scholarly journals Dynamics analysis of a delayed virus model with two different transmission methods and treatments

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Tongqian Zhang ◽  
Junling Wang ◽  
Yuqing Li ◽  
Zhichao Jiang ◽  
Xiaofeng Han
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Reproduction Number ◽  
Dynamics Analysis ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Local Hopf Bifurcation ◽  
Virus Model ◽  
The Stability ◽  
The Basic Reproduction Number

AbstractIn this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

Dynamics of the impact of Twitter with time delay on the spread of infectious diseases

2018 ◽  
Vol 11 (05) ◽  
pp. 1850067 ◽  
Author(s):  
Maoxing Liu ◽  
Yuting Chang ◽  
Haiyan Wang ◽  
Benxing Li
Keyword(s):  
Infectious Disease ◽  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Reproduction Number ◽  
Behavioral Changes ◽  
The Public ◽  
The Stability ◽  
The Impact ◽  
The Basic Reproduction Number

In this paper, a mathematical model to study the impact of Twitter in controlling infectious disease is proposed. The model includes the dynamics of “tweets” which may enhance awareness of the disease and cause behavioral changes among the public, thus reducing the transmission of the disease. Furthermore, the model is improved by introducing a time delay between the outbreak of disease and the release of Twitter messages. The basic reproduction number and the conditions for the stability of the equilibria are derived. It is shown that the system undergoes Hopf bifurcation when time delay is increased. Finally, numerical simulations are given to verify the analytical results.

scholarly journals Simple MSEIR Model for Measles Transmission

2019 ◽  
pp. 1-11
Author(s):  
Mojeeb Al-Rahman EL-Nor Osman ◽  
Appiagyei Ebenezer ◽  
Isaac Kwasi Adu
Keyword(s):  
Mathematical Model ◽  
Basic Reproduction Number ◽  
Recovery Rate ◽  
Birth Rate ◽  
Reproduction Number ◽  
Progression Rate ◽  
Asymptotically Stable ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria.  The basic reproduction number  is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out.  The disease was locally asymptotically stable if  and unstable if  . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

scholarly journals Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Tao Dong ◽  
Xiaofeng Liao ◽  
Huaqing Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Virus ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
Virus Model ◽  
Theoretical Results

By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.

scholarly journals Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

2021 ◽  
Vol 3 (1) ◽  
pp. 16-25
Author(s):  
Adin Lazuardy Firdiansyah
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Interesting Phenomenon ◽  
Type Iii ◽  
Interior Equilibrium ◽  
The Stability

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases.  In the end, several numerical solutions are presented to check the analytical results.

scholarly journals On Oscillatory Pattern of Malaria Dynamics in a Population with Temporary Immunity

2007 ◽  
Vol 8 (3) ◽  
pp. 191-203 ◽  
Author(s):  
J. Tumwiine ◽  
J. Y. T. Mugisha ◽  
L. S. Luboobi
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Oscillatory Pattern ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number ◽  
Temporary Immunity

We use a model to study the dynamics of malaria in the human and mosquito population to explain the stability patterns of malaria. The model results show that the disease-free equilibrium is globally asymptotically stable and occurs whenever the basic reproduction number,R0is less than unity. We also note that whenR0>1, the disease-free equilibrium is unstable and the endemic equilibrium is stable. Numerical simulations show that recoveries and temporary immunity keep the populations at oscillation patterns and eventually converge to a steady state.

STABILITY OF AN AGE-STRUCTURED SEIS EPIDEMIC MODEL WITH INFECTIVITY IN INCUBATIVE PERIOD

2010 ◽  
Vol 03 (03) ◽  
pp. 299-312 ◽  
Author(s):  
SHU-MIN GUO ◽  
XUE-ZHI LI ◽  
XIN-YU SONG
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Stability Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured ◽  
The Stability ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an age-structured SEIS epidemic model with infectivity in incubative period is formulated and studied. The explicit expression of the basic reproduction number R0 is obtained. It is shown that the disease-free equilibrium is globally asymptotically stable if R0 < 1, at least one endemic equilibrium exists if R0 > 1. The stability conditions of endemic equilibrium are also given.

scholarly journals Modeling Computer Virus and Its Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Mei Peng ◽  
Xing He ◽  
Junjian Huang ◽  
Tao Dong
Keyword(s):  
Basic Reproduction Number ◽  
Stable Equilibrium ◽  
Reproduction Number ◽  
Computer Virus ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
Virus Model ◽  
Virus Spreading ◽  
The Basic Reproduction Number

Based on that the computer will be infected by infected computer and exposed computer, and some of the computers which are in suscepitible status and exposed status can get immunity by antivirus ability, a novel coumputer virus model is established. The dynamic behaviors of this model are investigated. First, the basic reproduction numberR0, which is a threshold of the computer virus spreading in internet, is determined. Second, this model has a virus-free equilibriumP0, which means that the infected part of the computer disappears, and the virus dies out, andP0is a globally asymptotically stable equilibrium ifR0<1. Third, ifR0>1then this model has only one viral equilibriumP*, which means that the computer persists at a constant endemic level, andP*is also globally asymptotically stable. Finally, some numerical examples are given to demonstrate the analytical results.

scholarly journals Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Wei Tan ◽  
Jianguo Gao ◽  
Wenjun Fan
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Center Manifold ◽  
Distributed Delay ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Bifurcation Parameter ◽  
Normal Form Theorem ◽  
The Stability ◽  
Variation Tendency

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

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 A fractional-order love dynamical model with time delay for synergic couple : Stability analysis and Hopf bifurcation

2021 ◽  
Author(s):  
SANTOSHI PANIGRAHI ◽  
Sunita Chand ◽  
S Balamuralitharan
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Order Model ◽  
Fractional Order Model ◽  
The Stability ◽  
The Impact

We investigate the fractional order love dynamic model with time delay for synergic couples in this manuscript. The quantitative analysis of the model has been done where the asymptotic stability of the equilibrium points of the model have been analyzed. Under the impact of time delay, the Hopf bifurcation analysis of the model has been done. The stability analysis of the model has been studied with the reproduction number less than or greater than 1. By using Laplace transformation, the analysis of the model has been done. The analysis shows that the fractional order model with a time delay can sufficiently improve the components and invigorate the outcomes for either stable or unstable criteria. In this model, all unstable cases are converted to stable cases under neighbourhood points. For all parameters, the reproduction ranges have been described. Finally, to illustrate our derived results numerical simulations have been carried out by using MATLAB. Under the theoretical outcomes from parameter estimation, the love dynamical system is verified.

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