Geoffry Mercer

Combustion waves for gases (Le = 1) and solids (Le→∞)

The traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a premixed solid fuel are revisited using a simpler (than previously) non–dimensional temperature. It turns out to be possible to carry out asymptotic calculations for flame speed and the agreement with corresponding numerical calculations is remarkably good. In each case the uniqueness of the speed is considered using phase plane methods, with a little effort to determine the nature of the ‘cold’ critical point. Consideration of the stability of the travelling combustion wave fronts suggests a period doubling route to chaos for the premixed solid fuel (as the exothermicity is decreased) and corresponds with previous work using different non–dimensional temperature and parameters.

An oscillatory route to extinction for solid fuel combustion waves due to heat losses

A numerical method is used to show that heat loss increases can lead to a period–doubling route to the cessation of propagation of solid fuel combustion. Oscillatory combustion waves are found in certain regions of the parameter space. The behaviour of these oscillatory waves becomes more complex as the heat loss is increased until extinction of the combustion reaction occurs. Large excursions in temperature, above the adiabatic temperature, are possible in the non–adiabatic case close to this extinction point.

Finite difference schemes for multilayer diffusion

Although numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

Modelling the transmission dynamics of dengue in the presence of Wolbachia

Use of the bacterium Wolbachia is an innovative new strategy designed to break the cycle of dengue transmission. There are two main mechanisms by which Wolbachia could achieve this: by reducing the level of dengue virus in the mosquito and/or by shortening the host mosquitos lifespan. However, although Wolbachia shortens the lifespan, it also gives a breeding advantage which results in complex population dynamics. This study focuses on the development of a mathematical model to quantify the effect on human dengue cases of introducing Wolbachia into the mosquito population. The model consists of a compartment-based system of first-order differential equations; seasonal forcing in the mosquito population is introduced through the adult mosquito death rate. The analysis focuses on a single dengue outbreak typical of a region with a strong seasonally-varying mosquito population. We found that a significant reduction in human dengue cases can be obtained provided that Wolbachia-carrying mosquitoes persist when competing with mosquitoes without Wolbachia. Furthermore, using the Wolbachia strain WMel reduces the mosquito lifespan by at most 10% and allows them to persist in competition with non-Wolbachia-carrying mosquitoes. Mosquitoes carrying the WMelPop strain, however, are not likely to persist as it reduces the mosquito lifespan by up to 50%. When all other effects of Wolbachia on the mosquito physiology are ignored, cytoplasmic incompatibility alone results in a reduction in the number of human dengue cases. A sensitivity analysis of the parameters in the model shows that the transmission probability, the biting rate and the average adult mosquito death rate are the most important parameters for the outcome of the cumulative proportion of human individuals infected with dengue.

Evans function stability of non-adiabatic combustion waves

In this paper we investigate the linear stability and properties of the planar travelling non–adiabatic combustion front for the cases of zero and non–zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as ‘fast’ and ‘slow’. The Evans function approach extended by the compound–matrix method is employed to numerically solve the linear–stability problem for the travelling–wave solution. We demonstrate that the ‘slow’ branch of the solutions is unstable, whereas the ‘fast’ branch can be stable or exhibits Hopf or Bogdanov–Takens instability, depending on the parameter values.

Effective reproduction numbers are commonly overestimated early in a disease outbreak.

Reproduction numbers estimated from disease incidence data can give public health authorities valuable information about the progression and likely size of a disease outbreak. Here, we show that methods for estimating effective reproduction numbers commonly give overestimates early in an outbreak. This is due to many factors including the nature of outbreaks that are used for estimation, incorrectly accounting for imported cases and outbreaks arising in subpopulations with higher transmission rates. Awareness of this bias is necessary to correctly interpret estimates from early disease outbreak data.

