Murray E. Alexander

Modelling strategies for controlling SARS outbreaks

Severe acute respiratory syndrome (SARS), a new, highly contagious, viral disease, emerged in China late in 2002 and quickly spread to 32 countries and regions causing in excess of 774 deaths and 8098 infections worldwide. In the absence of a rapid diagnostic test, therapy or vaccine, isolation of individuals diagnosed with SARS and quarantine of individuals feared exposed to SARS virus were used to control the spread of infection. We examine mathematically the impact of isolation and quarantine on the control of SARS during the outbreaks in Toronto, Hong Kong, Singapore and Beijing using a deterministic model that closely mimics the data for cumulative infected cases and SARS–related deaths in the first three regions but not in Beijing until mid–April, when China started to report data more accurately. The results reveal that achieving a reduction in the contact rate between susceptible and diseased individuals by isolating the latter is a critically important strategy that can control SARS outbreaks with or without quarantine. An optimal isolation programme entails timely implementation under stringent hygienic precautions defined by a critical threshold value. Values below this threshold lead to control, but those above are associated with the incidence of new community outbreaks or nosocomial infections, a known cause for the spread of SARS in each region. Allocation of resources to implement optimal isolation is more effective than to implement sub–optimal isolation and quarantine together. A community–wide eradication of SARS is feasible if optimal isolation is combined with a highly effective screening programme at the points of entry.

A wavelet-based method for improving signal-to-noise ratio and contrast in MR images

Magnetic resonance (MR) images acquired with fast measurement often display poor signal-to-noise ratio (SNR) and contrast. With the advent of high temporal resolution imaging, there is a growing need to remove these noise artifacts. The noise in magnitude MR images is signal-dependent (Rician), whereas most de-noising algorithms assume additive Gaussian (white) noise. However, the Rician distribution only looks Gaussian at high SNR. Some recent work by Nowak employs a wavelet-based method for de-noising the square magnitude images, and explicitly takes into account the Rician nature of the noise distribution. In this article, we apply a wavelet de-noising algorithm directly to the complex image obtained as the Fourier transform of the raw k-space two-channel (real and imaginary) data. By retaining the complex image, we are able to de-noise not only magnitude images but also phase images. A multiscale (complex) wavelet-domain Wiener-type filter is derived. The algorithm preserves edges better when the Haar wavelet rather than smoother wavelets, such as those of Daubechies, are used. The algorithm was tested on a simulated image to which various levels of noise were added, on several EPI image sequences, each of different SNR, and on a pair of low SNR MR micro-images acquired using gradient echo and spin echo sequences. For the simulated data, the original image could be well recovered even for high values of noise (SNR approximately 0 dB), suggesting that the present algorithm may provide better recovery of the contrast than Nowaks method. The mean-square error, bias, and variance are computed for the simulated images. Over a range of amounts of added noise, the present method is shown to give smaller bias than when using a soft threshold, and smaller variance than a hard threshold; in general, it provides a better bias-variance balance than either hard or soft threshold methods. For the EPI (MR) images, contrast improvements of up to 8% (for SNR = 33 dB) were found. In general, the improvement in contrast was greater the lower the original SNR, for example, up to 50% contrast improvement for SNR of about 20 dB in micro-imaging. Applications of the algorithm to the segmentation of medical images, to micro-imaging and angiography (where the correct preservation of phase is important for flow encoding to be possible), as well as to de-noising time series of functional MR images, are discussed.

A Vaccination Model for Transmission Dynamics of Influenza

Despite the availability of preventive vaccines and public health vaccination programs, influenza inflicts substantial morbidity, mortality, and socio-economic costs and remains a major public heal...

Emergence of drug resistance: implications for antiviral control of pandemic influenza.

Given the danger of an unprecedented spread of the highly pathogenic avian influenza strain H5N1 in humans, and great challenges to the development of an effective influenza vaccine, antiviral drugs will probably play a pivotal role in combating a novel pandemic strain. A critical limitation to the use of these drugs is the evolution of highly transmissible drug-resistant viral mutants. Here, we develop a mathematical model to evaluate the potential impact of an antiviral treatment strategy on the emergence of drug resistance and containment of a pandemic. The results show that elimination of the wild-type strain depends crucially on both the early onset of treatment in indexed cases and population-level treatment. Given the probable delay of 0.5–1 day in seeking healthcare and therefore initiating therapy, the findings indicate that a single strategy of antiviral treatment will be unsuccessful at controlling the spread of disease if the reproduction number of the wild-type strain exceeds 1.4. We demonstrate the possible occurrence of a self-sustaining epidemic of resistant strain, in terms of its transmission fitness relative to the wild-type, and the reproduction number . Considering reproduction numbers estimated for the past three pandemics, the findings suggest that an uncontrollable pandemic is likely to occur if resistant viruses with relative transmission fitness above 0.4 emerge. While an antiviral strategy is crucial for containing a pandemic, its effectiveness depends critically on timely and strategic use of drugs.

Unsupervised hierarchical image segmentation with level set and additive operator splitting

This paper presents an unsupervised hierarchical segmentation method for multi-phase images based on a single level set (2-phase) method and the semi-implicit additive operator splitting (AOS) scheme which is stable, fast, and easy to implement. The method successively segments image subregions found at each step of the hierarchy using a decision criterion based on the variance of intensity across the current subregion. The segmentation continues until a specified number of levels has been reached. The segmentation information for sub-images at each stage is stored in a tree data structure, and is used for reconstructing the segmented images. The method avoids the complicated governing equations of the multi-phase segmentation approach, and appears to converge in fewer iterations. The method can easily be parallelized because the AOS scheme decomposes the equations into a sequence of one dimensional systems.

BIFURCATION ANALYSIS OF AN SIRS EPIDEMIC MODEL WITH GENERALIZED INCIDENCE

An SIRS epidemic model, with a generalized nonlinear incidence as a function of the number of infected individuals, is developed and analyzed. Extending previous work, it is assumed that the natural immunity acquired by infection is not permanent but wanes with time. The nonlinearity of the functional form of the incidence of infection, which is subject only to a few general conditions, is biologically justified. The stability analysis of the associated equilibria is carried out, and the threshold quantity (

A non-standard numerical scheme for a generalized Gause-type predator–prey model ☆

\R

A Delay Differential Model for Pandemic Influenza with Antiviral Treatment

) that governs the disease dynamics is derived. It is shown that

A Positivity-preserving Mickens-type Discretization of an Epidemic Model

\R

The Registration of MR Images using Multiscale Robust Methods

, called the basic reproductive number, is independent of the functional form of the incidence. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms are derived for the different types of bifurcation that the model undergoes, including Hopf, saddle-node, and Bogdanov--Takens. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such a...