Initiation of HIV therapy

In this paper, we numerically show that the dynamics of the HIV system is sensitive to both the initial condition and the system parameters. These phenomena imply that the system is chaotic and exhibits a bifurcation behavior. To control the system, we propose to initiate an HIV therapy based on both the concentration of the HIV-1 viral load and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population. If the concentration of the HIV-1 viral load is higher than a threshold, then the first type of therapy will be applied. If the concentration of the HIV-1 viral load is lower than or equal to the threshold and the ratio of the CD4 lymphocyte population to the CD8 lymphocyte population is greater than another threshold, then the second type of therapy will be applied. Otherwise, no therapy will be applied. The advantages of the proposed control strategy are that the therapy can be stopped under certain conditions, while the state variables of the overall system is asymptotically stable with fast convergent rate, the concentration of the controlled HIV-1 viral load is monotonic decreasing, as well as the positivity constraint of the system states and that of the dose concentration is guaranteed to be satisfied. Computer numerical simulation results are presented for an illustration

Nonlinear Systems: Asymptotic Methods, Stability, Chaos, Control, And Optimization

[No abstract available]201

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

Some Remarks about the Complexity of Epidemics Management

Recent outbreaks of Ebola, H1N1 and other infectious diseases have shown that the assumptions underlying the established theory of epidemics management are too idealistic. For an improvement of procedures and organizations involved in fighting epidemics, extended models of epidemics management are required. The necessary extensions consist in a representation of the management loop and the potential frictions influencing the loop. The effects of the non-deterministic frictions can be taken into account by including the measures of robustness and risk in the assessment of management options. Thus, besides of the increased structural complexity resulting from the model extensions, the computational complexity of the task of epidemics management - interpreted as an optimization problem - is increased as well. This is a serious obstacle for analyzing the model and may require an additional pre-processing enabling a simplification of the analysis process. The paper closes with an outlook discussing some forthcoming problems

