In this paper, we study the numerical approximation of a size-structured population model whose dependency on the environment is managed by the evolution of a vital resource. We show that this is a difficult task: some numerical methods are not suitable for a long-time integration. We analyze the reasons for the failure.

Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially, and, in the second one, both processes occur simultaneously. Here we follow the second approach and give a necessary and sufficient condition for the instability of the disease-free equilibrium in generic networks under the assumption of limited (or frequency-dependent) transmission. Moreover, for uncorrelated networks and under the assumption of non-limited (or density-dependent) transmission, we give a bounding interval for the dominant eigenvalue of the Jacobian matrix of the model equations around the disease-free equilibrium. Finally, for this latter case, we study numerically the prevalence of the infection across the metapopulation as a function of the patch connectivity.

Approximate aggregation techniques consist of introducing certain approximations that allow one to reduce a complex system involving many coupled variables obtaining a simpler ʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allow one to extract information about the complex original system in terms of the behavior of the reduced one. Often, the feature that allows one to carry out such a reduction is the presence of different time scales in the system under consideration. In this work we deal with aggregation techniques in stochastic discrete time models and their application to the study of multiregional models, i.e., of models for an age structured population distributed amongst different spatial patches and in which migration between the patches is usually fast with respect to the demography (reproduction-survival) in each patch. Stochasticity in population models can be of two kinds: environmental and demographic. We review the formulation and the main properties of the dynamics of the different models for populations evolving in discrete time and subjected to the effects of environmental and demographic stochasticity. Then we present different stochastic multiregional models with two time scales in which migration is fast with respect to demography and we review the main relationships between the dynamics of the original complex system and the aggregated simpler one. Finally, and within the context of models with environmental stochasticity in which the environmental variation is Markovian, we make use these techniques to analyze qualitatively the behavior of two multiregional models in which the original complex system is intractable. In particular we study conditions under which the population goes extinct or grows exponentially.

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values R0 and R∗ and further obtained that the disease will be permanent when R0 > 1 and the disease will be going to extinct when R∗ < 1. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

The CDC launched the National Plan to Eliminate Syphilis from the USA in October 1999 [4]. In order to reach this goal, a good understanding of the transmission dynamics of the disease is necessary. Based on a SIRS model Breban et al. [3] provided some evidence that supports the feasibility of the plan proving that no recurring outbreaks should occur for syphilis. We study in this work a syphilis model that includes partial immunity and vaccination. This model suggests that a backward bifurcation very likely occurs for the real-life estimated epidemiological parameters for syphilis. This may explain the resurgence of syphilis after mass treatment [21]. Occurrence of backward bifurcation brings a new challenge for the plan of the CDC’s –striking a balance between treatment of early infection, vaccination development and health education. Our models suggest that the development of an effective vaccine, as well as health education that leads to enhanced biological and behavioral protection against infection in high-risk populations, are among the best ways to achieve the goal of elimination of syphilis in the USA.

For an SI type endemic model with one host and two parasite strains, we study the stability of the endemic coexistence equilibrium, where the host and both parasite strains are present. Our model, which is a system of three ordinary differential equations, assumes complete cross-protection between the parasite strains and reduced fertility and increased mortality of infected hosts. It also assumes that one parasite strain is exclusively vertically transmitted and cannot persists just by itself. We give several sufficient conditions for the equilibrium to be locally asymptotically stable. One of them is that the horizontal transmission is of density-dependent (mass-action) type. If the horizontal transmission is of frequency-dependent (standard) type, we show that, under certain conditions, the equilibrium can be unstable and undamped oscillations can occur. We support and extend our analytical results by numerical simulations and by two-dimensional plots of stability regions for various pairs of parameters.

A simple model of biological evolution of community food webs is introduced. This model is based on the niche model, which is known to generate model food webs that are very similar to empirical food webs. The networks evolve by speciation and extinction. Co-extinctions due to the loss of all prey species are found to play a major role in determining the longterm shape of the food webs. The central aim is to design the model such that the characteristic parameters of the niche model food webs remain in realistic intervals. When the mutation rule is chosen accordingly, it is found that food webs with a complex, biologically meaningful structure emerge and that the statistics of extinction events agrees well with that observed in the paleontological data.

In order to study the impact of fishing on a grouper population, we propose in this paper to model the dynamics of a grouper population in a fishing territory by using structured models. For that purpose, we have integrated the natural population growth, the fishing, the competition for shelter and the dispersion. The dispersion was considered as a consequence of the competition. First we prove, that the grouper stocks may be less sensitive to the removal of large male individuals if female population are totally protected. Second, we show that fishing does not disturb the demographic structure of the population. Finally, we prove that female selective fisheries have the potential of drastically reduce reproductive rates. We also prove that male fishing decreases competition and then increases the total population number.

An optimal control problem is studied for a predator-prey system of PDE, with a logistic growth rate of the prey and a general functional response of the predator. The control function has two components. The purpose is to maximize a mean density of the two species in their habitat. The existence of the optimal solution is analyzed and some necessary optimality conditions are established. The form of the optimal control is found in some particular cases.

Tuberculosis (TB) is the leading cause of death among individuals infected with the hepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. We formulate and analyze a deterministic mathematical model which incorporates of the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only and TB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the TB-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for TB, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcation phenomenon. Through simulations, we mainly find that i) the two diseases will co-exist whenever their partial reproductive numbers exceed unity; (ii) the increased progression rate due to exogenous reinfection from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and (iii) the increased progression rates from acute stage to chronic stage of HBV infection have increased the prevalence levels of HBV and TB prevalences.

We have carried out a polysystem analysis of the population dynamics by using a variety of hypotheses and their respective models based on different system interpretations of the phenomenon under investigation. Each of the models supplements standard dynamic equations for explaining the effects observed. A qualitative model-based analysis is made of the age-specific male mortality for a Siberian industrial city. The study revealed the tendencies for background mortality to increase with age and over time, which characterizes in an integral manner the influence of socio-ecological factors on the decline in population viability. It is shown that these tendencies are similar for different years and for different population age groups.