
# 7.1: Finding Equilibrium Points

[ "article:topic", "authorname:hsayama", "equilibrium points", "Susceptible-Infected-Recovered (SIR) model" ]

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Finding equilibrium points of a continuous-time model $$\frac{dx}{dt} = G(x)$$ can be done in the same way as for a discrete-time model, i.e., by replacing all $$x$$’s with $$x_{eq}$$’s (again, note that these could be vectors). This actually makes the left hand side zero, because $$x_{eq}$$ is no longer a dynamical variable but just a static constant. Therefore, things come down to just solving the following equation

$0=G(x_{eq})$

with regard to $$x_{eq}$$. For example, consider the following logistic growth model:

$\frac{dx}{dt} =rx \left(1-\dfrac{x}{K} \right) \label{7.1}$

Replacing all the $$x$$’s with $$x_{eq}$$’s, we obtain

$0 =rx_{eq} \left(1-\dfrac{x}{K} \right) \label{7.3}$

$x_{eq} =0, K \label{7.4}$

It turns out that the result is the same as that of its discrete-time counterpart(see Eq.(5.1.6)).

Exercise $$\PageIndex{1}$$

Find the equilibrium points of the following model:

$\frac{dx}{dt} =x^{2} -rx +1 \label{7.5}$

Exercise $$\PageIndex{2}$$: Simple Pendulum

Find the equilibrium points of the following model of a simple pendulum:

$\frac{d^{2} \theta}{dt^{2}} = -\frac{g}{L} \sin{\theta}$

Exercise $$\PageIndex{3}$$: Susceptible-Infected-Recovered model

The following model is called a Susceptible-Infected-Recovered (SIR) model, a mathematical model of epidemiological dynamics. $$S$$ is the number of susceptible individuals, $$I$$ is the number of infected ones, and $$R$$ is the number of recovered ones. Find the equilibrium points of this model.

\begin{align} \frac{dS}{dt} &= -aSI \label{7.7} \\[5pt] \frac{dI}{dt} &= aSI -bI \label{7.8} \\[5pt] \frac{dR}{dt} &=bI \label{7.9} \end{align}