A numerical efficient splitting method for the solution of two dimensional susceptible infected recovered epidemic model of whooping cough dynamics: Applications in bio-medical engineering. (July 2020) - Ahmed, Nauman; Ali, Mubasher; Rafiq, M.; Khan, Ilyas; Nisar, Kottakkaran Sooppy; Rehman, M.A.; Ahmad, M.O.
Highlights: Reaction-diffusion whooping cough epidemic model in two space dimension is investigated for numerical study. A structure preserving numerical algorithm based on operator splitting finite difference discretization is designed with very good computational efficiency. The proposed technique preserves all the important properties of epidemic model like positivity and stability of two equilibrium point of continuous model. Numerical simulations verify all the attributes of the designed numerical technique. Abstract: Background and Objective The positivity property of the non-linear dynamical systems is one of the essential features in different fields of bio-medical engineering, science and many more. The state variables, involving in the models, describing the natural phenomenon such as concentration, density and population size etc. must be positive. Therefore, the computing techniques used to solve the system of non-linear differential equations must be consisted with the continuous nature of the models. But, unfortunately there are some existing techniques in the literature that do not preserve the positivity property, especially for the multi-space dimensional models. So there is a gap in the literature that should be filled up, by constructing the positivity preserving numerical algorithms. In this study, we consider a susceptible-infected-recovered (SIR) reaction diffusion epidemic model in two space dimensions from biomedical engineering and solved numerically to observe the behavior of the model. Since the state variables involved in this system are population densities therefore we design a novel computational method which is time efficient because of its splitting structure and holds the positivity as well as other important structure of epidemic system. Methods Three different computational techniques are designed to examine the numerical solution of SIR model of infectious disease. Two approaches are well-known existing computing methods named as forward Euler finite difference (FD) method and backward Euler operator splitting finite difference (OS-FD) method. The third approach is operator splitting nonstandard finite difference (OS-NSFD) method which is devised by using the NSFD rules. Results The proposed OS-NSFD technique retains efficiently the stability of equilibria as well as the positivity. Graphical behavior depicts that the existing computing methods can not get success to preserve the structure of the epidemic system of whooping cough dynamics. At the same time OS-NSFD computing method is proven to be reliable and suitable for the system of bio-medical engineering mathematically and graphically. Conclusion A reliable and novel computing technique is developed for the solution of two dimensional reaction diffusion problem. This technique preserves all the imperative characteristics of the model under study. Also the time efficiency of this method makes it easy to find the solution of physical system in two space dimension. The comparison with other techniques shows the efficacy and reliability of the designed technique.