Modelling the effect of seasonal influenza vaccination on the risk of pandemic influenza infection

BackgroundRecent studies have suggested that vaccination with seasonal influenza vaccine resulted in an apparent higher risk of infection with pandemic influenza H1N1 2009. A simple mathematical model incorporating strain competition and a hypothesised temporary strain-transcending immunity is constructed to investigate this observation. The model assumes that seasonal vaccine has no effect on the risk of infection with pandemic influenza.ResultsResults of the model over a range of reproduction numbers and effective vaccination coverage confirm this apparent increased risk in the Northern, but not the Southern, hemisphere. This is due to unvaccinated individuals being more likely to be infected with seasonal influenza (if it is circulating) and developing hypothesised temporary immunity to the pandemic strain. Because vaccinated individuals are less likely to have been infected with seasonal influenza, they are less likely to have developed the hypothesised temporary immunity and are therefore more likely to be infected with pandemic influenza. If the reproduction number for pandemic influenza is increased, as it is for children, an increase in the apparent risk of seasonal vaccination is observed. The maximum apparent risk effect is found when seasonal vaccination coverage is in the range 20-40%.ConclusionsOnly when pandemic influenza is recently preceded by seasonal influenza circulation is there a modelled increased risk of pandemic influenza infection associated with prior receipt of seasonal vaccine.

Period doubling and chaotic transient in a model of chain-branching combustion wave propagation

The propagation of planar combustion waves in an adiabatic model with two-step chain-branching reaction mechanism is investigated. The travelling combustion wave becomes unstable with respect to pulsating perturbations as the critical parameter values for the Hopf bifurcation are crossed in the parameter space. The Hopf bifurcation is demonstrated to be of a supercritical nature and it gives rise to periodic pulsating combustion waves as the neutral stability boundary is crossed. The increase of the ambient temperature is found to have a stabilizing effect on the propagation of the combustion waves. However, it does not qualitatively change the behaviour of the travelling combustion waves. Further increase of the bifurcation parameter leads to the period-doubling bifurcation cascade and a chaotic regime of combustion wave propagation. The chaotic regime has a transient nature and the combustion wave extinguishes when the bifurcation parameter becomes sufficiently large. For Lewis numbers of fuel close to unity, the parameter regions where pulsating solutions exist become very close to each other and this makes it difficult to experimentally observe the period-doubling. It is shown that the average velocity of pulsating waves is less than the speed of the travelling wave for the same parameter values.

DYNAMICS OF HUMAN MILK EXTRACTION: A COMPARATIVE STUDY OF BREAST FEEDING AND BREAST PUMPING

We describe a mathematical model of the flow and deformation in a human teat. Our aim is to compare the theoretical milk yield during infant breast feeding with that obtained through the use of a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk, whereas breast pumps use a cyclic pattern of suction only. Our model is based on quasi-linear poroelasticity whereby the teat is modelled as a cylindrical porous elastic material saturated with fluid. We impose a cyclic axial suction pressure difference across the teat and impose a radial compressive force moving along the teat which mimics infant suckling. This is compared to the case of cyclic and steady pumping only which models the action of breast pumps. The results illustrate that there is an optimal time to apply the compressive force during the suction cycle that will increase the flow rate in our theoretical teat. The model and results may be of use in the future design of effective breast pumps.

Comparing Breastfeeding and Breast Pumps Using a Computer Model

There is a role for computer models in increasing the understanding of milk extraction from the human teat. A computer model can be used to investigate aspects of extracting milk from the human teat which are not feasible using clinical experiments. In this paper, the behavior of the human teat during an infant suckling and with the use of a breast pump is modeled. The model is used to (1) identify the role of suction and the peristaltic motion of the tongue during suckling and (2) compare the volume of milk extracted by an infant breastfeeding with that obtained using a breast pump. Infants use a peristaltic motion of the tongue, along with some suction, to extract milk. Breast pumps use a cyclic pattern of suction only. In the model, the human teat is represented as a cylindrical porous elastic material saturated with fluid. We mimic an infant suckling by imposing both suction and a peristaltic force in the computer model of the human teat. This is compared to the effect of suction only, which models the action of breast pumps. The results demonstrate that there is an optimal time to apply the peristaltic force during the suction cycle which will increase the milk volume. The model and results may be of use in the future design of effective breast pumps.