Candace K Sleeman

Stochastic processes in epidemiology : HIV/AIDS, other infectious diseases and computers

This text deals with the mathematical and statistical techniques underlying the models used to understand the population dynamics of not only HIV/AIDS, but also of other infectious diseases. Attention is given to the development of strategies for the prevention and control of the international epidemic within the frameworks of the models. The text incorporates stochastic and deterministic formulations within a unifying conceptual framework.

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks.

Abstract Use of the final size distribution of minor outbreaks for the estimation of the reproduction numbers of supercritical epidemic processes has yet to be considered. We used a branching process model to derive the final size distribution of minor outbreaks, assuming a reproduction number above unity, and applying the method to final size data for pneumonic plague. Pneumonic plague is a rare disease with only one documented major epidemic in a spatially limited setting. Because the final size distribution of a minor outbreak needs to be normalized by the probability of extinction, we assume that the dispersion parameter (k) of the negative-binomial offspring distribution is known, and examine the sensitivity of the reproduction number to variation in dispersion. Assuming a geometric offspring distribution with k=1, the reproduction number was estimated at 1.16 (95% confidence interval: 0.97–1.38). When less dispersed with k=2, the maximum likelihood estimate of the reproduction number was 1.14. These estimates agreed with those published from transmission network analysis, indicating that the human-to-human transmission potential of the pneumonic plague is not very high. Given only minor outbreaks, transmission potential is not sufficiently assessed by directly counting the number of offspring. Since the absence of a major epidemic does not guarantee a subcritical process, the proposed method allows us to conservatively regard epidemic data from minor outbreaks as supercritical, and yield estimates of threshold values above unity.

An algorithmic synthesis of the deterministic and stochastic paradigms via computer intensive methods.

In this paper, an approach to synthesizing the deterministic and stochastic paradigms, via computer intensive methods, is presented within the framework of a stochastic model of a HIV/AIDS epidemic in a population of homosexuals. Because of dependence among members of a population, the problem of determining threshold conditions was approached by systematically embedding a system of differential equations in a stochastic process and determining if the Jacobian matrix of this system is stable or not stable, when evaluated at a disease free equilibrium. It has been shown in numerous Monte Carlo simulation experiments that if this matrix is not stable, then an epidemic will develop in a population with positive probability, following the introduction of infectives into a population of susceptibles. This technique was used to search for points in the parameter space such that an epidemic would develop in a population of susceptibles, following the entrance of one or more infectious recruits during any time interval with small probability. Such recurrent rare events are of interest in the studying the emergence of new diseases, involving the transmission of a virus from a species that has evolved resistance to it to another species that lacks resistance.

Stochastic processes in genetics and evolution : computer experiments in the quantification of mutation and selection

Introduction of Mathematical Probability Linkage and Recombination at Multiple Linker Loci Nucleotide Substitution Models Formulated as Markov Processes in Continuous Time Genealogies, Coalescence and Self Regulating Branching Processes Two Sex Multi-type Self Regulating Branching Processes in Evolutionary Genetics Selected Topics from Molecular Genetics Detecting Genomic Signal of Selection Suggestions of Further Research, Reading and Viewing and other papers.

A NEW DESIGN OF STOCHASTIC PARTNERSHIP MODELS FOR EPIDEMICS OF SEXUALLY TRANSMITTED DISEASES WITH STAGES

A problem of importance in modelling epidemics of sexually transmitted diseases is the development of mathematical structures accommodating sexual and other contacts among members of a population. Because these models may be complex, it is often necessary to use computer intensive methods in their analysis, which raises questions on the design of computer models. In this paper a new approach to designing models sexual contacts is presented within the context of a stochastic model accommodating the formation and dissolution of partnerships in heterosexual populations. Emphasis will be placed on the development of algorithms with a view towards developing software to implement computer intensive methods. Unlike previous formulations, rather than using rejection methods in Monte Carlo simulations to impose necessary constraints on random functions describing partnership formation, in the new formulation all constraints are satisfied with probability one.

Simulating the Emergence and Survival of Mutations Using a Self Regulating Multitype Branching Processes

It is difficult for an experimenter to study the emergence and survival of mutations, because mutations are rare events so that large experimental population must be maintained to ensure a reasonable chance that a mutation will be observed. In his famous book, The Genetical Theory of Natural Selection, Sir R. A. Fisher introduced branching processes into evolutionary genetics as a framework for studying the emergence and survival of mutations in an evolving population. During the lifespan of Fisher, computer technology had not advanced to a point at which it became an effective tool for simulating the phenomenon of the emergence and survival of mutations, but given the wide availability of personal desktop and laptop computers, it is now possible and financially feasible for investigators to perform Monte Carlo Simulation experiments. In this paper all computer simulation experiments were carried out within a framework of self regulating multitype branching processes, which are part of a stochastic working paradigm. Emergence and survival of mutations could also be studied within a deterministic paradigm, which raises the issue as to what sense are predictions based on the stochastic and deterministic models are consistent. To come to grips with this issue, a technique was used such that a deterministic model could be embedded in a branching process so that the predictions of both the stochastic and deterministic compared based on the same assigned values of parameters.

