Calibration of Population Growth Mathematical Models by Using Time Series

  • Francisco Novoa-MuñozEmail author
  • Sergio Contreras Espinoza
  • Aníbal Coronel Pérez
  • Ian Hess Duque
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 301)


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.


Calibration Inverse problems Time series SIR models 



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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Francisco Novoa-Muñoz
    • 1
    Email author
  • Sergio Contreras Espinoza
    • 1
  • Aníbal Coronel Pérez
    • 2
  • Ian Hess Duque
    • 2
  1. 1.GMA, Departamento de Estadística, Facultad de CienciasUniversidad del Bío-BíoConcepciónChile
  2. 2.GMA, Departamento de Ciencias Básicas, Facultad de CienciasUniversidad del Bío-BíoChillánChile

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