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.
Estimating the Effective Reproduction Number and Variables of Disease Models for the COVID-19 Epidemic