Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories

Since certain species of domestic poultry and poultry are the main food source in many countries, the outbreak of avian influenza, such as H7N9, is a serious threat to the health and economy of those countries. This can be considered as the main reason for considering the preventive ways of avian influenza. In recent years, the disease has received worldwide attention, and a large variety of different mathematical models have been designed to investigate the dynamics of the avian influenza epidemic problem. In this paper, two fractional models with logistic growth and with incubation periods were considered using the Liouville-Caputo and the new definition of a nonlocal fractional derivative with the Mittag-Leffler kernel. Local stability of the equilibria of both models has been presented. For the Liouville-Caputo case, we have some special solutions using an iterative scheme via Laplace transform. Moreover, based on the trapezoidal product-integration rule, a novel iterative method is utilized to obtain approximate solutions for these models. In the Atangana-Baleanu-Caputo sense, we studied the uniqueness and existence of the solutions, and their corresponding numerical solutions were obtained using a novel numerical method. The method is based on the trapezoidal product-integration rule. Also, we consider fractal-fractional operators to capture self-similarities for both models. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law and the Mittag-Leffler function. These models were solved numerically via the Adams-Bashforth-Moulton and Adams-Moulton scheme, respectively. We have performed many numerical simulations to illustrate the analytical achievements. Numerical simulations show very high agreement between the acquired and the expected results.