Application of Non-singular Kernel in a Tumor Model with Strong Allee Effect

We obtain the analytical solutions in implicit form of a tumor cell population differential equation with strong Allee effect. We consider the ordinary case and then a fractional version. Some particular cases are plottedThe research of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant PID2020-113275GB-I00, and by Xunta de Galicia, grant ED431C 2019/02. Subhas Khajanchi acknowledges the financial support from the Department of Science and Technology (DST), Govt. of India, under the Scheme “Fund for Improvement of S &T Infrastructure (FIST)” [File No. SR/FST/MS-I/2019/41]. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureS

Antiretroviral therapy of HIV infection using a novel optimal type-2 fuzzy control strategy

Abstract The human immunodeficiency virus (HIV), as one of the most hazardous viruses, causes destructive effects on the human bodies' immune system. Hence, an immense body of research has focused on developing antiretroviral therapies for HIV infection. In the current study, we propose a new control technique for a fractional-order HIV infection model. Firstly, a fractional model of the HIV model is investigated, and the importance of the fractional-order derivative in the modeling of the system is shown. Afterward, a type-2 fuzzy logic controller is proposed for antiretroviral therapy of HIV infection. The developed control scheme consists of two individual controllers and an aggregator. The optimal aggregator modifies the output of each individual controller. Simulations for two different strategies are conducted. In the first strategy, only reverse transcriptase inhibitor (RTI) is used, and the superiority of the proposed controller over a conventional fuzzy controller is demonstrated. Lastly, in the second strategy, both RTI and protease inhibitors (PI) are used simultaneously. In this case, an optimal type-2 fuzzy aggregator is also proposed to modify the output of the individual controllers based on optimal rules. Simulations results demonstrate the appropriate performance of the designed control scheme for the uncertain system

A review of mechanistic learning in mathematical oncology

Mechanistic learning, the synergistic combination of knowledge-driven and data-driven modeling, is an emerging field. In particular, in mathematical oncology, the application of mathematical modeling to cancer biology and oncology, the use of mechanistic learning is growing. This review aims to capture the current state of the field and provide a perspective on how mechanistic learning may further progress in mathematical oncology. We highlight the synergistic potential of knowledge-driven mechanistic mathematical modeling and data-driven modeling, such as machine and deep learning. We point out similarities and differences regarding model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Then, organizing combinations of knowledge- and data-driven modeling into four categories (sequential, parallel, intrinsic, and extrinsic mechanistic learning), we summarize a variety of approaches at the interface between purely data- and knowledge-driven models. Using examples predominantly from oncology, we discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. We see that mechanistic learning, with its intentional leveraging of the strengths of both knowledge and data-driven modeling, can greatly impact the complex problems of oncology. Given the increasing ubiquity and impact of machine learning, it is critical to incorporate it into the study of mathematical oncology with mechanistic learning providing a path to that end. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input dat

A Filippov tumor-immune system with antigenicity

A threshold strategy model is proposed to demonstrate the extinction of tumor load and the mobilization of immune cells. Using Filippov system theory, we consider global dynamics and sliding bifurcation analysis. It was found that an effective model of cell targeted therapy captures more complex kinetics and that the kinetic behavior of the Filippov system changes as the threshold is altered, including limit cycle and some of the previously described sliding bifurcations. The analysis showed that abnormal changes in patients' tumor cells could be detected in time by using tumor cell-directed therapy appropriately. Under certain initial conditions, exceeding a certain level of tumor load (depending on the patient) leads to different tumor cell changes, that is, different post-treatment effects. Therefore, the optimal control policy for tumor cell-directed therapy should be individualized by considering individual patient data

A comparative study of the dynamics of a three-disk dynamo system with and without time delay

The disk dynamo plays an important role in studying the geodynamo and much research works have been devoted to the understanding of dynamo dynamics. This paper further investigates an extended disk dynamo system having three coupled conducting disks and incorporates the interaction-induced time delay in the dynamic governing equations. By carrying out a comparative analysis, the dynamic behaviors of the coupled three-disk dynamo system with and without time delay are studied to explore novel and complex nonlinear dynamic phenomena in the coupled delayed dynamo system. It is found that the double Hopf bifurcations can be induced in the time-delayed dynamo system. Three different topological structures of the unfolding are obtained under different time delays. Accordingly, it is shown that the novel dynamic behaviors, including quasi-periodic torus, three-dimensional torus and the coexistence of multiple attractors, can appear in the time-delayed dynamo system. Furthermore, by performing the continuation analysis on the periodic orbit generated from the Hopf bifurcation of equilibrium, some new coexistence patterns, e.g., the coexistence of periodic orbits and chaos, the coexistence of quasi-periodic orbits and chaos, are observed in the dynamo system with time delay. Based on the obtained results, it is believed that the inclusion of time delay in the modelling of the three-disk dynamo system is necessary and meaningful for developing an in-depth understanding of dynamo dynamics. Finally, the results of theoretical analyses are verified by the numerical simulations