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The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviors, or others.To include a sufficiently general disease model, we first build up from first principles a discrete-time susceptible–exposed–infectious–recovered–susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number R0.Then, we propose a general full model that includes sequentially the two processes at different time scales and proceed to its analysis through a reduced model. The basic reproduction number R‾0 of the reduced system gives a good approximation of R0 of the full model since it serves at analyzing its asymptotic behavior.As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.

Traditional biomedical approaches treat diseases in isolation, but the importance of synergistic disease interactions is now recognized. As a first step we present and analyze a simple coinfection model for two diseases simultaneously affecting a population. The host population is affected by the primary disease, a long-term infection whose dynamics is described by a SIS model with demography, which facilitates individuals acquiring a second disease, secondary (or opportunistic) disease. The secondary disease is instead a short-term infection affecting only the primary infected individuals. Its dynamics is also represented by a SIS model with no demography. To distinguish between short- and long-term infection the complete model is written as a two-time-scale system. The primary disease acts at the slow time scale while the secondary disease does at the fast one, allowing a dimension reduction of the system and making its analysis tractable. We show that an opportunistic disease outbreak might change drastically the outcome of the primary epidemic process, although it does among the outcomes allowed by the primary disease. We have found situations in which either acting on the opportunistic disease transmission or recovery rates or controlling the susceptible and infected population size allows eradicating/promoting disease endemicity.

In this work we present a reduction result for discrete-time systems with two time scales. In order to be valid, previous results in the field require some strong hypotheses that are difficult to check in practical applications. Roughly speaking, the iterates of a map as well as their differentials must converge uniformly on compact sets. Here, we eliminate the hypothesis of uniform convergence of the differentials at no significant cost in the conclusions of the result. This new result is then used to extend to non-linear cases the reduction of some population discrete models involving processes acting at different time scales. In practical cases, some processes that occur at a fast time scale are often only measured at slow time intervals, notably mortality. For a general class of linear models that include such a kind of processes, it has been shown that a more realistic approach requires the re-scaling of those processes to be considered at the fast time scale. We develop the same type of re-scaling in some non-linear models and prove the corresponding reduction results. We also provide an application to a particular model of a structured population in a two-patch environment.

Context Mediterranean managed dry-edge pine forests maintain biodiversity and supply key ecosystem services but are threatened by climate change and are highly vulnerable to desertification. Forest management through its effect on stand structure can play a key role on forest stability in response to increasing aridity, but the role of forest structure on drought resilience remains little explored. Objectives To investigate the role of tree growth and forest structure on forest resilience under increasing aridity and two contrasting policy-management regimes. We compared three management scenarios; (i) “business as usual”-based on the current harvesting regime and increasing aridity—and two scenarios that differ in the target forest function; (ii) a “conservation scenario”, oriented to preserve forest stock under increasing aridity; and (iii), a “productivity scenario” oriented to maintain forest yield under increasingly arid conditions. Methods The study site is part of a large-homogeneous pine-covered landscape covering sandy flatlands in Central Spain. The site is a dry-edge forest characterized by a lower productivity and tree density relative to most Iberian Pinus pinaster forests. We parameterized and tested an analytical size-structured forest dynamics model with last century tree growth and forest structure historical management records. Results Under current management (Scenario-i), increasing aridity resulted in a reduction of stock, productivity, and maximum mean tree size. Resilience boundaries differed among Scenario-ii and -Scenario-iii, revealing a strong control of the management regime on resilience via forest structure. We identified a trade-off between tree harvest size and harvesting rate, along which there were various possible resilient forest structures and management regimes. Resilience boundaries for a yield-oriented management (Scenario-iii) were much more restrictive than for a stock-oriented management (Scenario-ii), requiring a drastic decrease in both tree harvest size and thinning rates. In contrast, stock preservation was feasible under moderate thinning rates and a moderate reduction in tree harvest size. Conclusions Forest structure is a key component of forest resilience to drought. Adequate forest management can play a key role in reducing forest vulnerability while ensuring a long-term sustainable resource supply. Analytical tractable models of forest dynamics can help to identify key mechanisms underlying drought resilience and to design management options that preclude these social-ecological systems from crossing a tipping point over a degraded alternate state.

In this work we present a reduction result for discrete time systems with two time scales. In order to be valid, previous results in the field require some strong hypotheses that are difficult to check in practical applications. Roughly speaking, the iterates of a map as well as their differentials must converge uniformly on compact sets. Here, we eliminate the hypothesis of uniform convergence of the differentials at no significant cost in the conclusions of the result. This new result is then used to extend to nonlinear cases the reduction of some population discrete models involving processes acting at different time scales. In practical cases, some processes that occur at a fast time scale are often only measured at slow time intervals, notably mortality. For a general class of linear models that include such kind of processes, it has been shown that a more realistic approach requires the re-scaling of those processes to be considered at the fast time scale. We develop the same type of re-scaling in some nonlinear models and prove the corresponding reduction results. We also provide an application to a particular model of a structured population in a two-patch environment.

In this work we address the analysis of discrete-time models of structured metapopulations subject to environmental stochasticity. Previous works on these models made use of the fact that migrations between the patches can be considered fast with respect to demography (maturation, survival, reproduction) in the population. It was assumed that, within each time step of the model, there are many fast migration steps followed by one slow demographic event. This assumption allowed one to apply approximate reduction techniques that eased the model analysis. It is however a questionable issue in some cases since, in particular, individuals can die at any moment of the time step. We propose new non-equivalent models in which we re-scale survival to consider its effect on the fast scale. We propose a more general formulation of the approximate reduction techniques so that they also apply to the proposed new models. We prove that the main asymptotic elements in this kind of stochastic models, the Stochastic Growth Rate (SGR) and the Scaled Logarithmic Variance (SLV), can be related between the original and the reduced systems, so that the analysis of the latter allows us to ascertain the population fate in the first. Then we go on to considering some cases where we illustrate the reduction technique and show the differences between both modelling options. In some cases using one option represents exponential growth, whereas the other yields extinction.

In this work we develop a discrete model of competing species affected by a common parasite. We analyze the influence of the fast development of the shared disease on the community dynamics. The model is presented under the form of a two time scales discrete system with four variables. Thus, it becomes analytically tractable with the help of the appropriate reduction method. The 2-dimensional reduced system, that has the same the asymptotic behaviour of the full model, is a generalization of the Leslie-Gower competition model. It has the unfrequent property in this kind of models of including multiple equilibrium attractors of mixed type. The analysis of the reduced system shows that parasites can completely alter the outcome of competition depending on the parasite's basic reproductive number R0. In some cases, initial conditions decide among several exclusion or coexistence scenarios.

The main aim of the work is to present a general class of two time scales discrete-time epidemic models. In the proposed framework the disease dynamics is considered to act on a slower time scale than a second different process that could represent movements between spatial locations, changes of individual activities or behaviours, or others. To include a sufficiently general disease model, we first build up from first principles a discrete-time Susceptible-Exposed-Infectious-Recovered-Susceptible (SEIRS) model and characterize the eradication or endemicity of the disease with the help of its basic reproduction number R0. Then, we propose a general full model that includes sequentially the two processes at different time scales, and proceed to its analysis through a reduced model. The basic reproduction number R0 of the reduced system gives a good approximation of the R0 of the full model since it serves at analyzing its asymptotic behaviour. As an illustration of the proposed general framework, it is shown that there exist conditions under which a locally endemic disease, considering isolated patches in a metapopulation, can be eradicated globally by establishing the appropriate movements between patches.

Ministerio para la Transición Ecológica y el Reto Demográfico

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.