Models to capture the potential for disease transmission in domestic sheep flocks

Successful control of livestock diseases requires an understanding of how they spread amongst animals and between premises. Mathematical models can offer important insight into the dynamics of disease, especially when built upon experimental and/or field data. Here the dynamics of a range of epidemiological models are explored in order to determine which models perform best in capturing real-world heterogeneities at sufficient resolution. Individual based network models are considered together with one- and two-class compartmental models, for which the final epidemic size is calculated as a function of the probability of disease transmission occurring during a given physical contact between two individuals. For numerical results the special cases of a viral disease with a fast recovery rate (foot-and-mouth disease) and a bacterial disease with a slow recovery rate (brucellosis) amongst sheep are considered. Quantitative results from observational studies of physical contact amongst domestic sheep are applied and results from the differently structured flocks (ewes with newborn lambs, ewes with nearly weaned lambs and ewes only) compared. These indicate that the breeding cycle leads to significant changes in the expected basic reproduction ratio of diseases. The observed heterogeneity of contacts amongst animals is best captured by full network simulations, although simple compartmental models describe the key features of an outbreak but, as expected, often overestimate the speed of an outbreak. Here the weights of contacts are heterogeneous, with many low weight links. However, due to the well-connected nature of the networks, this has little effect and differences between models remain small. These results indicate that simple compartmental models can be a useful tool for modelling real-world flocks; their applicability will be greater still for more homogeneously mixed livestock, which could be promoted by higher intensity farming practices.