Introduction

The probability with which the human immunodeficiency virus (HIV) is transmitted from an infected to a susceptible individual during the course of one or more unprotected sexual contacts plays an important role in models of the AIDS epidemic. Several studies found no association between the number of contacts with a given partner and transmission, and some modellers have therefore preferred to use transmission rates per partnership rather than per contact. However, an analysis of data from the California Partners' Study (Padian et al. 1990) indicated the presence of an association, although not consistent with a constant probability of transmission per contact. Similar data from a European study have been analyzed to investigate further the relationship between the number of unprotected sexual contacts and the probability of transmission of HIV.

Methods

Data on 563 HIV-infected subjects (index cases) and their stable heterosexual partners were collected at study entry (between March 1987 and March 1991) and at 6-monthly intervals thereafter (European Study Group on Heterosexual Transmission of HIV 1992). For each couple, the number of unprotected sexual contacts was estimated using the reported frequency of contacts and of condom use, both before and after any reported change in behaviour, together with an estimate of the length of the period during which the partner was at risk. This latter was determined as the duration of the relationship prior to the date of HIV test of the partner and from either the date of infection of the index case, when known (rarely the case), or January 1982 (or one of several alternative dates).

Introduction

The development and use of ONCHOSIM for studying epidemiology and control of onchocerciasis is a joint effort of the Onchocerciasis Control Programme (OCP) and the Department of Public Health of the Erasmus University, Rotterdam. ONCHOSIM uses the so-called microsimulation technique for modelling stochastic systems (Habbema et al. 1995). This technique is characterized by mimicking individual life histories of humans and – in the case of ONCHOSIM – parasites. Biological factors and characteristics of control measures can be modelled in detail. Output of microsimulation models can be detailed (age- and sex-specific tables for comparison with detailed data sets) and simple (trends in prevalence during control). New insights can readily be incorporated by redefining relationships in the model and adapting the computer program which is used to perform simulations with the model.

A model that is built and quantified using the ONCHOSIM computer program has two types of assumptions. One concerns the deep model of the transmission cycle and disease process of onchocerciasis. The other concerns the description of the relevant characteristics (‘experimental setting’) of the village or region under study and of the control measures. The degree of complexity of both the deep model and the descriptive part depends on the aims of the model use, the available data and other considerations. When compared to most other current epidemiological models, the most pronounced difference is probably the level of detail in which the descriptive part is modelled. This is a reflection of the primary reason for involvement of modelling in OCP: supporting evaluation and decision making in a particular control programme (Remme et al. 1995).

Introduction

The relationship between virulence and transmissibility is an important theme in analysis of host-parasite interactions in natural populations (Anderson and May 1991). However, there are few studies of the effects of parasite virulence on the population dynamics of major infectious diseases of humans. Data from the era of malaria therapy (James 1932, Covell 1951) and recent molecular studies (reviewed by Marsh 1992) indicate that there may be considerable variation in virulence within the Plasmodium falciparum parasite, which causes over 1 million malaria deaths each year. In this paper we explore the population dynamic and genetic implications of such proposed parasite diversity to ask whether they may explain some of the now well-defined epidemiological features of malarial disease.

In African children, amongst whom the great majority of malaria deaths occur, Plasmodium falciparum malaria can be clinically resolved into ‘mild’ and ‘severe’ types. This distinction describes a clear, and readily recognisable, clinical differentiation of malaria into a majority (about 99%) of uncomplicated cases with a very low mortality (« 1%), and a small number of severe cases with a mortality of 10-20% under treatment (Brewster 1990). Furthermore, severe malarial disease manifests as either severe malarial anaemia or cerebral malaria, both pathologically distinct from mild malaria. Hence, this classification is not just an arbitrary division of a continuum of disease severity, but reflects a clear bimodality in the severity of malarial disease.

There is strong evidence that host genetic susceptibility influences the clinical outcome of malarial infection; the immunological, nutritional and sociological status of the host may also be of varying degrees of importance (reviewed by Greenwood et al. (1991)).

The fact that AIDS is mainly a sexually transmitted disease has brought human sexual behaviour into the focus of attention and with it the underlying social structure of the population. The problem of how to incorporate the determinants of the sexual contact structure into a mathematical model of disease transmission has been one of the central questions in AIDS-modelling in recent years. While most of this work up to now has been based on the methodology of differential equations, lately there has been some interest in so-called network models. The basic idea of the network approach is that a population and its sexual contact structure can be described by a graph, where the vertices represent individuals and the edges existing sexual relations.

A simulation model based on the network approach has been developed in Kretzschmar et al. (1990,1994). The model describes a stochastic pair formation and dissolution process in a heterosexual population. Infection can be transmitted in contacts between an infected and a susceptible individual. A major problem in analyzing results from network simulations is the question of what are the appropriate quantities to measure and compare. I have chosen, amongst others, to look at the degree distribution of the ‘cumulative’ network over a given time of observation, because this can be determined with a certain accuracy in sociological surveys. One can then study how the number of infected individuals in the course of the epidemic depends on the mean and variance of this degree distribution.

