  This paper deals with the problem of estimating variables in nonlinear models
for the spread of disease and its application to the COVID-19 epidemic. First
unconstrained methods are revisited and they are shown to correspond to the
application of a linear filter followed by a nonlinear estimate of the
effective reproduction number after a change-of-coordinates. Unconstrained
methods often fail to keep the estimated variables within their physical range
and can lead to unreliable estimates that require aggressively smoothing the
raw data. In order to overcome these shortcomings a constrained estimation
method is proposed that keeps the model variables within pre-specified
boundaries and can also promote smoothness of the estimates. Constrained
estimation can be directly applied to raw data without the need of
pre-smoothing and the associated loss of information and additional lag. It can
also be easily extended to handle additional information, such as the number of
infected individuals. The resulting problem is cast as a convex quadratic
optimization problem with linear and convex quadratic constraints. It is also
shown that both unconstrained and constrained methods when applied to death
data are independent of the fatality rate. The methods are applied to public
death data from the COVID-19 epidemic.
