Numerical simulation of spatio-temporal spread of an infectious disease utilizing a collocation method based on local radial basis functions

Abstract

The main goal of this research paper is to propose a computational approach for solving mixed Hammerstein integral equations, which are used to model the spread of epidemics over time and geographical regions. The proposed method first discretizes the temporal direction of these integral equations via local radial basis functions (LRBFs). Subsequently, the solution is approximated utilizing the discrete collocation scheme together with shape functions derived from LRBFs that are constructed based on scattered points distributed throughout the spatial domain. In fact, the offered method in this study adopts a selective approach by employing a limited number of nodes instead of considering all points within the solution domain. To calculate the integrals involved in the offered algorithm, the Gauss–Legendre integration method is utilized. Due to its characteristic of not requiring mesh generation on the solution domain, the method presented in this paper can be classified as a meshless approach. It offers computational efficiency by utilizing fewer resources compared to widely used radial basis functions, making it suitable for computers with limited memory capacity. The error estimation and convergence rate of the technique are also provided. The effectiveness and efficiency of the new approach are demonstrated through illustrative examples presented in the paper.