Rate of Convergence to the Disease Free Equilibrium for Multi-Population SIS Networks in the Critical Case

A networked version of the Susceptible-Infected-Susceptible (SIS) deterministic epidemic model is studied. Existing results establish that convergence to an equilibrium occurs exponentially fast, except in the critical case, when the basic reproduction number is equal to 1. This letter uses nonlinear systems and center manifold theory to establish that with such a reproduction number, convergence occurs at a linear rate. Numerical simulations help to illustrate the results.