SATUMA JUNKICHI

We present a systematic approach to the construction of discrete analogues for differential systems. Our method is tailored to first-order differential equations and relies on a formal linearization, followed by a Pade-like rational approximation of an exponential evolution operator. We apply our method to a host of systems for which there exist discretization results obtained by what we call the 'intuitive' method and compare the discretizations obtained. A discussion of our method as compared to one of the Mickens is also presented. Finally we apply our method to a system of coupled Riccati equations with emphasis on the preservation of the integrable character of the differential system.

We examine two integrable discrete lattice equations obtained by Levi and Yamilov. We show that the first one is a form of the lattice KdV equation already obtained by Hirota and Tsujimoto, while the second one is a discrete form of mKdV. We present the Miura transformations between the various equations involved, including the more familiar potential form of the lattice mKdV.

We examine two integrable discrete lattice equations obtained by Levi and Yamilov. We show that the first one is a form of the lattice KdV equation already obtained by Hirota and Tsujimoto, while the second one is a discrete form of mKdV. We present the Miura transformations between the various equations involved, including the more familiar potential form of the lattice mKdV.

We present two models for an epidemic where the individuals are infective over it fixed period of time and which never becomes endemic i.e., no infective individuals remain after the epidemic has run its course. The first model is based oil it delay-difference scheme. We show that, as a function of the delay (which corresponds to the Period of infectiveness) the percentage of non-infected population varies over a wide range. We present also a variant of our model where the recovery rate follows a Poisson law and obtain it discrete version of the SIR model. We estimate the percentage of non-infected population in the two models, show that they lead to almost the same values and present an explanation of this fact. The second model is based oil the assumption that the infection is spread by carriers. Under the hypothesis that the carriers are relatively long-lived, and that the number of the infected ones is a relatively small fraction of the total population of potential carriers, we show that the model reduces to the same version of the discrete SIR obtained by our first model. (C) 2007 Elsevier Ltd. All rights reserved.

We present two models for an epidemic where the individuals are infective over it fixed period of time and which never becomes endemic i.e., no infective individuals remain after the epidemic has run its course. The first model is based oil it delay-difference scheme. We show that, as a function of the delay (which corresponds to the Period of infectiveness) the percentage of non-infected population varies over a wide range. We present also a variant of our model where the recovery rate follows a Poisson law and obtain it discrete version of the SIR model. We estimate the percentage of non-infected population in the two models, show that they lead to almost the same values and present an explanation of this fact. The second model is based oil the assumption that the infection is spread by carriers. Under the hypothesis that the carriers are relatively long-lived, and that the number of the infected ones is a relatively small fraction of the total population of potential carriers, we show that the model reduces to the same version of the discrete SIR obtained by our first model. (C) 2007 Elsevier Ltd. All rights reserved.