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Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

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  • Enrique Lozano-Ochoa
  • Jorge Fernando Camacho
  • Cruz Vargas-De-León

Abstract

We study an epidemiological mathematical model formulated in terms of an ODE system taking into account both social and nonsocial contagion risks of obesity. Analyzing first the case in which the model presents only the effect due to social contagion and using qualitative methods of the stability analysis, we prove that such system has at the most three equilibrium points, one disease-free equilibrium and two endemic equilibria, and also that it has no periodic orbits. Particularly, we found that when considering (the basic reproductive number) as a parameter, the system exhibits a backward bifurcation: the disease-free equilibrium is stable when and unstable when , whereas the two endemic equilibria appear from (a specific positive value reached by and less than unity), one being asymptotically stable and the other unstable, but for values, only the former remains inside the feasible region. On the other hand, considering social and nonsocial contagion and following the same methodology, we found that the dynamic of the model is simpler than that described above: it has a unique endemic equilibrium point that is globally asymptotically stable.

Suggested Citation

  • Enrique Lozano-Ochoa & Jorge Fernando Camacho & Cruz Vargas-De-León, 2017. "Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-12, January.
    • Handle: RePEc:hin:jnddns:1084769
    DOI: 10.1155/2017/1084769
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    References listed on IDEAS

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    1. Arenas, Abraham J. & González-Parra, Gilberto & Jódar, Lucas, 2009. "Periodic solutions of nonautonomous differential systems modeling obesity population," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1234-1244.
    2. González-Parra, Gilberto & Acedo, L. & Villanueva Micó, Rafael-J. & Arenas, Abraham J., 2010. "Modeling the social obesity epidemic with stochastic networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(17), pages 3692-3701.
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