Nonparametric Inferences for the Hazard Function with Right Truncation

Incompleteness is a major feature of time-to-event data. As one type of incompleteness, truncation refers to the unobservability of the time-to-event variable because it is smaller (or greater) than the truncation variable. A truncated sample always involves left and right truncation. Left truncation has been studied extensively while right truncation has not received the same level of attention. In one of the earliest studies on right truncation, Lagakos et al. (1988) proposed to transform a right truncated variable to a left truncated variable and then apply existing methods to the transformed variable. The reverse-time hazard function is introduced through transformation. However, this quantity does not have a natural interpretation. There exist gaps in the inferences for the regular forward-time hazard function with right truncated data. This dissertation discusses variance estimation of the cumulative hazard estimator, one-sample log-rank test, and comparison of hazard rate functions among finite independent samples under the context of right truncation. First, the relation between the reverse- and forward-time cumulative hazard functions is clarified. This relation leads to the nonparametric inference for the cumulative hazard function. Jiang (2010) recently conducted a research on this direction and proposed two variance estimators of the cumulative hazard estimator. Some revision to the variance estimators is suggested in this dissertation and evaluated in a Monte-Carlo study. Second, this dissertation studies the hypothesis testing for right truncated data. A series of tests is developed with the hazard rate function as the target quantity. A one-sample log-rank test is first discussed, followed by a family of weighted tests for comparison between finite $K$-samples. Particular weight functions lead to log-rank, Gehan, Tarone-Ware tests and these three tests are evaluated in a Monte-Carlo study. Finally, this dissertation studies the nonparametric inference for the hazard rate function for the right truncated data. The kernel smoothing technique is utilized in estimating the hazard rate function. A Monte-Carlo study investigates the uniform kernel smoothed estimator and its variance estimator. The uniform, Epanechnikov and biweight kernel estimators are implemented in the example of blood transfusion infected AIDS data

The mathematical modelling of the transmission dynamics of HIV/AIDS and the impact of antiviral therapies

Thesis presented for the degree of Doctor of Philosophy, Department of Electronics and Mathematics, Faculty of Science, Technology and Design, the University of LutonThis thesis is concerned with the structure, analysis and numerical solution of the mathematical models used to estimate the transmission dynamics of the Human Immunodeficiency Virus (HIV)) the causative agent of Acquired Immune Deficiency Syndrome (AIDS). Investigations show that the devised deterministic mathematical models in term of system of first-order non-linear ordinary differential equations (ODEs) follow the stochastic nature of the problem at any time. In this thesis a generic form of the deterministic mathematical models is introduced which mirrors the transmission dynamics of HIV/AIDS in populations with different states of affairs, which leads to the division of large-scale and complex mathematical models. When analysing and;or solving a large-scale system of ODEs numerically, the key element in speeding up the process is selecting the maximum possible time step. This work introduces some new techniques used to estimate the maximum possible time step, avoiding the appearance of chaos and divergence in the solution when they are not features of the system. The solution to these mathematical models are presented graphically and numerically, aiming to identify the effect of the anti-HIV therapies and sex education in controlling the disease. The numerical results presented in this thesis indicate that lowering the average number of sexual partners per year is more effective in controlling the disease than the current anti-HIV treatments. For the purpose of this study the mathematical software 'Mathematica 3.0' was used to solve the system of differential equations, modelling HIV/AIDS propagation. This package also provided the graphical detail incorporated in the thesis

