John Cross, epidemic theory, and mathematically modeling the Norwich smallpox epidemic of 1819

In this paper, we reintroduce Dr. John Cross' neglected and unusually complete historical data set describing a smallpox epidemic occurring in Norwich, England in 1819. We analyze this epidemic data in the context of early models of epidemic spread including the Farr-Evans-Brownlee Normal law, the Kermack-McKendrick square Hyperbolic Secant and SIR laws, along with the modern Volz-Miller random-network law. We show that Cross' hypothesis of susceptible pool limitation is sufficient to explain the data under the SIR law, but requires parameter estimates differing from the modern understanding of smallpox epidemiology or large errors in Cross' data collection. We hypothesize that these discrepancies are due to the mass-action hypothesis in SIR theory, rather than significant errors by Cross, and use Volz-Miller theory to support this. Our analysis demonstrates the difficulties arising in inference of attributes of the disease from death incidence data and how model hypotheses impact these inferences. Our study finds that, combined with Volz-Miller modeling theory, Cross' death incidence data and population observations give smallpox attributes which largely cohere to those used in modern smallpox models.