Local approximation of a metapopulation's equilibrium.

We consider the approximation of the equilibrium of a metapopulation model, in which a finite number of patches are randomly distributed over a bounded subset [Formula: see text] of Euclidean space. The approximation is good when a large number of patches contribute to the colonization pressure on any given unoccupied patch, and when the quality of the patches varies little over the length scale determined by the colonization radius. If this is the case, the equilibrium probability of a patch at z being occupied is shown to be close to [Formula: see text], the equilibrium occupation probability in Levins's model, at any point [Formula: see text] not too close to the boundary, if the local colonization pressure and extinction rates appropriate to z are assumed. The approximation is justified by giving explicit upper and lower bounds for the occupation probabilities, expressed in terms of the model parameters. Since the patches are distributed randomly, the occupation probabilities are also random, and we complement our bounds with explicit bounds on the probability that they are satisfied at all patches simultaneously.

Simulating the component counts of combinatorial structures.

This article describes and compares methods for simulating the component counts of random logarithmic combinatorial structures such as permutations and mappings. We exploit the Feller coupling for simulating permutations to provide a very fast method for simulating logarithmic assemblies more generally. For logarithmic multisets and selections, this approach is replaced by an acceptance/rejection method based on a particular conditioning relationship that represents the distribution of the combinatorial structure as that of independent random variables conditioned on a weighted sum. We show how to improve its acceptance rate. We illustrate the method by estimating the probability that a random mapping has no repeated component sizes, and establish the asymptotic distribution of the difference between the number of components and the number of distinct component sizes for a very general class of logarithmic structures.

Connecting deterministic and stochastic metapopulation models.

In this paper, we study the relationship between certain stochastic and deterministic versions of Hanski's incidence function model and the spatially realistic Levins model. We show that the stochastic version can be well approximated in a certain sense by the deterministic version when the number of habitat patches is large, provided that the presence or absence of individuals in a given patch is influenced by a large number of other patches. Explicit bounds on the deviation between the stochastic and deterministic models are given.

Individual and patch behaviour in structured metapopulation models.

Density dependent Markov population processes with countably many types can often be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, provided that the total population size is large. They also exhibit diffusive stochastic fluctuations on a smaller scale about this deterministic path. Here, it is shown that the individuals in such processes experience an almost deterministic environment. Small groups of individuals behave almost independently of one another, evolving as Markov jump processes, whose transition rates are prescribed functions of time. In the context of metapopulation models, we show that 'individuals' can represent either patches or the individuals that migrate among the patches; in host-parasite systems, they can represent both hosts and parasites.

Estimating the fitness effect of an insertion sequence.

Since its discovery, mobile DNA has fascinated researchers. In particular, many researchers have debated why insertion sequences persist in prokaryote genomes and populations. While some authors think that insertion sequences persist only because of occasional beneficial effects they have on their hosts, others argue that horizontal gene transfer is strong enough to overcome their generally detrimental effects. In this study, we model the long-term fate of a prokaryote cell population, of which a small proportion of cells has been infected with one insertion sequence per cell. Based on our model and the distribution of IS5, an insertion sequence for which sufficient data is available in 525 fully sequenced proteobacterial genomes, we show that the fitness cost of insertion sequences is so small that they are effectively neutral or only slightly detrimental. We also show that an insertion sequence infection can persist and reach the empirically observed distribution if the rate of horizontal gene transfer is at least as large as the fitness cost, and that this rate is well within the rates of horizontal gene transfer observed in nature. In addition, we show that the time needed to reach the observed prevalence of IS5 is unrealistically long for the fitness cost and horizontal gene transfer rate that we computed. Occasional beneficial effects may thus have played an important role in the fast spreading of insertion sequences like IS5.

Emergent multicellular life cycles in filamentous bacteria owing to density-dependent population dynamics.

Filamentous bacteria are the oldest and simplest known multicellular life forms. By using computer simulations and experiments that address cell division in a filamentous context, we investigate some of the ecological factors that can lead to the emergence of a multicellular life cycle in filamentous life forms. The model predicts that if cell division and death rates are dependent on the density of cells in a population, a predictable cycle between short and long filament lengths is produced. During exponential growth, there will be a predominance of multicellular filaments, while at carrying capacity, the population converges to a predominance of short filaments and single cells. Model predictions are experimentally tested and confirmed in cultures of heterotrophic and phototrophic bacterial species. Furthermore, by developing a formulation of generation time in bacterial populations, it is shown that changes in generation time can alter length distributions. The theory predicts that given the same population growth curve and fitness, species with longer generation times have longer filaments during comparable population growth phases. Characterization of the environmental dependence of morphological properties such as length, and the number of cells per filament, helps in understanding the pre-existing conditions for the evolution of developmental cycles in simple multicellular organisms. Moreover, the theoretical prediction that strains with the same fitness can exhibit different lengths at comparable growth phases has important implications. It demonstrates that differences in fitness attributed to morphology are not the sole explanation for the evolution of life cycles dominated by multicellularity.

