Sixth-order adaptive non-uniform grids for singularly perturbed boundary value problems

In this paper, a sixth order adaptive non-uniform grid has been developed for solving a singularly perturbed boundary-value problem (SPBVP) with boundary layers. For this SPBVP with a small parameter in the leading derivative, an adaptive finite difference method based on the equidistribution principle, is adopted to establish 6th order of convergence. To achieve this supra-convergence, we study the truncation error of the discretized system and obtain an optimal adaptive non-uniform grid. Considering a second order three-point central finite-difference scheme, we develop sixth order approximations by a suitable choice of the underlying optimal adaptive grid. Further, we apply this optimal adaptive grid to nonlinear SPBVPs, by using an extra approximations of the nonlinear term and we obtain almost 6th order of convergence. Unlike other adaptive non-uniform grids, our strategy uses no pre-knowledge of the location and width of the layers. We also show that other choices of the grid distributions lead to a substantial degradation of the accuracy. Numerical results illustrate the effectiveness of the proposed higher order adaptive numerical strategy for both linear and nonlinear SPBVPs

Nitazoxanide Stimulates Autophagy and Inhibits mTORC1 Signaling and Intracellular Proliferation of Mycobacterium tuberculosis

Tuberculosis, caused by Mycobacterium tuberculosis infection, is a major cause of morbidity and mortality in the world today. M. tuberculosis hijacks the phagosome-lysosome trafficking pathway to escape clearance from infected macrophages. There is increasing evidence that manipulation of autophagy, a regulated catabolic trafficking pathway, can enhance killing of M. tuberculosis. Therefore, pharmacological agents that induce autophagy could be important in combating tuberculosis. We report that the antiprotozoal drug nitazoxanide and its active metabolite tizoxanide strongly stimulate autophagy and inhibit signaling by mTORC1, a major negative regulator of autophagy. Analysis of 16 nitazoxanide analogues reveals similar strict structural requirements for activity in autophagosome induction, EGFP-LC3 processing and mTORC1 inhibition. Nitazoxanide can inhibit M. tuberculosis proliferation in vitro. Here we show that it inhibits M. tuberculosis proliferation more potently in infected human THP-1 cells and peripheral monocytes. We identify the human quinone oxidoreductase NQO1 as a nitazoxanide target and propose, based on experiments with cells expressing NQO1 or not, that NQO1 inhibition is partly responsible for mTORC1 inhibition and enhanced autophagy. The dual action of nitazoxanide on both the bacterium and the host cell response to infection may lead to improved tuberculosis treatment