Search

A novel crossover model for monkeypox disease that incorporates Ψ-Caputo fractional derivatives is presented here, where we use a simple nonstandard kernel function Ψ(t). We can be obtained the Caputo and Caputo–Katugampola derivatives as special cases from the proposed derivative. The crossover dynamics model defines four alternative models: fractal fractional order, fractional order, variable order, and fractional stochastic derivatives driven by fractional Brownian motion over four time intervals. The Ψ-nonstandard finite difference method is designed to solve fractal fractional order, fractional order, and variable order mathematical models. Also, the nonstandard modified Euler Maruyama method is used to study the fractional stochastic model. A comparison between Ψ-nonstandard finite difference method and Ψ-standard finite difference method is presented. Moreover, numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions.

The monkeypox virus (MPXV) is what causes monkeypox (MPX) disease, which is comparable to both smallpox and cowpox. Using classical, fractional-order and complex order differential equations, we offer a deterministic mathematical model of the monkeypox virus in this study to research its possible breakouts in United States. The complex order derivative makes the fractional order derivative and the integer order derivative more common when the imaginary part of the complex order equals zero and when the real part in complex order derivatives is zero in this case, the new behaviour appears that doesn't appear in integer and fractional order derivatives. Eight nonlinear complex order differential equations make up this model consisting of humans and rodents population sizes. The population of humans Nh is divided into five different classes. The population of rodent Nr is divided into three classes, and the derivatives are described in the Atangana-Baleanu-Caputo sense and Mittag-Leffler kernels are employed. Numerical methods to simulate complex order systems such as The standard and nonstandard two-step Lagrange interpolation methods are utilised to fit the model. The basic reproduction number of the model is given. The stability of the suggested model's disease-free equilibrium point is shown. Finally, we study the duration and monkeypox outbreak's transmission pattern in the United States from June 13 to Sep 16, 2022, numerical simulations to illustrate our findings are presented. The results show that keeping diseased people apart from the general population decreases the spread of disease.

In this artical, we present a novel complex order nonlinear mathematical model of HIV/AIDS and Malaria co-infection with multi-delay. The proposed model is formulated as a system of twelve complex order differential equations. The complex order derivative is defined in the Atangana-Baleanu-Caputo sense. Two numerical methods are utilized for simulating the determined system; standard two-step Lagrange interpolation method and the non-standard two step Lagrange interpolation method. In order to establish numerical simulations, the theoretical results and comparative studies are given.