Consider the following differential equations
$${dS \over dt} = \lambda-\beta SI-\mu S+\theta I$$
$${dI \over dt} =\beta SI-(\mu +d)I-\theta I$$
In all papers that I have read it is only mentioned that
$$\Omega = \left\{ (S,I) : I\geq 0, S \geq 0, S+I \leq {\lambda \over \mu} \right\}$$
is positively invariant. How can I explicitly show that the set $\Omega $ is positively invariant?
The boundary of the set (which is a triangle in the $(S,I)$-plane) consists of three line segments: \begin{align} \ell_1 &= \left\{ (S,I) \,\middle\vert\, I = 0 ,\, 0 < S < \frac{\lambda}{\mu} \right\},\\ \ell_2 &= \left\{ (S,I) \,\middle\vert\, 0 < I < \frac{\lambda}{\mu},\, S = 0 \right\},\\ \ell_3 &= \left\{ (S,I) \,\middle\vert\, S + I = \frac{\lambda}{\mu}\right\}. \end{align} A set is positive invariant if you can't escape it, i.e. if you start in the set, you stay in that set for all positive time. Now, the only way to escape such a set is to flow out of it -- at the boundaries! Therefore, the only thing you have to prove is that the flow at the boundaries of the set doesn't point out of the set at any point on the boundary. Pointing inwards everywhere is of course sufficient, but it's ok if the flow is tangential to the boundary; only if there is outwards pointing flow, the set isn't positive invariant anymore.
