scholarly journals Global Dynamics of Secondary DENV Infection with Diffusion

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
A. M. Elaiw ◽  
A. S. Alofi
Keyword(s):  
Global Solutions ◽  
Denv Infection ◽  
Nonlinear Pdes ◽  
Target Cells ◽  
Global Dynamics ◽  
Spatial Mobility ◽  
Existence And Stability ◽  
Infected Cells ◽  
In The Beginning ◽  
Theoretical Results

During the past eras, many mathematicians have paid their attentions to model the dynamics of dengue virus (DENV) infection but without taking into account the mobility of the cells and DENV particles. In this study, we develop and investigate a partial differential equations (PDEs) model that describes the dynamics of secondary DENV infection taking into account the spatial mobility of DENV particles and cells. The model includes five nonlinear PDEs describing the interaction among the target cells, DENV-infected cells, DENV particles, heterologous antibodies, and homologous antibodies. In the beginning, the well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive three threshold parameters which govern the existence and stability of the four equilibria of the model. We study the global stability of all equilibria based on the construction of suitable Lyapunov functions and usage of Lyapunov–LaSalle’s invariance principle (LLIP). Last, numerical simulations are carried out in order to verify the validity of our theoretical results.

scholarly journals Analysis of an HTLV/HIV dual infection model with diffusion

2021 ◽  
Vol 18 (6) ◽  
pp. 9430-9473
Author(s):  
A. M. Elaiw ◽  
◽  
N. H. AlShamrani ◽  
◽  
Keyword(s):  
Lyapunov Functions ◽  
Global Solutions ◽  
Dual Infection ◽  
Steady States ◽  
Infection Model ◽  
Well Posedness ◽  
Pde Model ◽  
Existence And Stability ◽  
Infected Cells ◽  
Theoretical Results

<abstract><p>In the literature, several HTLV-I and HIV single infections models with spatial dependence have been developed and analyzed. However, modeling HTLV/HIV dual infection with diffusion has not been studied. In this work we derive and investigate a PDE model that describes the dynamics of HTLV/HIV dual infection taking into account the mobility of viruses and cells. The model includes the effect of Cytotoxic T lymphocytes (CTLs) immunity. Although HTLV-I and HIV primarily target the same host, CD$ 4^{+} $T cells, via infected-to-cell (ITC) contact, however the HIV can also be transmitted through free-to-cell (FTC) contact. Moreover, HTLV-I has a vertical transmission through mitosis of active HTLV-infected cells. The well-posedness of solutions, including the existence of global solutions and the boundedness, is justified. We derive eight threshold parameters which govern the existence and stability of the eight steady states of the model. We study the global stability of all steady states based on the construction of suitable Lyapunov functions and usage of Lyapunov-LaSalle asymptotic stability theorem. Lastly, numerical simulations are carried out in order to verify the validity of our theoretical results.</p></abstract>

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150050
Author(s):  
Demou Luo ◽  
Qiru Wang
Keyword(s):  
Growth Rate ◽  
Allee Effect ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Growth ◽  
Trivial Equilibrium ◽  
Existence And Stability ◽  
Stable Equilibria ◽  
Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

scholarly journals Global Stability of Humoral Immunity HIV Infection Models with Chronically Infected Cells and Discrete Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-25
Author(s):  
A. M. Elaiw ◽  
N. A. Alghamdi
Keyword(s):  
Hiv Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Sufficient Conditions ◽  
Humoral Immune ◽  
Infection Models ◽  
Existence And Stability ◽  
Discrete Delays ◽  
Infected Cells ◽  
Theoretical Results

We study the global stability of three HIV infection models with humoral immune response. We consider two types of infected cells: the first type is the short-lived infected cells and the second one is the long-lived chronically infected cells. In the three HIV infection models, we modeled the incidence rate by bilinear, saturation, and general forms. The models take into account two types of discrete-time delays to describe the time between the virus entering into an uninfected CD4+T cell and the emission of new active viruses. The existence and stability of all equilibria are completely established by two bifurcation parameters,R0andR1. The global asymptotic stability of the steady states has been proven using Lyapunov method. In case of the general incidence rate, we have presented a set of sufficient conditions which guarantee the global stability of model. We have presented an example and performed numerical simulations to confirm our theoretical results.

scholarly journals Global Properties of a General Latent Pathogen Dynamics Model with Delayed Pathogenic and Cellular Infections

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
A. M. Elaiw ◽  
A. A. Almatrafi ◽  
A. D. Hobiny ◽  
K. Hattaf
Keyword(s):  
Global Stability ◽  
Infection Model ◽  
Global Dynamics ◽  
Dynamics Model ◽  
Pathogen Dynamics ◽  
Global Properties ◽  
Latently Infected ◽  
Infected Cells ◽  
Theoretical Results ◽  
Pathogenic Infection

This paper studies the global dynamics of a general pathogenic infection model with two ways of infections. The effect of antibody immune response is analyzed. We incorporate three discrete time delays and both latently infected cells and actively infected cells. The infection rate and production and clearance/death rates of the cells and pathogens are given by general functions. We determine two threshold parameters to investigate the global stability of three equilibria. We use Lyapunov method to establish the global stability. We support our theoretical results by numerical simulations.

