Dengue fever is increasing in geographical range, spread by invasion of its vector mosquitoes. The trade in second-hand tires has been implicated as a factor in this process because they act as mobile reservoirs of mosquito eggs and larvae. Regional transportation of tires can create linkages between rural areas with dengue and disease-free urban areas, potentially giving rise to outbreaks even in areas with strong local control measures. In this work we sought to model the dynamics of mosquito transportation via the tire trade, in particular to predict its role in causing unexpected dengue outbreaks through vertical transmission of the virus across generations of mosquitoes. We also aimed to identify strategies for regulating the trade in second-hand tires, improving disease control. We created a mathematical model which captures the dynamics of dengue between rural and urban areas, taking into account the movement and storage time of tires, and mosquito diapause. We simulate a series of scenarios in which a mosquito population is introduced to a dengue-free area via movement of tires, either as single or multiple events, increasing the likelihood of a dengue outbreak. A persistent disease state can be induced regardless of whether urban conditions for an outbreak are met, and an existing endemic state can be enhanced by vector input. Finally we assess the potential for regulation of tire processing as a means of reducing the transmission of dengue fever using a specific case study from Puerto Rico. Our work demonstrates the importance of the second-hand tire trade in modulating the spread of dengue fever across regions, in particular its role in introducing dengue to disease-free areas. We propose that reduction of tire storage time and control of their movement can play a crucial role in containing dengue outbreaks.
A regularization method is proposed for the polynomial approximation of a function from its approximated values in a fixed family of nodes. As a regularization parameter we consider the number of nodes. We present explicit expressions for the optimal number of nodes in terms of the original error of the approximated values of the function. These problems appear frequently in studying inverse problems and when a smoothing technique should be applied to a series of numerical data. We obtain estimation of the approximation error by means of discrete versions of a convolution operators with polynomial kernels, and we observe the differences between the use of positive and non positive kernels.Some numerical examples are provided to illustrate the efficiency and computational performance of the method. They also help us to compare different criteria for the construction of polynomial approximations.
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