Disease Dynamics in the presence of a reservoir: A case study of bovine tuberculosis in UK cattle industry

Abstract

Bovine tuberculosis (bTB) is one of the most complex, persistent and controversial problems facing the British cattle industry. For the last 20 years, increasing incidence rates have resulted in bTB becoming endemic in much of England (especially the southwest). Imperfect control strategies used to mitigate bTB effects often lead to cyclic disease behaviour, where once a farm is cleared of bTB infected cattle, the disease will continue to remerge some time later. This thesis explores one of bTB’s primary open questions, what are the quantitative proportions each pathway contributes to persistent reinfection? The current data sets regarding residual disease are imperfect due to data gaps, high variability in disease parameters, and conflicts between data sets. These factors obscure the actual underlying mechanics of bTB within a herd, preventing the construction of optimal control strategies. This thesis uses mathematical modelling to examine the primary disease mechanisms that result in residual disease remaining on a farm; latency, wildlife reservoir, and environmental contamination. Each of these disease mechanisms forms a chapter of the thesis, by focusing on each mechanism we can understand how they affect disease dynamics. After examining these mechanisms separately, this thesis considers their combined effect by numerically simulating bTB within a herd. The first mathematical model investigates how imperfect testing and bTB’s long latency period permits infection to remain on the farm. The highly variable latency period suggests that cattle may potentially be infected years before becoming infectious themselves. Addition- ally, imperfect disease diagnostic tools permit further transmission, as undiagnosed infected hosts may further spread bTB before being discovered and removed. The mathematical models derived from examining this mechanisms uses a non-markovian exposure period, extending the exponentially distributed parameters to the Erlang distribution. The highly flexible and adjustable Erlang distribution means that we can further incorporate the variable nature of the latency period by adjusting the number of exposure compartments. In this chapter, we construct the SEnTIRC model and perform mathematical analysis on the associated set of differential equations, examining the long term behaviour of solutions, system equilibria, and the threshold value for the system (R0). Afterwhich, we further extend the SEnTIRC model, creating the SEnTmIRC model, which has Erlang distributed parameters for the entire latency period. Lastly, this chapter finishes by exploring how altering the latency period distribution affects the long term disease dynamics of our model. The next mechanism examined through mathematical modelling is wildlife reservoirs, exploring how they affect inter-herd disease dynamics. The interconnected disease dynamics between a herd and wildlife reservoirs create a continuous cycle of unobserved spill-over and spill-back between the two populations. This chapter studies this relationship through the construction of a multi-host system. The analysis of this model is similar to the previous chapter, as we examine the long term behaviour of solutions, system equilibria, and the threshold value for the system (R0). However, the interconnected system dynamics permit the system to backward bifurcate, a cumbersome phenomenon from a public health and control perspective. If the system undergoes backward bifurcation, the normal threshold (R0) of the system may not be enough to reduce the disease presence, as a further critical threshold is created. Further control measures must therefore be implemented to reduce the disease dynamics under this new lower threshold value. The nature and feasibility of backward bifurcation are therefore explored and discussed for this disease wildlife model. The last mechanism explored is bTB’s environmental contamination and its effect on disease transmission within the farm. The bacterium M. bovis can contaminate soil, troughs, cattle feed, hay, and various other materials for substantial periods of time. Even though the ecological literature heavily discusses environmental contamination, especially from the perspective of how the wildlife reservoirs and cattle interact, very little of the modelling literature discusses this component. This chapter poses and analyses a mathematical model incorporating en- vironmental contamination as a component. Similar to the previous models, this model is analysed in terms of its long term behaviour of solutions, system equilibria, and the threshold value for the system (R0), where numerical simulations are also presented to give a more complete representation of the model dynamics. The last model uses simulation techniques to investigate how the different model mech- anisms expressed throughout this thesis work in conjunction. The other mechanisms were contemplated and considered through the lens of dynamical systems. If all model mechanisms were considered in conjunction, the resulting complexity of the set of differential equations would make the system very complicated to analyse, if not impossible, with currently avail- able methods. However, through the use of simulations and their techniques, much of the underlying complexity can be mitigated. This simulation model explores three main research themes; badger contribution, culling rate, and residual disease. This simulation examines the associated wildlife reservoir’s contribution to disease dynamics by considering how the system reacts if the wildlife reservoir is fully and partially excluded. The following section examines how the testing and detection strategy works by varying the test sensitivity and the associated culling rate. Lastly, we attempt to quantify the different pathways in which residue disease is left on the farm. Governments and Public Health officials can only construct optimal control strategies by completely understanding how bTB spreads and transmits. Quantifying these mechanisms will shed further light on these residual disease pathways, providing researchers with a clearer understanding of how to best mitigate bTB’s effect. Only through the construction of better control mechanisms can we possibly eradicate the most complex, persistent and controversial problems facing the British cattle industry.