Modeling of epidemics of infectious diseases using differential equations has a long history and many applications to various diseases, most recently to the COVID-19 pandemic. It is through the wide application of mathematical modeling during the COVID-19 pandemics, we realized that standard differential equations models, simple or complex, have a tendency to significantly over-predict (or over-project) the size of an epidemic. In searching for reasons for this ``annoying’’ tendency, many questions can be asked, one of them is: should mathematical models be validated before projection are made?

The traditional belief is that mathematical models of epidemics are built based on the best science of mechanisms of disease transmission and spread, and should be trusted. The resulting dynamical system would truthfully predict the disease transmission dynamics, after the calibration using disease data. However, if the disease data used for calibration are positive case reports, which is a proportion of the true number of infections, then we are using a deterministic model with information from partial observations of the state variables. As it turns out (I will show in this talk), when only partial observations are available for model calibration, our mathematical model can behave very badly, and model validation becomes crucial before reliable predictions can be made.