Temporal and Spacial Studies of Infectious Diseases: Mathematical Models and Numerical Solvers

Abstract

The SIR model is a classical model characterizing the spreading of infectious diseases. This model describes the time-dependent quantity changes among Susceptible, Infectious, and Recovered groups. By introducing space-depend effects such as diffusion and creation in addition to the SIR model, the Fisher's model is in fact a more advanced and comprehensive model. However, the Fisher's model is much less popular than the SIR model in simulating infectious disease numerically due to the difficulties from the parameter selection, the involvement of 2-d/3-d spacial effects, the configuration of the boundary conditions, etc. This paper aim to address these issues by providing numerical algorithms involving space and time finite difference schemes and iterative methods, and its open-source Python code for solving the Fisher's model. This 2-D Fisher's solver is second order in space and up to the second order in time, which is rigorously verified using test cases with analytical solutions. Numerical algorithms such as SOR, implicit Euler, Staggered Crank-Nicolson, and ADI are combined to improve the efficiency and accuracy of the solver. It can handle various boundary conditions subject to different physical descriptions. In addition, real-world data of Covid-19 are used by the model to demonstrate its practical usage in providing prediction and inferences.