Reading the entire paper isn't required. At the top of page 2168 in the journal, the author splits the region in three cases (C1, C2, C3) as a consequence of (16). This is understood for proving global stability of the disease-free equilibrium.
Next, they went on to prove the endemic equilibrium, the same technique is applied as the disease free equilibrium but this time we have a set of inequalities labelled (22). They then consider 9 cases (C1–C9) but I don't know why they split the region in that particular way. Any ideas? Or was it a typo?