  Given the ongoing Covid-19 pandemic, it is of interest to understand how the
infections spread as the combined result of measures taken by central planners
(governments) and individual behavior. In this work, the spread of Covid-19 is
modelled as a differentiable game between the planner and population with
appropriate disease spread dynamical equations. We first characterise the
equilibrium dynamics of only the population with modifed
Susceptible-Infected-Recovered (SIR) equations to highlight the qualitative
nature of the equilbrium. Using this result, we formulate the joint equilibrium
exposure profile between the planner and population. Additionally, as in case
of Covid-19, the role of asymptomatic carriers, inadequacies in testing,
contact tracing and quarantining can lead to a significant underestimate of the
true infected numbers as compared to just the detected numbers. Therefore, it
is vital to model the true infected numbers within the context of choices made
by individuals within the population. To incorporate this, we extend our
framework by modifying the dynamics to include additional sub-compartments of
`undetected infected' and `detected infected' in the disease dynamics. The
individuals make their own estimates of the total infected from the detected
numbers and base their strategies on those estimates. We show that these
considerations lead to a retarded optimal control problem for the players. We
present some simulation results based on these results to demonstrate how
population behavior, planner control, detection rates and trust in the reported
numbers play a key role in how the disease spreads.
