A beta-Poisson model for infectious disease transmission

Outbreaks of emerging and zoonotic infections represent a substantial threat to human health and well-being. These outbreaks tend to be characterised by highly stochastic transmission dynamics with intense variation in transmission potential between cases. The negative binomial distribution is commonly used as a model for transmission in the early stages of an epidemic as it has a natural interpretation as the convolution of a Poisson contact process and a gamma-distributed infectivity. In this study we expand upon the negative binomial model by introducing a beta-Poisson mixture model in which infectious individuals make contacts at the points of a Poisson process and then transmit infection along these contacts with a beta-distributed probability. We show that the negative binomial distribution is a limit case of this model, as is the zero-inflated Poisson distribution obtained by combining a Poisson-distributed contact process with an additional failure probability. We assess the beta-Poisson model’s applicability by fitting it to secondary case distributions (the distribution of the number of subsequent cases generated by a single case) estimated from outbreaks covering a range of pathogens and geographical settings. We find that while the beta-Poisson mixture can achieve a closer to fit to data than the negative binomial distribution, it is consistently outperformed by the negative binomial in terms of Akaike Information Criterion, making it a suboptimal choice on parsimonious grounds. The beta-Poisson performs similarly to the negative binomial model in its ability to capture features of the secondary case distribution such as overdispersion, prevalence of superspreaders, and the probability of a case generating zero subsequent cases. Despite this possible shortcoming, the beta-Poisson distribution may still be of interest in the context of intervention modelling since its structure allows for the simulation of measures which change contact structures while leaving individual-level infectivity unchanged, and vice-versa.Author summary: The early stages of epidemics are characterised by dramatic variations in the number of new cases generated by each infectious individual, with some cases generating no new infections and some “superspreading” cases generating disproportionately large numbers of subsequent cases. In this study we introduce a mathematical model based on a two-step interpretation of infectious disease transmission: infectious individuals make a random number of contacts according to some fixed contact distribution and then infect their contacts with an infection probability which is unique to that specific infectious individual. This model has the advantage of generalizing more commonly used models of early epidemic dynamics, while allowing for policy analyses which assess the impact of measures which impact social contact behaviour and infectiousness across contacts separately. We find that while our model performs at least as well as pre-existing models in modelling individual-level capacity to generate new infections, the extra mathematical complexity our model introduces is not justified by commonly-used measures of parsimony. This suggests that our model could be applicable in specific policy settings but does not offer a substantial improvement over past approaches in a purely observational setting.