Non-hyperbolic disease free equilibrium

Question

There are compartmental models that have constant total population therefore zero eigenvalues in their jacobians such as https://www.nature.com/articles/s41591-020-0883-7 , https://www.nature.com/articles/s41598-021-91114-5. Since these models have zero eigenvalues as stated, the Hartman-Grobman theorem cannot be used to linearize them at the disease free equilibrium, nevertheless both of these papers linearized their model at their own DFE and analyzed the models' stability. My question is that how can these models (in general constant total population size models) with zero eigenvalues in their jacobians justify linearizing their models at DFE and analyzing their stability? There aren't any implication on both of these paper about this so I guess it is fairly a common and simple thing to do. Did they use something like center manifold theorem to justify their linearization at non-hyperbolic equilibrium point? If so how did they do that since these are very complicated topics, can someone show the way to do it? Thank you very much.