The rates of globalization and growth of the human population puts ever increasing pressure on the agricultural sector to intensify and grow more complex, and with this intensification comes an increased risk of outbreaks of infectious livestock diseases. At the same time, and for the same reasons, the detrimental effect that humans have on other species with which we share the environment has never been more apparent, as the current rates of species loss from ecological communities rival those of ancient mass extinction events. In order to find ways to lessen the effects of and eventually solve such problems we need ways to quantify the risks involved, something that can be difficult when for instance the sheer size or sensitivity of the systems makes practical experimentation unsuitable. For these situations mathematical models have become invaluable tools due to their flexibility and noninvasiveness. This thesis presents four works involving the quantification of risk in livestock epidemic and ecological contexts using mathematical models. Two of them deal with extinctions of species within model ecological communities, and how species interactions play a role in the identity of the lost species following perturbations to specific species (Papers I and II). The other two regard how the spatial layout of the underlying population of livestock premises affect the risk of foot and mouth disease outbreaks among farms in the USA, and how models of such outbreaks can be optimized to improve their usefulness (Papers III and IV).

Ecological communities consist of species and the often intricate pattern of interactions between them. These interspecies connections can propagate effects caused by disturbances in one end of the network, through the community via the links, to other parts of the network. In some cases, a reduction in the abundance of one species can cause the extinction of a second species before the first species disappears, something called functional extinction. Despite this, many conservation efforts revolve around simply keeping populations of single species at a high enough level for their own survival. In a model setting, the study of Paper I explores and attempts to quantify how common such functional extinctions are in relation to the alternative outcome that a perturbed species itself becomes extinct. This is done by first constructing stable model food webs describing predator-prey interactions of up to 50 species, parameterized through allometric relationships between metabolic processes and body size. Then the smallest amount of extra mortality that can be applied to each and every species in the web before any species become extinct is determined. The study shows that in these model communities, more often than not (>80%) another species, rather than the species that is subjected to the additional mortality will be the one to become extinct first.

The approach of Paper I is taken further in Paper II by applying the same methodology to ecological networks that include mixtures of both antagonistic (predator-prey) and mutualistic (e.g. pollination and seed dispersal) interactions. The results further reinforce the findings of Paper I, and show that ecological networks containing a mixture of antagonistic and mutualistic interactions are more sensitive to functional extinctions than purely antagonistic or purely mutualistic ones, an important finding considering the diversity of interaction types in natural systems. Furthermore, the type of species found to have the lowest threshold before becoming functionally extinct were those with a mixture of interaction types, such as pollinating insects. Both Paper I and II consolidate the notion that when doing conservation work it is important to have the entire community in mind by considering the population sizes that are viable from a multi-species perspective, rather than just focusing on the minimum population sizes that are viable for the individual species.

In Papers III and IV the focus changes somewhat, from models of ecological systems to models of how infectious livestock disease spread between farms in spatially explicit contexts. For this kind of model, information about the spatial distribution of the hosts is of course crucial, but not always readily available. In the USA, the only available information about livestock premises demography is aggregated at the county scale, meaning that the spatial distribution of the premises within each county is unknown. However, a method exists to simulate realistic stochastic spatial configurations of premises using a set of predictor variables, such as topology, climate and roads. An alternative approach that have been used previously is to assume a uniformly random spatial distribution of premises within each county. But to what extent does the choice between these two methods affect the model’s evaluation of the risk of disease outbreaks? In Paper III, this is analyzed specifically for foot and mouth disease. Through simulated outbreaks and by looking at the reproductive ratio of the disease, the outbreak dynamics within the two different spatial configurations of premises are compared. The results show that there is a clear difference in the risk of outbreaks between them, with the non-uniform distributions showing a general pattern of higher outbreak risk. However this difference is dependent on the size and geographic location of the county that the outbreak start in with larger counties in the west of the US showing a stronger effect.

When running numerical simulations with large scale models such as the one used in Paper III, a considerable amount of replication is usually necessary in order to account for the high degree of stochasticity inherent to the problem. Even further replication is required when performing sensitivity analyses of model parameters or when exploring different scenarios, for instance when trying to determine the optimal control strategy for a disease. For this reason, the amount and quality of results that can be produced by such studies can quickly become limited by the availability of computational resources. Finding ways to optimize the computations involved with regard to simulation time is therefore of great value as it can be directly related to the robustness of the results. In Paper IV, an efficient optimization method for the kind of kernel-based local disease spread model used in paper III is presented. The method revolves around constructing a grid structure that is overlaid on top of the farm landscape and dividing the infection process into two steps, first evaluating if any farms within one of the grid squares can become infected given an over-estimation of the probability of infection, and then only if so, evaluate actual infection of a subset of the farms within the receiving square. The method is compared to similar published methods and is shown to be more efficient in most cases, while also being easy to implement and understand. Furthermore, while other methods often involve approximations of the transmission process in order to improve computational speed, the method of Paper IV is shown to be exact. This is a major advantage, since with an approximative method the extent to which the results are affected by the simplification is unknown unless the effect of the approximation is explicitly quantified. In most cases, such quantification would require extensive simulations with the unsimplified approach, something which of course may not be feasible.