This work studies a mathematical model for the dynamics of Chagas disease, a parasitic disease that affects humans and domestic mammals throughout rural areas in Central and South America. It presents a modified version of the model found in Spagnuolo et al. [A model for Chagas disease with controlled spraying, J. Biol. Dyn. 5 (2011), pp. 299-317] with a delayed logistic growth term, which captures an overshoot, beyond the vector carrying capacity, in the total vector population when the blood meal supply is large. It studies the steady states of the system in the case of constant coefficients without spraying, and the analysis shows that for given-averaged parameters, the endemic equilibrium is stable and attracting. The numerical simulations of the model dynamics with time-dependent coefficients are shown when interruptions in the annual insecticide spraying cycles are taken into account. Simulations show that when there are spraying schedule interruptions, spraying may become ineffective when the blood meal supply is large.

Let (R, m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {R n } of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {R n } n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S = n≥0 R n . Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the R n . We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S = T ∩ V . If S is archimedian, then the complete integral closure S * of S has the form S * = W ∩ T , where W is the rank 1 valuation overring of V .

Let (R, m) be a regular local ring of dimension at least 2. For each valuation domain birationally dominating R, there is an associated sequence {R n } of local quadratic transforms of R. We consider the case where this sequence {R n } n≥0 is infinite and examine properties of the integrally closed local domain S = n≥0 R n in the case where S is not a valuation domain. For this sequence, there is an associated boundary valuation ring V = n≥0 i≥n V i , where V i is the order valuation ring of R i . There exists a unique minimal proper Noetherian overring T of S. T is the regular Noetherian UFD obtained by localizing outside the maximal ideal of S and S = V ∩ T . In the present paper, we define functions w and e, where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals. We describe V explicitly in terms of w and e and prove that V is either rank 1 or rank 2. We define an invariant τ associated to S that is either a positive real number or +∞. If τ is finite, then S is archimedean and T is not local. In this case, the function w defines the rank 1 valuation overring W of V and W dominates S. The rational dependence of τ over w(T × ) determines whether S is completely integrally closed and whether V has rank 1. We give examples where S is completely integrally closed. If τ is infinite, then S is non-archimedean and T is local. In this case, the function e defines the rank 1 valuation overring E of V . The valuation ring E is a DVR and E dominates T , and in certain cases we prove that E is the order valuation ring of T .

Let {R n , m n } n≥0 be an infinite sequence of regular local rings with R n+1 birationally dominating R n and m n R n+1 a principal ideal of R n+1 for each n. We examine properties of the integrally closed local domain S = n≥0 R n .

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