| Abstract: |
| Our paper started as a review of three related topics. One is a recent proof of the celebrated next generation matrix (NGM) method of
mathematical epidemiology (ME), which was shown recently
to be an outcome of the fact that the invariance/siphon property fundamental in positive ODEs and in particular in chemical reaction networks (CRN) forces the Jacobian at any reasonable boundary equilibrium to have a triangular block form. The second topic are seven results pertaining to a family of bilinear models with rank one NGM introduced by Fall, Iggidr, Sallet and Bonzi, which utilize the explicit eigenvectors of the NGM to compute the unique endemic equilibrium (EE), and Lyapunov functions at both the disease free equilibrium (DFE) and EE.
Here we clarified that the Bonzi-Iggidr-Sallet bilinear models with rank one NGM
may be classified in two classes, with slight variations in the eigenvector formulas, and that extensions in the presence of feedback from infectious to susceptible are possible. The third topic are results of Shuai and Van den Driessche (2013) which essentially deal with the same DFE -EE stability exchange in the non-rank one case, when the eigenvectors are not explicit.
Beyond the review, we identified another positive ODE/CRN concept, that of minimal siphons lattice, which might be useful in ME, in the direction for studying multi-strain models with multiple boundary equilibria. This work is ongoing, but it yielded already several results, as well as an algorithm implemented in our Mathematica package Epid-CRN at https://github.com/florinav/EpidCRNmodels, which partitions the parameter space into regions where only one boundary point may be locally asymptotically stable (LAS) (the so-called competitive
exclusion partition (CEP).
All the examples in the paper have accompanying notebooks, available at
the same address. |
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