The effect of seasonal host birth rates on disease persistence.

In this paper, we add seasonality to the birth rate of an SIR model with density dependence in the death rate. We find that disease persistence can be explained by considering the average value of the seasonal term. If the basic reproductive ratio R(0)>1 with this average value then the disease will persist and if R(0)<1 with this average value then the disease will die out. However, if the underlying non-seasonal model displays oscillations towards the equilibrium then the dynamics of the seasonal model can become more complex. In this case, the seasonality can interact with the underlying oscillations, resonate and the population can display a range of complex behaviours including chaos. We discuss these results in terms of two examples, Cowpox in bank voles and Rabbit Haemorrhagic disease in rabbits.

The effect of seasonal host birth rates on population dynamics: the importance of resonance.

Many of the simple mathematical models currently in use often fail to capture important biological factors. Here we extend current models of insect-pathogen interactions to include seasonality in the birth rate. In particular, we consider the SIR model with self-regulation when applied to specific cases--rabbit haemorrhagic disease and fox rabies. In this paper, we briefly summarize the results of the model with a constant time-independent birth rate, a, which we then replace with the time dependent birth rate a(t), to investigate how this effects the dynamics of the host population. We can split parameter space into an area in which the model without seasonality has no oscillations, in which case a simple averaging rule predicts the behaviour. Alternatively, in the area where oscillations to the equilibrium do occur in the non-seasonal model, disease persistence is more complicated and we get more complex dynamical behaviour in this case. We apply resonance techniques to discover the structure of the subharmonic modes of the SIR model with self-regulation. We then look at whether many biological systems are likely to display these "resonant" dynamics and find that we would expect them to be widespread.

Identification of appropriate primitive polynomials to avoid cross-contamination in multifocal electroretinogram responses.

Abstract-The basis of the multifocal/electroretinogram is the use of a decimated m-sequence for simultaneous and independent stimulation of many areas of the visual pathway. The purpose of this study was to investigate the effects of cross-contamination from higher orders of the response. A series of primitive polynomials were found by construction of finite fields. The first-order ERG response was formed by cross-correlation of m-sequence with the physiological response. A second-order response was formed by investigation of particular flash sequences of the stimulation sequence and cross-correlation of a second-order m-sequence with the physiological response. Zech logarithms were used to identify cross-contamination between the various first and second-order sequences. Tables of good and bad primitive polynomials were constructed for degrees 12-16, and the effects of window length and decimation length were examined. When the sequence was decimated into 128 areas, and a window of length 16 was examined, cross-contamination occurred in all sequences generated from primitive polynomials of degree less than or equal to 12, but in only 26% of degree 14, and 5.6% of degree 16. A photodiode (artificial eye) was used in an experiment to construct trace arrays showing responses from 61 individual areas. Additional waveforms were present on the trace array when the experiment was carried out with a bad primitive polynomial. The use of finite field theory to generate primitive polynomials and Zech logarithm analysis allowed prediction of which primitive polynomials were suitable for m-sequence generation for multifocal electroretinography. Practical investigations supported the theoretical analysis. This has important implications for developers of multifocal electrophysiology systems.