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    Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells
    (Pergamon-Elsevier Science Ltd, 2020) Naik, Parvaiz Ahmad; Owolabi, Kolade M.; Yavuz, Mehmet; Zu, Jian
    Mathematical models in epidemiology have been studied in the literature to understand the mechanism that underlies AIDS-related cancers, providing us with a better insight towards cancer immunity and viral oncogenesis. In this study, we propose a dynamical fractional order HIV-1 model in Caputo sense which involves the interactions between cancer cells, healthy CD4(+)T lymphocytes, and virus infected CD4(+)T lymphocytes leading to chaotic behavior. The model has been investigated for the existence and uniqueness of its solution via fixed point theory, while the unique non-negative solution remains bounded within the biologically feasible region. The stability analysis of the model is performed and the biological relevance of the equilibria is also discussed in the paper. The numerical simulations are obtained under different instances of fractional order alpha. It is observed that, as the fractional power decreases from 'one' the chaotic behavior becomes more and more attractive. The existence of chaotic attractors for various species interaction has been observed in 2D and 3D cases. The time series evolution of the species show ing different distributions under different fractional order alpha. The results show that order of the fractional derivative has a significant effect on the dynamic process. (c) 2020 Elsevier Ltd. All rights reserved.
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    Complex dynamics of a discrete-time Bazykin-Berezovskaya prey-predator model with a strong Allee effect
    (Elsevier, 2022) Naik, Parvaiz Ahmad; Eskandari, Zohreh; Yavuz, Mehmet; Zu, Jian
    The present paper investigates the critical normal form coefficients for the one-parameter and two-parameter bifurcations of a discrete-time Bazykin-Berezovskaya prey-predator model. Based on the critical coefficients, it can be determined which scenario corresponds to each bifurcation. Further, for a better representation of the study, the complex dynamics of the model are investigated theoretically and numerically using MatcotM, which is a Matlab package. Some graphical representations of the model are presented to verify the obtained results. The outcome of the study reveals that the model undergoes multiple bifurcations including period-doubling, Neimark-Sacker, and strong resonance bifurcations. (C) 2022 Elsevier B.V. All rights reserved.
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    Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan
    (Springer Heidelberg, 2020) Naik, Parvaiz Ahmad; Yavuz, Mehmet; Qureshi, Sania; Zu, Jian; Townley, Stuart
    Coronaviruses are a large family of viruses that cause different symptoms, from mild cold to severe respiratory distress, and they can be seen in different types of animals such as camels, cattle, cats and bats. Novel coronavirus called COVID-19 is a newly emerged virus that appeared in many countries of the world, but the actual source of the virus is not yet known. The outbreak has caused pandemic with 26,622,706 confirmed infections and 874,708 reported deaths worldwide till August 31, 2020, with 17,717,911 recovered cases. Currently, there exist no vaccines officially approved for the prevention or management of the disease, but alternative drugs meant for HIV, HBV, malaria and some other flus are used to treat this virus. In the present paper, a fractional-order epidemic model with two different operators called the classical Caputo operator and the Atangana-Baleanu-Caputo operator for the transmission of COVID-19 epidemic is proposed and analyzed. The reproduction number R-0 is obtained for the prediction and persistence of the disease. The dynamic behavior of the equilibria is studied by using fractional Routh-Hurwitz stability criterion and fractional La Salle invariant principle. Special attention is given to the global dynamics of the equilibria. Moreover, the fitting of parameters through least squares curve fitting technique is performed, and the average absolute relative error between COVID-19 actual cases and the model's solution for the infectious class is tried to be reduced and the best fitted values of the relevant parameters are achieved. The numerical solution of the proposed COVID-19 fractional-order model under the Caputo operator is obtained by using generalized AdamsBashforth-Moulton method, whereas for the Atangana-Baleanu-Caputo operator, we have used a new numerical scheme. Also, the treatment compartment is included in the population which determines the impact of alternative drugs applied for treating the infected individuals. Furthermore, numerical simulations of the model and their graphical presentations are performed to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary-order derivative.
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    The role of prostitution on HIV transmission with memory: A modeling approach
    (Elsevier, 2020) Naik, Parvaiz Ahmad; Yavuz, Mehmet; Zu, Jian
    HIV is a topic that has been greatly discussed and researched due to its impact on human population. Many campaigns have been put into place, and people have been made aware of the various effects of the disease. This paper considers a fractional-order HIV epidemic model with the inclusion of prostitution in the population and its consequences on the disease transmission. The model describes the spread of HIV disease in a system consisting of a population of susceptible males and the female sex workers. The focus is on the spread of HIV by female sex workers through prostitution, because in the present world sexual transmission is the major cause of the HIV trans-mission. The fractional derivatives are taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Positivity and boundedness of the solution and the stability conditions of the fractional-order system are determined. Moreover, model equilibria are determined, and their sta-bility analysis are considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle's invariant principle. The threshold quantity namely basic reproduction number R-0 is cal-culated and analyzed for the disease status. On the basis of R-0, the disease progress can be deter-mined i.e., the population is free from the disease if R-0 < 1 and disease spreads in the population if R-0 > 1. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, numerical simulations of the model and their graphical pre-sentations are performed to visualize the effectiveness of our theoretical results and to observe the impact of the arbitrary order derivative. The results obtained show the effectiveness and strength of the applied L1 scheme. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).