A comprehensive mathematical analysis of fractal–fractional order nonlinear re-infection model

Abstract

In recent times, various real word problems including infectious diseases, engineering problems, and chemical processes, etc are modelled by using differential equations with fractals and fractional orders. The mentioned area provides a powerful tool to investigate the aforesaid problems from different perspectives like analysis, numerical investigation, and qualitative theory. Similarly, one of the global issue is devoted to re-infection of viral diseases like COVID-19. The said behaviour has produced very saviour impact on human life worldwide. Also, this is very critical for health system of a country as well as for its economics situation. To study the aforesaid phenomenon of re-infection, a hybrid type fractal–fractional three compartments model is formulated. The basic results related to boundedness, positivity, equilibrium points, basic reproductive number, and sensitivity analysis of the basic threshold are included in the theoretical aspect. Additionally, the Volterra–Lyapunov approach is used to establish both local and global stability analysis under fractals and fractional derivative. Furthermore, utilizing techniques from numerical functional analysis and fixed point theory, the existence and Hyers–Ulam stability of the solutions are also examined. A numerical algorithm based on Lagrange’s interpolation polynomials is extended to simulate the results of various compartments. Graphical interpretations corresponding to different fractals–fractional order values are presented. For verification of our numerical results, a comparison is provided between the real and simulated data. Here it should be kept in mind that the differential operator of fractals–fractional order is considered with Mittag-Leffler kernel. The mentioned operator is more general among the available fractals fractional differential operators in literature.