scholarly journals Dynamics Analysis of an Epidemiological Model with Media Impact and Two Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Lianwen Wang ◽  
Yong Li ◽  
Liuyong Pang
Keyword(s):  
Reproduction Number ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Dynamics Analysis ◽  
Epidemic Spread ◽  
Media Impact ◽  
Timely Response ◽  
Two Delays ◽  
The Impact

This paper is concerned with exploring the global dynamics of SEIR epidemic model with media impact, which incorporates latency and relapse delays. The permanence of the model is carefully discussed. By suitable Lyapunov functionals, we establish the global stability of the equilibria. It is found that the basic reproduction number completely determines the threshold dynamics of the SEIR model. Finally, the impact of media on the epidemic spread is studied, which reveals that timely response of media and individuals may play a more key role in disease control.

scholarly journals Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Mahmoud A. Ibrahim ◽  
Attila Dénes
Keyword(s):  
Zika Virus ◽  
Reproduction Number ◽  
Virus Disease ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

scholarly journals Threshold Dynamics of a Huanglongbing Model with Logistic Growth in Periodic Environments

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Wang ◽  
Shujing Gao ◽  
Yueli Luo ◽  
Dehui Xie
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Logistic Growth ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

We analyze the impact of seasonal activity of psyllid on the dynamics of Huanglongbing (HLB) infection. A new model about HLB transmission with Logistic growth in psyllid insect vectors and periodic coefficients has been investigated. It is shown that the global dynamics are determined by the basic reproduction numberR0which is defined through the spectral radius of a linear integral operator. IfR0< 1, then the disease-free periodic solution is globally asymptotically stable and ifR0> 1, then the disease persists. Numerical values of parameters of the model are evaluated taken from the literatures. Furthermore, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproduction number to the changes of average and amplitude values of the recruitment function of citrus are shown. Finally, some useful comments on controlling the transmission of HLB are given.

scholarly journals Global dynamics of a Huanglongbing model with a periodic latent period

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yan Hong ◽  
Xiuxiang Liu ◽  
Xiao Yu
Keyword(s):  
Reproduction Number ◽  
Threshold Value ◽  
Transmission Model ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Temperature Dependent ◽  
Candidatus Liberibacter ◽  
Effects Of Temperature ◽  
Disease Free ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>Huanglongbing (HLB) is a disease of citrus that caused by phloem-restricted bacteria of the Candidatus Liberibacter group. In this paper, we present a HLB transmission model to investigate the effects of temperature-dependent latent periods and seasonality on the spread of HLB. We first establish disease free dynamics in terms of a threshold value <inline-formula><tex-math id="M1">\begin{document}$ R^p_0 $\end{document}</tex-math></inline-formula>, and then introduce the basic reproduction number <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> and show the threshold dynamics of HLB with respect to <inline-formula><tex-math id="M3">\begin{document}$ R^p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. Numerical simulations are further provided to illustrate our analytic results.</p>

A THEORETICAL ASSESSMENT OF THE EFFECTS OF DISTRIBUTED DELAY ON THE TRANSMISSION DYNAMICS OF HEPATITIS B

2015 ◽  
Vol 23 (03) ◽  
pp. 423-455
Author(s):  
P. MOUOFO TCHINDA ◽  
JEAN JULES TEWA ◽  
BOULECHARD MEWOLI ◽  
SAMUEL BOWONG
Keyword(s):  
Drug Therapy ◽  
Hepatitis B ◽  
Basic Reproduction Number ◽  
Disease Transmission ◽  
Time Delays ◽  
Reproduction Number ◽  
Distributed Delay ◽  
Global Dynamics ◽  
The Impact ◽  
The Basic Reproduction Number

In this paper, we investigate the global dynamics of a system of delay differential equations which describes the interaction of hepatitis B virus (HBV) with both liver and blood cells. The model has two distributed time delays describing the time needed for infection of cell and virus replication. We also include the efficiency of drug therapy in inhibiting viral production and the efficiency of drug therapy in blocking new infection. We compute the basic reproduction number and find that increasing delays will decrease the value of the basic reproduction number. We study the sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Our analysis reveals that the model exhibits the phenomenon of backward bifurcation (where a stable disease-free equilibrium (DFE) co-exists with a stable endemic equilibrium when the basic reproduction number is less than unity). Numerical simulations are presented to evaluate the impact of time-delays on the prevalence of the disease.

scholarly journals Global Dynamics of an SVEIR Model with Age-Dependent Vaccination, Infection, and Latency

2018 ◽  
Vol 2018 ◽  
pp. 1-21 ◽  
Author(s):  
Rodrigue Yves M’pika Massoukou ◽  
Suares Clovis Oukouomi Noutchie ◽  
Richard Guiem
Keyword(s):  
Infectious Diseases ◽  
Differential Equations ◽  
Reproduction Number ◽  
Coupled System ◽  
Hyperbolic Partial Differential Equations ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Age Dependent ◽  
Induced Immunity ◽  
The Basic Reproduction Number

Vaccine-induced protection is substantial to control, prevent, and reduce the spread of infectious diseases and to get rid of infectious diseases. In this paper, we propose an SVEIR epidemic model with age-dependent vaccination, latency, and infection. The model also considers that the waning vaccine-induced immunity depends on vaccination age and the vaccinated individuals fall back to the susceptible class after losing immunity. The model is a coupled system of (hyperbolic) partial differential equations with ordinary differential equations. The global dynamics of the model is established through construction of appropriate Lyapunov functionals and application of Lasalle’s invariance principle. As a result, the global stability of the infection-free equilibrium and endemic equilibrium is obtained and is fully determined by the basic reproduction number R0.

