Statistical estimation of parameters in a disease transmission model: analysis of a Cryptosporidium outbreak

Population dynamic models, commonly used tools in the study of epidemics and other complex population processes, are implicit non-linear mathematical equations. Inference based on such models can be difficult due to the problems associated with high dimensional parameters that may be non-identified and complex likelihood functions that are difficult to maximize. To address a problem of non-identifiability due to collinearity of parameter estimates in a mathematical model of the 1993 Milwaukee Cryptosporidium parvum outbreak, we examined the utility of a constrained profile likelihood approach. This method was used to study two parameters of interest from the mathematical model: (i). the rate of secondary transmission; (ii). the proportional increase in primary transmission due to water treatment failure. The estimated values of these parameters were shown to depend strongly on poorly understood aspects of Cryptosporidium epidemiology such as asymptomatic proportion and the population immune status. Our analysis demonstrated that the combination of a disease transmission model and a constrained profile likelihood procedure provides an effective approach for inference and estimation of important parameters regulating infectious disease outbreaks.