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Modeling population dynamics with random initial conditions

Gilberto González-Parra
Grupo de Matemática Multidisplinar
Universidad de los Andes, Mérida, Venezuela
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Juan Carlos Cortés
Instituto de Matemática Multidisciplinar
Universidad Politécnica de Valencia, España
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Rafael Jacinto Villanueva
Instituto de Matemática Multidisciplinar
Universidad Politécnica de Valencia, España
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Francisco José Santonja
Departamento de Estadística e Investigación Operativa
Universidad de Valencia, España
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  • Abstract
    In this paper a random differential equation system modeling population dynamics is investigated by means of the statistical moments equation. Monte Carlo simulations are performed in order to compare with the statistical moments equation approach. The randomness in the model appears due to the uncertainty on the initial conditions. The model is a nonlinear differential equation system with random initial conditions and is based on the classical SIS epidemic model. By assuming different probability distribution functions for the initial conditions of different classes of the population we obtain the mean and variance of the stochastic process representing the proportion of these classes at any time. The results show that the theoretical approach of moments equation agrees very well with Monte Carlo numerical results and the solutions converge to the equilibrium point independently of the probability distribution function of the initial conditions.
  • Keywords: Random differential equation, Statistical moments equation, Stochastic process, Population dynamics, SIS epidemic model, Monte Carlo method.
  • AMS Subject classifications: 37N25, 65C05, 81T80, 91D10, 91B74, 93A30.