This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model.
Owing to the COVID-19 pandemic, which broke out in December 2019 and is still disrupting human life across the world, attention has been recently focused on the study of epidemic mathematical models able to describe the spread of the disease. The number of people who have received vaccinations is a new state variable in the COVID-19 model that this paper introduces to further the discussion of the subject. The study demonstrates that the proposed compartment model, which is described by differential equations of integer order, has two fixed points, a disease-free fixed point and an endemic fixed point. The global stability of the disease-free fixed point is guaranteed by a new theorem that is proven. This implies the disappearance of the pandemic, provided that an inequality involving the vaccination rate is satisfied. Finally, simulation results are carried out, with the aim of highlighting the usefulness of the conceived COVID-19 compartment model.
Nowadays, a lot of research papers are concentrating on the diffusion dynamics of infectious diseases, especially the most recent one: COVID-19. The primary goal of this work is to explore the stability analysis of a new version of the SEIR model formulated with incommensurate fractional-order derivatives. In particular, several existence and uniqueness results of the solution of the proposed model are derived by means of the Picard–Lindelöf method. Several stability analysis results related to the disease-free equilibrium of the model are reported in light of computing the so-called basic reproduction number, as well as in view of utilising a certain Lyapunov function. In conclusion, various numerical simulations are performed to confirm the theoretical findings.
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