Stability in an age-structured metapopulation model

We present a discrete model for a metapopulation of a single species with overlapping generations. Based on the dynamical behavior of the system in absence of dispersal, we have shown that a migration mechanism which depends only on age can not stabilize a previously unstable homogeneous equilibrium, but can drive a stable uncoupled system to instability if the migration rules are strongly related to age structure.     

A statistical distribution with an unbounded hazard function and its application to a theory from demography.

We introduce a statistical distribution with hazard function mu(t) proportional to (psi-t)beta-1 for 0 less than or equal to t less than psi and shape parameter beta satisfying 0 less than beta less than 1. This hazard function is suggested by a theory of aging in demography. We discuss properties of this distribution and maximum likelihood estimates of its parameters. Mixture distributions are considered to account for pooling across dissimilar populations. This model is compared with the Gompertz and generalized Pareto distributions and used to estimate a finite limit on human lifespan based on the survival of a group of female centenarians.     

Mathematical hazard models of mortality: an alternative to model life tables

A five-parameter competing hazard model of the age pattern of mortality is described, and methods of fitting it to survivorship, death rate, and age structure data are developed and presented. The methods are then applied to published life table and census data to construct life tables for a Late Woodland population, a Christian period Nubian population, and the Yanomama. The advantage of this approach over the use of model life tables is that the hazard model facilitates life-table construction without imposing a particular age pattern of mortality on the data. This development makes it possible to use anthropological data to extend the study of human variation in mortality patterns to small populations.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

The application of an age-structured model to the north Aegean anchovy fishery: an evaluation of different management measures

The objective of this paper is the integration of existing biological and fishery knowledge of anchovy into a unified modelling framework in order to advance our understanding of species' population dynamics under different fishing strategies. The model simulates the anchovy biomass by combining an age-specific growth equation and a continuous age-structured population model based on the McKendrick-Von Foerster equation. Model predictions were compared to the biomass estimates and annual catches during the period 2003-2008. The present work provided direct evidence for the significance of the prespawning period as a critical life period for the management of anchovy stock in the Aegean Sea. It was found that the introduction of additional management measures could increase the profits in the long run for the fishery. However, for these to become apparent they will require a minimum of four years. Results also indicated that the reduction of fishing mortality directed at the spawning stock (recruitment overfishing) and the selective harvesting of younger individuals may be a plausible means of increasing stock's total anchovy biomass. Finally, as a criterion of long-term population survival, we have considered the mathematical notation of persistence. The numerical criteria of persistence in the present model indicated that the anchovy population could be considered viable.     

The Chapman-Enskog procedure for an age-structured population model: initial, boundary and corner layer corrections

We consider a mathematical model of an age-structured population of some fisheries (for example, anchovies, sardines or soles). Two time scales are involved in the problem: the fast time scale for the migration dynamics and the slow time scale for the demographic process. At a first step, we study the so called 'aggregated' system by means of the semigroups theory. Then, we study the asymptotic behaviour of the model by using the Chapman-Enskog procedure. In particular, we study initial, boundary and corner layer effects in order to obtain the exact initial and boundary conditions the approximated solution has to satisfy.     

An age-structured epidemic model of rotavirus with vaccination

The recent approval of a rotavirus vaccine in Mexico motivates this study on the potential impact of the use of such a vaccine on rotavirus prevention and control. An age-structured model that describes the rotavirus transmission dynamics of infections is introduced. Conditions that guarantee the local and global stability analysis of the disease-free steady state distribution as well as the existence of an endemic steady state distribution are established. The impact of maternal antibodies on the implementation of vaccine is evaluated. Model results are used to identify optimal age-dependent vaccination strategies. A convergent numerical scheme for the model is introduced but not implemented. This paper is dedicated to Prof. K. P. Hadeler, who continues to push the frontier of knowledge in mathematical biology.     

An age-structured model to evaluate the potential of novel malaria-control interventions: a case study of fungal biopesticide sprays

It has recently been proposed that mosquito vectors of human diseases, particularly malaria, may be controlled by spraying with fungal biopesticides that increase the rate of adult mortality. Though fungal pathogens do not cause instantaneous mortality, they can kill mosquitoes before they are old enough to transmit disease. A model is developed (i) to explore the potential for fungal entomopathogens to reduce significantly infectious mosquito populations, (ii) to assess the relative value of the many different fungal strains that might be used, and (iii) to help guide the tactical design of vector-control programmes. The model follows the dynamics of different classes of adult mosquitoes with the risk of mortality due to the fungus being assumed to be a function of time since infection (modelled using the Weibull distribution). It is shown that substantial reductions in mosquito numbers are feasible for realistic assumptions about mosquito, fungus and malaria biology and moderate to low daily fungal infection probability. The choice of optimal fungal strain and spraying regime is shown to depend on local mosquito and malaria biology. Fungal pathogens may also influence the ability of mosquitoes to transmit malaria and such effects are shown to further reduce vectorial capacity.     

Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia

Cyclical neutropenia (CN) is a rare hematopoietic disorder in which the patient's neutrophil level drops to extremely low levels for a few days approximately every three weeks. CN is effectively treated with granulocyte colony stimulating factor (G-CSF), which is known to interfere with apoptosis in neutrophil precursors and to consequently increase the circulating neutrophil level. However, G-CSF treatment usually fails to eliminate the oscillation. In this study, we establish an age-structured model of hematopoiesis, which reduces to a set of four delay differential equations with specific forms of initial functions. We numerically investigate the possible stable solutions of the model equations with respect to changes in the parameters as well as the initial conditions. The results show that the hematopoietic system possesses multistability for parameters typical of the normal healthy state. From our numerical results, decreasing the proliferation rate of neutrophil precursors or increasing the stem cell death rate are two possible mechanisms to induce cyclical neutropenia, and the periods of the resulting oscillations are independent of the changing parameters. We also discuss the dependence of the model solution on the initial condition at normal parameter values corresponding to a healthy state. Using insight from our results we design a hybrid treatment method that is able to abolish the oscillations in CN.     

Different demography of friends and strangers: an experiment on the impact of kinship and familiarity in Clethrionomys glareolus

Summary We examined demographic effects of familiarity and relatedness in the bank vole Clethrionomys glareolus (Schreber) in four 0.5-ha enclosures in Central Finland. In two enclosures were mature voles which had overwintered together and some of their mature off-spring (hereafter referred to as Friends), and in the other two individuals of the same species captured from different localities near the study area (Strangers). The experiment lasted from June to September. The populations of Friends reached densities twice as high as those of Strangers with a significantly higher rate of recruitment and survival of the young. This may have been due to mutual familiarity decreasing antagonism towards the juveniles. The conflicting results obtained from studies of Clethrionomys and Microtus are discussed. We believe that these genera represent behavioural adaptations to different habitats and ways of life. Most behavioural population regulation hypotheses are based on studies of Microtus. We conclude that these results should be applied with great caution to other rodent genera.     

Public vaccination policy using an age-structured model of pneumococcal infection dynamics

Public health professionals are charged with the task of designing prevention programs for the effective control of biologically intricate infectious diseases at a population level. The effective vaccination of a population for pneumococcal diseases (infections caused by Streptococcus pneumoniae) remains a relevant question in the scientific community. It is complicated by heterogeneity in individuals’ responses to exposure to the bacterium and their responses to vaccination. Due to these complexities, most modelling efforts in this area have been on the cellular/bacteria level. Here, we introduce an age-structured SEIS-type model of pneumococcal diseases and their vaccination. We discuss the use of this framework in predicting the impact of vaccine strategies, with pneumococcal diseases as an example. Using parameter values reasonable for a developed country, we discuss the effects of targeting the colonization and/or infection stages on the age profiles of morbidity in a population.     

The effective population size of an age-structured population with a sex-linked locus

Let a population have the same age distribution and age-specific sex ratios at times 0, 1, 2,..., and let M, F, and L, respectively, be the numbers of males and females in the youngest age group and the generation interval. It can then be shown that if there is a sex-linked locus the fixation probabilities of a neutral allele are respectively 1/3LM or 1/3LF if the allele first appears in one newborn male or in one newborn female. The effective population size can then be derived. It is the same as for a population with discrete generations having the same means, variances, and covariances of male and female progeny during a lifetime and the same number of individuals entering the population per generation.     

The demography of the annual halophyte Spergularia marina on a baltic seashore meadow

The survivorship and reproduction of the annual halophyte Spergularia marina was studied during four years on three different sites on a Baltic seashore meadow. The survivorship showed variation both between years and between sites. The mortality rate was not density-dependent in spite of very high densities in some years. At two of the sites the seed output was related to density, but individuals differed in their reproductive capacity. Limitation of the flower production was the main reason for this difference. At the third site the length of the flowering season determined the reproductive output.     

Frailty modeling for spatially correlated survival data,with application to infant mortality in Minnesota

The use of survival models involving a random effect or 'frailty' term is becoming more common. Usually the random effects are assumed to represent different clusters, and clusters are assumed to be independent. In this paper, we consider random effects corresponding to clusters that are spatially arranged, such as clinical sites or geographical regions. That is, we might suspect that random effects corresponding to strata in closer proximity to each other might also be similar in magnitude. Such spatial arrangement of the strata can be modeled in several ways, but we group these ways into two general settings: geostatistical approaches, where we use the exact geographic locations (e.g. latitude and longitude) of the strata, and lattice approaches, where we use only the positions of the strata relative to each other (e.g. which counties neighbor which others). We compare our approaches in the context of a dataset on infant mortality in Minnesota counties between 1992 and 1996. Our main substantive goal here is to explain the pattern of infant mortality using important covariates (sex, race, birth weight, age of mother, etc.) while accounting for possible (spatially correlated) differences in hazard among the counties. We use the GIS ArcView to map resulting fitted hazard rates, to help search for possible lingering spatial correlation. The DIC criterion (Spiegelhalter et al., Journal of the Royal Statistical Society, Series B 2002, to appear) is used to choose among various competing models. We investigate the quality of fit of our chosen model, and compare its results when used to investigate neonatal versus post-neonatal mortality. We also compare use of our time-to-event outcome survival model with the simpler dichotomous outcome logistic model. Finally, we summarize our findings and suggest directions for future research.     

