  A very large class of ODE epidemic models (2.2) discussed in this paper
enjoys the property of admitting also an integral renewal formulation, with
respect to an "age of infection kernel" a(t) which has a matrix exponential
form (3.2). We observe first that a very short proof of this fact is available
when there is only one susceptible compartment, and when its associated "new
infections" matrix has rank one. In this case, a(t) normalized to have integral
1, is precisely the probabilistic law which governs the time spent in all the
"infectious states associated to the susceptible compartment", and the
normalization is precisely the basic replacement number. The Laplace transform
(LT) of a(t) is a generalization of the basic replacement number, and its
structure reflects the laws of the times spent in each infectious state.
Subsequently, we show that these facts admit extensions to processes with
several susceptible classes, provided that all of them have a new infections
matrix of rank one. These results reveal that the ODE epidemic models
highlighted below have also interesting probabilistic properties.
