Selecting a Lyapunov function for a SEI model Consider the system:
$$\frac{dS}{dt} =\nu N -\frac{\beta S I}{N} -\nu S$$
$$\frac{dE}{dt} =\frac{\beta S I}{N} -(\sigma+\nu)E $$
$$\frac{dI}{dt} = \sigma E -\nu I$$
where $N=S+E+I$.
Removing degeneracy we have the following reduced system:
$$\frac{dS}{dt} = -\beta S I +\nu -\nu S$$
$$\frac{dI}{dt} = \sigma  -\sigma S -(\sigma+\nu) I$$
where $\mathcal{R}_0 = \frac{\beta \sigma}{\nu(\nu+\sigma)}$.
The equilibrium point:
\begin{align*}
e_2 : \left( S_2^*, E_2^*, I_2^*\right)&= \left(\frac{\nu\left(\sigma + \nu\right)}{\beta \sigma},\frac{\nu^2 \left( \sigma + \nu \right) \left( \frac{\beta \sigma}{\nu\left(\sigma+\nu\right)} -1 \right) }{\beta\sigma\left(\sigma +\nu \right)} , \frac{\nu\sigma \left( \sigma + \nu \right) \left( \frac{\beta \sigma}{\nu\left(\sigma+\nu\right)} -1 \right) }{\beta\sigma\left(\sigma +\nu \right)}\right)\\[1ex]
\end{align*}
I've been looking for a (global) Lyapunov function to prove global stability for the equilibrium point $e_2$ but I haven't found one that works with this specific system, Any ideas?
A similar question was asked here: Searching for a Lyapunov function for a SIRS model
 A: I believe there is no Lyapunov functional for this ODE system.
Nevertheless, one can prove the global stability of this fixed point by using
information one gathers from  nullclines of this system, which are the curves
where $\dot S=0$ and $\dot I=0$.
From the reduced system, we have the $\frac{dS}{dt}=0$ nullcline given by
$$
-\beta S I +\nu -\nu S=0,
$$
this corresponds to the curve
$$
I=\frac{\nu}{\beta}\left(\frac1S-1\right).
$$
Above it $\frac{dS}{dt}<0$ while below it $\frac{dS}{dt}>0$.
The  nullcline with $\frac{dI}{dt}=0$ is given by
$$
\sigma  -\sigma S -(\sigma+\nu) I=0.
$$
The straight line is
$$
S+\left(1+\frac\nu\sigma\right)I=1.
$$
Below it: $\frac{dI}{dt}>0$, while above it: $\frac{dI}{dt}<0$.
Due to the normalization $S+E+I=1$, hence, all states start in the region
$\Omega$ (greenish shaded area), which we define as below the line $S+I=1$ and in the first quadrant.
Defining the velocity of the flow given in the reduced system as
$$\vec v=(-\beta S I +\nu -\nu S, \sigma  -\sigma S -(\sigma+\nu) I),$$
On the boundary $I=0$ and $0\leq S\leq1$, one finds that $\frac{dS}{dt}=\nu(1-S)\geq 0$ and
$\frac{dI}{dt}=\sigma(1-S)\geq0$, hence the trajectories cannot leave the region
$\Omega$ there.
On the boundary $S=0$ and $0\leq I\leq1$, one finds that $\vec
v\cdot(1,0)=\nu\geq0$, hence the trajectories cannot leave the region
$\Omega$ there either.
Finally, on the boundary $S+I=1$, whose normal vector is
$\hat n=(\frac1{\sqrt{2}}, \frac1{\sqrt{2}})$, $\vec v\cdot \hat
n=-\frac1{\sqrt{2}}\beta SI\leq0$.
Hence, again no trajectories can leave region $\Omega$ across this boundary.
Therefore, we  conclude that $\Omega$ is an invariant region under the flow
defined by the reduced ODE system.
One can show that the shaded blue area between the nullclines is an attractor.
We call it attractor $A$.
Furthermore, there are no other fixed points in $A$ besides the unstable
fixed point $e_1=(1,0)$ and the stable fixed point
$e_2=\left(\frac{\nu\left(\sigma + \nu\right)}{\beta \sigma},
\frac\sigma{\sigma+\nu} -\frac\nu\beta \right)$.
Also, since the trajectories cannot leave region $A$ since on the nullcline 1 (lower boundary of $A$) the flow points upwards and on
nullcline 2 (upper boundary of $A$) the flow points leftwards.
In addition, in region $A$
$\frac{dS}{dt}\leq0$ and $\frac{dI}{dt}\geq0$ for all time, then periodic
solutions (limit cycles) are ruled out.
Furthermore, due to the Poincaré-Bendixon theorem (PBT), all trajectories in that region
with converge to $e_2$.
We don't even need to use the PBT inside region $A$, if we notice that in each vertical cut of it, the flow points from left to right towards fixed-point 2. Take a cut at $S=S_0$ for example, then we obtain $\vec v\cdot (-1,0)=\beta S_0 I -\nu +\nu S_0\geq0$. See the trajectories in Fig. 2, starting at one such cut, illustrating this
point.
Below nullclines 1 and 2, we have $\frac{dS}{dt}>0$ and $\frac{dI}{dt}>0$, hence
no fixed-points and no limit cycles exist there.
In region $\Omega$, below nullcline 1 and above nullcline 2, we have
$\frac{dS}{dt}>0$ and $\frac{dI}{dt}<0$, hence, there are no fixed-points nor
limit cycles therein.
Finally, in region $\Omega$ above both nullclines, we have $\frac{dS}{dt}<0$ and
$\frac{dI}{dt}<0$, hence there are no fixed points nor any periodic orbits
there.


