Quantum Chaos: Spectral Analysis of Floquet Operators

The Floquet operator, defined as the time-evolution operator over one period, plays a central role in the work presented in this thesis on periodically perturbed quantum systems. Knowledge of the spectral nature of the Floquet operator gives us information on the dynamics of such systems. The work presented here on the spectrum of the Floquet operator gives further insight into the nature of chaos in quantum mechanics. After discussing the links between the spectrum, dynamics and chaos and pointing out an ambiguity in the physics literature, I present a number of new mathematical results on the existence of different types of spectra of the Floquet operator. I characterise the conditions for which the spectrum remains pure point and then, on relaxing these conditions, show the emergence of a continuous spectral component. The nature of the continuous spectrum is further analysed, and shown to be singularly continuous. Thus, the dynamics of these systems are a candidate for classification as chaotic. A conjecture on the emergence of a continuous spectral component is linked to a long standing number-theoretic conjecture on the estimation of finite exponential sums.Comment: PhD Thesis (2004-5), 176 pages. Copyright and publication information entere

Spatial information allows inference of the prevalence of direct cell-to-cell viral infection

The role of direct cell-to-cell spread in viral infections - where virions spread between host and susceptible cells without needing to be secreted into the extracellular environment - has come to be understood as essential to the dynamics of medically significant viruses like hepatitis C and influenza. Recent work in both the experimental and mathematical modelling literature has attempted to quantify the prevalence of cell-to-cell infection compared to the conventional free virus route using a variety of methods and experimental data. However, estimates are subject to significant uncertainty and moreover rely on data collected by inhibiting one mode of infection by either chemical or physical factors. These methods assume that this inhibition process fully eliminates its target mode of infection while exactly preserving the dynamics of the other. In this work, we provide a framework for estimating the prevalence of cell-to-cell infection from data which is experimentally obtainable without the need for additional interventions, and two standard mathematical models for viral dynamics with the two modes of infection. We provide guidance for the design of relevant experiments and mathematical tools for accurately inferring the prevalence of cell-to-cell infection

Incorporating population dynamics into household models of infectious disease transmission

Most household models of disease transmission assume static household distributions. Although this is a reasonable simplification for assessing vaccination strategies at a single point in time or over the course of an outbreak, it has considerable drawbacks for assessing long term vaccination policies or for predicting future changes in immunity. We demonstrate that household models that include births, deaths and movement between households can show dramatically different patterns of infection and immunity to static population models. When immunity is assumed to be life-long, the pattern of births by household size is the key driver of infection, suggesting that the influx of susceptibles has most impact on infection risk in the household. In a comparison of 12 countries, we show that both the crude birth rate and the mean household size affect the risk of infection in households

The impact of the internet on the practice of general practitioners and community pharmacists in Northern Ireland

Objective The objective of this study was to gain an insight into the use of the internet for practice-related purposes by community pharmacists and general practitioners (GPs) in Northern Ireland, and to gather information about their experiences relating to patients and the internet. Method A postal questionnaire survey of all community pharmacies (n=522) and all GPs practising in Northern Ireland (n=1081). Results A total of 542 completed questionnaires were returned, giving an overall response rate of 34%. The majority of respondents had access to the internet in their workplace, and approximately 60% of respondents in each profession accessed health-related websites on up to five occasions per week. Of those who did not access health-related websites, lack of time was the main reason cited. The most popular sites for both professions were online journals. Significant differences were found in the activities undertaken by the two professions whilst online. Significantly more GPs than community pharmacists reported searching for disease-related (non-drug) information, using web-based disease management tools or reading online journal articles. Few respondents reported recommending websites to patients, although significantly more GPs than pharmacists did so. Significantly more pharmacists had been approached or felt challenged by patients who had downloaded information from the internet. GPs were more likely to communicate with colleagues about patients by email but neither profession reported frequent correspondence with patients by email. Conclusions Both professions used the internet regularly as a source of health-related information and both had to deal with 'internet-informed', (or sometimes misinformed) patients. Community pharmacists were more likely to feel challenged by these patients and GPs sometimes had to deal with unnecessarily worried patients or patients with unrealistic expectations. Both professions will have to change working practices to accommodate the impact of the internet. This will have significant future training implications

