  The classic SIR model of epidemic dynamics is solved completely by
quadratures, including a time integral transform expanded in a series of
incomplete gamma functions. The model is also generalized to arbitrary
time-dependent infection rates and solved explicitly when the control parameter
depends on the accumulated infections at time $t$. Numerical results are
presented by way of comparison. Autonomous and non-autonomous generalizations
of SIR for interacting regions are also considered, including non-separability
for two or more interacting regions. A reduction of simple SIR models to one
variable leads us to a generalized logistic model, Richards model, which we use
to fit Mexico's COVID-19 data up to day number 134. Forecasting scenarios
resulting from various fittings are discussed. A critique to the applicability
of these models to current pandemic outbreaks in terms of robustness is
provided. Finally, we obtain the bifurcation diagram for a discretized version
of Richards model, displaying period doubling bifurcation to chaos.
