  The non-population conserving SIR (SIR-NC) model to describe the spread of
infections in a community is proposed and studied. Unlike the standard SIR
model, SIR-NC does not assume population conservation. Although similar in form
to the standard SIR, SIR-NC admits a closed form solution while allowing us to
model mortality, and also provides different, and arguably a more realistic,
interpretation of the model parameters. Numerical comparisons of this SIR-NC
model with the standard, population conserving, SIR model are provided.
Extensions to include imported infections, interacting communities, and models
that include births and deaths are presented and analyzed. Several numerical
examples are also presented to illustrate these models. Two control problems
for the SIR-NC epidemic model are presented. First we consider the continuous
time model predictive control in which the cost function variables correspond
to the levels of lockdown, the level of testing and quarantine, and the number
of infections. We also include a switching cost for moving between lockdown
levels. A discrete time version that is more amenable to computation is then
presented along with numerical illustrations. We then consider a
multi-objective and multi-community control where we can define multiple cost
functions on the different communities and obtain the minimum cost control to
keep the value function corresponding to these control objectives below a
prescribed threshold.