Compound processes as models for clumped parasite data.

Compound processes are proposed as models for the acquisition of hydatid cysts in sheep, caused by the parasite Echinococcus granulosus. The hypothesis of a clumped infection process against single ingestions is tested and it is shown that the clump-based approach provides a more accurate description of the two data sets investigated. Models with simple and mixed Poisson incidence processes and different clump size distributions are compared. A mixed Poisson incidence process with a zero-truncated negative binomial distribution for the clump sizes is shown to give an adequate description, suggesting that the acquisition of hydatid cysts in the sheep population is heterogeneous, and that the clump sizes are aggregated. The estimates of the parameters derived from the data take plausible values. The average infection rate and the clump size distribution are comparable in both data sets. Goodness-of-fit measures indicate that the model fits the data reasonably well.

Estimation of the transmission dynamics of Theileria equi and Babesia caballi in horses.

For the evaluation of the epidemiology of Theileria equi and Babesia caballi in a herd of 510 horses in SW Mongolia, several mathematical models of the transmission dynamics were constructed. Because the field data contain information on the presence of the parasite (determined by PCR) and the presence of antibodies (determined by IFAT), the models cater for maternal protection with antibodies, susceptible animals, infected animals and animals which have eliminated the parasite and also allow for age-dependent infection in susceptible animals. Maximum likelihood estimation procedures were used to estimate the model parameters and a Monte Carlo approach was applied to select the best fitting model. Overall, the results are in line with previous experimental work, and add evidence that the epidemiology of T. equi differs from that of Babesia spp. The presented modelling approach provides a useful tool for the investigation of some vector-borne diseases and the applied model selection procedure avoids asymptotical assumptions that may not be adequate for the analysis of epidemiological field data.

Convergence of a structured metapopulation model to Levins's model.

We consider a structured metapopulation model describing the dynamics of a single species, whose members are located in separate patches that are linked through migration according to a mean field rule. Our main aim is to find conditions under which its equilibrium distribution is reasonably approximated by that of the unstructured model of Levins (1969). We do this by showing that the (positive) equilibrium distribution converges, as the carrying capacity of each population goes to infinity together with appropriate scalings on the other parameters, to a bimodal distribution, consisting of a point mass at 0, together with a positive part which is closely approximated by a shifted Poisson centred near the carrying capacity. Under this limiting regime, we also give simpler approximate formulae for the equilibrium distribution. We conclude by showing how to compute persistence regions in parameter space for the exact model, and then illustrate all our results with numerical examples. Our proofs are based on Stein's method.

The duration of a sequence of epidemics.

The typical duration of an epidemic in a sequence of linearly ordered populations shows a surprising nonmonotonic behaviour with respect to population size, which was noted by Swinton (1998) [Bull. Math. Biol., 60, 215-230]. This paper gives the sketch of a proof of the phenomenon.

Simplified estimation of Widmark "r" values by the method of Forrest.

To simplify blood alcohol calculations, tables are presented giving Widmark 'r' values estimated by the method of ARW Forrest (Journal of the Forensic Science Society 1986; 27: 249-252).

Recognizing very distant sequence relationships among proteins by family profile analysis.

Family profile analysis (FPA), described in this paper, compares all available homologous amino acid sequences of a target family with the profile of a probe family while conventional sequence profile analysis (Gribskov M, Lüthy R, Eisenberg D. Meth Enzymol 1990;183:146-159) considers only a single target sequence in comparison with the probe family. The increased input of sequence information in FPA expands the range for sequence-based recognition of structural relationships. In the FPA algorithm, Zscores of each of the target sequences, obtained from a probe profile search over all known amino acid sequences, are averaged and then compared with the scores for sequences of 100 reference families in the same probe family search. The resulting F-Zscore of the target family, expressed in "effective standard deviations" of the mean Zscores of the reference families, with value above a threshold of 3.5 indicates a statistically significant evolutionary relationship between the target and probe families. The sensitivity of FPA to sequence information was tested with several protein families where distant relationships have been verified from known tertiary protein architectures, which included vitamin B6-dependent enzymes, (beta/alpha)8-barrel proteins, beta-trefoil proteins, and globins. In comparison to other methods, FPA proved to be significantly more sensitive, finding numerous new homologies. The FPA technique is not only useful to test a suspected relationship between probe and target families but also identifies possible target families in profile searches over all known primary structures.

Mathematical model of a virus-neutralizing immunglobulin response.

