Before revision:  In the present paper , those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, namely , piece-wisely,  generalized logistic functions: they {\em   unveil a useful feature, that in fact  is also owned by the (numerical) solutions of the exact equations {\em  {\em   .
After revision:  In the present paper , those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, namely , piece-wisely, generalized logistic functions: they {\em unveil a useful feature, that in fact is also owned by the (numerical) solutions of the exact equations {\em {\em , that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn't appear much discussed in analogous works .