The epidemiology of infectious human diseases

Understanding and controlling the spread of infections is of vital importance to society, and in the past century the epidemiology of human disease has become a subject in its own right. Theory and applicable techniques have been developed to study both the evolution of disease within individual people and the transmission of infections through populations. Mathematics has an important role to play in these studies, which raise challenging problems ranging from broad theoretical issues to specific practical ones, and in recent years there have been significant advances in developing and analysing mathematical models of disease progression. For example, in human diseases in particular, the problems of modelling population heterogeneity are especially important.

Over the last decade there has been a great deal of work concerned with HIV and AIDS. This has been concentrated mainly in two areas: the statistical estimation of various parameters associated with HIV infection (for example, the probability of vertical transmission; the description of the incubation period from infection to clinical disease; the estimation from reported AIDS cases of the number of people infected), and the description of transmission of HIV within and between populations (for example, the characterisation of networks of risk behaviour; the impact of different control strategies). To an extent, the growth of studies in this area has become divorced from the study of other infections, and therefore one of the primary purposes of this volume is to bring together work on modelling a wide range of human diseases so as to encourage cross-fertilisation between AIDS related research and research of the epidemiology of other infections.

Introduction

AIDS continues to place enormous demands on health-care resources and it is essential for public health planning that useful estimates are available of current and future numbers of individuals at different stages of HIV disease. People with HIV infection are eligible to receive treatments at ever earlier stages of the disease, and accurate estimates are required to ensure adequate resources are available. People sick with advanced HIV disease may be in need of special care. Estimates are also crucial for developing policy on awareness campaigns and intervention programs, as well as for investigating the value of needle exchange and other prevention, including vaccination, programs.

Many unanswered questions about the epidemic are essentially statistical in nature, for despite efforts over the past decade to improve both the collection and quality of data on HIV and AIDS, the data are still often incomplete, and there remain large gaps in our knowledge on many key epidemiological parameters.

In particular, the infectivity of HIV is a fundamental unknown and there is uncertainty about the incubation period and its space-time trends. The available data are therefore an incomplete description of phenomena which are, on the whole, relatively poorly understood, and predictions of the epidemic based on the available data are subject to considerable uncertainty. This uncertainty makes AIDS grimly interesting to statisticians, but the prediction problems have been forced upon us because of their practical urgency, regardless of whether or not we can solve them. The role of markers such as CD4 cell counts, IgA and other markers in HIV disease is currently receiving considerable attention by AIDS researchers and statisticians. A further major uncertainty is that of treatment efficacy.

We explore the fitting of a class of hierarchical regression models to longitudinal CD4 lymphocyte count data from the Edinburgh City Hospital cohort. This mainly drug-using (IDU) cohort provides an excellent resource for the study of HIV disease progression for several reasons: seroconversions have been estimated for a large proportion of the cohort on the basis of stored sera retrospectively tested for HIV antibodies and knowledge of needle-sharing behaviour; immunological monitoring has been thorough since 1985 with blood taken at most clinic visits and regular attendance behaviour encouraged; immunological measurements are considered accurate from quality control comparisons between UK laboratories.

Thus we are able to consider a set of 164 seropositives who have well-estimated seroconversions and at least 10 CD4 counts each to the end of 1991; 102 of these subjects have at least 15 counts and 51 have at least 20 giving good longitudinal marker series. We also have checking data from 1992 which are not used in the initial modelling but which are used later to compare the models we fit.

Our basic model is a hierarchical regression model for the square root of CD4 count which we find to decay in a plausibly linear fashion. We fit the model using Markov chain Monte Carlo techniques, specifically the Gibbs sampler. This approach is easily implemented and takes in its stride the highly unbalanced time ‘design’ of the data which would cause great problems in conventional modelling. Our model could also be described as a random effects growth curve and bears similarities to recent work by Lange et al. (1992).

Back-calculation methods have been widely used to reconstruct the past history of the HIV epidemic and to provide short-term predictions of AIDS incidence, on the basis of reported AIDS cases, knowledge of the incubation period distribution and assumptions on the shape of the HIV infection curve.

Within the back-calculation framework, a great variety of different model assumptions and modelling approaches have been employed at each stage of the process (infection, incubation and reporting). Considerable uncertainty exists about the appropriate form for each stage. For example only information on first half of the incubation distribution is available, and knowledge of the effect and extent of AIDS prophylaxis and treatment is still limited. Furthermore, the history of HIV incidence can only be inferred indirectly.

