From enzyme kinetics to epidemiological models with Michaelis–Menten contact rate: Design of nonstandard finite difference schemes

We consider the basic SIR epidemiological model with the Michaelis–Menten formulation of the contact rate. From the study of the Michaelis–Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert WW function and schemes obtained by using Mickens’s rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms. We compare and investigate the performance of the two types of schemes by showing that they are dynamically consistent with the continuous model. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.