There is a great deal of interest among malariologists about the effect of antigenic diversity on the transmission dynamics of malaria (Day and Marsh 1991). Complementing other modelling studies on this effect at a population level (Gupta et al. 1994), we explore the within-host dynamics of blood stage malaria in a single individual infected with a basket of different parasite strains. By generalising a previous model (Anderson et al. 1989) to account for this situation and calibrating it against the observed data on the time course of malaria infection, we are able to include the effects of the host immune response despite the difficulty of obtaining quantitative data on this component.

We intend to use our model to investigate the effects of different types of immune response: for example the relative success of a fast, quickly decaying strain-specific response combined with a slowly generated but persistent strain-transcending response.

We will discuss whether combinations of strain specific and strain transcending responses to a basket of infections at the blood stage of malaria account for the observed patterns of parasitaemia in malaria endemic areas.

Following Anderson et al. (1989), and similarly to Hellriegel (1992), we define x(t) as the number of uninfected erythrocytes, m(t) as the number of free merozoites and y(t) as the number of infected red blood cells. We assume that red blood cells are produced at a constant rate Λ and die at a per-capita rate μ with a mean life-span of l/μ. Free merozoites infect unparasitised cells at a rate βxm; these cells rupture after a mean time of 1/α to produce r more free merozoites.

Hepatitis B virus (HBV) is one of the most common viral infections in many parts of the world. There are an estimated 300 million carriers of the virus worldwide, each of whom has a high probability of suffering chronic liver disease. There is a safe and effective vaccine against HBV which does not interfere with other vaccines commonly given in childhood and would therefore appear to be ideally suited to mass cohort immunisation. However the vaccine is expensive compared to other Expanded Programme of Immunisation vaccines and there is evidence that vaccine induced immunity declines with time. As the epidemiology of HBV is complex (see below) the outcome of mass immunisation is difficult to predict. The aim of this work is to use a mathematical model of the transmission of HBV to aid the design of vaccination programmes in developing countries.

The epidemiology of HBV has a number of interesting features which complicate the dynamics of infection. Infection with HBV can lead to long-term carriage of the virus. Furthermore the propensity for individuals to develop this chronic carrier state is related to the age at infection in a highly nonlinear manner. The probability of developing the chronic carrier state is highest amongst infants (approximately 0.9), then rapidly declines, and levels off in late childhood so that older children and adults have approximately a 1 in 10 chance of becoming carriers if infected. The epidemiological study of HBV is further complicated by its modes of transmission.

Introduction

Mixing patterns in multi-group populations are now recognized to have an important role in the population dynamics of disease (Hethcote and Yorke 1984, Sattenspiel 1987b, Anderson et al 1990). Initially in response to the resurgence of gonorrhea and later with the rapid growth of the AIDS epidemic, selective mixing has become a major focus for epidemiological modelers. Various methods for summarizing the structure of selective mixing have been proposed (Gupta and Anderson 1989, Blythe et al. 1991, Koopman et al. 1991, Morris 1991). Simulation studies show that these effects can be both strong and variable (Hyman and Stanley 1988, Haraldsdottir et al. 1992, Morris 1995), and that they can bias the estimates of other epidemiological parameters if they are not taken into account (Koopman et al. 1991). Analytic expressions for the effect of mixing on the reproductive rate (or number) of a disease and the definition of core groups are beginning to be developed (Diekmann et al 1990, Jacquez et al 1993).

One of the major issues in modeling the mixing patterns of a multi-group population concerns the solution of multiple matching constraints in nonequilibrium populations. Constraints are imposed by the symmetry inherent in contact processes, i.e., if I meet you, then you have to meet me. This is a generalized version of the ‘two-sex problem’ familiar to demographers. In its classical form this problem arises in life table modeling when births are projected on the basis of two-sex populations. The birth process implies a matching process between the age-structured populations of males and females, and these constraints become complicated when vital dynamics are considered (Pollard 1948, Schoen 1982).

The long incubation period of AIDS, with variation in infectiousness over its course, has emphasized the need to model progression of the disease process. The models used for progression of HIV infection to AIDS have generally been staged Markov models that imply a one-way progression from infection to AIDS to death and so do not allow for temporary remissions in the progression of the disease. Such models have negative exponential distributions for the transit times in a stage and independence of transit times in successive stages (Longini et al. 1992, Longini et al. 1991). In our studies to estimate transmission probabilities from data on the Chicago MACS cohort, by stage of infection, we found it necessary to examine progression in the cohort.