On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

By a straightforward generalisation, we extend the work of Combescure from rank-1 to rank-N perturbations. The requirement for the Floquet operator to be pure point is established and compared to that in Combescure. The result matches that in McCaw. The method here is an alternative to that work. We show that if the condition for the Floquet operator to be pure point is relaxed, then in the case of the delta-kicked Harmonic oscillator, a singularly continuous component of the Floquet operator spectrum exists. We also provide an in depth discussion of the conjecture presented in Combescure of the case where the unperturbed Hamiltonian is more general. We link the physics conjecture directly to a number-theoretic conjecture of Vinogradov and show that a solution of Vinogradov's conjecture solves the physics conjecture. The result is extended to the rank-N case. The relationship between our work and the work of Bourget on the physics conjecture is discussed.Comment: 25 pages, published in Journal of Mathematical Physic

Diagnosis and Antiviral Intervention Strategies for Mitigating an Influenza Epidemic

BACKGROUND: Many countries have amassed antiviral stockpiles for pandemic preparedness. Despite extensive trial data and modelling studies, it remains unclear how to make optimal use of antiviral stockpiles within the constraints of healthcare infrastructure. Modelling studies informed recommendations for liberal antiviral distribution in the pandemic phase, primarily to prevent infection, but failed to account for logistical constraints clearly evident during the 2009 H1N1 outbreaks. Here we identify optimal delivery strategies for antiviral interventions accounting for logistical constraints, and so determine how to improve a strategy's impact. METHODS AND FINDINGS: We extend an existing SEIR model to incorporate finite diagnostic and antiviral distribution capacities. We evaluate the impact of using different diagnostic strategies to decide to whom antivirals are delivered. We then determine what additional capacity is required to achieve optimal impact. We identify the importance of sensitive and specific case ascertainment in the early phase of a pandemic response, when the proportion of false-positive presentations may be high. Once a substantial percentage of ILI presentations are caused by the pandemic strain, identification of cases for treatment on syndromic grounds alone results in a greater potential impact than a laboratory-dependent strategy. Our findings reinforce the need for a decentralised system capable of providing timely prophylaxis. CONCLUSIONS: We address specific real-world issues that must be considered in order to improve pandemic preparedness policy in a practical and methodologically sound way. Provision of antivirals on the scale proposed for an effective response is infeasible using traditional public health outbreak management and contact tracing approaches. The results indicate to change the transmission dynamics of an influenza epidemic with an antiviral intervention, a decentralised system is required for contact identification and prophylaxis delivery, utilising a range of existing services and infrastructure in a "whole of society" response

Predicting the outcomes of new short-course regimens for multidrug-resistant tuberculosis using intrahost and pharmacokinetic-pharmacodynamic modelling

Background: Short-course regimens for multi-drug resistant tuberculosis (MDR-TB) are urgently needed. Limited data suggest that bedaquiline (BDQ), when used in conjunction with other drugs, improves treatment outcomes and potentially shorten MDR-TB treatment duration to less than six months. Further assessment on the efficacy of short-course BDQ-containing regimens is required before recommendations can be made about its value in MDR-TB treatment. Mathematical models combining drug pharmacokinetics-pharmacodynamics (PK-PD) with the intrahost immune response can provide a platform to investigate different dosing strategies to identify highly effective regimens. Materials/methods: A mathematical model was developed to mimic the human immune response to TB. Major elements of the immune response to TB including macrophages, cytokines and lymphocytes were incorporated. This model was then combined with a PK-PD model to simulate various short-course BDQ-containing regimens, and estimate their anti-mycobacterial effects. These regimens consisted of an initial intensive phase with BDQ, moxifloxacin (MXF), clofazimine (CFZ), pyrazinamide (PZA), isoniazid (INH) and kanamycin (KNM), followed by a continuation phase with BDQ, MXF, CFZ and PZA. Various durations of treatment were investigated and a comparative analysis of their efficacy was undertaken in order to identify highly effective regimens. Results: We found that treatment duration for MDR-TB can be reduced to just 18 weeks while still maintaining a very high treatment success rate (100% for daily BDQ for two weeks during the intensive phase, or 95% when BDQ is given daily for one week during the intensive phase). The estimated time to bacterial clearance of these regimens ranges from 27 to 33 days. Achieving optimal exposure early, in the first four weeks of treatment, is critical for successful treatment. Intermittent dosing of MXF (three times weekly or weekly) does not compromise treatment efficacy. Conclusions: This study represents a novel approach to the global challenge of MDR-TB. Our study shows that MDR-TB treatment could potentially be further shortened to four months with BDQ. The findings provide the justification for empirical evaluation of short-course BDQ-containing regimens. If BDQ-containing regimens are found to improve outcomes then we anticipate clear cost-savings and a subsequent improvement in the efficiency of national TB programs