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Calibration of Population Growth Mathematical Models by Using Time Series

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Selected Contributions on Statistics and Data Science in Latin America (FNE 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 301))

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Abstract

In this paper, we study the problem of coefficients identification in population growth models. We consider that the dynamics of the population is described by a system of ordinary differential equations of susceptible-infective-recovered (SIR) type, and we assume that we have a discrete observation of infective population. We construct a continuous observation by applying time series and an appropriate fitting to the discrete observation data. The identification problem consists in the determination of different parameters in the governing equations such that the infective population obtained as solution of the SIR system is as close as to the observation. We introduce a reformulation of the calibration problem as an optimization problem where the objective function and the restriction are given by the comparison in the \(L_2\)-norm of theoretical solution of the mathematical model and the observation, and the SIR system governing the phenomenon, respectively. We solve numerically the optimization problem by applying the gradient method where the gradient of the cost function is obtained by introducing an adjoint state. In addition, we consider a numerical example to illustrate the application of the proposed calibration method.

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Acknowledgements

We thank to research projects DIUBB 172409 GI/C and FAPEI at U. del Bío-Bío, Chile. AC thanks to the research project DIUBB 183309 4/R at U. del Bío-Bío, Chile. IH thanks to the program “Becas de doctorado” of Conicyt-Chile. Ian Hess and Francisco Novoa-Muñoz would thank the support of the program “Fortalecimiento del postgrado” of the project “Instalación del Plan Plurianual UBB 2016–2020”

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Authors and Affiliations

  1. GMA, Departamento de Estadística, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile

    Francisco Novoa-Muñoz & Sergio Contreras Espinoza

  2. GMA, Departamento de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Chillán, Chile

    Aníbal Coronel Pérez & Ian Hess Duque

Authors
  1. Francisco Novoa-Muñoz
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  2. Sergio Contreras Espinoza
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  3. Aníbal Coronel Pérez
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  4. Ian Hess Duque
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Corresponding author

Correspondence to Francisco Novoa-Muñoz .

Editor information

Editors and Affiliations

  1. Department of Environmental Sciences, Informatics and Statistics, Ca' Foscari University of Venice, Venice, Italy

    Isadora Antoniano-Villalobos

  2. Department of Probability and Statistics, IIMAS, UNAM, Mexico City, Mexico

    Ramsés H. Mena

  3. Department of Statistics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    Manuel Mendoza

  4. Department of Mathematics, Facultad de Ciencias, UNAM, Mexico City, Mexico

    Lizbeth Naranjo

  5. Department of Statistics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    Luis E. Nieto-Barajas

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Novoa-Muñoz, F., Espinoza, S.C., Pérez, A.C., Duque, I.H. (2019). Calibration of Population Growth Mathematical Models by Using Time Series. In: Antoniano-Villalobos, I., Mena, R., Mendoza, M., Naranjo, L., Nieto-Barajas, L. (eds) Selected Contributions on Statistics and Data Science in Latin America. FNE 2018. Springer Proceedings in Mathematics & Statistics, vol 301. Springer, Cham. https://doi.org/10.1007/978-3-030-31551-1_8

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