The Multicenter AIDS Cohort Study (MACS) involves 4 cohorts of male homosexuals recruited in 1984 in 4 cities: Baltimore, Chicago, Los Angeles and Pittsburgh (Kaslow et al. 1987). Approximately every 6 months, the participants had physical examinations, had blood drawn and filled out a questionnaire on sexual practices. We examine progression in the Chicago MACS cohort which consisted of 1020 individuals at the start of the study. We present data on the first to twelfth waves of examinations, covering the period 1984–90.

Cumulative plots of seropositivity for HIV-1 show that approximately 40% of the Chicago cohort was HIV(+) by the first wave and that about 70 more seroconversions occurred from wave 1 to wave 12. The experience of the other cohorts was similar. Thus roughly 85% of the infections occurred before the first wave of examinations.

We develop a transmission model to examine three facets of prophylactic vaccine failure: take (the vaccine may only work in a fraction of those who are vaccinated), degree (the vaccine may only reduce and not eliminate the probability of infection upon exposure), and duration (the vaccine may only confer protection for a limited time period).

We demonstrate:

1. how to derive a summary measure of vaccine imperfection;

2. how to calculate the critical vaccination coverage that is required to eradicate an HIV epidemic;

3. how to assess the potential impact of different types of imperfect prophylactic vaccines (i.e. vaccines that fail in different way) in both clinical trials and mass vaccination campaigns.

We present analytic and scenario results, the latter being based upon parameter values that are derived from the HIV epidemic in gay men in San Francisco, California. For further details see McLean and Blower (1993).

References

McLean, A.R., and Blower, S.M. (1993) ‘Imperfect vaccines and herd immunity to HIV’, Proc. Roy. Soc. Land. B253, 9.

Discussion

Garnett To what extent do the results of Blower and McLean depend on their assumption of an exponential decay in loss of vaccine-induced immunity?

Reply The initial HIV vaccine model that we have presented is based upon reasonable biological assumptions and is also based upon what is known about the mechanism of action of prophylactic vaccines for other diseases. The data that are available from (non-HIV) clinical trials with long follow-up periods, although fairly limited, suggest generally that vaccine-induced immunity is either life-long or tends to decay exponentially.

Introduction

Among diseases with long development times – from inception to diagnosis, from diagnosis to death, or both – feature breast and cervical cancer, endstage organ failure and cardiovascular disease; all four have aspects in common with HIV disease, but its transmissibility by many routes sets it apart from the rest.

Breast and cervical cancers are detectable when asymptomatic (or precancerous) by screening; treatment of screen-detected lesions saves lives. Blood pressure lowering drugs reduce the incidence of stroke and coronary heart disease. An HIV antibody positive test leads to consideration of personal measures to prevent onward transmission of HIV disease and, from clinical management, to a better quality and length of HIV infected life, but as yet no cure. Cervical cancer, like HIV disease, is viral in origin and sexual transmission is implicated in its spread. Transplantation shares with HIV disease an immunological basis, recency, and unusual intensity of patient monitoring through laboratory markers.

Section 2 reviews the statistical problems posed by these other four applications, all of high public health or political profile, before consideration in Section 3 of chronic disease processes generally, and the transmissibility of HIV disease. Section 4 focusses on three data analytic themes in HIV disease – progression markers, incubation distribution and infectivity – and briefly reviews how they have been tackled. Future statistical directions in HIV disease are outlined in Section 5 with emphasis on transmission study design and overview, and on the non-proportionality over time of covariate influences. This includes unmeasured (frailty) as well as measured (and appropriately parametrized) covariates.

What's so special about within-host dynamics?

When an epidemiology conference hosts a session on ‘within-host dynamics’, three questions immediately come to a discussant's mind: ‘Why are we doing this?’, ‘What are we doing here?’ and ‘What difference does being within a host make?’ The first of these three questions is answered by the quality of the papers presented in this session. There are many fascinating questions about the pathogenesis of infectious diseases, and about the dynamics of host responses to infectious organisms. These questions often involve highly nonlinear interactions between host and pathogen within the host organism. The rigour and clarity of thought required by mathematical description of such interactions is a great aid in developing an intuitive understanding of which processes are important, and of what patterns those processes might generate.

