The effects of spatial heterogeneity in population dynamics

The dynamics of a population inhabiting a heterogeneous environment are modelled by a diffusive logistic equation with spatially varying growth rate. The overall suitability of an environment is characterized by the principal eigenvalue of the corresponding linearized equation. The dependence of the eigenvalue on the spatial arrangement of regions of favorable and unfavorable habitat and on boundary conditions is analyzed in a number of cases.Research supported by National Science Foundation grant #DMS 88-02346     

Kin groups and trait groups: population structure and epidemic disease selection

A Monte Carlo simulation based on the population structure of a small-scale human population, the Semai Senoi of Malaysia, has been developed to study the combined effects of group, kin, and individual selection. The population structure resembles D.S. Wilson's structured deme model in that local breeding populations (Semai settlements) are subdivided into trait groups (hamlets) that may be kin-structured and are not themselves demes. Additionally, settlement breeding populations are connected by two-dimensional stepping-stone migration approaching 30% per generation. Group and kin-structured group selection occur among hamlets the survivors of which then disperse to breed within the settlement population. Genetic drift is modeled by the process of hamlet formation; individual selection as a deterministic process, and stepping-stone migration as either random or kin-structured migrant groups. The mechanism for group selection is epidemics of infectious disease that can wipe out small hamlets particularly if most adults become sick and social life collapses. Genetic resistance to a disease is an individual attribute; however, hamlet groups with several resistant adults are less likely to disintegrate and experience high social mortality. A specific human gene, hemoglobin E, which confers resistance to malaria, is studied as an example of the process. The results of the simulations show that high genetic variance among hamlet groups may be generated by moderate degrees of kin-structuring. This strong microdifferentiation provides the potential for group selection. The effect of group selection in this case is rapid increase in gene frequencies among the total set of populations. In fact, group selection in concert with individual selection produced a faster rate of gene frequency increase among a set of 25 populations than the rate within a single unstructured population subject to deterministic individual selection. Such rapid evolution with plausible rates of extinction, individual selection, and migration and a population structure realistic in its general form, has implications for specific human polymorphisms such as hemoglobin variants and for the more general problem of the tempo of evolution as well.     

The epidemic threshold of vector-borne diseases with seasonality

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R ₀ which is adapted to periodic environments. This R ₀ is estimated numerically for the epidemic in Chichaoua; 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor 3.76.     

Patterns in the effects of infectious diseases on population growth

An infectious disease may reduce or even stop the exponential growth of a population. We consider two very simple models for microparasitic and macroparasitic diseases, respectively, and study how the effect depends on a contact parameter K. The results are presented as bifurcation diagrams involving several threshold values of . The precise form of the bifurcation diagram depends critically on a second parameter , measuring the influence of the disease on the fertility of the hosts. A striking outcome of the analysis is that for certain ranges of parameter values bistable behaviour occurs: either the population grows exponentially or it oscillates periodically with large amplitude.The work of this author was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG)     

A cytomic approach reveals population heterogeneity of Cupriavidus necator in response to harmful phenol concentrations

The understanding of functions of cells within microbial populations or communities is certainly needed for existing and novel cytomic approaches which grip the individual scale. Population behaviour results from single cell performances and is caused by the individual genetic pool, history, life cycle states and microenvironmental surroundings. Mimicking natural impaired environments, the paper shows that the Gram-negative Betaproteobacterium Cupriavidus necator dramatically altered its population heterogeneity in response to harmful phenol concentrations. Multiparametric flow cytometry was used to follow variations in structural cellular parameters like chromosome contents and storage materials. The functioning of these different cell types was resolved by ensuing proteomics after the cells' spatial separation by cell sorting, finding 11 proteins changed in their expression profile, among them elongation factor Tu and the trigger factor. At least one third of the individuals clearly underwent starving states; however, simultaneously these cells prepared themselves for entering the life cycle again. Using cytomics to recognise individual structure and function on the microbial scale represents an innovative technical design to describe the complexity of such systems, overcoming the disadvantage of small cell volumes and, thus, to resolve bacterial strategies to survive harmful environments by altering population heterogeneity.     

The effect of contact heterogeneity and multiple routes of transmission on final epidemic size

Heterogeneity in the number of potentially infectious contacts amongst members of a population increases the basic reproduction ratio (R(0)) and markedly alters disease dynamics compared to traditional mean-field models. Most models describing transmission on contact networks only account for one specific route of transmission. However, for many infectious diseases multiple routes of transmission exist. The model presented here captures transmission through a well defined network of contacts, complemented by mean-field type transmission amongst the nodes of the network that accounts for alternative routes of transmission. The impact of these combined transmission mechanisms on the final epidemic size is investigated analytically. The analytic predictions for the purely mean-field case and the transmission through the network-only case are confirmed by individual-based network simulations. There is a critical transmission potential above which an increased contribution of the mean-field type transmission increases the final epidemic size while an increased contribution of the transmission through the network decreases it. Below the critical transmission potential the opposite effect is observed.     

