ Since the condition $$I=0$$ is enough to have $$\frac{dI}{dt} = \frac{dS}{dt} = 0$$, you have a sufficient condition. Thus the equilibriums are given by: $$(S^*,I^*,R^*) = (K, 0, N-K),$$ for any $$0\leq K \leq N$$.

• Mhm, I was being stupid again. Anyway thanks :)
– user644376
Commented Mar 10, 2021 at 15:49
• Hmm technically your solution is correct however since this is an epidemic model, we cant have zero infected individuals with positive recovered individuals simultaneously because this would be nonsensical. So I believe the only equilibrium is the disease-free one.
– user644376
Commented Mar 10, 2021 at 16:01
• @Math Why not? If your epidemic vanishes, i.e. $I(t)\rightarrow 0$ as $t\rightarrow\infty$, but $I(t_0)>0$ you would then have had a nonzero recovered population.
– Rem
Commented Mar 10, 2021 at 16:54
• because $I=0$ implies $R=0$
– user644376
Commented Mar 15, 2021 at 12:08