The application of genetic algorithms in behavioural ecology, illustrated with a model of anti-predator vigilance

We develop a genetic algorithm (GA) approach to a well-known model of vigilance behaviour in a group of animals. We first demonstrate that the GA approach can provide a good match to analytic solutions to the original model. We demonstrate that a GA can be used to find the evolutionarily stable strategies in a model relevant to behavioural ecology where the fitness of each strategy is determined by the frequencies of different strategies in the population. We argue that the GA implementation demonstrates the combination of assumptions used to generate analytic solution to the original model can only be simultaneously satisfied under relatively restrictive conditions on the ecology of the species involved; specifically that group membership is very fluid but group size is conserved over timescales of individual foraging bouts. We further explore the sensitivity of model predictions to alternative choices in the implementation of the GA, and present advice for implementation and presentation of similar models. In particular, we emphasise the need for care in measuring the predictions of such models, so as to capture the intrinsic behaviour of the system and not the remnant of often arbitrarily chosen initial conditions. We also emphasise the potential for GA models to be more transparent about model assumptions regarding underlying biology than analytic models.     

Using singular perturbations to reduce an epidemiological model: application to bovine viral diarrhoea virus within-herd spread

Studying the spread of a pathogen in a managed metapopulation such as cattle herds in a geographical region often requires to take into account both the within- and between-herd transmission dynamics. This can lead to high-dimensional metapopulation systems resulting from the coupling of several within-herd transmission models. To tackle this problem, we aim in this paper at reducing the dimension of a tractable but realistic dynamical system reproducing the within-herd spread. The context chosen to illustrate our purpose is bovine viral diarrhoea virus (BVDV) transmission in a cattle herd structured in two age classes and several epidemiological states, including two infectious states (transiently and persistently infected). Different time scales, corresponding to the epidemiological and demographic processes, are identified which allow to build a reduced model. Singular perturbation technique is used to prove that, under some non-restrictive conditions on parameter values, the behaviour of the original system is quite accurately approximated by that of the reduced system. Simulations are also performed to corroborate the approximation quality. Our study illustrates the methodological interest of using singular perturbations to reduce model complexity. It also rigorously proves the biologically intuitive assumption that transiently infected individuals can be neglected in a homogeneous population, when capturing the global dynamics of BVDV spread.     

Dynamics of an age-structured population with Allee effects and harvesting

We propose a discrete-time, age-structured population model to study the impact of Allee effects and harvesting. It is assumed that survival probabilities from one age class to the next are constants and fertility rate is a function of weighted total population size. Global extinction is certain if the maximal growth rate of the population is less than one. The model can have multiple attractors and the asymptotic dynamics of the population depend on its initial distribution if the maximal growth rate is larger than one. An Allee threshold depending on the components of the unstable interior equilibrium is derived when only the last age class can reproduce. The population becomes extinct if its initial population distribution is below the threshold. Harvesting on any particular age class can decrease the magnitude of the possible stable interior equilibrium and increase the magnitude of the unstable interior equilibrium simultaneously.     

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations

In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.     

A stochastic-covariate failure model with an application to case-control analysis

A stochastic process X(t) is periodically stationary (and ergodic) if, for every k> or =1 and every (t(1),ellipsis,t(k)) in R(k), the sequence of random vectors (X(t(1)+n),ellipsis,X(t(k)+n))n=0,+1, ellipsis, is stationary (and ergodic). For such an ergodic process, let T be a positive random variable defined on the sample space of the process, representing a time of failure. The local failure-rate function is assumed to be of the form up(x),-infinity0 is a small number, tending to 0; and, for each u,T=T(u) is the corresponding failure-time. It is shown that X(T(u)) and uT(u) have, for u-->0, a limiting joint distribution and are, in fact, asymptotically independent. The marginal distributions are explicitly given. Let Y be a random variable whose distribution is the limit of that of X(T(u)). Under the hypothesis that p(x) is unknown or of known functional form but with unknown parameters, it is shown how p(x) can be estimated on the basis of independent copies of the random variable Y. The results are applied to the analysis of a case-control study featuring a 'marker' process X(t) and an 'event-time' T. The event in the study is considered to be particularly rare, and this is reflected in the assumption u-->0. The control-distribution is identified with the average marginal distribution of the (periodically stationary) marker process X(t), and the case-distribution is identified with that of Y. The particular application is a biomedical trial to determine the risk of stroke in terms of the level of an anticoagulant in the blood of the patient.     

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.