Modelling HIV/AIDS epidemic in Nigeria

Nigeria is one of the countries most affected by the HIV/AIDS pandemic, third only to India and South Africa. With about 10% of the global HIV/AIDS cases estimated to be in the country, the public health and socio-economic implications are enormous. This thesis has two broad aims: the first is to develop statistical models which adequately describe the spatial distribution of the Nigerian HIV/AIDS epidemic and its associated ecological risk factors; the second, to develop models that could reconstruct the HIV incidence curve, obtain an estimate of the hidden HIV/AIDS population and a short term projection for AIDS incidence and a measure of precision of the estimates. To achieve these objectives, we first examined data from various sources and selected three sets of data based on national coverage and minimal reporting delay. The data sets are the outcome of the National HIV/AIDS Sentinel Surveillance Survey conducted in 1999, 2001, 2003 and 2005 by the Federal Ministry of Health; the outcome of the survey of 1057 health and laboratory facilities conducted by the Nigerian Institute of Medical Research in 2000; and case by case HIV screening data collected from an HIV/AIDS centre of excellence. A thorough review of methods used by WHO/UNAIDS to produce estimates of the Nigerian HIV/AIDS scenario was carried out. The Estimation and Projection Package (EPP) currently being used for modelling the epidemic partitions the population into at-risk, not-at-risk and infected sub-populations. It also requires some parameter input representing the force of infection and behaviour or high risk adjustment parameter. It may be difficult to precisely ascertain the size of these population groups and parameters in countries as large and diverse as Nigeria. Also, the accuracy of vital rates used in the EPP and Spectrum program is doubtful. Literature on ordinary back-calculation, nonparametric back-calculation, and modified back-calculation methods was reviewed in detail. Also, an indepth review of disease mapping techniques including multilevel models and geostatistical methods was conducted. The existence of spatial clusters was investigated using cluster analysis and some measure of spatial autocorrelation (Moran I and Geary c coefficients, semivariogram and kriging) applied to the National HIV/AIDS Surveillance data. Results revealed the existence of spatial clusters with significant positive spatial autocorrelation coefficients that tended to get stronger as the epidemic developed through time. GAM and local regression fit on the data revealed spatial trends on the north-south and east - west axis. Analysis of hierarchical, spatial and ecological factor effects on the geographical variation of HIV prevalence using variance component and spatial multilevel models was performed using restricted maximum likelihood implemented in R and empirical and full Bayesian methods in WinBUGS. Results confirmed significant spatial effects and some ecological factors were significant in explaining the variation. Also, variation due to various levels of aggregation was prominent. Estimates of cumulative HIV infection in Nigeria were obtained from both parametric and nonparametric back-calculation methods. Step and spline functions were assumed for the HIV infection curve in the parametric case. Parameter estimates obtained using 3-step and 4-step models were similar but the standard errors of these parameters were higher in the 4-step model. Estimates obtained using linear, quadratic, cubic and natural splines differed and also depended on the number and positions of the knots. Cumulative HIV infection estimates obtained using the step function models were comparable with those obtained using nonparametric back-calculation methods. Estimates from nonparametric back-calculation were obtained using the EMS algorithm. The modified nonparametric back-calculation method makes use of HIV data instead of the AIDS incidence data that are used in parametric and ordinary nonparametric back-calculation methods. In this approach, the hazard of undergoing HIV test is different for routine and symptom-related tests. The constant hazard of routine testing and the proportionality coefficient of symptom-related tests were estimated from the data and incorporated into the HIV induction distribution function. Estimates of HIV prevalence differ widely (about three times higher) from those obtained using parametric and ordinary nonparametric back-calculation methods. Nonparametric bootstrap procedure was used to obtain point-wise confidence interval and the uncertainty in estimating or predicting precisely the most recent incidence of AIDS or HIV infection was noticeable in the models but greater when AIDS data was used in the back-projection model. Analysis of case by case HIV screening data indicate that of 33349 patients who attended the HIV laboratory of a centre of excellence for the treatment of HIV/AIDS between October 2000 and August 2006, 7646 (23%) were HIV positive with females constituting about 61% of the positive cases. The bulk of infection was found in patients aged 15-49 years, about 86 percent of infected females and 78 percent of males were in this age group. Attendance at the laboratory and the proportion of HIV positive tests witnessed a remarkable increase when screening became free of charge. Logistic regression analysis indicated a 3-way interaction between time period, age and sex. Removing the effect of time by stratifying by time period left 2-way interactions between age and sex. A Correction factor for underreporting was ascertained by studying attendance at the laboratory facility over two time periods defined by the cost of HIV screening. Estimates of HIV prevalence obtained from corrected data using the modified nonparametric back-calculation are comparable with UN estimates obtained by a different method. The Nigerian HIV/AIDS pandemic is made up of multiple epidemics spatially located in different parts of the country with most of them having the potential of being sustained into the future given information on some risk factors. It is hoped that the findings of this research will be a ready tool in the hands of policy makers in the formulation of policy and design of programs to combat the epidemic in the country. Access to data on HIV/AIDS are highly restricted in the country and this hampers more in-depth modelling of the epidemic. Subject to data availability, we recommend that further work be done on the construction of stratification models based on sex, age and the geopolitical zones in order to estimate the infection intensity in each of the population groups. Uncertainties surrounding assumptions of infection intensity and incubation distribution can be minimized using Bayesian methods in back-projection

