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The problem of foot and mouth disease (FMD) is of serious concern to the livestock sector in most nations, especially in developing countries. This paper presents the formulation and analysis of a deterministic model for the transmission dynamics of FMD through a contaminated environment. It is shown that the key parameters that drive the transmission of FMD in a contaminated environment are the shedding, transmission, and decay rates of the virus. Using numerical results, it is depicted that the host-to-host route is more severe than the environmental-to-host route. The model is then transformed into an optimal control problem. Using the Pontryagin's Maximum Principle, the optimality system is determined. Utilizing a gradient type algorithm with projection, the optimality system is solved for three control strategies: optimal use of vaccination, environmental decontamination, and a combination of vaccination and environmental decontamination. Results show that a combination of vaccination and environmental decontamination is the most optimal strategy. These results indicate that if vaccination and environmental decontamination are used optimally during an outbreak, then FMD transmission can be controlled. Future studies focusing on the control measures for the transmission of FMD in a contaminated environment should aim at reducing the transmission and the shedding rates, while increasing the decay rate.

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In this thesis, a single serotype (j = 1) and patch (i = 1) ordinary difference equation (ODE) model is formulated and analysed for the effects of direct and transplacental transmission on the probability of bluetongue virus (BTV) persistence. Using the next generation approach, the basic reproduction number (R0) is determined. When R0 < 1, the model exhibits a backward bifurcation indicating that the virus persists. When R0 > 1, a continuous-time Markov chain (CTMC) model derived from the ODE model is used to estimate the probability of BTV persistence. By approximating the CTMC model with a multitype branching process, it is shown that both direct and transplacental transmission can have a large effect on the probability of persistence in regions with temperature T < 12◦C and a small effect for those with T > 12◦C. The ODE and CTMC models are extended to include r serotypes and n patches with the aim of determining the effects of midge movement on the outbreak and coexistence of multiple BTV serotypes in an environment divided into patches depending on the risk of infection. An estimate for the probability of a major outbreak of two BTV serotypes in two patches is obtained by approximating the CTMC model with a multitype branching process. It is shown that without movement a major outbreak occurs in the high-risk patch, but with cattle or midge movement it occurs in both patches. When a major outbreak occurs, numerical simulations of the ODE model illustrate possible coexistence in both patches if the patches are connected by midge or cattle movement. The multi-patch single-serotype ODE model is then modified as an optimal control problem to evaluate the effectiveness of vaccination, quarantine, insecticide spraying and the use of repellent control strategies in reducing the within- and between-patches transmission. By using optimal control theory, the effectiveness of these strategies is established. In a single patch, vaccination, insecticide spraying and the use of a repellent are all highly effective in minimising transmission, but the most costeffective is vaccination. In patches connected by host and midge movements, if any of these controls is applied in the high-risk patch, a disease-free status is achieved in both patches, but if implemented in the low-risk patch, it is not attained in any patch. If hosts and midges move, quarantine has no effect, but for no midge movement, the effect can be large in the low-risk patch if it’s internally imposed.

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