ABSTRACT
Biomathematics has become one of the most significant areas of research as a result of interdisciplinary study. Chronic diseases sometimes referred to as non-communicable and communicable diseases, are conditions that develop over an extended period as a result of different factors like genetics, lifestyle, and environment. The most important common types of disease are cardiovascular, alcohol, cancer, and diabetes. More than three-quarters of the world's (31.4 million) deaths occur in low- and middle-income nations, which are disproportionately affected by different infections. Fractional Calculus is a prominent topic for research within the discipline of Applied Mathematics due to its usefulness in solving problems in many different branches of science, engineering, and medicine. Recent researchers have identified the importance of mathematical tools in various disease models as being very useful to study the dynamics with the help of fractional and integer calculus modeling. Due to the complexity of the underlying connections, both deterministic and stochastic epidemiological models are founded on an inadequate understanding of the infectious network. Over the past several years, the use of different fractional operators to model the problem has grown, and it is now a common way to study how epidemics spread. Recently, researchers have actively considered fractional calculus to study different diseases like COVID-19, cancer, TB, HIV, dengue fever, diabetes, cholera, pine welts, smoking and heart attacks, etc. With the help of fractional operator, we modified a mathematical model for the dynamical transmission, analysis, treatment, vaccination, and precaution leveling necessary to mitigate the negative impact of illness on society in the long run, overcoming the memory effect without defining or considering others parameters. In this review paper, we considered all the recent studies based on the fractional modeling of infectious and non-infectious diseases with different fractional operators such as Caputo, Caputo Fabrizio, ABC, and constant proportional with Caputo, etc. This review paper aims to bring all the information together by considering different fractional operators and their uses in the field of infectious disease modeling. The steps taken to accomplish the goal were developing a mathematical model, identifying the equilibrium point, figuring out the minimal reproductive number, and assessing the stability around the equilibrium point. For future direction, we consider the cancer model to study the growth cells of cancer and the impact of therapy to control infections. An equilibrium solution and an analysis of the behavior dynamics of the cell spread with treatment in the form of chemotherapy were obtained. The simulation shows that the population of cancer cells is influenced by the pace of cancer cell growth with the Caputo fractional derivative. The acquired results show how effective and precise the suggested approach is in helping to better understand how chemotherapy works. Chemotherapy medications have been found to increase immunity against particular cancer by reducing the number of tumor cells. Further, we suggested some future work directions with the help of the new hybrid fractional operator. Our innovative methodology might have significant effects on global stakeholders, policymakers, and national health systems. The current strategies for controlling outbreaks and the vaccination and prevention policies that have been implemented would benefit from a more accurate representation of the dynamics of contagious diseases, which necessitates the development of highly complex mathematical models. Microorganisms, interactions between individuals or groups, and environmental, social, economic, and demographic factors on a broader scale are all examples.
ABSTRACT
Recently, Atangana proposed new operators by combining fractional and fractal calculus. These recently proposed operators, referred to as fractal–fractional operators, have been widely used to study complex dynamics. In this paper, the COVID-19 model is considered via Atangana–Baleanu fractal-fractional operator. The Lyapunov stability for the model is derived for first and second derivative. Numerical results have developed through Lagrangian-piecewise interpolation for the different fractal–fractional operators. We develop numerical outcomes through different differential and integral fractional operators like power-law, exponential law, and Mittag-Leffler kernel. To get a better outcome of the proposed scheme, numerical simulation is made with different kernels having the memory effects with fractional parameters. [ FROM AUTHOR] Copyright of Fractals is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)
ABSTRACT
Coronavirus and its associated comorbidities have been the drivers of many deaths across the globe in recent times. Individuals with underlying medical conditions are at higher risk of becoming critically ill and developing complications if they are infected with the Coronavirus. In this paper, a Caputo-Fabrizio fractionalorder model of coronavirus disease with comorbidity is formulated to access the impact of comorbidity diseases on COVID-19 transmission using both a fractional-order as well as a stochastic approach. Exponential law is utilized to present the existence and uniqueness of solutions using the fixed-point theory. The fractional stochastic approach is adopted to examine the same model to explore the random effect. Numerical simulations are used to support the theoretical results and the simulation results suggest that the increase of comorbidity development and the fractional-order derivative factor simultaneously increases the prevalence of the infection and the spread of the disease. The fractional stochastic numerical results suggest that the prediction of infection rate is more stochastic than deterministic. © 2022 the author(s).