Estimation of the burden of people living with Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome (HIV/AIDS) in Kerala state, India.

Background: Worldwide, 36.7 million people were infected with Acquired Immunodeficiency Syndrome (AIDS) by the end of 2015. Over the period 2007 to 2015, there was a declining trend in the prevalence of adult Human Immunodeficiency Virus (HIV) in the state of Kerala, India. The current study aims to find a suitable statistical modelling technique for the distribution of HIV incubation time and predict the cumulative number of AIDS cases. Materials and Methods: The requisite data were obtained from the Kerala State AIDS Control Society (KSACS) for the years 2007 to 2015. To assess the distribution of HIV incubation time, the data of 22 HIV-infected Keralite patients were retrieved from the medical records of a teaching hospital. Data included age, gender, and incubation time. The back-calculation method was utilized to predict the cumulative HIV/AIDS cases. Results: The estimated total cumulative AIDS cases in Kerala for the years 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 and 2015 were found to be 35,777, 48,944, 62,039, 45,669, 45,668, and 43,605, 42,377, 39,362, 37,617, 39,583, 25,414 respectively using back-calculation method with Weibull (2) incubation time distribution. The mean incubation time of the total HIV cases (male and female) was 4.4 years which indicates a rapid progression of the disease in the state of Kerala. Conclusion: The back-calculation method is a powerful tool to estimate the cumulative frequency of AIDS cases; which predicted a declining HIV trend among Keralites. Moreover, the Weibull distribution is the best fitted distribution for HIV incubation time in our population

Modeling disease transmission dynamics with random data and heavy tailed random effects: the Zika case

This work was supported by Research Fund of the Recep Tayyip Erdogan University.In this study, we investigate a compartmental model of Zika Virus transmission under random effects. Random effects enable the analysis of random numerical characteristics of transmission, which cannot be modeled through deterministic equations. Data obtained from Zika studies in the literature are used along with heavy tailed random effects to obtain new random variables for the parameters of the deterministic model. Finally, simulations of the model are carried out to analyze the random dynamics of Zika Virus transmission. Deterministic results are compared with results from the simulations of the random system to underline the advantages of a random modeling approach. It is shown that the random model provides additional results for disease transmission dynamics such as results for standard deviation and coefficients of variation, making it a valuable alternative to deterministic modeling. Random results suggest around 90% - 120% coefficient of variation for the random model underlining the fact that the randomness should not be ignored for the transmission of this disease.Publisher's Versio

A mathematical analysis of the financial and medical impact of hepatitis C among drug users in Perth, Western Australia

The ability of public health planners continues to be hampered by uncertainties encountered with transmissible diseases. Key epidemiological factors such as, how many Western Australian injecting drug users are hepatitis C seropositive or will become infected, duration of intravenous drug use, the intensity of infection, the fraction of those infected that will develop end-stage disease and after how long a period, all combine to limit the ability of a mathematical model to predict future trends. These models can, however, provide information about certain epidemiological parameters and identify data required to predict future trends. They can be applied to make predictions about the course of infection in the individual and provide a guide to the interpretation of the observed data. This research aims to develop a model of the transmission and spread of hepatitis C, adapting existing models used to predict the spread of HIV and AIDS in one and two sex communities. This model will be used to demonstrate the dynamics and incidence of hepatitis C infection among injecting drug users in Perth, Western Australia. Predictions derived from the model will then be used to undertake an analysis of the cost of treating those with hepatitis C and cirrhosis related complications, resulting in a prediction of the financial impact of hepatitis C on the Western Australian community