The subject matter of the four talks: two on HIV, one on malaria and one on schistosomiasis is probably a fair representation of the field. The enigma of HIV's pathogenesis has prompted many theoretical (and empirical) investigations. Nowak's theory is one elegant example of the numerous theories proposed to explain the long period between infection with HIV and illness with AIDS (reviewed in McLean 1993). In contrast to the care and rigour with which Nowak's theory has been expounded, some of the ‘verbal theories’ of HIV's pathogenesis are classic examples of why biologists ought to make mathematical models; so that they can see when the predictions made by their verbal models simply cannot be matched up with the patterns they aim to explain. A cogent argument for the use of mathematical models in an exploratory fashion by biologists is given by Hillis (1993).

Introduction

If a fraction of a population is vaccinated, the spread of the infective agent is slowed down and consequently the incidence of infection for non-vaccinated persons is reduced. If the vaccine itself carries some risk then the risk of illness for a non-vaccinated person can drop below that for a vaccinated one. This occurs when the spread of infection has been greatly reduced by vaccination. It then becomes questionable whether people will agree to be vaccinated and whether, therefore, an infectious disease can be eliminated by vaccination on a voluntary basis. With smallpox vaccination it was shown that in the final years of the campaign more cases of illness were caused in the US by vaccination than by infections (CDC 1971) and nowadays there is a lively discussion about the oral poliomyelitis vaccines which have been incriminated in causing more paralytic cases in the US than the rare wild viruses do (Beale 1990, Begg et al 1987, Cossart 1977, McBean and Modlin 1987). Fine and Clarkson (1986) were the first to compare the risk of illness of vaccinated persons with that of non-vaccinated ones from a theoretical point of view. To estimate the incidence of infection that results from a given vaccination coverage, they made arbitrary assumptions which imply that an infection can only be eliminated if 100 percent of the population are effectively immunized. Moreover, they did not take into consideration an age-specific conditional probability of illness or death upon infection. Many of the so-called ‘childhood diseases’ tend to be more serious in adults than in infants.

Introduction

Hydatid disease is caused by accidental infection with the intermediate stages of tapeworms of Echinococcus species, principally Echinococcus granulosus and E.multilocularis. The adult worm parasitises the small intestine of carnivores, usually Canidae, and sheds proglottids containing eggs which pass with the host's faeces. If the eggs are then ingested by a herbivore they develop into a larval stage (cyst) within the liver or other viscera. The cycle is completed when the carnivore consumes the herbivore, ingesting a mature cyst. Echinococcus granulosus originated in a wolf-deer life-cycle, and has evolved in dogs and sheep and other domesticated and wild animals. The definitive hosts of E.multilocularis are foxes, and the intermediate hosts small rodents such as voles and lemmings. Echinococcus granulosus is ubiquitous but E.multilocularis is confined to the Northern Hemisphere. Cystic hydatid disease in man is caused by the larval form of E.granulosus. Surgery provides a cure in 50–90% of cases, but recurrence is common. Alveolar hydatid disease caused by E.multilocularis results in metastases throughout the soft organs. Until recently it was invariably fatal, but chemotherapy may retard the proliferation of cysts. For further information on the epidemiology of echinococcosis and hydatid disease see Roberts and Gemmell (1994). This paper presents two examples where models of the dynamics of Echinococcus species have been used to investigate control policies.

Echinococcus granulosusin farmed animals in New Zealand.

The first case of cystic hydatid disease in New Zealand was recorded in 1862. The annual number of cases peaked at 7.2 per 100,000 in 1946, and then declined to 0.37 per 100,000 in 1987, largely due to the control programme that was initiated in 1959. From that time all dogs were subjected to regular chemotherapy to remove tapeworms.

Comments on Heterogeneity Aspects in Mathematical Epidemiology

To begin with I remark generally upon two important aspects of communication in mathematical epidemiology, nomenclature and interpretation of formulae (Sections 1 and 2), and then upon the tension between simple and sophisticated models (Section 3). Finally possible quantitative and qualitative effects of heterogeneity on the basic reproduction ratio in epidemic models are discussed (Section 4).

