Dengue is one of the most important infectious diseases in the world, in terms of death and economic cost. Hence, the modeling of dengue is of great importance to help us understand the dynamics disease, and interfering with its spreading mathematical by the proposition of control methods. In this work, control strategies in an attempt to eliminate the Aedes aegypti mosquito, as well as proposals for the vaccination campaign are evaluated. In our mathematical model, the mechanical control is accomplished through the environmental support capacity affected by a discrete function that represents removal of breeding. Chemical control is carried out from the use of insecticide and larvicidal. The efficacious of vaccination is studied through the removal of a fraction of individuals, proportional to the rate of vaccination, from the susceptible compartment and its transfer to the recovered compartment. Our major find is that the dengue fever epidemic is only eradicated with the use of an immunizing vaccine because control measures directed against the vector are not enough to halt disease spreading. Even where the infected mosquitoes are eliminated from the system, susceptible mosquitoes are still present, and infected humans cause dengue fever to reappear in the human population.
Microorganisms and plants very commonly release toxic secondary chemical compounds (allelochemicals) that inhibit or kill sensitive strains or individuals from their own or other species. In this work we study a model that describes two species interacting through allelopathic suppression and competing for resources. Employing linear stability analysis, the conditions for coexistence or extinction of species in spatially homogeneous systems were determined. We found that the borders between the regimes of bistability, coexistence, and the extinction of the weaker by the stronger competitor, are altered by allelopathic interactions. In addition, traveling wave solutions for one species invasion were obtained considering the spatially explicit nature of the model. Our findings indicate that the minimum speed of the invasion wavefronts depends primarily on the competition coefficients and the parameters characterizing the species' functional responses to their allelochemicals. As a general rule, the species provided with the most effective chemical weapons dominates the population dynamics. Finally, we found a tristability at the coexistence region due to the combination of allelopathy and patchy population distributions in space. So, our model provides a distinct mechanism, independent of social behaviors, that produces such unexpected tristability impossible in classical competition models involving one-to-one individual interactions.
Evidence shows that diversity and spatial distributions of biological communities are largely driven by the race of living organisms in their adaptation to chemicals synthesized by their neighbors. In this report, the emergence of mathematical models on pure spatial self-organization induced by biochemical suppression (allelopathy) and competition between species were investigated through numerical analysis. For both random and patched initial spatial distributions of species, we demonstrate that warfare survivors are self-organized on the landscape in Turing-like patterns driven by diffusive instabilities of allelochemicals. These patterns are simple; either all species coexist at low diffusion rates or are massively extinct, except for a few at high diffusivities, but they are complex and biodiversity-sustained at intermediate diffusion rates. “Defensive alliances” and ecotones seem to be basic mechanisms that sustain great biodiversity in our hybrid cellular automata model. Moreover, species coexistence and extinction exhibit multi-stationarity.
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