## Section: New Results

### Robustness and ${\mathcal{R}}_{0}$

We have obtained new results about the relation between Robustness and the basic reproduction number ${\mathcal{R}}_{0}$.

It is now well admitted that the basic reproduction ratio ${\mathcal{R}}_{0}$ is a key concept in mathematical epidemiology and the literature devoted to this concept is now quite important, see [20] , [40] , [19] , [22] , [23] , [24] , [26] , [28] , [30] , [34] and references therein.

This number is a threshold parameter for bifurcation of an epidemic system : for a general compartmental disease transmission model, if ${\mathcal{R}}_{0}<1$, the disease free equilibrium (DFE) is locally asymptotically stable; whereas, if ${\mathcal{R}}_{0}>1$, the DFE is unstable.

It is said in some papers that ${\mathcal{R}}_{0}$ is a measure to gauge the amount of uniform effort needed to eliminate infection from a population [22] , [24] , [25] , [31] , [30] .

The concept of robustness, coming from control theory, is associated to uncertainty. Usually the parameters of a system are known within a certain margin. A question is, how some properties, e.g. stability, can be ascertained with uncertainty on the parameters. In control theory “stability margin” is an important concept. Another way to formulate this problem is to analyze the effect of perturbations, unstructured or structured. This problem is also related to the so-called pseudo-spectrum [36] , [37] , [35] .

We found that the basic reproduction number of an epidemic system is not an accurate gauge of the distance from the Jacobian $J$ of this system, computed at the disease free equilibrium, to the set of stable matrices (if $J$ in unstable), respectively to the set of unstable matrices (if $J$ is stable). The same conclusion arises for another indicator, introduced by Heestebeck et al. [24] , [31] , [30] , the type-reproduction number.