Consider this paper:

Hai-Feng Huo and Li-Xiang Feng, "Global stability for an HIV/AIDS epidemic model with different latent stages and treatment", Applied Mathematical Modelling, Volume 37, Issue 3, February 2013, Pages 1480-1489, MR3002236, Zbl 1351.34044.

Everything is understood apart from page 7 (page 1486 in the journal). Why/how did the author come to the conclusion that the largest invariant set is the singleton $\{(1,1,1,1)\}$? Is it because this singleton kills the variable terms in the Lyapunov function?

Addendum:

The authors then go on to say by LaSalle's invariance principle, the equilibrium point is globally stable, how/why can they conclude in this manner?

**EDIT**:

As per Martin M. W. comment:

Here is the equivalent system(please double check):

\begin{align} \dot x&= x[\beta_1 I_2^*(\frac{1}{x}-z)+\beta_2J^*(\frac{1}{x}-u)+\mu(\frac{1}{x}-1)]\\[1ex] \dot y &= y[\frac{p \beta_1 S^* I_2^*}{I_1^*}(\frac{xz}{y}-1)+\frac{q \beta_2 S^* J^*}{I_1^*}(\frac{xu}{y}-1)+\frac{\xi_1 J^*}{I_1^*}(\frac{u}{y}-1)]\\[1ex] \dot z&= z[(1-p)\beta_1 S^*(x-1)+(1-q)\frac{\beta_2 S^* J^*}{I_2^*}(\frac{xu}{z}-1)+\frac{\epsilon I_1^*}{I_2^*}(\frac{y}{z}-1)+\frac{\xi_2 J^*}{I_2^*}(\frac{u}{z}-1)]\\[1ex] \dot u&= u[\frac{p_1 I_2^*}{J^*}(\frac{z}{u}-1)] \end{align}

I omitted the fifth equation as we can decouple this from the original system.

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