The maintenance of strain structure in populations of recombining infectious agents.

Using mathematical models that combine population genetic and epidemiological processes, we resolve the paradox that many important pathogens appear to persist as discrete strains despite the constant exchange of genetic material. We show that dominant polymorphic determinants (that is, those that elicit the most effective immune responses) will be organized into nonoverlapping combinations as a result of selection by the host immune system, thereby defining a set of discrete independently transmitted strains. By analysing 222 isolates of Neisseria meningitidis, we show that two highly polymorphic epitopes of the outer membrane protein PorA exist in nonoverlapping combinations as predicted by this general framework. The model indicates that dominant polymorphic determinants will be in linkage disequilibrium, despite frequent genetic exchange, even though they may be encoded by several unlinked genes. This suggests that the detection of nonrandom associations between epitope regions can be employed as a novel strategem for identifying dominant polymorphic antigens.

Limit cycles in predator-prey communities.

Essentially all models that have been proposed for predator-prey systems are shown to possess either a stable point equilibrium or a stable limit cycle. This stable limit cycle, an explicitly nonlinear feature, is commonly overlooked in conventional analyses of these models. Such a stable limit cycle provides a satisfying explanation for those animal communities in which populations are observed to oscillate in a rather reproducible periodic manner.

Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos.

Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.

How many species are there on Earth?

This article surveys current answers to the factual question posed in the title and reviews the kinds of information that are needed to make these answers more precise. Various factors affecting diversity are also reviewed. These include the structure of food webs, the relative abundance of species, the number of species and of individuals in different categories of body size, along with other determinants of the commonness and rarity of organisms.

Extinction and the loss of evolutionary history.

Extinction episodes, such as the anthropogenic one currently under way, result in a pruned tree of life. But what fraction of the underlying evolutionary history survives when k of n species in a taxon are lost? This is relevant both to how species loss has translated into a loss of evolutionary history and to assigning conservation priorities. Here it is shown that approximately 80 percent of the underlying tree of life can survive even when approximately 95 percent of species are lost, and that algorithms that maximize the amount of evolutionary history preserved are not much better than choosing the survivors at random. Given the political, economic, and social realities constraining conservation biology, these findings may be helpful.

Infectious diseases and population cycles of forest insects.

The regulation of natural populations of invertebrate hosts by viral, bacterial, protozoan, or helminth infections is discussed, using models that combine elements of conventional epidemiology (where the host population is assumed constant) with dynamic elements drawn from predator-prey studies; the apparent absence of acquired immunity in invertebrates simplifies the analysis. Highly pathogenic infections, with long-lived infective stages, tend to produce cyclic behavior in their host populations. The models give an explanation of the 9- to 10-year population cycles of the larch bud moth (Zeiraphera diniana) in the European Alps and suggest that microsporidian protozoan and baculovirus infections may be responsible for the 5- to 12-year population cycles observed in many temperate forest insects.

Ecology. Species-area relations in tropical forests.

A power law called the species-area relationship describes the finding that the number of species is proportional to the size of the area in which they are found, raised to an exponent (usually, a number between 0.2 and 0.3). In their Perspective, May and Stumpf discuss new results from a survey of five tropical forest census areas containing a total of a million trees. They explain how this large data set can be used to fine-tune the existing power law so that it provides a better prediction of species diversity in small census samples.

Species turnover rates on islands: dependence on census interval.

Measurements of species turnover in island bird communities demonstrate two trends with increasing census interval t: (i) Apparent turnover rates T decrease greatly with t, and (ii) the coefficient of variation of T decreases asymptotically to a constant value. These effects are predicted by a statistical model whose parameters are the immigration and extinction probabilities of each species. Available bird censuses at intervals of decades underestimate turnover rates by about an order of magnitude.

Harvesting natural populations in a randomly fluctuating environment.

As harvesting effort and yield are increased, animal populations that are being harvested for sustained yield will take longer to recover from environmentally imposed disturbances. One consequence is that the coefficient of variation (the relative variance) of the yield increases as the point of maximum sustained yield (MSY) is approached. When overexploitation has resulted in a population smaller than that for MSY, high effort produces a low average yield with high variance. These observations accord with observed trends in several fish and whaling industries. We expect these effects to be more pronounced for a harvesting strategy based on constant quotas than for one based on constant effort. Although developed in a MSY context, the conclusions also apply if the aim is to maximize the present value of (discounted) net economic revenue.

