Calibration of Population Growth Mathematical Models by Using Time Series

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 L2-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.