Zero Singularity Research Articles

Numerical and bifurcation analyses for a population model of HIV chemotherapy

A competitive implicit finite-difference method will be developed and used for the solution of a non-linear mathematical model associated with the administration of highly-active chemotherapy to an HIV-infected population aimed at delaying progression to disease. The model, which assumes a non-constant transmission probability, exhibits two steady states; a trivial steady state (HIV-infection-free population) and a non-trivial steady state (population with HIV infection). Detailed stability and bifurcation analyses will reveal that whilst the trivial steady state only undergoes a static bifurcation (single zero singularity), the non-trivial steady state can not only exhibit static and dynamic (Hopf) bifurcations, but also a combination of two types of bifurcation (a double zero singularity). Although the Gauss–Seidel-type method to be developed in this paper is implicit by construction, it enables the various sub-populations of the model to be monitored explicitly as time t tends to infinity. Furthermore, the method will be seen to be more competitive (in terms of numerical stability) than some well-known methods in the literature. The method is used to determine the impact of the chemotherapy treatment by comparing the population sizes at equilibrium of the treated and untreated infecteds.

A Spectral Theory for Hybrid Modulation

Certain properties of periodic signals are defined in terms of the zeros and singularities of associated analytic functions of a complex time variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> . This algebraic approach is a generalization of analytic signal theory, and leads to the conception of hybrid modulation as the superposition of two <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -plane zero-singularity (ZS) patterns associated with amplitude- and angle-modulating signals, respectively. It is shown that important spectral properties of the modulated signal, such as band limitation, are explicit in the resultant pattern. Signal design is then interpreted in terms of ZS manipulation and placement. The theory is applied in a unified approach to compatible singlesideband (CSSB) modulation systems. It is shown that two types of proposed CSSB systems give rise to essentially nonband-limited output signals. The relation between conventional and single-sideband (SSB) angle modulation is also discussed in terms of their characteristic ZS patterns.