Directly transmitted infections diseases: control by vaccination.

Mathematical models for the dynamics of directly transmitted viral and bacterial infections are guides to the understanding of observed patterns in the age specific incidence of some common childhood diseases of humans, before and after the advent of vaccination programs. For those infections that show recurrent epidemic behavior, the interepidemic period can be related to parameters characterizing the infection (such as latent and infectious periods and the average age of first infection); this relation agrees with the data of a variety of childhood diseases. Criteria for the eradication of a disease are given, in terms of the proportion of the population to be vaccinated and the age-specific vaccination schedule. These criteria are compared with a detailed analysis of the vaccination programs against measles and whooping cough in Britain, and estimates are made of the levels of protection that would be needed to eradicate these diseases.

Antigenic diversity thresholds and the development of AIDS.

Longitudinal studies of patients infected with HIV-1 reveal a long and variable incubation period between infection and the development of AIDS. Data from a small number of infected patients show temporal changes in the number of genetically distinct strains of the virus throughout the incubation period, with a slow but steady rise in diversity during the progression to disease. A mathematical model of the dynamic interaction between viral diversity and the human immune system suggests the existence of an antigen diversity threshold, below which the immune system is able to regulate viral population growth but above which the virus population induces the collapse of the CD4+ lymphocyte population. The model suggests that antigenic diversity is the cause, not a consequence, of immunodeficiency disease. The model is compared with available data, and is used to assess how the timing of the application of chemotherapy or immunotherapy influences the rate of progress to disease.

Management of multispecies fisheries.

With the overexploitation of many conventional fish stcocks, and growing interest in harvesting new kinds of food from the sea, there is increasing need for managers of fisheries to take account of interactions among species. In particular, as Antarctic krill-fishing industries grow, there is a need to agree upon sound principles for managing the Southern Ocean ecosystem. Using simple models, we discuss the way multispecies food webs respond to the harvesting of species at differrent trophic levels. These biological and economic insights are applied to a discussion of fisheries in the Southern Otean and the North Sea and to enunciate some for harvesting in multispecies systems.

The population biology of the interaction between HIV-1 and HIV-2: coexistence or competitive exclusion?

BACKGROUND: The emergence and rapid world-wide spread of HIV provides an unusual opportunity for the study of the evolution and maintenance of virulence in a major human pathogen. OBJECTIVE: To analyse the available biological and epidemiological data on the pathogenicity, transmissibility and antigenic similarity of HIV-1 and HIV-2, and use simple mathematical models of competition between the two viral types within a defined host community. RESULTS AND CONCLUSIONS: Analysis revealed a positive association between pathogenicity and reproductive success. A mathematical model of the concomitant transmission of the two viruses suggests that HIV-1 will competitively displace HIV-2 in the longer term in areas where both viruses are being transmitted within the same sexually active population.

Networks of sexual contacts: implications for the pattern of spread of HIV.

This paper examines the influence of sexual contact patterns (mixing matrices) on the pattern of the AIDS epidemic in a male homosexual community via numerical studies of a mathematical model of the transmission dynamics of HIV. A discussion is presented of the range of possible structures of networks of sexual contacts with extremes of assortative (within sexual activity groups) and disassortative (between sexual activity groups) mixing. The assortative mixing extreme is shown to generate the most rapid growth in the incidence of infection in the early stages of the epidemic while the disassortative extreme is shown to generate the epidemic of the largest magnitude over a long period. High within-group mixing (assortative) may generate multi-peak epidemics. The results are discussed in the context of both the interpretation of observed patterns of the spread of HIV and the acquisition of data on sexual contact patterns.

The evolutionary dynamics of HIV-1 quasispecies and the development of immunodeficiency disease.

