Integrating genealogical and dynamical modelling to infer escape and reversion rates in HIV epitopes

The rates of escape and reversion in response to selection pressure arising from the host immune system, notably the cytotoxic T-lymphocyte (CTL) response, are key factors determining the evolution of HIV. Existing methods for estimating these parameters from cross-sectional population data using ordinary differential equations (ODE) ignore information about the genealogy of sampled HIV sequences, which has the potential to cause systematic bias and over-estimate certainty. Here, we describe an integrated approach, validated through extensive simulations, which combines genealogical inference and epidemiological modelling, to estimate rates of CTL escape and reversion in HIV epitopes. We show that there is substantial uncertainty about rates of viral escape and reversion from cross-sectional data, which arises from the inherent stochasticity in the evolutionary process. By application to empirical data, we find that point estimates of rates from a previously published ODE model and the integrated approach presented here are often similar, but can also differ several-fold depending on the structure of the genealogy. The model-based approach we apply provides a framework for the statistical analysis of escape and reversion in population data and highlights the need for longitudinal and denser cross-sectional sampling to enable accurate estimate of these key parameters

Mathematical models for the transmission dynamics of HIV and its progression to AIDS in Ireland

Despite advances m understanding the basic biology of HIV the aetiological agent of AIDS, medica1, public health and health education planning is plagued by uncertainties Mathematical models of the dynamics of HIV transmission and its progression to AIDS can clarify what data must be collected in order to predict future prevalence, make predictions about the likely effect of future intervention pobcies and provide predictions for several decades ahead. The motivation of this research is to provide reliable estimates of the incidence of HIV infection and AIDS in the Irish population. In Chapters 1 and 2 we discuss the background to the disease in Ireland and the role of mathematical modelling in the spread AIDS. From this we show where key epidemiological data is lacking and how models to date have concentrated on the spread of the disease within the homosexual population. In Chapter 3 we describe the adjustment of the number of AIDS cases to allow for reporting delays Subsequently we consider the solution of the integral equation models generated by the back-projection method for the adjusted AIDS cases. In Chapter 4 we improve upon the estimates of the incidence of HIV infection found in Chapter 3 by evaluating the integral arising in back-projection, in terms of a gamma function plus a remainder in the form of a series in t. We also provide error bounds for the remainder. This new solution allows us to predict new and more reliable estimates of the level of HIV infection m Ireland. In Chapter 5 we provide estimates of the minimum number of deaths from AIDS, based on the number of AIDS cases known to the Department of Health and the distribution of the length of survival times after the onset of AIDS. The results of a HIV transmission survey are presented in Chapter 6 These provide detailed information on the habits and behaviour of those at risk of HIV infection and allow us to derive preliminary model parameters. Finally in Chapter 7 we develop and implement a nonlinear deterministic differential equation model for the spread of HIV and its progression to AIDS m the Irish IVDU and homosexual populations. We examine the effects of likely intervention policies on the extent and spread of the disease and we make recommendations based on our thesis findings

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS

The use of a formal sensitivity analysis on epidemic models with immune protection from maternally acquired antibodies

This paper considers the outcome of a formal sensitivity analysis on a series of epidemic model structures developed to study the population level effects of maternal antibodies. The analysis is used to compare the potential influence of maternally acquired immunity on various age and time domain observations of infection and serology, with and without seasonality. The results of the analysis indicate that time series observations are largely insensitive to variations in the average duration of this protection, and that age related empirical data are likely to be most appropriate for estimating these characteristics

Birth/birth-death processes and their computable transition probabilities with biological applications

Birth-death processes track the size of a univariate population, but many biological systems involve interaction between populations, necessitating models for two or more populations simultaneously. A lack of efficient methods for evaluating finite-time transition probabilities of bivariate processes, however, has restricted statistical inference in these models. Researchers rely on computationally expensive methods such as matrix exponentiation or Monte Carlo approximation, restricting likelihood-based inference to small systems, or indirect methods such as approximate Bayesian computation. In this paper, we introduce the birth(death)/birth-death process, a tractable bivariate extension of the birth-death process. We develop an efficient and robust algorithm to calculate the transition probabilities of birth(death)/birth-death processes using a continued fraction representation of their Laplace transforms. Next, we identify several exemplary models arising in molecular epidemiology, macro-parasite evolution, and infectious disease modeling that fall within this class, and demonstrate advantages of our proposed method over existing approaches to inference in these models. Notably, the ubiquitous stochastic susceptible-infectious-removed (SIR) model falls within this class, and we emphasize that computable transition probabilities newly enable direct inference of parameters in the SIR model. We also propose a very fast method for approximating the transition probabilities under the SIR model via a novel branching process simplification, and compare it to the continued fraction representation method with application to the 17th century plague in Eyam. Although the two methods produce similar maximum a posteriori estimates, the branching process approximation fails to capture the correlation structure in the joint posterior distribution