Determination of threshold conditions for a non-linear stochastic partnership model for heterosexually transmitted diseases with stages

When comparing the performance of a stochastic model of an epidemic at two points in a parameter space, a threshold is said to have been crossed when at one point an epidemic develops with positive probability; while at the other there is a tendency for an epidemic to become extinct. The approach used to find thresholds in this paper was to embed a system of ordinary non-linear differential equations in a stochastic process, accommodating the formation and dissolution of marital partnerships in a heterosexual population, extra-marital sexual contacts, and diseases such as HIV/AIDS with stages. A symbolic representation of the Jacobian matrix of this system was derived. To determine whether this matrix was stable or non-stable at a particular parameter point, the Jacobian was evaluated at a disease-free equilibrium and its eigenvalues were computed. The stability or non-stability of the matrix was then determined by checking if all real parts of the eigenvalues were negative. By writing software to repeat this process for a selected set of points in the parameter space, it was possible to develop search engines for finding points in the parameter space where thresholds were crossed. The results of a set of Monte Carlo simulation experiments were reported which suggest that, by combining the stochastic and deterministic paradigms within a single formulation, it was possible to obtain more informative interpretations of simulation experiments than if attention were confined solely to either paradigm.

A computer exploration of some properties of non-linear stochastic partnership models for sexually transmitted diseases with stages.

In this paper, branching process approximations to non-linear stochastic partnership models for sexually transmitted diseases in heterosexual populations were used to find points in the parameter space such that an epidemic would occur. At selected points in the parameter space, samples of Monte Carlo realizations of the process were computed and analyzed statistically to gain insights into the stochastic evolution of epidemics seeded by one infective single female and male. Non-linear difference equations were embedded in the stochastic processes, making it possible to compare trajectories computed according to the deterministic model with those computed from samples of Monte Carlo realizations. From these trajectories it was shown that stochastic fluctuations may have a profound effect on the long-term evolution of an epidemic, and examples demonstrate that an investigator may be misled if a deterministic model alone were used to project an epidemic, particularly when there is a significant probability of extinction.

Simulating the Emergence of Mutations and Their Subsequent Evolution in an Age-Structured Stochastic Self-Regulating Process with Two Sexes

The stochastic process under consideration is intended to be not only part of the working paradigm of evolutionary and population genetics but also that of applied probability and stochastic processes with an emphasis on computer intensive methods. In particular, the process is an age-structured self-regulating multitype branching process with a genetic component consisting of an autosomal locus with two alleles for females and males. It is within this simple context that mutation will be quantified in terms of probabilities that a given allele mutates to the other per meiosis. But, unlike many models that are currently being used in mathematical population genetics, in which natural selection is often characterized in terms of parameters called fitness by genotype or phenotype, in this paper the parameterization of submodules of the model provides a framework for characterizing natural selection in terms of some of its components. One of these modules consists of reproductive success that is quantified in terms of the total expected number of offspring a female contributes to the population throughout her fertile years. Another component consists of survival probabilities that characterize an individual’s ability to compete for limited environmental resources. A third module consists of a parametric function that expresses the probabilities of survival in a birth cohort of individuals by age for both females and males. A forth module of the model as an acceptance matrix of conditional probabilities such female may show a preference for the genotype or phenotype as her male sexual partner. It is assumed that any force of natural selection acts at the level of the three genotypes under consideration for each sex. By assigning values of the parameters in each of the modules under consideration, it is possible to conduct Monte Carlo simulation experiments designed to study the effects of each component of selection separately or in any combination on a population evolving from a given initial population over some specified period of time.

A methodological study on fitting a nonlinear stochastic model of the AIDS epidemic in Philadelphia

A nonlinear stochastic model accommodating heterogeneous risk behavior and recruitment was fit to Philadelphia public health data adjusted for delays in reporting. The methodological study that was performed resulted in a finding of significant clinical importance. Namely, that the variable time from infection with HIV to seroconversion may be longer than reported in the literature. The finding challenges our understanding of the progression of HIV disease and has profound public health implications. Although the model and the software incorporated several risk categories, such as heterosexual males and females among others, those examined in this paper are confined to white male homosexual/bisexual intravenous drug users and white male homosexual/bisexual nonintravenous drug users. The results of this paper demonstrate that it is possible to fit a rather complex stochastic model to public health data, using computer intensive methods, and thus, more completely reflect the diversity of human behavior represented by the defined risk groups of the data.