Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction-diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs. The new capabilities are then applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is each replaced by the spectral fractional Laplacian). Our study reveals some common effects, which contributes to a better understanding of fractional diffusion in generic reaction-diffusion systems. In particular, we investigate the changes in snaking bifurcation diagrams and also study the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order can induce very significant qualitative and quantitative changes in global bifurcation structures

The effects of fecundity, mortality and distribution of the initial condition in phenological models

Pest phenological models describe the cumulative flux of the individuals into each stage of the life cycle of a stage-structured population. Phenological models are widely used tools in pest control decision making. Despite the fact that these models do not provide information on population abundance, they share some advantages with respect to the more sophisticated and complex demographic models. The main advantage is that they do not require data collection to define the initial conditions of model simulation, reducing the effort for field sampling and the high uncertainty affecting sample estimates. Phenological models are often built considering the developmental rate function only. To the aim of adding more realism to phenological models, in this paper we explore the consequences of improving these models taking into consideration three additional elements: the age distribution of individuals which exit from the overwintering phase, the age- and temperature-dependent profile of the fecundity rate function and the consideration of a temperature-dependent mortality rate function. Numerical simulations are performed to investigate the effect of these elements with respect to phenological models considering development rate functions only. To further test the implications of different models formulation, we compare results obtained from different phenological models to the case study of the codling moth (Cydia pomonella) a primary pest of the apple orchard. The results obtained from model comparison are discussed in view of their potential application in pest control decision support

double blind placebo controlled randomized trial on low dose azithromycin prophylaxis in patients with primary antibody deficiencies

Background Lacking protective antibodies, patients with primary antibody deficiencies (PADs) experience frequent respiratory tract infections, leading to chronic pulmonary damage. Macrolide prophylaxis has proved effective in patients with chronic respiratory diseases. Objective We aimed to test the efficacy and safety of orally administered low-dose azithromycin prophylaxis in patients with PADs. Methods We designed a 3-year, double-blind, placebo-controlled, randomized clinical trial to test whether oral azithromycin (250 mg administered once daily 3 times a week for 2 years) would reduce respiratory exacerbations in patients with PADs and chronic infection–related pulmonary diseases. The primary end point was the number of annual respiratory exacerbations. Secondary end points included time to first exacerbation, additional antibiotic courses, number of hospitalizations, and safety. Results Eighty-nine patients received azithromycin (n = 44) or placebo (n = 45). The number of exacerbations was 3.6 (95% CI, 2.5-4.7) per patient-year in the azithromycin arm and 5.2 (95% CI, 4.1-6.4) per patient-year in the placebo arm (P = .02). In the azithromycin group the hazard risk for having an acute exacerbation was 0.5 (95% CI, 0.3-0.9; P = .03), and the hazard risk for hospitalization was 0.5 (95% CI, 0.2-1.1; P = .04). The rate of additional antibiotic treatment per patient-year was 2.3 (95% CI, 2.1-3.4) in the intervention group and 3.6 (95% CI, 2.9-4.3) in the placebo group (P = .004). Haemophilus influenzae and Streptococcus pneumoniae were the prevalent isolates, and they were not susceptible to macrolides in 25% of patients of both arms. Azithromycin's safety profile was comparable with that of placebo. Conclusion The study reached the main outcome centered on the reduction of exacerbation episodes per patient-year, with a consequent reduction in additional courses of antibiotics and risk of hospitalization