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle

[EN] The purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (T,N,∝) -cut set of an intuitionistic fuzzy set. We intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. Moreover, concrete results on existence of the solution of a delay differential equation and a system of Riemann-Liouville Cauchy type problems have been derived. In addition, we also present illustrative examples to substantiate the usability of our main result.Tabassum, R.; Azam, A.; Mohammed, SS. (2019). Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle. Applied General Topology. 20(2):449-469. https://doi.org/10.4995/agt.2019.11683SWORD449469202H. M. Abu-Donia, Common fixed points theorems for fuzzy mappings in metric space under $varphi$-contraction condition, Chaos Solitons & Fractals 34 (2007), 538-543. https://doi.org/10.1016/j.chaos.2005.03.055A. Z. Al-Abedeen, Existence theorem on differential equation of generalized order, Al-Rafidain J. Sci. Mosul University, Iraq, 1 (1976), 95-104.Y. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and Its Applications, Birkhäuser, Basel (1997), 7-22. https://doi.org/10.1007/978-3-0348-8910-0_2H. L. Arora and J. G. Alshamani, Stability of differential equations of noninteger order through fixed point in the large, Indian J. Pure Appl. Math. 11, no. 3 (1980), 307-313.K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20, no. 1 (1986), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3A. Azam, M. Arshad and P. Vetro, On a pair of fuzzy $varphi$-contractive mappings, Mathematical and Computer Modelling 52, no. 1 (2010), 207-214. https://doi.org/10.1016/j.mcm.2010.02.010A. Azam and M. Rashid, A fuzzy coincidence theorem with applications in a function space, Journal of Intelligent and Fuzzy Systems 27, no. 4 (2014), 1775-1781.A. Azam, R. Tabassum and M. Rashid, Coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications, Journal of Mathematical Analysis 8, no. 4 (2017), 56-77.A. Azam and R. Tabassum, Existence of common coincidence point of intuitionistic fuzzy maps, Journal of Intelligent and Fuzzy Systems 35 (2018), 4795-4805. https://doi.org/10.3233/JIFS-18411J. S. Bae, Fixed point theorems for weakly contractive multivalued maps, Journal of Mathematical Analysis and Applications 284, no. 2 (2003), 690-697. https://doi.org/10.1016/S0022-247X(03)00387-1I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory and Applications 2006 (2006), 1-7. https://doi.org/10.1155/FPTA/2006/74503M. A. Al-Bassam, Some existence theorems on differential equations of generalized order, J. Reine Angew. Math. 218, no. 1 (1965), 70-78. https://doi.org/10.1515/crll.1965.218.70V. Berinde, Approximating fixed points of weak contractions, Fixed Point Theory 4 (2003), 131-142.S. M. Ciupe, B. L. de Bivort, D. M. Bortz and P. W. Nelson, Estimates of kinetic parameters from HIV patient data during primary infection through the eyes of three different models, Math. Biosci., to appear.K. Cooke, Y. Kuang and B. Li, Analyses of an antiviral immune response model with time delays, Canad. Appl. Math. Quart. 6, no. 4 (1998), 321-354.K. L. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999), 332-352. https://doi.org/10.1007/s002850050194P. Z. Daffer and H. Kaneko, Fixed points of generalized contractive multi-valued mappings, Journal of Mathematical Analysis and Applications 192, no. 2 (1995), 655-666. https://doi.org/10.1006/jmaa.1995.1194S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117, no. 2 (2001), 209-213. https://doi.org/10.1016/S0165-0114(98)00235-8D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Appl. 204, no. 2 (1996), 609-625. https://doi.org/10.1006/jmaa.1996.0456S. Heilpern, Fuzzy mappings and fixed point theorems, Journal of Mathematical Analysis and Applications 83, no. 2 (1981), 566-569. https://doi.org/10.1016/0022-247X(81)90141-4Z. Jia, L. Amselang and P. Gros, Content-based image retrieval from a large image database, Pattern Recognition 11, no. 5 (2008), 1479-1495. https://doi.org/10.1016/j.patcog.2007.06.034A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy 98, no. 1 (2009), 35-39. https://doi.org/10.1016/j.homp.2008.10.003A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 2006.S. Konjik, L. Oparnica and D. Zorica, Waves in viscoelastic media described by a linear fractional model, Integral Transforms Spec. Funct. 22 (2011), 283-291. https://doi.org/10.1080/10652469.2010.541039A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Nauka, Moscow, 1968.D. F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, J. Comput. Syst. Sci. 70 (2005), 73-85. https://doi.org/10.1016/j.jcss.2004.06.002D. Martinetti, V. Janis and S. Montes, Cuts of intuitionistic fuzzy sets respecting fuzzy connectives, Information Sciences 232 (2013), 267-275. https://doi.org/10.1016/j.ins.2012.12.026S. B. Nadler Jr, Multi-valued contraction mappings, Pacific Journal of Mathematics 30, no. 2 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includesan intracellular delay. Math. Biosci. 163 (2000), 201-215. https://doi.org/10.1016/S0025-5564(99)00055-3B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis 4, no. 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1A. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives : Theory and Applcations, Gordon and Breach Science Publishers, Switzerland, 1993.A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comp. 68, no. 1 (1995), 15-25. https://doi.org/10.1016/0096-3003(94)00080-NA. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems, I. Appl. Anal. 78, no. 1-2 (2001), 153-192. https://doi.org/10.1080/00036810108840931P. Turchin and A. D. Taylor, Complex dynamics in ecological time series, Ecology 73 (1992), 289-305. https://doi.org/10.2307/1938740D. Valerjo, D. Machadoa and J. T. Kryakova, Some pioneers of the applications of fractional calculus, Fract. Calc. Appl. Anal. 17 (2014), 552-578. https://doi.org/10.2478/s13540-014-0185-1B. Vielle and G. Chauvet, Delay equation analysis of human respiratory stability, Math. Biosci. 152, no. 2 (1998), 105-122. https://doi.org/10.1016/S0025-5564(98)10028-7M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol. 47, no. 3 (2003), 270-294. https://doi.org/10.1007/s00285-003-0211-0Y. H. Shen, F. X. Wang and W. Chen, A note on intuitionistic fuzzy mappings, Iranian Journal of Fuzzy Systems 9, no. 5 (2012), 63-76.L. A. Zadeh, Fuzzy sets, Information and Control 8, no. 3 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-

Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps

As extension of Fuzzy Cognitive Maps are now introduced the Neutrosophic Cognitive Map