Diseases carried by non-flying insects have not yet been intensively investigated mathematically. In this thesis we model two of these diseases, "Louse-Borne Relapsing Fever (LBRF)" and "Chagas Disease". Various aspects of each disease are considered using five mathematical models.

In Chapter 2, we present a detailed derivation and analysis of a model consisting of seven coupled delay differential equations for Louse Borne Relapsing Fever (LBRF), a disease transmitted from human to human by the body louse "Pediculus humanus humanus. Delays model the latency stages of LBRF in humans and lice, which vary in duration from individual to individual, and are therefore modelled using distributed delays with relatively general kernels. A particular feature of the transmission of LBRF to a human is that it involves the death of the louse, usually by crushing which has the effect of releasing the infected body fluids of the dead louse onto the hosts skin. Careful attention is paid to this aspect. We obtain results on existence, positivity, boundedness, linear and nonlinear stability, and persistence. We also derive a basic reproduction number R_0 for the model.

In Chapter 3 we add seasonality to the LBRF model of Chapter 2 to incorporate weather-dependent characteristics of LBRF transmission which required careful re-derivation of some terms. The target state is then a disease-free periodic solution and sufficient conditions for its existence are presented. Results are backed up with numerical simulations.

Chapter 4 presents an ODE model for Chagas disease modelling both human and vector (kissing bugs) population compartments. The model incorporates vertical transmission and both the acute and chronic phases of Chagas in humans. We present results on the positivity and boundedness of solutions and linear stability analysis in terms of a basic reproduction number R_0. Lyapunov analysis shows that Chagas disease can be globally eradicated under certain conditions. However, an endemic steady state can exist under different conditions. We also investigate the persistence of Chagas disease in both human and vector populations. We establish both weak and strong disease persistence under the condition R_0˃1 provided vectors are present. Again, numerical simulations confirm the results.

In Chapter 5, we use a reaction-diffusion system to study the spatial spread of Chagas disease due to the slow movement of the non-flying vectors that use small mammals as vehicles. We study the travelling wave-front solutions and their speed using linearised theory. We confirm that this analysis correctly predicts the speed of the travelling wave using numerical simulations of the initial value problem. The system is therefore linearly determinate. Results on positivity and boundedness of solutions of the reaction-diffusion system are also established.

Finally, in Chapter 6, we improve the Chagas model established in Chapter 4 by incorporating age-structure. We incorporate age dependence in the human population only. We study the disease-free steady state of the age-structured model and its stability, and establish a formula for the basic reproduction number R_0^{as}, showing that the disease-free steady state is locally stable when R_0^{as}˂1 and unstable when R_0^{as}˃1.