A spatial temporal data set of dengue in Peru from 1994-2008 was made available to us by the Ministry of Health of Peru and the analyses of its spatio-temporal patterns motivated the work in this dissertation. We found that aggregated reported data masked the spatio-temporal patterns of dengue over this window in time. A series of models are presented in this dissertation in order to identify mechanisms that capture observed patterns. We have in fact identified a framework capable of capturing dengue outbreaks in Peru. Deterministic and stochastic single and two-patch models are introduced and some of their properties identified via some mathematical analyses complemented with extensive simulations. We find that the asymptotics of the mean field model (final size epidemic), while useful, mask critical details that are central to control and public health policies. We introduce a stochastic migration model that allows us to construct a family of distributions of time T, which the first infected individual leaves the "home" patch, and estimate the variance along the CDF. A two patch model, where each person has a positive probability of being in a patch alternate to his home location, shows the effect coupling coefficients will have on the time between the epidemic peaks. The inclusion of seasonality and human demographics to the two patch model leads to reoccurring outbreaks, as seen in the data. The two approaches of modeling migration, though different mathematically, complement each other in explaining what factors affect the spread of dengue. Finally, optimal control methods are incorporated into our models. We consider what strategy should be used if the objective is to minimize the total number of infected individuals, at a minimal cost, during a fixed time interval. When control is applied to a patch with the basic reproductive number R0 below a certain threshold, it effectively stops the epidemic. Also, control will delay the spread of dengue from one patch to another.
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