Analysis of a nonstandard computer method to simulate a nonlinear stochastic epidemiological model of coronavirus-like diseases. (June 2021) - Macías-Díaz, J.E.; Raza, Ali; Ahmed, Nauman; Rafiq, Muhammad
Highlights: A stochastic model for the propagation of diseases is deduced from epidemiological assumptions. The reproductive number and equilibria of the deterministic system are calculated. A nonstandard scheme to solve the stochastic system is proposed and theoretically analyzed. The simulations show that the scheme is epidemiologically more robust than other approaches. Abstract: Background and objective: We propose a nonstandard computational model to approximate the solutions of a stochastic system describing the propagation of an infectious disease. The mathematical model considers the existence of various sub-populations, including humans who are susceptible to the disease, asymptomatic humans, infected humans and recovered or quarantined individuals. Various mechanisms of propagation are considered in order to describe the propagation phenomenon accurately. Methods: We propose a stochastic extension of the deterministic model, considering a random component which follows a Brownian motion. In view of the difficulties to solve the system exactly, we propose a computational model to approximate its solutions following a nonstandard approach. Results: The nonstandard discretization is fully analyzed for positivity, boundedness and stability. It is worth pointing out that these properties are realized in the discrete scenario and that they are thoroughly established herein using rigorous mathematical arguments. We provide some illustrative computational simulations to exhibit the main computational features of this approach. Conclusions: The results show that the nonstandard technique is capable of preserving the distinctive characteristics of the epidemiologically relevant solutions of the model, while other (classical) approaches are not able to do it. For the sake of convenience, a computational code of the nonstandard discrete model may be provided to the readers at their requests.