The original purpose of this article was to modify the original SIR equations to allow for a direct source of infection (without which the original equations would have no solutions unless one starts with an already infected population) and also to see to what extent one could obtain multiple outbreaks of an infectious disease. In the course of developing the basic ideas several other factors arose to take prominent roles.Perhaps one of the more salient factors is the point that choosing an arbitrary time to change conditions from say a lock-down for the population to a less stringent social behavior, such as allowing partial or complete opening of businesses and schools, etc. should be based on knowledge of the disease and its evolution. Such decisions are usually made by politicians who have less than full information concerning the consequences of their actions. Several examples are given to illustrate these points.

While there are many models of epidemic evolution perhaps the basis for such models finds itself in the lumped behavior expressed through the so-called SIR model (Susceptible, Infectious, Recovered) from which spring many related models. This paper discusses multiple analytic solutions to that equation including those that are available in closed analytic form and those for which at least one final integral has to be done numerically, so-called quasi-analytic solutions. The solutions are intrinsically time-dependent of course. The hope is that such an investigation will lead to a better understanding of when and how models can be of use in studying the dynamical evolution of diseases including, perhaps, the great influenza pandemic of 1918 together with later pandemics and epidemics not excluding the Covid-19 pandemic of the present day.