The complexity of the total model has prevented a formal treatment of uncertainty in model formulation and parameter estimation. For example, parameters of the incubation distribution are usually fixed. Further complexities are added through use of other sources of data, such as seroprevalence estimates (and their inherent imprecision). Informal sensitivity analyses and bootstrapping have provided partial answers to the effects of uncertainty on AIDS projections, but a formal treatment of uncertainty demands a new approach to estimation. In particular, a Bayesian framework is indicated, since informative prior distributions on some parameters would allow useful compromises between assuming complete ignorance about their values, and fixing them absolutely.

We describe the basics of AIDS back calculation, reviewing model assumptions and generalisations. We motivate our approach to model building, and propose estimation through Markov chain Monte Carlo (MCMC). We show some results for the epidemic in England and Wales.

Introduction

Vaccines activate the immune system so that it is hoped that the response of the host to subsequent infections will be less harmful for the host. Leaving aside the mechanisms by which vaccination works, one can observe three changes in the host-pathogen interaction, i.e. compared to a non-vaccinated individual, a vaccinated individual has: (1) a lower probability of becoming infected when exposed (reduced susceptibility), (2) fewer clinical signs when it is infected (clinical protection), and (3) less infectivity when it is infected (reduced infectivity).

For a first vaccine evaluation, reduced infectivity would not be considered as a positive effect of the vaccine, because it is not of direct benefit to the individual receiving the vaccine. It is important to take reduced infectivity into account, when estimating the combined effect of reduced susceptibility and clinical protection, i.e. vaccine efficacy, because reduced infectivity lessens the exposure of the other individuals in the population. Thus for the estimation of vaccine efficacy one should compensate for differential exposure, and reduced infectivity is a nuisance parameter.

It is especially important to consider how vaccines differ in their effect on infectivity whenever several vaccines are available, that all have similar efficacy. It is known for some vaccines, for example those against measles and polio, that although in vaccinated individuals clinical signs are either absent or very mild, vaccinated individuals are susceptible and they can replicate and excrete virus (infectivity). This implies that virus could circulate in vaccinated populations and therefore it is important to quantify and compare the amount of virus circulation in groups of individuals vaccinated with various vaccines.

Introduction

MHC antigens and the immune system

Since the report by Mitchison in the early 1950s demonstrating that cellmediated, rather than humoral, immunity played a greater role in tumour rejection (Mitchison, 1953), its primacy in tumour rejection has become an increasingly accepted mechanism by most immunologists. This has mainly been attributed to specific immunity involving MHC antigens as restriction element and to a lesser extent the non-specific immunity involving LAK/NK activity (Jabrane-Ferrat et al., 1990; Mule et al., 1984).

It has long been established that the MHC antigens are an individual's fingerprint and they exist in two forms, the class I and class II antigens. Zinkernagel & Doherty (1979) showed that they act as associative molecules for presentation of non-self antigens to CTLs and TH cells, respectively. The critical role of CTLs for regulating resistance to viral infection (McMichael et al., 1977) as well as in graft and tumour rejection in experimental models (Hui, Grosveld & Festenstein, 1984; Wallich et al., 1985) has previously been reported. TH cells have been shown to act mainly as an immune amplifier, since their stimulation results in the production of a series of immunoreactive cytokines, such as IL-2, which in turn are critical for initiating immune responses, including activation of CTLs (Greenberg et al., 1988).

Introduction

The growth cycles and oncogenic properties of the murine and human retroviruses are inextricably linked to the immune system. Those viruses that cause leukaemias, lymphomas or immunodeficiencies do so by infecting and often activating immune cells. Conversely, those retroviruses that induce solid tumours must evolve ways to aid the host cell in evading the cellular immune system. One major molecular mechanism by which these retroviruses can either activate or evade the immune system is by control of MHC class I antigen expression in the cells they infect. An effect of murine retrovirus infection on MHC antigen expression was first suspected in the late 1970s, when it was observed that thymocytes obtained from animals several weeks after infection with leukaemia viruses appeared to express higher levels of MHC class I antigens than thymic cells from control animals. Conversely, down-regulation of MHC expression on solid tumours induced by oncogene-containing (sarcoma) retroviruses had also been observed. Because of the experimental constraints of these in vivo systems, however, proof of a causal relationship between retrovirus infection and MHC regulation was lacking. More recent studies have demonstrated a direct action of retroviruses on MHC gene regulation and have begun to elucidate the ways in which these compact viruses, with only 6000–10 000 bases of coding sequence, regulate the histocompatibility antigen expression of their host cells.

Murine leukaemia virusesy

The murine retroviruses can be broadly divided into two classes: the leukaemia viruses and the sarcoma viruses.