We present a mathematical model to simulate the kinetics of B-cell activation and the virus-neutralizing immunoglobulin response in the spleen of mice after infection with vesicular stomatitis virus (VSV). Our model combines data from in vitro experiments and in vivo kinetic observations. A system of eight nonlinear differential equations was used in the computer experiments and numerically solved. The isotype switch from IgM to IgG in the presence of T-cell help was modelled by a time variable function, used as a parameter. The model solutions indicate fast kinetics of the generation of VSV-neutralizing IgM antibodies within 2-3 days post immunization peaking on day 5 at a serum concentration of approximately 80 microgram ml-1 IgM, which is the equivalent to about 10% of the total IgM serum concentration. The frequency of virus-specific B cells increases about 1000-fold within the first 4 days after immunization. Protective levels of VSV-neutralizing IgG antibodies (>/=10 microgram ml-1) are reached within 5 to 6 days post immunization. Fitting the model solution to the experimentally observed neutralizing serum titers suggests an increase in the neutralizing activity of IgGs occurring between days 5 and 8 post-infection. The model indicates that less than 10 VSV-specific B cells have to be triggered daily to maintain protective IgG serum titers during the memory phase.

Modeling the transmission of schistosomiasis: an introductory view.

The transmission of schistosomiasis can be modeled at various levels of complexity in terms of systems of mathematical equations. This paper shows how such models can be set up, and stresses the importance of incorporating the right basic assumptions from the outset. The use of models for evaluating possible control strategies is illustrated in the context of a simple prevalence model of transmission. Some limitations and uncertainties involved in modeling schistosomiasis are also indicated.

A host-parasite model yielding heterogeneous parasite loads.

Many models of parasitic infections lead to an approximately Poisson distribution of parasites among hosts, in stark contrast to the highly over-dispersed distributions that are usually encountered in practice. In this paper, a model is analyzed which, while assuming all individuals to be alike, can still lead to a very heterogeneous distribution of parasites among the host population. The model can be viewed as a very simple mean field interacting particle system, with the particles corresponding to the individual hosts, which behaves like an associated deterministic system when the number of hosts is large. The deterministic system describes the evolution over time of the proportions of the population with different parasite loads and its equilibria are interpreted as typical distributions of parasites among hosts. Despite its simplicity, the model is complicated enough mathematically to leave a number of open problems.

Modeling the overdispersion of parasite loads.

Many epidemic models lead to an approximately Poisson distribution of parasites among hosts. This is at variance with observation, where heavy overdispersion is the rule. A simple model is proposed that, while treating individuals alike, nonetheless gives rise to highly variable parasite loads.

GC/MS analysis of propylated barbiturates.

A simple and reproducible method for the analysis of barbiturates by GC/MS after derivatization with dimethylformamide dipropyl acetal is reported. The method is readily adapted to screening, confirmation, and quantitation.

Aspects of line-fitting in bivariate allometric analyses.

One of the fundamental problems involved in analyses of the scaling effects of body size (allometric analysis is the choice of an appropriate best-fit line in bivariate logarithmic plots. Following a discussion of some basic aspects of allometric analysis, the tow mai procedures for the determination of a best-fit line - the least-squares regression and the major axis - are examined with respect to their different properties and underlying models. It is important to distinguish intraspecific from interspecific scaling and to recognize the distinction between use of a best-fit line to define a relationship and use of the line for prediction. An alternative model to the bivariate normal distribution, referred to as the 'extruded normal distribution', is presented and its implications are examined with respect to two test cases (scaling of basal metabolic rate in human males; scaling of population density in mammals).

The importance of age and water contact patterns in relation to Schistosoma haematobium infection.

Data obtained by Dalton & Pole in 1978 were analysed in order to assess the relative importance of age and water contact patterns in explaining the variations in level of infection with Schistosoma haematobium observed among their study population. It was found that age and sex were the only two significant factors, and that the information about an individual's water contact pattern contained in the data was of no further help in predicting his level of infection. This conclusion is in marked opposition to that reached by Dalton & Pole.

Macdonald's model and the transmission of bilharzia.

The paper considers a model for the transmission of bilharzia based on Macdonald's assumptions, in the light of data observed in the field. It is shown, in particular, that the threshold parameter governing whether or not an endemic cycle can be established is closely related to the proportion of infected snails in a community, and that this proportion is normally observed to be rather smaller than is compatible with the model. By considering more sophisticated models, allowing for the latent period of infection in the snails, and also for spatial and seasonal heterogeneity, the effective proportion of infected snails, from the point of view of Macdonald's model, is shown to be rather larger, and expressions are given whereby it can be evaluated from observable quantities. However, for the data from Malirong which are taken as illustration, it is also demonstrated that an even more plausible threshold value is obtained from a simple model incorporating human immunity in addition to the assumptions of Macdonald's model, and that, if this model were reasonable, human immunity would appear to be the most important factor in controlling the level of the disease in Malirong.