Theory and algorithm: An optimal control approach for parameter identification of turing patterns with respect to an epidemic model with Allee Effects on both continuous space and complex networks

Abstract

In this paper, a parameter identification method based on optimal control theory is used to identify the parameters of the Turing pattern in the reaction–diffusion system. In terms of theory, we firstly conduct linear stability analysis on an epidemic disease compartment model, and make some simple derivations on the necessary conditions for Turing pattern to appear. Next, the well-posedness of the direct problem and the existence of optimal solutions to the parameter identification inverse problem are proved. For the first-order necessary conditions, we formally derive the general form of the variational inequality with respect to the cost function and reaction–diffusion system also in general forms. In terms of algorithm and simulation: to accelerate iteration speed of the algorithm, the Adam algorithm is adopted to replace the traditional gradient descent algorithm, which greatly increases the convergence speed and completes the identification task of up to seven parameters. Regarding the global convergence of the algorithm, by iterating on the already stable pattern, we achieve the effect of global convergence and overcome the defect that the old method is sensitive to the initial value. We also try to combine theory with practice, and reproduce a natural pattern with spatially heterogeneous parameters. Furthermore, we present similar identification methods for parameter identification for time-delay systems and systems on complex networks.