A modified mathematical model for HIV transmission, AIDS and intervention strategies in Ireland

The aim of this project was to modify transmission models previously developed to describe HIV transmission in Ireland through more precise estimates of parameters delineating sexual and needle sharing behaviour and AIDS related mortality, with a view to examining the extent of spread from IVDUs to the general heterosexual population. To provide predictions on HIV transmission and AIDS cases up to the year 2010 and to examine the sensitivity of the predictions to changes in the key parameters under different assumptions on intervention strategies. Essential data of relevance to HIV transmission were acquired (through the cooperation of the Department of Health, the AIDS Resource Centre, the Drug Treatment Centre, the Ana Liffey Drug Project, and St James’s Hospital, Dublin) and analysed. The parameters obtained from the descriptive and survival analyses were used in a refined deterministic model of HIV transmission which was solved numerically. An attempt at a solution using perturbation methods was undertaken. Results provided a plausible range of projected new HIV infections and AIDS cases up to the year 2010. The model projects that if present behaviour continues, approximately 1009 new infections may be expected in Dublin by the year 2000, and 1496 new infections by 2010. Small changes in the values of key parameters induce significant changes in projected trends, particularly in the longer term. However all projections point to the fact that •Non-IVDU women are particularly susceptible to HIV infection •Early introduction of behaviour change makes a significant difference to the growth of the epidemic •Expanding the Drug Treatment System could help to significantly reduce the spread of HIV •Central collation of data is essential if rational implementation of intervention plans is to be achieve

The Role of RANTES (CCL5) In The Immune Response to Bacterial Infections

Background: Sepsis is a lethal condition that causes organ dysfunction due to a dysregulated host response to an infection. It is one of the leading causes of morbidity and mortality globally with approximately 6 million deaths occurring every year. Transfusion-associated sepsis (T-AS) can occur through the transfusion of contaminated blood components, commonly platelet concentrates (PCs). The “cytokine storm” is thought to be the main pathophysiology behind sepsis and the cytokines secreted within bacterial contaminated PCs could play a key role in the development of T-AS in recipients. This study investigated the effect of bacterially-primed platelet releasates on the activation status of neutrophils and monocytes, with the aim of understanding the link between platelet product contamination, the cytokine storm, and the severity of clinical symptoms in transfusion recipients. Findings: The cytometric bead array assay demonstrated that RANTES was significantly elevated in comparison with IL-1β, IL-6, and TNF-α when platelet rich plasma samples were incubated with planktonic and biofilm forms of Staphylococcus epidermidis and Serratia marcescens. Flow cytometric analysis of the surface markers CD54, CD11b and CD66b on U937 and HL-60 cell lines demonstrated no changes in expression when treated platelet releasates that had been primed with S. epidermidis and wild type (WT) Escherichia coli. Yet treating these cells with platelet releasates primed with multi-resistant E. coli (planktonic or biofilm form) caused a highly significant upregulation in CD54 expression. Platelet-free plasma alone was found to cause a degree of upregulation of all three surface markers in some cases suggesting minor immunomodulatory activity of plasma proteins. Despite its significant release from bacterially-primed platelets, recombinant RANTES (rhRANTES) at different concentrations did not induce significant changes in surface expression of the activation markers on U937 and HL-60 cells however, variable effects on CD54 and CD11b expression were seen on primary human neutrophils. Curiously, CD66b appeared to be downregulated with increasing exogenous rhRANTES concentration. Conclusions: The results may indicate that RANTES requires synergy with additional cytokines/chemokines in order to induce activation of neutrophils and monocytes and cannot act in isolation. Bacterial virulence appears to play a role in the resultant cytokine profiles of platelets evidenced by the stark difference in cell activation caused by multi-resistant E. coli-primed platelets in comparison with WT E. coli and S. epidermidis

Selected Papers in Combinatorics - a Volume Dedicated to R.G. Stanton

Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions