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.     

A moment closure model for sexually transmitted disease transmission through a concurrent partnership network

A moment closure model of sexually transmitted disease spread through a concurrent partnership network is developed. The model employs pair approximations of higher-order correlations to derive equations of motion in terms of numbers of pairs and singletons. The model is derived from an underlying stochastic process of partnership network formation and disease transmission. The model is analysed numerically; and the final size and time evolution are considered for various levels of concurrency, as measured by the concurrency index kappa3 of Kretzschmar and Morris. Additionally, a new way of calculating R0 for spatial network models is developed. It is found that concurrency significantly increases R0 and the final size of a sexually transmitted disease, with some interesting exceptions.     

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.     

A study of a Malthusian parameter in relation to some stochastic models of human reproduction

The paper is divided into six sections and is devoted to a study of a Malthusian parameter in relation to some stochastic models of human reproduction. In Section 1, some of the motivations underlying the study are discussed, and in Section 2 some literature on the stochastic model of population growth underlying the foundations of the paper is briefly reviewed. Section 3, which lays the foundations for the study of a more complicated model in Section 4, is devoted to the study of the Malthusian parameter in relation to a stochastic model of human reproduction formulated as a terminating renewal process. In Section 4 the Malthusian parameter is studied in relation to a terminating Markov renewal model of human reproduction, stemming from the work of Perrin and Sheps (1964). Among the mathematical results of independent interest in this section is a complete spectral decomposition of the Laplace-Stieltjes transform of the semi-Markov transition matrix in the model of Perrin and Sheps. Section 5 is devoted to the discussion of a mathematical method which allows accomodating in the model the time taken by an individual to reach reproductive age, and Section 6 ends the paper by supplying bounds for the Malthusian parameter which are valid under quite general conditions. Possible applications of the results in evaluating what influences a population policy may have on population growth are also discussed.     

Some techniques in building and validating stochastic models of human reproduction

Stochastic models of human reproduction are beginning to play significant roles in the evaluation of family planning programs. A class of stochastic processes called absorbing, agedependent, semi-Markov processes frequently arises in the construction of such models. The paper begins with a discussion of some technicalities regarding absorbing, age-dependent, semi-Markov processes. Then, an algorithm due to Littman, which makes possible the computerization of this class of stochastic processes, is presented. Briefly, Littman’s algorithm provides an efficient method for numerically solving systems of renewal type integral equations, provided the system does not contain a large number of equations. After setting down a concrete model for a large clinical trial of intrauterine devices conducted in Taiwan, the paper concludes with a discussion of a method for validating the model based on the data collected in the clinical trial. Presented at the Society for Mathematical Biology Meeting, University of Pennsylvania, Philadelphia, August 19–21, 1976.     

On the stochastic theory of compartments: A semi-Markov approach for the analysis of theK-compartmental systems

In this paper a semi-Markov process approach is developed to analyse stochastic compartmental systems using straightforward probabilistic arguments. Explicit expressions for several characteristics of thek-compartmental systems with a Poisson process input are derived and various models found in the literature arising from biological applications are generalised here using the semi-Markov process technique.     

Mathematical models of eye movements in reading: a possible role for autonomous saccades

 An efficient method for the exact numerical simulation of semi-Markov processes is used to study minimal models of the control of eye movements in reading. When we read a text, typical sequences of fixations form a rather complicated trajectory – almost like a random walk. Mathematical models of eye movement control can account for this behavior using stochastic transition rules between few discrete internal states, which represent combinations of certain stages of lexical access and saccade programs. We show that experimentally observed fixation durations can be explained by residence-time-dependent transition probabilities. Stochastic processes with this property are known as semi-Markov processes. For our numerical simulations we use the minimal process method (Gillespie algorithm), which is an exact and efficient simulation algorithm for this class of stochastic processes. Within this mathematical framework, we study different forms of coupling between eye movements and shifts of covert attention in reading. Our model lends support to the existence of autonomous saccades, i.e., the hypothesis that initiations of saccades are not completely determined by lexical access processes. Received: 21 March 2000 / Accepted in revised form: 10 January 2001     

The guinea pig as a model of infectious diseases

The words 'guinea pig' are synonymous with scientific experimentation, but much less is known about this species than many other laboratory animals. This animal model has been used for approximately 200 y and was the first to be used in the study of infectious diseases such as tuberculosis and diphtheria. Today the guinea pig is used as a model for a number of infectious bacterial diseases, including pulmonary, sexually transmitted, ocular and aural, gastrointestinal, and other infections that threaten the lives of humans. Most studies on the immune response to these diseases, with potential therapies and vaccines, have been conducted in animal models (for example, mouse) that may have less similarity to humans because of the large number of immunologic reagents available for these other species. This review presents some of the diseases for which the guinea pig is regarded as the premier model to study infections because of its similarity to humans with regard to symptoms and immune response. Furthermore, for diseases in which guinea pigs share parallel pathogenesis of disease with humans, they are potentially the best animal model for designing treatments and vaccines. Future studies of immune regulation of these diseases, novel therapies, and preventative measures require the development of new immunologic reagents designed specifically for the guinea pig.     

Toward a unified theory of sexual mixing and pair formation.

Sexually transmitted diseases such as gonorrhea, syphilis, herpes, and AIDS are driven and maintained in populations by epidemiological and sociological factors that are not completely understood. One such factor is the way in which people mix sexually. In this paper, we outline a unified approach to modeling sexual mixing structures, where such structures are defined in terms of a set of axioms for a finite number of distinct groups of people. Theorems for homosexual, heterosexual, and arbitrary group mixing are presented, leading to a representation of all mixing structures defined by the axioms. The representation and its parameters are interpreted in terms of intergroup affinities for sexual mixing. The use of the approach in sexually transmitted disease modeling is discussed.     

Higher variability in the number of sexual partners in males can contribute to a higher prevalence of sexually transmitted diseases in females

By examining published, empirical data we show that men and women consistently differ in the shape of the distribution of the number of sexual partners. The female distribution is always relatively narrow—variance is low—with a big majority of women having a number of partners close to the average. The male distribution is much wider—variance is high—with many men having few sex partners and many others having more partners than most females.Using stochastic modelling we demonstrate that this difference in variance is, in principle, sufficient to cause a difference in the gender prevalence of sexually transmitted diseases: compared to the situation where the genders have identical sex partner distributions, men will reach a lower equilibrium value, while women will stay at the same level (meaning that female prevalence becomes higher than male). We carefully analyse model behaviour and derive approximate expressions for equilibrium prevalences in the two different scenarios. We find that the size of the difference in gender prevalence depends on the variance ratio (the ratio between the variances of the male and female sex partner distributions), on the expected number of life-time partners, and on the probability of disease transmission. We note that in addition to humans, the variance phenomenon described here is likely to play a role for sexually transmitted diseases in other species also.We also show, again by examining published, empirical data, that the female to male prevalence ratio increases with the overall prevalence of a sexually transmitted disease (i.e., the more widespread the disease, the more women are affected). We suggest that this pattern may be caused by the effect described above in highly prevalent sexually transmitted diseases, while its impact in low-prevalence epidemics is surpassed by the action of high-risk individuals (mostly males).     

The Impact of Contact Tracing in Clustered Populations

The tracing of potentially infectious contacts has become an important part of the control strategy for many infectious diseases, from early cases of novel infections to endemic sexually transmitted infections. Here, we make use of mathematical models to consider the case of partner notification for sexually transmitted infection, however these models are sufficiently simple to allow more general conclusions to be drawn. We show that, when contact network structure is considered in addition to contact tracing, standard “mass action” models are generally inadequate. To consider the impact of mutual contacts (specifically clustering) we develop an improvement to existing pairwise network models, which we use to demonstrate that ceteris paribus, clustering improves the efficacy of contact tracing for a large region of parameter space. This result is sometimes reversed, however, for the case of highly effective contact tracing. We also develop stochastic simulations for comparison, using simple re-wiring methods that allow the generation of appropriate comparator networks. In this way we contribute to the general theory of network-based interventions against infectious disease.     

Modeling AIDS as a function of other sexually transmitted disease.

The spread of AIDS, as with any sexually transmitted disease, will depend on the pattern of sexual activity. Both the proportion of the population who have high partner exchange rates and the extent to which that proportion interacts with the remainder of the population are likely to be important determinants of the AIDS epidemic. However, it does not seem likely that surveys could obtain sufficiently reliable information of this nature for use in an accurate model of the AIDS epidemic. On the other hand, such information is implicitly contained in the epidemiology of other sexually transmitted diseases (STDs). Therefore a method is suggested of calculating the parameters of a model of the AIDS epidemic by comparing it with the epidemiology of another STD. The result is a model that predicts the likelihood of infection by the AIDS virus as a function of time and an individual's history of STD. It is suggested that further work along these lines may lead to a quantitative approach to assessing the importance of various STDs as cofactors in the spread of AIDS.     

A semi-Markov model for survival data with covariates

Clinical trials are often concerned with the evaluation of two or more time-dependent stochastic events and their relationship. The information on covariates for individuals in the studies is valuable in assessing the survival function. This paper develops a multistate stochastic survival model which incorporates covariates. It is assumed that the underlying process follows a semi-Markov model. The proportional hazards techniques are applied to estimate the force of transition in the process. The maximum likelihood estimators are derived along with the survival function for competing risks problems. An application is given to analyzing the survival of patients in the Stanford Heart Transplant Program.     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency

Deterministic differential equation models indicate that partnership concurrency and non-homogeneous mixing patterns play an important role in the spread of sexually transmitted infections. Stochastic discrete-individual simulation studies arrive at similar conclusions, but from a very different modeling perspective. This paper presents a stochastic discrete-individual infection model that helps to unify these two approaches to infection modeling. The model allows for both partnership concurrency, as well as the infection, recovery, and reinfection of an individual from repeated contact with a partner, as occurs with many mucosal infections. The simplest form of the model is a network-valued Markov chain, where the network's nodes are individuals and arcs represent partnerships. Connections between the differential equation and discrete-individual approaches are constructed with large-population limits that approximate endemic levels and equilibrium probability distributions that describe partnership concurrency. A more general form of the discrete-individual model that allows for semi-Markovian dynamics and heterogeneous contact patterns is implemented in simulation software. Analytical and simulation results indicate that the basic reproduction number R(0) increases when reinfection is possible, and the epidemic rate of rise and endemic levels are not related by 1-1/R(0), when partnerships are not point-time processes.     

Coexistence and specialization of pathogen strains on contact networks

The coexistence of different pathogen strains has implications for pathogen variability and disease control and has been explained in a number of different ways. We use contact networks, which represent interactions between individuals through which infection could be transmitted, to investigate strain coexistence. For sexually transmitted diseases the structure of contact networks has received detailed study and has been shown to be a vital determinant of the epidemiological dynamics. By using analytical pairwise models and stochastic simulations, we demonstrate that network structure also has a profound influence on the interaction between pathogen strains. In particular, when the population is serially monogamous, fully cross-reactive strains can coexist, with different strains dominating in network regions with different characteristics. Furthermore, we observe specialization of different strains in different risk groups within the network, suggesting the existence of diverging evolutionary pressures.     

Human papilloma virus associated with genital infection

Genital human papillomavirus (HPV) infections are among the most common sexually transmitted diseases. HPV is associated with a spectrum of diseases ranging from benign vulgar verrucae and condylomata accuminata to malignant cancers of the cervix, vulva, anus and penis. Genital HPV is in most cases transmitted sexually, but non-sexual routes of transmission, such as perinatal and autoinoculation, are possible. Men can be a reservoir of the virus that lives in latent or subclinical form on genital mucosa. Such an asymptomatic infection may be an oncogenic factor in the development of cervical cancer Colposcopic examination of the genitalia after the application of 3-5% acetic acid is a reliable method for the identification of subclinical HPV infection. Successful therapy of anogenital warts is characterized by their complete clearance, as well as by the lack of recurrence. Current treatments do not reliably eradicate HPV infections. The diagnosis and therapy of HPV infection in men is potentially beneficial because the eradication of penile HPV infection may decrease the reservoir of the virus.     

Fatal or Harmless: Extreme Bistability Induced by Sterilizing,Sexually Transmitted Pathogens

Models of sexually transmitted infections have become a fixture of mathematical epidemiology. A common attribute of all these models is treating reproduction and mating, and hence pathogen transmission, as uncoupled events. This is fine for humans, for example, where only a tiny fraction of sexual intercourses ends up with having a baby. But it can be a deficiency for animals in which mating and giving birth are tightly coupled, and mating thus mediates both reproduction and pathogen transmission. Here, we model dynamics of sterilizing, sexually transmitted infections in such animals, assuming structural consistency between the processes of reproduction and pathogen transmission. We show that highly sterilizing, sexually transmitted pathogens trigger bistability in the host population. In particular, the host population can end up in two extreme alternative states, disease-free persistence and pathogen-driven extinction, depending on its initial state. Given that sterilizing, sexually transmitted infections that affect animals are abundant, our results might implicate an effective pest control tactic that consists of releasing the corresponding pathogens, possibly after genetically enhancing their sterilization power.     

Parasite evolution and extinctions

We examine the evolution of diseases that show the frequency‐dependent transmission process that is commonly applied to sexually and vector‐transmitted infections. As is commonly found, the basic reproductive ratio (R₀) of the parasite is maximized by evolution. This has important implications, as it implies that for a wide range of circumstances diseases that show frequency‐dependent transmission may be selected to evolve towards driving their hosts to extinction. This contrasts with the results obtained in spatially explicit models where although parasite‐driven host extinction may occur, it is unlikely to evolve. We further show that an evolutionary constraint between transmission and virulence is required for evolution to lead to an endemic coexistence of both the host and the disease. Furthermore, this constraint needs to be saturating, such that transmission is ‘bought’ at an increasing cost in terms of virulence, to avoid evolution to extinction.     

Modeling the effects of biologically incorporated radionuclides and chronic low-dose ionizing radiation exposure on both cancer and chronic non-cancer diseases

Efforts to model the health effects of low-dose ionizing radiation (IR) have often focused on cancer. Meanwhile, significant evidence links IR and age-associated non-cancer diseases. Modeling of such complex processes, which are not currently well understood, is a challenging problem. In this paper we briefly overview recent successful attempts to model cancer on a population level and propose how those models may be adapted to include the impact of IR and to describe complex non-cancer diseases. We propose three classes of models which we believe are well suited for the analysis of the health effects in human populations exposed to low-dose IR. These models use biostatistical/epidemiological techniques and mathematical formulas describing the biological mechanisms of the impact of IR on human health. They can combine data from multiple sources and from distinct levels of biological/population organization. The proposed models are intrinsically multivariate and non-linear and capture the dynamic aspects of health change.