Global dynamics of a cholera model with age structures and multiple transmission modes

2019 ◽  
Vol 12 (05) ◽  
pp. 1950051
Author(s):  
Xia Wang ◽  
Yuming Chen ◽  
Xinyu Song
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Global Attractors ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Transmission Modes ◽  
Multiple Transmission ◽  
The Basic Reproduction Number

In this paper, we propose and analyze a cholera model. The model incorporates both direct transmission (person-to-person transmission) and indirect transmission (contaminated environment-to-person transmission: hyper-infectivity and lower-infectivity). Moreover, we employ general nonlinear incidences and introduce infection age of infectious individuals and biological ages of pathogens in the environment. After considering the well-posedness of the system, we study the existence and local stability of steady states, which is determined by the basic reproduction number. To establish the attractivity of the infection steady state, we also get the uniform persistence and existence of compact global attractors. The main result is a threshold dynamics obtained by applying the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the infection-free steady state is globally asymptotically stable while when the basic reproduction number is larger than one, the infection steady state attracts each solution with nonzero infection force at some time point. The effect of multiple transmission modes on the disease dynamics is also discussed.

scholarly journals Global proprieties of a delayed epidemic model with partial susceptible protection

2021 ◽  
Vol 19 (1) ◽  
pp. 209-224
Author(s):  
Abdelheq Mezouaghi ◽  
◽  
Salih Djillali ◽  
Anwar Zeb ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Isolation Rate ◽  
Globally Asymptotically Stable ◽  
The Government ◽  
The Basic Reproduction Number

<abstract><p>In the case of an epidemic, the government (or population itself) can use protection for reducing the epidemic. This research investigates the global dynamics of a delayed epidemic model with partial susceptible protection. A threshold dynamics is obtained in terms of the basic reproduction number, where for $ R_0 &lt; 1 $ the infection will extinct from the population. But, for $ R_0 &gt; 1 $ it has been shown that the disease will persist, and the unique positive equilibrium is globally asymptotically stable. The principal purpose of this research is to determine a relation between the isolation rate and the basic reproduction number in such a way we can eliminate the infection from the population. Moreover, we will determine the minimal protection force to eliminate the infection for the population. A comparative analysis with the classical SIR model is provided. The results are supported by some numerical illustrations with their epidemiological relevance.</p></abstract>

scholarly journals Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yunguo Jin
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Reproduction Number ◽  
Stage Structure ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Differential Infectivity ◽  
The Stability ◽  
Theoretical Results

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.

scholarly journals Modeling the Impact of Lock-down on COVID-19 Spread in Malaysia

2020 ◽  
Author(s):  
Altahir A. Altahir ◽  
Nirbhay Mathur ◽  
Loshini Thiruchelvam ◽  
Ghulam E. Mustafa Abro ◽  
Syaimaa’ S. M. Radzi ◽  
...  
Keyword(s):  
Infectious Disease ◽  
Reproduction Number ◽  
Formal Analysis ◽  
Time Frame ◽  
Immature Stage ◽  
Epidemic Spread ◽  
Prevention Measures ◽  
Seir Model ◽  
The World ◽  
The Impact

AbstractAfter a breakdown notified in Wuhan, China in December 2019, COVID-19 is declared as pandemic diseases. To the date more than 13 million confirmed cases and more than half a million are dead around the world. This virus also attached Malaysia in its immature stage where 8718 cases were confirmed and 122 were declared as death. Malaysia responsibly controlled the spread by enforcing MCO. Hence, it is required to visualize the pattern of Covid-19 spread. Also, it is necessary to estimate the impact of the enforced prevention measures. In this paper, an infectious disease dynamic modeling (SEIR) is used to estimate the epidemic spread in Malaysia. The main assumption is to update the reproduction number Rt with respect to the implemented prevention measures. For a time-frame of five month, the Rt was assumed to vary between 2.9 and 0.3. Moreover, the manuscript includes two possible scenarios: the first will be the extension of the stricter measures all over the country, and the second will be the gradual lift of the lock-down. After implementing several stages of lock-down we have found that the estimated values of the Rt with respect to the strictness degree varies between 0.2 to 1.1. A continuous strict lock-down may reduce the Rt to 0.2 and accordingly the estimated active cases will be reduced to 20 by the beginning of September 2020. In contrast, the second scenario considers a gradual lift of the enforced prevention measures by the end of June 2020, here we have considered three possible outcomes according to the MCO relaxation. Thus, the estimated values of Rt = 0.7, 0.9, 1.1, which shows a rapid increase in the number of active cases. The implemented SEIR model shows a close resemblance with the actual data recorded from 10, March till 7, July 2020.Author summaryConceptualization, A.A.A; methodology, A.A.A, N.M; validation, A.A.A, N.M; formal analysis, A.A.A; investigation, N.M, A.A.A; resources, G.E.M.A, L.T; data collection, L.T, N.M; writing—original draft preparation, A.A.A, L.T, G.E.M.A, N.M; writing—review and editing, V.S.A, S.C.D, B.S.G, P.S, S.A.B.M.Z, N.M; visualization, N.M; supervision, V.S.A; project administration, V.S.A. All authors have read and agreed to the published version of the manuscript

scholarly journals Global dynamics of a delayed epidemic model with latency and relapse

2013 ◽  
Vol 18 (2) ◽  
pp. 250-263 ◽  
Author(s):  
Rui Xu
Keyword(s):  
Basic Reproduction Number ◽  
Local Stability ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Lyapunov Functionals ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

A mathematical model describing the transmission dynamics of an infectious disease with an exposed (latent) period, relapse and a saturation incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proven that if the basic reproduction number is less than unity, the diseasefree equilibrium is globally asymptotically stable and therefore the disease fades out; and if the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable and the disease becomes endemic.

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