scholarly journals Effect of Awareness Programs on the Epidemic Outbreaks with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Lixia Zuo ◽  
Maoxing Liu
Keyword(s):  
Infectious Disease ◽  
Growth Rate ◽  
Time Delay ◽  
Numerical Simulations ◽  
Direct Contact ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Implementation Rate ◽  
Awareness Programs ◽  
Dissemination Rate

An epidemic model with time delay has been proposed and analyzed. In this model the effect of awareness programs driven by media on the prevalence of an infectious disease is studied. It is assumed that pathogens are transmitted via direct contact between the susceptible and the infective populations and further assumed that the growth rate of cumulative density of awareness programs increases at a rate proportional to the infective population. The model is analyzed by using stability theory of differential equations and numerical simulations. Both equilibria have been proved to be globally asymptotically stable. The results we obtained and numerical simulations suggest the increasing of the dissemination rate and implementation rate can reduce the proportion of the infective population.

EFFECT OF AWARENESS PROGRAMS IN CONTROLLING THE PREVALENCE OF AN EPIDEMIC WITH TIME DELAY

2011 ◽  
Vol 19 (02) ◽  
pp. 389-402 ◽  
Author(s):  
A. K. MISRA ◽  
ANUPAMA SHARMA ◽  
VISHAL SINGH
Keyword(s):  
Mathematical Model ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Stability Theory ◽  
Direct Contact ◽  
Behavioral Changes ◽  
Nonlinear Mathematical Model ◽  
Awareness Programs

A nonlinear mathematical model with delay to capture the dynamics of effect of awareness programs on the prevalence of any epidemic is proposed and analyzed. It is assumed that pathogens are transmitted via direct contact between susceptibles and infectives. It is assumed further that cumulative density of awareness programs increases at a rate proportional to the number of infectives. It is considered that awareness programs are capable of inducing behavioral changes in susceptibles, which result in the isolation of aware population. The model is analyzed using stability theory of differential equations and numerical simulations. The model analysis shows that, though awareness programs cannot eradicate infection, they help in controlling the prevalence of disease. It is also found that time delay in execution of awareness programs destabilizes the system and periodic solutions may arise through Hopf-bifurcation.

scholarly journals Analysis of an HIV - HCV simultaneous infection model with time delay

2021 ◽  
pp. 1-11
Author(s):  
Adamu Shitu Hassan ◽  
Nafiu Hussaini
Keyword(s):  
Hepatitis C Virus ◽  
Hepatitis C ◽  
Time Delay ◽  
Numerical Simulations ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Globally Asymptotically Stable ◽  
Simultaneous Infection ◽  
Disease Free

A novel mathematical delay model for simultaneous infection of HIV and hepatitis C virus is formulated and dynamically analyzed. Basic properties of the model are established and proved. Basic reproductive threshold is systematically calculated as the maximum of three subthreshold parameters. A disease free equilibrium is determined to be globally asymptotically stable for all values of the delay when the threshold is less than unity. However, when the threshold is greater than one, endemic equilibrium emerged which is shown to be locally asymptotically stable for any length of delay. Although the delay has no effect on stabilities of equilibria points, however, it is found to reduce the infectivity of the viruses as the length of the delay is increased. Epidemiological interpretations of the results and numerical simulations illustrating them are given.

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.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

scholarly journals Threshold dynamics of a general delayed HIV model with double transmission modes and latent viral infection

2021 ◽  
Vol 7 (2) ◽  
pp. 2456-2478
Author(s):  
Xin Jiang ◽  
Keyword(s):  
Time Delay ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Model ◽  
Transmission Modes ◽  
Latently Infected ◽  
Vital Effect ◽  
Infected Cells

<abstract><p>In this paper, a general HIV model incorporating intracellular time delay is investigated. Taking the latent virus infection, both virus-to-cell and cell-to-cell transmissions into consideration, the model exhibits threshold dynamics with respect to the basic reproduction number $ \mathfrak{R}_0 $. If $ \mathfrak{R}_0 &lt; 1 $, then there exists a unique infection-free equilibrium $ E_0 $, which is globally asymptotically stable. If $ \mathfrak{R}_0 &gt; 1 $, then there exists $ E_0 $ and a globally asymptotically stable infected equilibrium $ E^* $. When $ \mathfrak{R}_0 = 1 $, $ E_0 $ is linearly neutrally stable and a forward bifurcation takes place without time delay around $ \mathfrak{R}_0 = 1 $. The theoretical results and corresponding numerical simulations show that the existence of latently infected cells and the intracellular time delay have vital effect on the global dynamics of the general virus model.</p></abstract>

scholarly journals Dynamical Behavior of a New Epidemiological Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zizi Wang ◽  
Zhiming Guo
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Endemic Equilibrium ◽  
Epidemiological Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

scholarly journals Modeling the Parasitic Filariasis Spread by Mosquito in Periodic Environment

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Yan Cheng ◽  
Xiaoyun Wang ◽  
Qiuhui Pan ◽  
Mingfeng He
Keyword(s):  
Sensitivity Analysis ◽  
Periodic Solution ◽  
Numerical Simulations ◽  
Parasitic Infection ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Periodic Environment ◽  
Disease Free

In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameterR0is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that whenR0<1, the disease-free periodic solution is globally asymptotically stable and whenR0>1by Poincaré map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters onR0, which provided references to seek optimal measures to control the transmission of lymphatic filariasis.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

GLOBAL DYNAMICS OF A CHOLERA MODEL WITH TIME DELAY

2013 ◽  
Vol 06 (01) ◽  
pp. 1250070 ◽  
Author(s):  
YUJIE WANG ◽  
JUNJIE WEI
Keyword(s):  
Time Delay ◽  
Numerical Simulations ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Threshold Value ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Periodic Oscillations ◽  
Theoretical Results

The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 < 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 > 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.

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>

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