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 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 examine a family of three-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement...

A method already introduced by the last two authors for finding the integrable cases of three-dimensional autonomous ordinary differential equations based on the Frobenius integrability theorem is described in detail. Using this method and computer algebra, the so-called three-dimensional...

We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one and reduce the system to a single differential equation....

Integrability in discrete-time systems can assume various forms. This work deals with mappings that can be linearized, that is, mappings the solution of which can be obtained from the solution of linear difference equations. We start with the discrete analogue of the Riccati equation and then...

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 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...

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are...

We use the dressing transformation in order to reconstruct one-dimensional Hamiltonians starting from their spectra. Whenever the given spectrum departs from oscillator-like local behaviour the resulting potential is fractal. An estimate of this fractal dimension is presented.