A minimal model coupling communicable and non-communicable diseases

This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leading cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, heterogeneous populations) allows the communicable disease to become endemic even if the basic reproduction number is less than $1$. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information.Comment: 19 pages, 5 figure

Reduction of Discrete Dynamical Systems with Applications to Dynamics Population Models

In this work we review the aggregation of variables method for discrete dynamical systems. These methods consist of describing the asymptotic behaviour of a complex system involving many coupled variables through the asymptotic behaviour of a reduced system formulated in terms of a few global variables. We consider population dynamics models including two processes acting at different time scales. Each process has associated a map describing its effect along its specific time unit. The discrete system encompassing both processes is expressed in the slow time scale composing the map associated to the slow one and the k-th iterate of the map associated to the fast one. In the linear case a result is stated showing the relationship between the corresponding asymptotic elements of both systems, initial and reduced. In the nonlinear case, the reduction result establishes the existence, stability and basins of attraction of steady states and periodic solutions of the original system with the help of the same elements of the corresponding reduced system. Several models looking over the main applications of the method to populations dynamics are collected to illustrate the general results

Discrete Models of Disease and Competition

The aim of this work is to analyze the influence of the fast development of a disease on competition dynamics. To this end we present two discrete time ecoepidemic models. The first one corresponds to the case of one parasite affecting demography and intraspecific competition in a single host, whereas the second one contemplates the more complex case of competition between two different species, one of which is infected by the parasite. We carry out a complete mathematical analysis of the asymptotic behavior of the solutions of the corresponding systems of difference equations and derive interesting ecological information about the influence of a disease in competition dynamics. This includes an assessment of the impact of the disease on the equilibrium population of both species as well as some counterintuitive behaviors in which although we would expect the outbreak of the disease to negatively affect the infected species, the contrary happens

A Minimal Model Coupling Communicable and Non-Communicable Diseases

This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leading cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, a dynamic heterogeneous population) allows the communicable disease to become endemic even if the basic reproduction number is less than 1. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the basic reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information

Fast Dispersal in Semelparous Populations

We consider a model for the dynamics of a semelparous age-structured population where individuals move among different sites. The model consists of a system of difference equations with two time scales. Individual movements are considered to be fast in comparison to demographic processes. We propose a general model with m + 1 age classes and n different sites. Demography is described locally by a general density dependent Leslie matrix. Dispersal for each age-class is defined by a stochastic matrix depending on the total numbers of individuals in each class. The (m + 1) × n dimensional two time scales system is approximately reduced to an m + 1 dimensional semelparous Leslie model. In the case of 2 age-classes and 2 sites with constant dispersal rates we consider the bifurcation that occurs at the trivial equilibrium using the inherent net reproductive number as the bifurcation parameter. We find that different dispersal strategies can change at the global level the local demographic outcome. This modeling framework can be further used to correctly embed different fast processes in population-level models

A Discrete Predator-Prey Ecoepidemic Model

In this work we present a discrete predator-prey ecoepidemic model. The predator-prey interactions are represented by a discrete Leslie-Gower model with prey intra-specific competition. The disease dynamics follows a discrete SIS epidemic model with frequency-dependent transmission. We focus on the case of disease only affecting prey though the case of a parasite of the predators is also presented. We assume that parasites provoke density- and trait-mediated indirect interactions in the predator-prey community that occur on a shorter time scale. This is included in the model considering that in each time unit there exist a number k of episodes of epidemic changes followed by a single episode of demographic change, all of them occurring separately. The aim of this work is examining the effects of parasites on the long-term prey-predators interactions. These interactions in the absence of disease are governed by the Leslie-Gower model. In the case of endemic disease they can be analyzed through a reduced predator-prey model which summarizes the disease dynamics in its parameters. Conditions for the disease to drive extinct the whole community are obtained. When the community keeps stabilized different cases of the influence of disease on populations sizes are presented

Fast Dispersal in Semelparous Populations

We consider a model for the dynamics of a semelparous age-structured population where individuals move among different sites. The model consists of a system of difference equations with two time scales. Individual movements are considered to be fast in comparison to demographic processes. We propose a general model with m + 1 age classes and n different sites. Demography is described locally by a general density dependent Leslie matrix. Dispersal for each age-class is defined by a stochastic matrix depending on the total numbers of individuals in each class. The (m + 1) × n dimensional two time scales system is approximately reduced to an m + 1 dimensional semelparous Leslie model. In the case of 2 age-classes and 2 sites with constant dispersal rates we consider the bifurcation that occurs at the trivial equilibrium using the inherent net reproductive number as the bifurcation parameter. We find that different dispersal strategies can change at the global level the local demographic outcome. This modeling framework can be further used to correctly embed different fast processes in population-level models