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18 pages, 579 KiB

Open AccessArticle

Global Stability of Multi-Strain SEIR Epidemic Model with Vaccination Strategy

by Zakaria Yaagoub and Karam Allali

Math. Comput. Appl. 2023, 28(1), 9; https://doi.org/10.3390/mca28010009 - 7 Jan 2023

Cited by 13 | Viewed by 3217

A three-strain SEIR epidemic model with a vaccination strategy is suggested and studied in this work. This model is represented by a system of nine nonlinear ordinary differential equations that describe the interaction between susceptible individuals, strain-1-vaccinated individuals, strain-1-exposed individuals, strain-2-exposed individuals, strain-3-exposed individuals, strain-1-infected individuals, strain-2-infected individuals, strain-3-infected individuals, and recovered individuals. We start our analysis of this model by establishing the existence, positivity, and boundedness of all the solutions. In order to show global stability, the model has five equilibrium points: The first one stands for the disease-free equilibrium, the second stands for the strain-1 endemic equilibrium, the third one describes the strain-2 equilibrium, the fourth one represents the strain-3 equilibrium point, and the last one is called the total endemic equilibrium. We establish the global stability of each equilibrium point using some suitable Lyapunov function. This stability depends on the strain-1 reproduction number

R_{0}^{1}

, the strain-2 basic reproduction number

R_{0}^{2}

, and the strain-3 reproduction number

R_{0}^{3}

. Numerical simulations are given to confirm our theoretical results. It is shown that in order to eradicate the infection, the basic reproduction numbers of all the strains must be less than unity. Full article

(This article belongs to the Special Issue Ghana Numerical Analysis Day)

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13 pages, 530 KiB

Open AccessArticle

HIV Dynamics with a Trilinear Antibody Growth Function and Saturated Infection Rate

by Fatima Ezzahra Fikri and Karam Allali

Math. Comput. Appl. 2022, 27(5), 85; https://doi.org/10.3390/mca27050085 - 8 Oct 2022

Cited by 1 | Viewed by 1853

Abstract

The objective of this paper is to study a new mathematical model describing the human immunodeficiency virus (HIV). The model incorporates the impacts of cytotoxic T lymphocyte (CTL) immunity and antibodies with trilinear growth functions. The boundedness and positivity of solutions for non-negative initial data are proved, which is consistent with biological studies. The local stability of the equilibrium is established. Finally, numerical simulations are presented to support our theoretical findings. Full article

(This article belongs to the Special Issue Ghana Numerical Analysis Day)

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14 pages, 1807 KiB

Open AccessArticle

Modeling the Adaptive Immune Response in an HBV Infection Model with Virus to Cell Transmission in Both Liver with CTL Immune Response and the Extrahepatic Tissue

by Fatima Ezzahra Fikri and Karam Allali

Math. Comput. Appl. 2022, 27(4), 65; https://doi.org/10.3390/mca27040065 - 28 Jul 2022

Cited by 2 | Viewed by 2180

Abstract

The objective of this paper is to investigate a mathematical model describing the infection of hepatitis B virus (HBV) in intrahepatic and extrahepatic tissues. Additionally, the model includes the effect of the cytotoxic T cell (CTL) immunity, which is described by a linear activation rate by infected cells. The positivity and boundedness of solutions for non-negative initial data are proven, which is consistent with the biological studies. The local stability of the equilibrium is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibrium is fulfilled by using appropriate Lyapanov functions. Finally, numerical simulations are performed to support our theoretical findings. It has been revealed that the fractional-order derivatives have no influence on the stability but only on the speed of convergence toward the equilibria. Full article

(This article belongs to the Special Issue Applied Mathematics (IOCAM 22))

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13 pages, 492 KiB

Open AccessFeature PaperArticle

Nonlinear Spatiotemporal Viral Infection Model with CTL Immunity: Mathematical Analysis

by Jaouad Danane, Karam Allali, Léon Matar Tine and Vitaly Volpert

Mathematics 2020, 8(1), 52; https://doi.org/10.3390/math8010052 - 1 Jan 2020

Cited by 5 | Viewed by 2368

Abstract

A mathematical model describing viral dynamics in the presence of the latently infected cells and the cytotoxic T-lymphocytes cells (CTL), taking into consideration the spatial mobility of free viruses, is presented and studied. The model includes five nonlinear differential equations describing the interaction among the uninfected cells, the latently infected cells, the actively infected cells, the free viruses, and the cellular immune response. First, we establish the existence, positivity, and boundedness for the suggested diffusion model. Moreover, we prove the global stability of each steady state by constructing some suitable Lyapunov functionals. Finally, we validated our theoretical results by numerical simulations for each case. Full article

(This article belongs to the Special Issue The Application of Mathematics to Physics and Nonlinear Science)

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16 pages, 553 KiB

Open AccessArticle

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

by Jaouad Danane and Karam Allali

High-Throughput 2018, 7(4), 35; https://doi.org/10.3390/ht7040035 - 19 Nov 2018

Cited by 12 | Viewed by 3018

Abstract

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin’s minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication. Full article

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18 pages, 891 KiB

Open AccessArticle

Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase

by Karam Allali, Jaouad Danane and Yang Kuang

Appl. Sci. 2017, 7(8), 861; https://doi.org/10.3390/app7080861 - 21 Aug 2017

Cited by 32 | Viewed by 6214

Abstract

A modified mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with cytotoxic T-lymphocytes (CTL) and infected cells in eclipse phase is presented and studied in this paper. The model under consideration also includes a saturated rate describing viral infection. First, the positivity and boundedness of solutions for nonnegative initial data are proved. Next, the global stability of the disease free steady state and the endemic steady states are established depending on the basic reproduction number

R_{0}

and the CTL immune response reproduction number

R_{C T L}

. Moreover, numerical simulations are performed in order to show the numerical stability for each steady state and to support our theoretical findings. Our model based findings suggest that system immunity represented by CTL may control viral replication and reduce the infection. Full article

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