Lie Point Symmetries and Commuting Flows for Equations on Lattices

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice

Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Supersymmetric KdV equation: Darboux transformation and discrete systems

For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the B\"{a}cklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related B\"{a}cklund transformation are used to construct integrable super differential-difference and difference-difference systems. The continuum limits of these discrete systems and of their Lax pairs are also considered.Comment: 13pages, submitted to Journal of Physics

Lie discrete symmetries of lattice equations

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlev\'e I equation and of the Toda lattice equation

Difference schemes with point symmetries and their numerical tests

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure

Can the UNAIDS 90-90-90 target be achieved? A systematic analysis of national HIV treatment cascades

Background In 2014, the Joint United Nations Programme on HIV and AIDS (UNAIDS) and partners set the ‘90-90-90 targets’; aiming to diagnose 90% of all HIV positive people, provide antiretroviral therapy (ART) for 90% of those diagnosed and achieve viral suppression for 90% of those treated, by 2020. This results in 81% of all HIV positive people on treatment and 73% of all HIV positive people achieving viral suppression. We aimed to analyse how effective national HIV treatment programmes are at meeting these targets, using HIV care continuums or cascades. Methods We searched for HIV treatment cascades for 196 countries in published papers, conference presentations, UNAIDS databases and national reports. Cascades were constructed using reliable, generalisable, recent data from national, cross-sectional and longitudinal study cohorts. Data were collected for four stages; total HIV positive people, diagnosed, on treatment and virally suppressed. The cascades were categorised as complete (four stages) or partial (3 stages), and analysed for ‘break points’ defined as a drop >10% in coverage between consecutive 90-90-90 targets. Results 69 country cascades were analysed (32 complete, 37 partial). Diagnosis (target one—90%) ranged from 87% (the Netherlands) to 11% (Yemen). Treatment coverage (target two—81% on ART) ranged from 71% (Switzerland) to 3% (Afghanistan). Viral suppression (target three—73% virally suppressed) was between 68% (Switzerland) and 7% (China). Conclusions No country analysed met the 90-90-90 targets. Diagnosis was the greatest break point globally, but the most frequent key break point for individual countries was providing ART to those diagnosed. Large disparities were identified between countries. Without commitment to standardised reporting methodologies, international comparisons are complex