Population models for diseases with no recovery

An S I epidemic model with a general shape of density-dependent mortality and incidence rate is studied. The asymptotic behaviour is global convergence to an endemic equilibrium, above a threshold, and to a disease-free equilibrium, below the threshold. The effect of vaccination is then examined.     

Non-linear age-dependent population diffusion

Summary A non-linear problem arising in the study of an age-dependent population diffusion is considered. Existence and uniqueness results together with a priori bounds for the growth of the population are obtained. Moreover the solutions are shown to depend continuously on the initial data.This work was done under the auspices of the G.N.A.F.A. of the National Research Council.     

若干具有非线性传染力的传染病模型的稳定性分析

讨论了具有常数迁入和非线性传染力的三类传染病模型,即SIRI模型,SIRI框架下的DS模型及SIR框架下的DI模型。给出了它们基本再生数R0的表达式,证明了R0≤1时无病平衡点是全局稳定的,同时证明了如果地方病平衡点存在,则必是全局稳定的结果（从而必唯一）对第一和第三个模型还给出了R0＞1时地方病平衡点的存在唯一性。     

Persistence in population models with demographic fluctuations

A persistence and extinction theory is developed through analytical studies of deterministic population models. Under hypotheses that require demographic parameters to fluctuate temporally, the populations may or may not oscillaate. Extinction, when it occurs, is asymptotic. An hierarchy of persistence criteria, based upon fluctuations measured by time average means, is derived. In some situations a threshold value is found to separate persistent population models from those that tend to extinction. Application of the persistence-extinction theory is to the problem of assessing effects of a toxic substance on a population when toxicant inputs to the environment and to resources are oscillatory.     

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.     

Clines in the presence of asymmetric migration

If a population, which consists of individuals having genetic variation at one locus, with two alleles A and a, evolves under the influence of migration and selection, gradients in the distribution of alleles may arise. We consider the effect of asymmetry in the migration and spatial dependence of the selection process, upon the emergence and stability of such gradients.     

Stochastic analogues of deterministic single-species population models

Although single-species deterministic difference equations have long been used in modeling the dynamics of animal populations, little attention has been paid to how stochasticity should be incorporated into these models. By deriving stochastic analogues to difference equations from first principles, we show that the form of these models depends on whether noise in the population process is demographic or environmental. When noise is demographic, we argue that variance around the expectation is proportional to the expectation. When noise is environmental the variance depends in a non-trivial way on how variation enters into model parameters, but we argue that if the environment affects the population multiplicatively then variance is proportional to the square of the expectation. We compare various stochastic analogues of the Ricker map model by fitting them, using maximum likelihood estimation, to data generated from an individual-based model and the weevil data of Utida. Our demographic models are significantly better than our environmental models at fitting noise generated by population processes where noise is mainly demographic. However, the traditionally chosen stochastic analogues to deterministic models--additive normally distributed noise and multiplicative lognormally distributed noise--generally fit all data sets well. Thus, the form of the variance does play a role in the fitting of models to ecological time series, but may not be important in practice as first supposed.     

Epidemic Models with Heterogeneous Mixing and Treatment

We consider a two-group epidemic model with treatment and establish a final size relation that gives the extent of the epidemic. This relation can be established with arbitrary mixing between the groups even though it may not be feasible to determine the reproduction number for the model. If the mixing of the two groups is proportionate, there is an explicit expression for the reproductive number and the final size relation is expressible in terms of the components of the reproduction number. We also extend the results to a two-group influenza model with proportionate mixing. Some numerical simulations suggest that (i) the assumption of no disease deaths is a good approximation if the disease death rate is small and (ii) a one-group model is a close approximation to a two-group model but a two-group model is necessary for comparing targeted treatment strategies. This research was supported by MITACS and an NSERC Research Grant.     

Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t+ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.This work was partially supported by the Centre National de la Recherche Scientifique through ATP 95939900     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Animal migration: linking models and data beyond taxonomic limits

An international workshop on animal migration was held at the Lorentz Center in Leiden, The Netherlands, 2–6 March 2009, bringing together leading theoreticians and empiricists from the major migratory taxa, aiming at the identification of cutting-edge questions in migration research that cross taxonomic borders.     

一类带有非线性感染率传染病模型的动力学性质（英文）

主要介绍了一类带有非线性感染率的传染病模型.并且证明了当基本再生数Ro≤1时,无病平衡点是全局稳定的,当基本再生数R_0〉1时,疾病持续.     

Using network models to approximate spatial point-process models

Spatial effects are fundamental to ecological and epidemiological systems, yet the incorporation of space into models is potentially complex. Fixed-edge network models (i.e. networks where each edge has the same fixed strength of interaction) are widely used to study spatial processes but they make simplistic assumptions about spatial scale and structure. Furthermore, it can be difficult to parameterize such models with empirical data. By comparison, spatial point-process models are often more realistic than fixed-edge network models, but are also more difficult to analyze. Here we develop a moment closure technique that allows us to define a fixed-edge network model which predicts the prevalence and rate of epidemic spread of a continuous spatial point-process epidemic model. This approach provides a systematic method for accurate parameterization of network models using data from continuously distributed populations (such as data on dispersal kernels). Insofar as point-process models are accurate representations of real spatial biological systems, our example also supports the view that network models are realistic representations of space.     

具有阶段结构和非线性接触率的SI传染病模型的渐近性态

研究了一类具有阶段结构和非线性接触率的传染病模型的渐近性态,得到了传染病最终消除和成为地方病的阀值．     

Stochastic models for cell motion and taxis

Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in response to a stimulus, termed taxis, has previously been modeled at a phenomenological level using the Keller-Segel diffusion equation. The Keller-Segel model cannot distinguish certain modes of taxis, and this motivates the introduction of a richer class of models which is nevertheless still amenable to statistical analysis. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation via maximum likelihood for a range of applicable models. One particular experimental situation, involving the effect of an electric field on cell behavior, is considered in detail. In this case, an Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model.     

Dynamic models of infectious diseases as regulators of population sizes

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.Research supported by Centers for Disease Control contract 200-87-0515. Support services provided at the University of Iowa Center for Advanced Studies     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.