scholarly journals A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

2014 ◽  
Vol 9 ◽  
pp. 53-68 ◽  
Author(s):  
B. El Boukari ◽  
N. Yousfi
Keyword(s):  
Reproduction Number ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Differential Equation Model ◽  
Globally Asymptotically Stable ◽  
Intracellular Delay ◽  
Immunodeficiency Virus ◽  
Delay Differential ◽  
Theoretical Results ◽  
Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeficiency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely defined by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay affects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

scholarly journals Dynamics of the Stochastic Belousov-Zhabotinskii Chemical Reaction Model

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 663
Author(s):  
Ying Yang ◽  
Daqing Jiang ◽  
Donal O’Regan ◽  
Ahmed Alsaedi
Keyword(s):  
Chemical Reaction ◽  
Existence And Uniqueness ◽  
Reaction Model ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
Globally Asymptotically Stable ◽  
Chemical Reaction Model ◽  
Theoretical Results

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we present some simulations to illustrate our theoretical results. We note that the unique equilibrium of the original ordinary differential equation model is globally asymptotically stable under appropriate conditions of the parameter value f, while the stochastic model is ergodic regardless of the value of f.

scholarly journals Optimal Strategies for Control of COVID-19: A Mathematical Perspective

Scientifica ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Baba Seidu
Keyword(s):  
Reproduction Number ◽  
Optimal Strategies ◽  
Cost Effective ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
Disease Free ◽  
The Basic Reproduction Number

A deterministic ordinary differential equation model for SARS-CoV-2 is developed and analysed, taking into account the role of exposed, mildly symptomatic, and severely symptomatic persons in the spread of the disease. It is shown that in the absence of infective immigrants, the model has a locally asymptotically stable disease-free equilibrium whenever the basic reproduction number is below unity. In the absence of immigration of infective persons, the disease can be eradicated whenever ℛ 0 < 1 . Specifically, if the controls u i ,   i = 1,2,3,4 , are implemented to 100% efficiency, the disease dies away easily. It is shown that border closure (or at least screening) is indispensable in the fight against the spread of SARS-CoV-2. Simulation of optimal control of the model suggests that the most cost-effective strategy to combat SARS-CoV-2 is to reduce contact through use of nose masks and physical distancing.

scholarly journals A delayed dynamical model for COVID-19 therapy with defective interfering particles and artificial antibodies

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanfei Zhao ◽  
Yepeng Xing
Keyword(s):  
Time Delay ◽  
Delay Differential Equations ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Defective Interfering Particles ◽  
Theoretical Predictions ◽  
Delay Differential ◽  
Disease Free ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we use delay differential equations to propose a mathematical model for COVID-19 therapy with both defective interfering particles and artificial antibodies. For this model, the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> is given and its threshold properties are discussed. When <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_0&lt;1 $\end{document}</tex-math></inline-formula>, the disease-free equilibrium <inline-formula><tex-math id="M3">\begin{document}$ E_0 $\end{document}</tex-math></inline-formula> is globally asymptotically stable. When <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ E_0 $\end{document}</tex-math></inline-formula> becomes unstable and the infectious equilibrium without defective interfering particles <inline-formula><tex-math id="M6">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> comes into existence. There exists a positive constant <inline-formula><tex-math id="M7">\begin{document}$ R_1 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M8">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> is globally asymptotically stable when <inline-formula><tex-math id="M9">\begin{document}$ R_1&lt;1&lt;\mathcal{R}_0 $\end{document}</tex-math></inline-formula>. Further, when <inline-formula><tex-math id="M10">\begin{document}$ R_1&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ E_1 $\end{document}</tex-math></inline-formula> loses its stability and infectious equilibrium with defective interfering particles <inline-formula><tex-math id="M12">\begin{document}$ E_2 $\end{document}</tex-math></inline-formula> occurs. There exists a constant <inline-formula><tex-math id="M13">\begin{document}$ R_2 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M14">\begin{document}$ E_2 $\end{document}</tex-math></inline-formula> is asymptotically stable without time delay if <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;R_1&lt;\mathcal{R}_0&lt;R_2 $\end{document}</tex-math></inline-formula> and it loses its stability via Hopf bifurcation as the time delay increases. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.</p>

scholarly journals Dynamical Analysis of an SEIRS Model With Incomplete Recovery and Relapse on Networks and Its Optimal Vaccination Control

