Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations |
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| Authors: | Jocirei D Ferreira Luz Myriam Echeverry Carlos A Peña Rincon |
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| Affiliation: | 1. Institute of Exact and Earth Science, Federal University of Mato Grosso, Barra do Gar?as, Mato Grosso, Brazil;2. Department of Mathematics, Universidad de los Andes, Santa Fe de Bogotá, Colombia;3. Department of Mathematics, Universidad Sergio Arboleda, Santa Fe de Bogotá, Colombia |
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| Abstract: | A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd. |
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| Keywords: | population dynamics toxoplasmosis model trans‐critical bifurcation reaction diffusion model local stability global stability |
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