While the association of particular major histocompatibility complex (MHC) haplotypes with human diseases has been extensively reviewed, we felt that the area of modulation of MHC antigen expression and its association with human and animal diseases was less well explored. These ideas were initially exchanged between the editors and some of the authors of this book at a most interesting meeting organized by the British Society for Immunology in Warwick in 1991 entitled ‘Viruses, Cytokines and the MHC’. Following this meeting, we realised that there was a need for a broad, interdisciplinary treatment of the area of MHC modulation that would be useful for both basic medical researchers and clinicians.

Expression of MHC class I and class II antigens follows a complex pathway from gene transcription to plasma membrane insertion and many steps can be stimulated or repressed leading to altered levels of cell surface MHC molecules. Therefore, to provide basic background information on the MHC, the genomic organization, antigen structure, biosynthesis and function and control of transcription of class I and II are considered first along with the important effects which cytokines and other extracellular agents can have on MHC antigen expression. Infection and oncogenic transformation of mammalian cells by viruses have provided powerful systems for analysing precise mechanisms of MHC antigen modulation and the relationship of this process to disease.

Introduction

To protect the individual from foreign agents, such as viruses and bacteria, mammals have evolved a sophisticated system that allows them to distinguish self from non-self. Self/non-self discrimination was first demonstrated in mammals by the rejection of foreign tissue grafts in mice (Gorer, 1936; Snell, 1958; Klein, 1975). The genetic loci involved in graft rejection were subsequently mapped to a region on chromosome 17 (Klein, 1975), which became known as the major histocompatibility complex (MHC). The human MHC, also known as the human leukocyte antigen (HLA) system, is located on chromosome 6 (van Someren et al, 1974).

The MHC occupies some 3.5 megabase pairs (Mb) of the genome and, in humans, approximately 75 genes (many still of unknown function) have been identified in this region (Trowsdale, Ragoussis & Campbell, 1991). The organization of the MHC is reviewed in detail in Chapter 2. The complex is often divided into three different classes of gene: I, II and III. There are multiple class I loci but the classical ‘transplantation antigens’ fall into three positions termed HLA-A, HLA-B and HLA-C in humans. The class II genes, encoded in the HLA-D region, encode proteins that help regulate the immune response to different antigens (Fig. 1.1). In early experiments that studied the genetic control of immunity to certain protein antigens, the immune response, whether strong or weak, was found to depend on particular alleles at loci within the MHC.

Introduction

Even before 1975, when the role of the MHC antigens as a ‘guidance mechanism’ for the immune system was discovered (Zinkernagel & Doherty, 1975), it was known that some tumour cells expressed abnormally low levels of MHC antigens and/or β2-m (Nilsson, Evsin & Welsh, 1974). In addition, there was ample experimental evidence that tumour cells could be ‘recognized’ and eliminated by the immune system, although early work had failed to take into account allogeneic recognition of transplantation antigens (Foley, 1953). By the 1950s it had been clearly shown that chemically induced, radiation-induced and spontaneously occurring tumours could express tumour antigens that could initiate and lead to ‘tumour rejection’, namely the tumour-associated transplantation antigens, TATAs (Gross, 1943). It was only in the late 1970s or early 1980s that these two strands of research could be put together and they have subsequently led to important advances in the clinical treatment of cancer.

These advances are related to the great insight of Paul Ehrlich, who postulated that the immune system acted as a surveillance system to detect changes within the body caused either by normal pathological events or by invading organisms (Ehrlich, 1909). It is now clear that the primary function of CTLs is to monitor cell surfaces for abnormal peptides presented by MHC class I antigens. T cells reactive to normal self peptides will have been made tolerant either by clonal deletion or clonal anergy.

Introduction

Infection with hepatitis B virus (HBV) can cause either an acute or a chronic hepatitis (Nielsen et al., 1971). Chronic infection is a major health problem affecting over 250 million people worldwide (Ganem & Varmus, 1987). Without treatment, chronic infection with HBV progresses to cirrhosis and/ or hepatocellular carcinoma in over 50% of infected patients. It is still not clear why some infected individuals develop a relatively mild, self-limiting hepatitis whilst others suffer from a prolonged, chronic infection. Recent research suggests that interferon-induced expression of MHC antigens on the surface of infected hepatocytes may play a key role in determining the outcome of infection with HBV.

Viral clearance in acute HBV infection

The majority of healthy adults infected with HBV develops a brief hepatitis, which resolves within a few months and is followed by elimination of the virus. A small percentage of infected adults (less than 5%) and the majority of infected neonates and children (greater than 90%) develop a chronic infection. Research in patients and chimpanzees acutely infected with HBV has provided an insight into the mechanisms underlying the normal eradication of HBV and these studies suggest that induction of MHC expression, by type I interferon (i.e. IFN-α/β), is a key factor in virus eradication.

Following infection with HBV, most adults show a marked increase in the serum concentration of circulating IFN (Ikeda, Lever & Thomas, 1986; Kato, Nakagawa & Kobayashi, 1986; Pignatelli et al., 1986).