We present a three-component model which could represent the reaction of the organism to pathogen invasion. A continuous-time (differential) model is constructed first. Its discrete analogue is then derived and is used for numerical simulations which show a great variety of behaviours. We also...

We present the discrete systems which result from the discrete Painlevé equations q-PVI and d-PV associated to the affine Weyl group E7(1). Two different procedures (“limits” and “degeneracies”) are used, giving rise to a host of new discrete Painlevé equations but also to some...

We present a simple model of population dynamics in the presence of an infection. The model is based on discrete-time equations for sane and infected populations in interaction and correctly describes the dynamics of the epidemic. We find that for some choices of the parameters, the model can...

We study a model of an epidemic where the individuals which are cured from the infection are not permanently immunised but have a finite probability of becoming reinfected. We show that the epidemic does not follow the usual pattern of growth and decay but rather oscillates towards an...

We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete form of this equation, the Gambier mapping, and present...

We apply the recently proposed integrability criterion for differential-difference systems (that blends the classical Painlevé analysis with singularity confinement for discrete systems) to a class of first-order differential-delay equations. Our analysis singles out the family of bi-Riccati...

We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painlevé property. In the continuous case, the solutions of the Gambier equation is obtained through a system of Riccati equations. The same holds true...

Starting from the standard form of the five discrete Painlevé equations we show how one can obtain (through appropriate limits) a host of new equations which are also the discrete analogues of the continuous Painlevé equations. A particularly interesting technique is the one based on the...

We present a special discrete analogue of the Gambier coupled-Riccati system where the coupling is introduced in an additive way. We show that there exist choices of the parameters for which the singularities of the mapping are confined. Computing the continuous limit of the mapping we show that...

We study the special limits of discrete Painlevé equations belonging to the q-PVI and d-PV families, when the independent variable goes to 1 or 0, respectively. We obtain discrete systems which are shown to be either contiguities of solutions of continuous Painlevé equations, usually of PVI...