**6** Publications

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PLoS One 2021 12;16(10):e0257975. Epub 2021 Oct 12.

High Institute of Public Health, Alexandria University, Alexandria, Egypt.

In this paper, a new mathematical model is formulated that describes the interaction between uninfected cells, infected cells, viruses, intracellular viral RNA, Cytotoxic T-lymphocytes (CTLs), and antibodies. Hence, the model contains certain biological relations that are thought to be key factors driving this interaction which allow us to obtain precise logical conclusions. Therefore, it improves our perception, that would otherwise not be possible, to comprehend the pathogenesis, to interpret clinical data, to control treatment, and to suggest new relations. This model can be used to study viral dynamics in patients for a wide range of infectious diseases like HIV, HPV, HBV, HCV, and Covid-19. Though, analysis of a new multiscale HCV model incorporating the immune system response is considered in detail, the analysis and results can be applied for all other viruses. The model utilizes a transformed multiscale model in the form of ordinary differential equations (ODE) and incorporates into it the interaction of the immune system. The role of CTLs and the role of antibody responses are investigated. The positivity of the solutions is proven, the basic reproduction number is obtained, and the equilibrium points are specified. The stability at the equilibrium points is analyzed based on the Lyapunov invariance principle. By using appropriate Lyapunov functions, the uninfected equilibrium point is proven to be globally asymptotically stable when the reproduction number is less than one and unstable otherwise. Global stability of the infected equilibrium points is considered, and it has been found that each equilibrium point has a specific domain of stability. Stability regions could be overlapped and a bistable equilibria could be found, which means the coexistence of two stable equilibrium points. Hence, the solution converges to one of them depending on the initial conditions.

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http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0257975 | PLOS |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC8509987 | PMC |

October 2021

Proc Inst Mech Eng H 2021 Mar 19;235(3):323-335. Epub 2020 Dec 19.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt.

The main objective of this research work is to develop an effective mathematical model of cardiac conduction system using a heterogeneous whole-heart model. The model is in the form of a system of modified Van der Pol and FitzHugh-Nagumo differential equations capable of describing the heart dynamics. The proposed model extends the range of normal and pathological electrocardiogram (ECG) waveforms that can be generated by the model. The effects of the respiratory sinus arrhythmia (RSA) and the Mayer waves (MW) are both incorporated to modulate the intrinsic frequency of the main oscillator that represents the sinoatrial node. Also, three pathological conditions are incorporated into the model. The heart rate variability (HRV) phenomenon is incorporated into the synthetic ECGs produced which yields valuable information about the cardiovascular health and the performance of the autonomic nervous system. The spectral analysis of the generated RR tachogram delivers power spectrums that resemble those obtained from real recordings. Also, the proposed model generates synthetic ECGs that characteristic the three considered pathological conditions, namely, the tall T wave, the ECG with U wave, and the Wolff-Parkinson-White syndrome. In general, the significance of this research work is in developing a mathematical model that represents the interactions between different pacemakers and allows analysis of cardiac rhythms. To show the effectiveness and the accuracy of the presented model, the results are compared to published results. The proposed model can be a useful tool to study the influences of different physiological conditions on the profile of the ECG. The synthetic ECG signals produced can be used as signal sources for the assessment of diagnostic ECG signal processing devices.

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http://dx.doi.org/10.1177/0954411920978052 | DOI Listing |

March 2021

Annu Int Conf IEEE Eng Med Biol Soc 2020 07;2020:2467-2470

A mathematical model, in the form of a system of nonlinear ordinary differential equations, that describes the interaction between tumor cells and effective immune cells is proposed. An exact solution cannot be found to this system like many other nonlinear systems. Yet, approximate analytical solution is explored. This solution should have a large interval of convergence to be acceptable because the interaction can take many days to reach its steady state. Power series method is used to obtain a series solution. In this process, some auxiliary variables are used to transform the system of equations to polynomial form. However, this solution has a small radius of convergence, therefore, Padé approximant method is used to extend the domain of convergence. Hence, the obtained approximate analytical solution is valid over a large interval and has a remarkable accuracy when compared with numerical solution.

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http://dx.doi.org/10.1109/EMBC44109.2020.9176529 | DOI Listing |

July 2020

Annu Int Conf IEEE Eng Med Biol Soc 2020 07;2020:2451-2454

A mathematically identical ordinary differential equations (ODEs) model was derived from a multiscale partial differential equations (PDEs) model of hepatitis c virus infection, which helps to overcome the limitations of the PDE model in clinical data analysis. We have discussed about basic properties of the system and found the basic reproduction number of the system. A condition for the local stability of the uninfected and the infected steady states is presented. The local stability analysis of the model shows that the system is asymptotically stable at the disease-free equilibrium point when the basic reproduction number is less than one. When the basic reproduction number is greater than one endemic equilibrium point exists, and the local stability analysis proves that this point is asymptotically stable. Numerical sensitivity analysis based on model parameters is performed and therefore the result describes the influence of each parameter on the basic reproduction number.

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http://dx.doi.org/10.1109/EMBC44109.2020.9176525 | DOI Listing |

July 2020

Comput Math Methods Med 2020 19;2020:7187602. Epub 2020 Feb 19.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria, Egypt.

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8T cells, but they neglected the role of CD4T cells. Other models were constructed to study the role of CD4T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4T cells, CD8T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge-Kutta method failed to solve it. Consequently, the "Adams predictor-corrector" method is used. The results reveal that the patient's immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4T cells can be an alternative to both CD8T cell therapy and cytokines in some cases. Moreover, CD4T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients' treatment intervention strategies.

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http://dx.doi.org/10.1155/2020/7187602 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7049850 | PMC |

January 2021

Anal Biochem 2016 Mar 18;496:50-4. Epub 2015 Dec 18.

Department of Engineering Mathematics and Physics, Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt.

In this article, a new approach--namely, the extended Parker-Sochacki method (EPSM)--is presented for solving the Michaelis-Menten nonlinear enzymatic reaction model. The Parker-Sochacki method (PSM) is combined with a new resummation method called the Sumudu-Padé resummation method to obtain approximate analytical solutions for the model. The obtained solutions by the proposed approach are compared with the solutions of PSM and the Runge-Kutta numerical method (RKM). The comparison proves the practicality, efficiency, and correctness of the presented approach. It serves as a basis for solving other nonlinear biochemical reaction models in the future.

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http://dx.doi.org/10.1016/j.ab.2015.11.017 | DOI Listing |

March 2016

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