Chagas disease is a major health problem in rural South and Central America where an estimated 8 to 11 million people are infected. It is a vector-borne disease caused by the parasite Trypanosoma cruzi, which is transmitted to humans mainly through the bite of insect vectors from several species of so-called kissing bugs or triatomines. One of the control measures to reduce the spread of the disease is insecticide spraying of housing units to prevent (re)infestation by the triatomines. However, (re)infestation of housing units by triatomines has been shown to occur as early as four to six months after insecticide-based control interventions. In this thesis, (re)infestation models that shed light on the effectiveness of the insecticide spraying are constructed and analyzed.

In Chapter 2 we introduce ODE-based mathematical models of the effect of insecticide spraying on triatomine (re)infestation. Conditions for existence and uniqueness of infestation-free and endemic equilibria are established for these models, with or without assuming migration of triatomines from sylvatic areas. Additionally, conditions for local as well as global stability of the equilibria are derived.

In Chapter 3 we describe a hysteresis-like effect when two different spraying rates lead to two different numbers of infested units at equilibrium. We prove that it occurs under fairly general conditions and under a variety of different modeling assumptions. These results have potentially important implications for designing cost- effective spraying strategies.

A mathematical model of intermittent spraying at fixed time intervals is introduced in Chapter 4. The model is based on a mixture of differential and difference equations. Conditions for the existence and uniqueness of a fixed point are established for this model. Numerical results indicate richer possibilities for the dynamics than in the ODE-based models of Chapter 2. In particular, periodic points of period 2 were discovered.

In Chapter 5 we compare the cost-effectiveness of spraying strategies: Spraying infested units only vs. spraying all units, and continuous vs. intermittent spraying. The results show that while spraying infested units only is more cost-effective than spraying all units at a fixed cost, there is no significant difference between cost-effectiveness of continuous and intermittent spraying.

In Chapter 6 we describe a more comprehensive Chagas disease model that takes into account all major known transmission and migration routes.

The models of Chapters 2 and 4 have three key rate parameters that need to be estimated from data. In Chapter 7 we explore a conjecture on infestation patterns that could in principle allow disentanglement of the rate of house-to-house infestation and the rate of infestation from sylvatic areas. The translation of the conjecture into a mathematical framework involves a certain centrality measure. We describe this measure and we report on an attempt to confirm the conjecture by statistical analysis of maps of 63 communities that showed patterns of infestation.