Fractal–fractional model and numerical scheme based on Newton polynomial for Q fever disease under Atangana–Baleanu derivative

Abstract

Scientists and researchers are increasingly interested in mathematical modelling of infectious diseases with non-integer order. It is self-evident that a fixed order can only characterize classical models in epidemiology, but models with fractional-order derivatives are not fixed order. When a derivative has a non-fixed order, it becomes more potent in representing real-world problems. Different fresh techniques concerning fractional operators, such as the exponential decay and the Mittag-Leffler kernel, have been proposed in recent years, which transcend the constraints of prior fractional-order derivatives. These novel operators are functional in modelling problems in science and engineering—a more recent operator in fractional calculus, known as the fractal–fractional operator. Fractal–fractional operators have not been frequently utilized to explore complex dynamics of infectious disease spread in an ecosystem. In this paper, a novel fractal–fractional operator is used to study the dynamics of Q fever in livestock and ticks. Fixed point theorem is used to prove the model’s existence and uniqueness under the Atangana–Baleanu (Mittag-Leffler kernel) fractal–fractional operator. A non-linear analysis is used to determine the model’s Hyers–Ulam stability. A numerical scheme for the Q fever disease using the newly constructed numerical scheme based on Newton polynomial is presented. The numerical simulation of the model shows some exciting dynamics of the disease in an ecosystem. The fractal–fractional dynamism shows that a change in the geometric pattern of nature also affects the spread of Q fever. For example, it is noticed that the fractal dimensions and fractional order produces different asymptotic stabilities for symptomatic livestock, Coxiella burnetii in the environment, susceptible and infected ticks. It is of a view that this study will also contribute to the rich dynamics of the fractal and the fractional perspective of disease spread in nature.