Modeling transmission of directly transmitted infectious diseases using colored stochastic Petri nets

In order to improve our understanding of directly transmitted pathogens within host populations, epidemic models should take into account individual heterogeneities as well as stochastic fluctuations in individual parameters. The associated cost results in an increasing level of complexity of the mathematical models which generally lack consistent formalisms. In this paper, we demonstrate that complex epidemic models could be expressed as colored stochastic Petri nets (CSPN). CSPN is a mathematical tool developed in computer science. The concept is based on the Markov Chain theory and on a standard well codified graphical formalism. This approach presents an alternative to other computer simulation methods since it offers both a theoretical formalism and a graphical representation that facilitate the implementation, the understanding and thus the replication or modification of the model. We explain how common concepts of epidemic models--such as the incidence function--can be easily translated into an individual based point of view in the CSPN formalism. We then illustrate this approach by using the well documented susceptible-infected model with recruitment and death.     

A two-stage model for the SIR outbreak: accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage

The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the randomized SI (susceptible/infected) model. We obtain the probability density function for the early behavior of the epidemic. This provides an appropriate distribution for the initial conditions with which to describe the subsequent deterministic evolution of the system. We apply the method of matching asymptotic expansions to link the two stages. This allows us to estimate the standard deviation of the number of infectives in the developed outbreak, and the statistical characteristics of the outbreak time. The potential trajectories caused by the stochastic nature of the contamination stage show greatest divergence at the initial and fade-out stages and coincide most tightly just after the peak of the epidemic. The time to the peak of the outbreak is not strongly dependent on the initial trajectory.     

我国特有树种长叶榧树的生物学特性与保护问题研究

在持续１０余年对我国特有珍贵树种长叶榧树的分布区、生长环境、生物学特性调查研究基础上,全面系统地提出了该树种保护措施与开发利用意见。     

Spatial and temporal heterogeneity in population dynamics of citrus aphids at a regional scale

A quick method of evaluating aphid infestations on young clementine trees is developed. It uses visual abundance classes. This method is used to compare the population dynamics of two aphid species in 11 orchards in Corsica (France). The curves describing the succession of the abundance indices on each sampling tree between April and July are compared using principal component analyses. The first three factorial axes illustrate the intensity, precocity and flatness of the infestation curves. For each species, a significant between-orchard variability is demonstrated by comparing the mean factorial coordinates of the sampling trees of the different orchards. The validity of such methods for analyzing the demographic strategies of phytophagous and entomophagous insects at a large scale is discussed.     

On moment closures for population dynamics in continuous space

A first-order moment closure, the mean-field assumption that organisms encounter one another in proportion to their spatial average densities, lies at the heart of much theoretical ecology. This assumption ignores all spatial information and, at the very least, needs to be replaced by a second-order closure to gain understanding of ecological dynamics in spatially structured populations. We describe a number of conditions that a second-order closure should satisfy and use these conditions to evaluate some closures currently available in the literature. Two conditions are particularly helpful in discriminating among the alternatives: that the closure should be positive, and that the dynamics should be unaltered when identical individuals are given different labels. On this basis, a class of closures we refer to as 'power-2' turns out to provide a good compromise between positivity and dynamical invariance under relabelling.     

Realistic population dynamics in epidemiological models: the impact of population decline on the dynamics of childhood infectious diseases. Measles in Italy as an example

Most contributions in the field of mathematical modelling of childhood infectious diseases transmission dynamics have focused on stationary or exponentially growing populations. In this paper an epidemiological model with realistic demography is used to investigate the impact of the non-equilibrium conditions typical of the transition to sustained below replacement fertility (BRF) recently observed in a number of western countries, upon the transmission dynamics of measles. The results depend on the manner we model the relation between the (changing) age distribution of the population and contacts. Under some circumstances the transitional ageing phase typical of BRF populations might complexly interact with epidemiological variables leading to (i) a substantial reduction in the amount of vaccination effort required for eliminating the disease; (ii) a significant magnification of the perverse impact of vaccination in terms of the burden of severe age related morbidity.     

The effect of population density on the growth of an animal population

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Contribution from the Department of Fisheries, Kyoto University.

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Contribution from the Entomological Laboratory, Kyoto University, No. 201.     

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7-2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.     

The effect of superspreading on epidemic outbreak size distributions

Recently, evidence has been presented to suggest that there are significant heterogeneities in the transmission of communicable diseases. Here, a stochastic simulation model of an epidemic process that allows for these heterogeneities is used to demonstrate the potentially considerable effect that heterogeneity of transmission will have on epidemic outbreak size distributions. Our simulation results agree well with approximations gained from the theory of branching processes. Outbreak size distributions have previously been used to infer basic epidemiological parameters. We show that if superspreading does occur then such distributions must be interpreted with care. The simulation results are discussed in relation to measles epidemics in isolated populations and in predominantly urban scenarios. The effect of three different disease control policies on outbreak size distributions are shown for varying levels of heterogeneity and disease control effort.     

Spatial heterogeneity, social structure and disease dynamics of animal populations

Social groupings, population dynamics and population movements of animals all give rise to spatio-temporal variations in population levels. These variations may be of crucial importance when considering the spread of infectious diseases since infection levels do not increase unless there is a sufficient pool of susceptible individuals. This paper explores the impact of social groupings on the potential for an endemic disease to develop in a spatially explicit model system. Analysis of the model demonstrates that the explicit inclusion of space allows asymmetry between groups to arise when this was not possible in the equivalent spatially homogeneous system. Moreover, differences in movement behaviours for susceptible and infected individuals gives rise to different spatial profiles for the populations. These profiles were not observed in previous work on an epidemic system. The results are discussed in an ecological context with reference to furious and dumb strains of infectious diseases.