Ambiguity in identifying parameters of an SIR model when fitting epidemic incidence data

When a new pathogen emerges, determining the key transmission parameters plays a crucial role in formulating public health policies and controlling the spread of the pathogen. It is important to note that not every parameter is "identifiable". A parameter of a model is said to be globally structurally identifiable from some kind of perfect data if it can be determined uniquely. Whether that parameter is identifiable depends on what kind of perfect data is available. In this work, we developed a new mathematical concept, the "decay-growth ratio", and using it, we prove that the basic reproduction number is "globally structurally identifiable" from the knowledge of this ratio, and with a bit more information, the duration of infectiousness can be determined as well. That, however, assumes perfect, noise-free data which in reality is unattainable. A parameter is said to be "practically identifiable" if it can be reliably estimated from finite, noisy data. Practical identifiability is inherently dependent on both the nature of the available data and the inferential methodology employed. We proved that neither the basic reproduction number nor the duration of infectiousness is practically identifiable from the common summary statistics of an outbreak, specifically its mean, standard deviation, and amplitude. In fact, we showed that given any outbreak, and given any value greater than one for the basic reproduction, there is an SIR solution with the same mean and standard deviation as the outbreak's. We further demonstrated how this result can be extended to more complex multi-compartment epidemic models. Moreover, we provided indistinguishable fits to real epidemic data with extremely different parameter choices. This insensitivity of fit quality to parameter choices means that traditional curve-fitting cannot reliably infer the key outbreak parameters from case data alone. Taken together, our results highlight fundamental limits on estimating epidemic parameters from incidence data, i.e., the rate of new cases, or from prevalence data, i.e., the number of infected people at a given time.