Abstract
Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville–Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper.
Highlights
The immune system is a real masterpiece that does extraordinary things every day while we do not realize such numerous activities
The job of the immune system is protecting our bodies against invading factors such as bacteria and viruses
We provide some novel approximate solutions to a computational model that formulates the interaction between tumor growth and the immune system, including several fractional and fractal operators
Summary
The immune system is a real masterpiece that does extraordinary things every day while we do not realize such numerous activities. In some cases the researchers have obtained desirable attractors, which were not achievable by common integer-order operators This fact highlights the importance of new derivative operators in other real-world models. Motivated by these achievements, especially following the work [24], we intend to investigate the model presented in equation (4) using some new efficient fractional and fractionalfractional operators. Especially following the work [24], we intend to investigate the model presented in equation (4) using some new efficient fractional and fractionalfractional operators To reach this goal, the subsequent parts of the paper are structured as follows.
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