scholarly journals Analysis of General Humoral Immunity HIV Dynamics Model with HAART and Distributed Delays

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 157 ◽  
Author(s):  
A. Elaiw ◽  
E. Elnahary
Keyword(s):  
Hiv Infection ◽  
The Body ◽  
Target Cells ◽  
Nonlinear Functions ◽  
Dynamics Model ◽  
Therapeutic Modalities ◽  
Hiv Dynamics ◽  
Highly Active ◽  
Infected Cells ◽  
Theoretical Results

This paper deals with the study of an HIV dynamics model with two target cells, macrophages and CD4 + T cells and three categories of infected cells, short-lived, long-lived and latent in order to get better insights into HIV infection within the body. The model incorporates therapeutic modalities such as reverse transcriptase inhibitors (RTIs) and protease inhibitors (PIs). The model is incorporated with distributed time delays to characterize the time between an HIV contact of an uninfected target cell and the creation of mature HIV. The effect of antibody on HIV infection is analyzed. The production and removal rates of the ten compartments of the model are given by general nonlinear functions which satisfy reasonable conditions. Nonnegativity and ultimately boundedness of the solutions are proven. Using the Lyapunov method, the global stability of the equilibria of the model is proven. Numerical simulations of the system are provided to confirm the theoretical results. We have shown that the antibodies can play a significant role in controlling the HIV infection, but it cannot clear the HIV particles from the plasma. Moreover, we have demonstrated that the intracellular time delay plays a similar role as the Highly Active Antiretroviral Therapies (HAAT) drugs in eliminating the HIV particles.

GLOBAL STABILITY OF A GENERAL VIRUS DYNAMICS MODEL WITH MULTI-STAGED INFECTED PROGRESSION AND HUMORAL IMMUNITY

2016 ◽  
Vol 24 (04) ◽  
pp. 535-560 ◽  
Author(s):  
A. M. ELAIW ◽  
N. H. ALSHAMRANI
Keyword(s):  
Humoral Immunity ◽  
Lyapunov Functions ◽  
Dynamical Behavior ◽  
Target Cells ◽  
Global Dynamics ◽  
Virus Dynamics ◽  
Nonlinear Functions ◽  
Dynamics Model ◽  
Number Formula ◽  
Infected Cells

In this paper, we propose an [Formula: see text]-dimensional nonlinear virus dynamics model that describes the interactions of the virus, uninfected target cells, [Formula: see text]-stages of infected cells and B cells. We assume that the incidence rate of infection, the generation and removal rates of all compartments are given by general nonlinear functions. We derive two threshold parameters, the basic reproduction number, [Formula: see text] and the humoral immunity number, [Formula: see text] and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. Utilizing Lyapunov functions and LaSalle’s invariance principle, the global asymptotic stability of all steady states of the model is proved. Numerical simulations are conducted for specific forms of the general functions in order to illustrate the dynamical behavior.

Global properties of nonlinear humoral immunity viral infection models

2015 ◽  
Vol 08 (05) ◽  
pp. 1550058 ◽  
Author(s):  
A. M. Elaiw ◽  
N. H. AlShamrani
Keyword(s):  
Viral Infection ◽  
Humoral Immunity ◽  
Nonlinear Models ◽  
Removal Rate ◽  
Lyapunov Theory ◽  
Target Cells ◽  
Global Dynamics ◽  
Virus Particles ◽  
Latently Infected ◽  
Infected Cells

In this paper, we consider two nonlinear models for viral infection with humoral immunity. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The second model is a modification of the first one by including the latently infected cells. The incidence rate, removal rate of infected cells, production rate of viruses and the latent-to-active conversion rate are given by more general nonlinear functions. We have established a set of conditions on these general functions and determined two threshold parameters for each model which are sufficient to determine the global dynamics of the models. The global asymptotic stability of all equilibria of the models has been proven by using Lyapunov theory and applying LaSalle's invariance principle. We have performed some numerical simulations for the models with specific forms of the general functions. We have shown that, the numerical results are consistent with the theoretical results.

scholarly journals Global Dynamics of a Stochastic Viral Infection Model with Latently Infected Cells

2021 ◽  
Vol 11 (21) ◽  
pp. 10484
Author(s):  
Chinnathambi Rajivganthi ◽  
Fathalla A. Rihan
Keyword(s):  
Viral Infection ◽  
Global Solutions ◽  
Infection Model ◽  
Global Dynamics ◽  
Large Intensity ◽  
Ergodic Distribution ◽  
Latently Infected ◽  
Infected Cells ◽  
White Noises ◽  
Viral Infection Model

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus.

scholarly journals Nonlinear Spatiotemporal Viral Infection Model with CTL Immunity: Mathematical Analysis

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 52
Author(s):  
Jaouad Danane ◽  
Karam Allali ◽  
Léon Matar Tine ◽  
Vitaly Volpert
Keyword(s):  
Nonlinear Differential Equations ◽  
Infection Model ◽  
Viral Dynamics ◽  
Spatial Mobility ◽  
Latently Infected ◽  
Infected Cells ◽  
Cellular Immune ◽  
Viral Infection Model ◽  
Theoretical Results ◽  
Positivity And Boundedness

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.

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