Nomenclature

Mathematical epidemiology is a scientific field where interdisciplinary collaboration is essential and, as part of this, communication between mathematicians and non-mathematicians (biologists, epidemiologists, etc.) is most important. One prerequisite for efficient and fruitful communication – in particular with people who are not specialists in mathematical epidemiology – is a joint nomenclature which tries to avoid using verbal expressions in ambiguous or misleading ways. But unfortunately there seems to persist some confusion about this, not only between persons who are specialists in different scientific fields, but occasionally even within fields.

One example of expressions in epidemiology which are quite misleading, but can be easily avoided, is random mixing and non-random mixing. Both terms assume that an infection is transmitted through contacts which are made at random (even if the mathematical model does not contain explicitly a stochastic formulation, but some deterministic counterpart). But whereas the first of these two terms intends to express that the population mixes homogeneously, and thus even contacts between individuals of distinct subpopulations are made uniformly, the latter particularly expresses that this is not the case. Since both types of mixing patterns involve random contacts, these two inappropriate verbal expressions should not be used.

Introduction

In general, little attention is given to homosexual role behaviour as a factor in the sexual transmission of HIV. Models that include variations in sexual behaviour are usually restricted to heterogeneity in sexual partner-change and the manner in which subpopulations mix. Following Trichopoulos et al (1988), Wiley and Herschkorn (1989), van Griensven et al. (1990), we lay emphasis on homosexual role behaviour (role separation) as a factor influencing the spread of HIV in homosexual populations. If there are large differences between the risks of receptive and insertive anal intercourse, with the latter carrying only minimal risk, then one may expect that changes in role behaviour distributions influence the spread of HIV. As pointed out by Trichopoulos et al. (1988), role separation is expected to reduce the spread of HIV since those who are practicing insertive intercourse would be at low risk and those practicing receptive intercourse would not be at very high risk because of the low prevalence of HIV among their sexual partners.

Based on this conjecture Wiley and Herschkorn (1989) constructed a theoretical model for exploring the effect of differentiation of roles in anal intercourse on the size of AIDS epidemics in homosexual populations. Under the assumption of no risk associated with insertive anal intercourse it was shown that epidemic intensity increases with increasing size of the dual-role (both insertive and receptive) subpopulation. Their paper, however, was not concerned with the analysis of specific data. Recently van Griensven et al. (1990) and van Zessen and van Griensven (1992) provided empirical evidence, using data from the first two cycles of the Amsterdam cohort, that homosexual role behaviour is a factor in the spread of HIV.

HIV infection is a major threat to the health of most nations in the world. Prevention of the spread of HIV requires major behavioural changes and the current evidence points to rather a gloomy picture. Prevention through vaccination is a hope for the future for developed countries but this hope has to be sustained with the knowledge that for the developing countries the expected large cost of vaccination would be a very major problem. Further, the travelling patterns of people mean that people from various countries will continue to be in contact. HIV infection is thus a challenging problem and efforts of a variety of people can contribute to the control of this threat.

The control of HIV infection and the care of HIV patients have to meet the challenges of complexity, uncertainty, variability, and limited resources. The modelling approach of Operational Research has grown from dealing with these challenges. This paper is mainly concerned with operational models for the care of HIV patients. The models can also provide some information for evaluating the effects of proposed preventive measures. The models can help two groups of users: Health Planners concerned with resource allocation and budgeting can use the models to obtain information about resource usage and costs over time; Clinical Staff interested in effective patient care and in the monitoring of resources used for patient care can obtain helpful quantitative information from the models.

Starting with an infected person, the natural history of HIV infection is characterised by uncertainty and variability. Clinical staff need quite detailed information about patients.

We use a deterministic model to study heterosexual HIV transmission. We focus on questions related to sexual partner selection across risk levels and the sensitivity of the model to the differences in infectivity between men and women. We neglect transmission into this purely heterosexual subpopulation from people who have been infected through other means, such as intravenous drug use or sex between men. As well, we neglect age, migration, and many other important features of the epidemic.

Modeling studies have shown that the AIDS epidemic is very sensitive to both the biological aspects of HIV infection and the human behaviors that spread HIV. They have demonstrated that the epidemic is sensitive to subtle features of the biology of HIV and human behavior, including the distribution of times from infection to AIDS, changes in infectiousness with duration of infection, and the distribution of partner acquisition-rates in the population (Hyman and Stanley 1989).