This paper presents a theory to explain the development of immunodeficiency disease after a long and variable incubation period of infection with HIV-1. Two assumptions are central to the theory: (1) mutation via reverse transcription during viral replication can generate viral strains resistant to neutralization by antibodies specific to earlier mutants in a particular host; (2) the virus can kill the CD4-positive lymphocytes that play a role in mounting an immunological attack directed at the virus. The theory is examined via the development of a mathematical model which reveals that an increasing number of antigenically distinct viral strains may overwhelm the immune system of the host. As the viral diversity increases beyond a certain level the immune system is unable to suppress the population growth of all the strains simultaneously. The intuitive explanation of this pattern of model behaviour lies in the assumption that each virus can kill CD4-positive lymphocytes that are specific to any of the viral strains, but each lymphocyte only directs immunological attack against a single viral strain.(ABSTRACT TRUNCATED AT 250 WORDS)

Superinfection and the evolution of parasite virulence.

Earlier ideas that parasites evolve toward becoming harmless to their hosts have, in recent years, given way to more analytic studies, focused on the 'basic reproductive rate', R0, of individual parasites. In general, the biology of the parasite life cycle will lead to constraining relations between virulence (parasite-associated host death or reduction in fertility) and transmissibility: the maximum R0 may then be attained by virulence being high, or low, or at some intermediate level, depending on the details of the constraining relations. Such studies have not generally included superinfection (where an already-infected host is infected by another parasite). Here we propose a general, but simple, model of superinfection, which is amenable to analytical treatment. In such models selection does not simply act to maximize R0; superinfection leads to selection for higher levels of virulence, highly polymorphic parasite populations and very complicated dynamics. We calculate the equilibrium distribution of parasite strains and the maximum level of virulence that can be maintained by superinfection. We also note the equivalence between our 'superinfection model' and recent approaches to the study of the meta-population dynamics of multi-species interactions.

Coinfection and the evolution of parasite virulence.

Analyses of the selection pressures acting on parasite virulence are made more complicated when individual hosts can simultaneously harbour many different strains or genotypes of a parasite. Here we explore the evolutionary dynamics of host-parasite associations in which individual hosts can be coinfected with many different parasite strains. (We take coinfection to mean that each strain transmits at a rate unaffected by the presence of others in the same host.) This study thus represents the opposite extreme to our earlier work on superinfection in which there is a dominance hierarchy such that only the most virulent strain present in a host is transmitted. For highly diverse populations of parasite strains, we find that such coinfection leads to selection for strains whose virulence-levels lie in a relatively narrow band close to the maximum consistent with the parasite's basic preproductive ratio, R0, exceeding unity.

The population dynamics of vertically and horizontally transmitted parasites.

We analyse a model of the transmission dynamics of a parasite transmitted both vertically and horizontally. The basic reproductive ratio (R0) of the parasite is shown to be a sum of horizontal and vertical components. We derive expressions for the equilibrium prevalence of infection for a mixture of horizontal and vertical transmission; prevalence can reach 100% if transmission is sufficiently high. At the endemic equilibrium, if prevalence is high, most transmission will in general be vertical, but horizontal transmission rates must be high to reach and stably maintain such an equilibrium. Surprisingly, for such parasites the highest equilibrium rates of vertical transmission are observed when horizontal transmission is very effective. We discuss the implications for assessing the importance of horizontal v. vertical transmission from field data, and we suggest some implications for the evolution of virulence.

Measles immunization strategies for an epidemiologically heterogeneous population: the Israeli case study.

Although the vaccine against measles has been routinely applied over a quarter of a century, measles is still an active disease in Israel. The January 1991 outbreak caused high morbidity in infant and adolescent populations and high mortality, especially among nomad Bedouins in the southern region of the country. The Bedouins form a small fraction of the total Israeli population (ca. 2%), but it is thought that they may experience significantly higher rates of transmission than the majority group. In this work we use deterministic compartmental mathematical models to define the optimal immunization strategy for a population consisting of a majority group characterized by low transmission rates and a minority group characterized by high transmission rates; this study allows both for transmission differences between the two groups, and for possible differences in the average cost (or difficulty) in reaching individuals for vaccination. Our analysis shows that the optimal vaccination policy for such a population involves different strategies for the two groups: a smaller fraction is to be vaccinated in the minority group if transmission in this group is not much larger than in the majority group, whereas, if the difference in transmission is very large, a higher proportion is to be vaccinated in the minority group. The advantage of this non-uniform vaccination policy is that it involves vaccination of a smaller fraction of the total population (and costs less, if there are differential costs between the groups), as compared with the proportion vaccinated under the conventional uniform vaccination policy.(ABSTRACT TRUNCATED AT 250 WORDS)