2021 ◽  
Author(s):  
Lei Zhang ◽  
Maoxing Liu ◽  
Qiang Hou ◽  
Boli Xie
Keyword(s):  
Control Strategy ◽  
Reproduction Number ◽  
Dynamical Analysis ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract For some infectious diseases, such as herpes and tuberculosis, there is incomplete recovery and relapse. These phenomena make them difficult to control. In consequence of this status, an SEIRS epidemic model with incomplete recovery and relapse on networks is established and the global dynamics is studied. The results show that when the basic reproduction number R 0 <=1 the disease-free equilibrium is globally asymptotically stable; when R 0 > 1, the endemic equilibrium is globally asymptotically stable. In addition, in consideration of vaccination control strategy, an SVEIRS model is introduced and the optimal control is solved. At last, the theoretical results are illustrated with numerical simulations.

Dynamic analysis of an SEIRS model with nonlinear infectivity on complex networks

2015 ◽  
Vol 09 (01) ◽  
pp. 1650009 ◽  
Author(s):  
Shouying Huang
Keyword(s):  
Basic Reproduction Number ◽  
Mathematical Analysis ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Globally Attractive ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we study the spreading of infections on complex heterogeneous networks based on an SEIRS epidemic model with nonlinear infectivity. By mathematical analysis, the basic reproduction number R0 is obtained. When R0 is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out, while R0 is greater than one, the disease-free equilibrium becomes unstable and the disease is permanent, and in the meantime there exists a unique endemic equilibrium which is globally attractive under certain conditions. Finally, the effects of various immunization schemes are studied. To verify our theoretical results, the corresponding numerical simulations are also included.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

scholarly journals A Stochastic TB Model for a Crowded Environment

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Sibaliwe Maku Vyambwera ◽  
Peter Witbooi
Keyword(s):  
Compartmental Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Equation Model ◽  
Ordinary Differential Equation Model ◽  
Differential Equation Model ◽  
The Stability ◽  
Crowded Environments ◽  
Disease Free

We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation model, and we impose stochastic perturbation. We prove the existence and uniqueness of positive solutions of a stochastic model. We introduce an invariant generalizing the basic reproduction number and prove the stability of the disease-free equilibrium when it is below unity or slightly higher than unity and the perturbation is small. Our main theorem implies that the stochastic perturbation enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to illustrate the analytical findings and the utility of the model.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

Mathematical Study of a Class of Epidemiological Models with Multiple Infectious Stages

2020 ◽  
Vol 21 (3-4) ◽  
pp. 259-274
Author(s):  
S. Bowong ◽  
A. Temgoua ◽  
Y. Malong ◽  
J. Mbang
Keyword(s):  
Theoretical Study ◽  
Reproduction Number ◽  
Communicable Disease ◽  
Transmission Dynamics ◽  
Asymptotically Stable ◽  
Epidemiological Models ◽  
Mathematical Study ◽  
Globally Asymptotically Stable ◽  
Disease Free

AbstractThis paper deals with the mathematical analysis of a general class of epidemiological models with multiple infectious stages for the transmission dynamics of a communicable disease. We provide a theoretical study of the model. We derive the basic reproduction number $\mathcal R_0$ that determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever $\mathcal R_0 \leq 1$, while when $\mathcal R_0 \gt 1$, the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is globally asymptotically stable. A case study for tuberculosis (TB) is considered to numerically support the analytical results.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

2020 ◽  
pp. 2050082
Author(s):  
Mehdi Lotfi ◽  
Azizeh Jabbari ◽  
Hossein Kheiri
Keyword(s):  
Mathematical Model ◽  
Asymptotic Stability ◽  
Stable Disease ◽  
Reproduction Number ◽  
Geometric Approach ◽  
Backward Bifurcation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Bifurcation Phenomenon ◽  
Disease Free

In this paper, we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection, in which the treatment is effective for a number of infectious individuals and it fails for some other infectious individuals who are being treated. We show that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity. Also, it is shown that under certain conditions the model cannot exhibit backward bifurcation. Furthermore, it is shown in the absence of re-infection, the backward bifurcation phenomenon does not exist, in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. The global asymptotic stability of the endemic equilibrium, when the associated reproduction number is greater than unity, is established using the geometric approach. Numerical simulations are presented to illustrate our main results.

Sign in / Sign up

Export Citation Format

Share Document