The male and female at-risk populations are divided into uninfected people, those infected with HIV but who have not yet developed AIDS, and the infecteds that have progressed to AIDS. We assume that the major characteristic that affects the probability of infection is the partner-acquisition rate, and distribute each of these populations according to a risk variable which determines this rate. NonAIDS infecteds are also distributed according to their duration of infection, and AIDS cases are distributed according to the duration of time since their diagnosis. People mature into a given risk group. They may change behavior, switching from one risk group to another.

In the first session of this conference, we have heard three very different papers on three very interesting topics. I wish, however, to claim the prerogative granted to me by the organizers to comment in detail on the paper by Dr. Gore and to make only brief reference to the other papers. This approach is primarily motivated by the prior availability of Dr, Gore's manuscript and is no reflection on the other presentations.

Dr. Gore has provided an impressive survey of data analysis methods which have been employed for the study of a variety of diseases with long development times. In my comments, I hope to elaborate on some of the issues raised rather than to offer specific criticisms.

Time to event regression models have played a major role in the analysis of longitudinal data. Dr. Gore has placed some stress on the need for further consideration of the covariate codings in such models, in particular with respect to HIV disease. I have five comments on this issue.

(1) When using time dependent variables, it is almost essential that lagged covariates be used. It is unlikely that, for example, interest is directed towards the predictive role of CD4 counts at the time of AIDS diagnosis. In an analysis of the Toronto Sexual Contact Cohort (TSCC) data (Coates et al. 1992), we adopted the approach of lagging immunological markers by one year so that the developed models used covariates of the form X(t – 1) rather than X(t). The need for this is sometimes not recognized because covariates are not updated continuously and therefore an effective lagging takes place because the last available measurement is used in the regression models.

Numerous factors influence the likelihood of contact between susceptible and infectious people, including participation in different social activities, cultural barriers such as membership of particular ethnic groups with associated customs, or separation due to geographic distance. These factors guarantee that contact among individuals within a population is distinctly nonrandom. Results from several theoretical studies show that nonrandom mixing among subgroups has many consequences for the outcome of epidemic spread, including affecting the time at which a disease is introduced into different subgroups and the speed of propagation and severity of an epidemic.

Most recent models for the spread of infectious diseases in human populations incorporate nonrandom patterns of mixing across subgroups and include a parameter for contact between groups that depends on the subgroups from which the susceptible and infective individuals derive. This parameter represents only the end result of the mixing process, leaving implicit the mechanism by which contact occurs. Here we describe a model that explicitly incorporates the mechanism for contact among individuals from different subgroups. Contact between individuals occurs as a result of the mobility of participants across either geographic or social space. Because it is simpler to visualize, we limit our discussion here to geographic mobility. Models for behavioral mobility are straightforward adaptations of this process (e.g. Sattenspiel and Castillo-Chavez 1990, Jacquez et al 1989).

Consider a population that is distributed among n regions. Individuals from region i leave the region at a rate σi per unit time. These visitors are then distributed among the n – 1 destinations with probabilities vij to each destination j.

Introduction

The human immunodeficiency virus (HIV) is the aetiological agent of the acquired immunodeficiency syndrome (AIDS). Despite intensive research during the past 9 years since the discovery of the virus, the epidemic continues to spread in the human population. Analysis of epidemiological data reveals a depressing picture for the worst afflicted regions such as sub-Saharan Africa, with increasing amounts of infection in the heterosexual population. In these regions it is likely that AIDS may result in population decline within a few decades if present trends continue (Anderson et al. 1991, Anderson and May 1991).

The course of HIV infections can be separated into three stages.

1. Acute clinical illness during primary HIV infection occurs in 50-70% of infected patients, starts generally 2-4 weeks after infection and lasts from 1-2 weeks (Tindall and Cooper 1991). The clinical manifestations are varied and include fever, neuropatic and dermatological symptoms. Virus can be isolated from infected blood cells, cell free plasma, cerebrospinal fluid and bone marrow cells. The high replication and widespread distribution of virus is followed by strong immunological responses, which result in a decrease of viral antigens to almost undetectable levels and a resolution of clinical symptoms.

2. The second, chronic, phase (8-10 years on average) is characterized by low levels of HIV expression and only small pathological changes. Patients are generally asymptomatic. CD4 cell concentrations are constant or slowly decreasing.

3. […]