scholarly journals Global Dynamics of a Host-Vector-Predator Mathematical Model

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fengyan Zhou ◽  
Hongxing Yao
Keyword(s):  
Mathematical Model ◽  
Predation Rate ◽  
Disease Suppression ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Vector Population ◽  
Threshold Conditions ◽  
Vector Theory ◽  
The Impact ◽  
Disease Free

A mathematical model which links predator-vector(prey) and host-vector theory is proposed to examine the indirect effect of predators on vector-host dynamics. The equilibria and the basic reproduction numberR0are obtained. By constructing Lyapunov functional and using LaSalle’s invariance principle, global stability of both the disease-free and disease equilibria are obtained. Analytical results show thatR0provides threshold conditions on determining the uniform persistence and extinction of the disease, and predator density at any time should keep larger or equal to its equilibrium level for successful disease eradication. Finally, taking the predation rate as parameter, we provide numerical simulations for the impact of predators on vector-host disease control. It is illustrated that predators have a considerable influence on disease suppression by reducing the density of the vector population.

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.

Global stability of an age-structured SVEIR epidemic model with waning immunity, latency and relapse

2017 ◽  
Vol 10 (03) ◽  
pp. 1750038 ◽  
Author(s):  
Lili Liu ◽  
Xianning Liu
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Epidemic Model ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Sharp Threshold ◽  
Age Dependent ◽  
Waning Immunity ◽  
Age Structured ◽  
Disease Free

The global dynamics of an SVEIR epidemic model with age-dependent waning immunity, latency and relapse are studied. Sharp threshold properties for global asymptotic stability of both disease-free equilibrium and endemic equilibrium are given. The asymptotic smoothness, uniform persistence and the existence of interior global attractor of the semi-flow generated by a family of solutions of the system are also addressed. Furthermore, some related strategies for controlling the spread of diseases are discussed.

scholarly journals Seasonality Impact on the Transmission Dynamics of Tuberculosis

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Yali Yang ◽  
Chenping Guo ◽  
Luju Liu ◽  
Tianhua Zhang ◽  
Weiping Liu
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Autonomous Systems ◽  
Positive Periodic Solution ◽  
Transmission Dynamics ◽  
Uniform Persistence ◽  
Globally Asymptotically Stable ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

scholarly journals Global Dynamics of a Discrete-Time MERS-Cov Model

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 563
Author(s):  
Mahmoud H. DarAssi ◽  
Mohammad A. Safi ◽  
Morad Ahmad
Keyword(s):  
Discrete Time ◽  
Global Attractivity ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Proposed Model ◽  
Threshold Conditions ◽  
Theoretical Results ◽  
Disease Free

In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. We proved that the DFE is globally asymptotically stable when R0≤1. Whenever R˜0>1, the proposed model has a unique endemic equilibrium that is globally asymptotically stable. The theoretical results are illustrated by a numerical simulation.

scholarly journals Impact of Hygiene on Malaria Transmission Dynamics: A Mathematical Model

2021 ◽  
Author(s):  
Temidayo Oluwafemi ◽  
Emmanuel Azuaba
Keyword(s):  
Mathematical Model ◽  
Malaria Transmission ◽  
Basic Reproduction Number ◽  
Human Population ◽  
Reproduction Number ◽  
Model System ◽  
Transmission Dynamics ◽  
The Impact ◽  
Disease Free ◽  
The Basic Reproduction Number

Malaria continues to pose a major public health challenge, especially in developing countries, 219 million cases of malaria were estimated in 89 countries. In this paper, a mathematical model using non-linear differential equations is formulated to describe the impact of hygiene on Malaria transmission dynamics, the model is analyzed. The model is divided into seven compartments which includes five human compartments namely; Unhygienic susceptible human population, Hygienic Susceptible Human population, Unhygienic infected human population , hygienic infected human population and the Recovered Human population  and the mosquito population is subdivided into susceptible mosquitoes  and infected mosquitoes . The positivity of the solution shows that there exists a domain where the model is biologically meaningful and mathematically well-posed. The Disease-Free Equilibrium (DFE) point of the model is obtained, we compute the Basic Reproduction Number using the next generation method and established the condition for Local stability of the disease-free equilibrium, and we thereafter obtained the global stability of the disease-free equilibrium by constructing the Lyapunov function of the model system. Also, sensitivity analysis of the model system was carried out to identify the influence of the parameters on the Basic Reproduction Number, the result shows that the natural death rate of the mosquitoes is most sensitive to the basic reproduction number.

scholarly journals Stability Analysis of an Epidemic Model with Infected Immigrants and Optimal Vaccination

2019 ◽  
Vol 8 (2) ◽  
pp. 3071-3077
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Epidemic Model ◽  
Susceptible Individual ◽  
Susceptible Population ◽  
Vaccination Strategies ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Impact ◽  
Disease Free

In this paper an SIR (Susceptible-infectious-recovered) epidemic model consisting of saturated incidence rate with vaccination to the susceptible individual in presence of infected immigrants is studied. Stabilities of disease free and endemic equilibrium are also analyzed. The impact of the infected immigrants in the spread of the illness in a populace is examined. A mathematical model has been used to investigate the inflow of the infected immigrants in a population who rapidly transmit the disease. By using appropriate vaccine level to the susceptible population, disease can be reduced. The main purpose of this work is minimizing the invectives and maximizes the recovered individuals. To attain this, apply optimal vaccination strategies by utilizing the pontryagin’s maximum principle (PMP). Speculative results are demonstrated through the numerical simulations

scholarly journals A Mathematical Model for the Prediction of the Impact of Coronavirus(COVID­19) and Social Distancing Effect

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Mathematical Model ◽  
Data Fitting ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Social Distancing ◽  
The World ◽  
Unique Style ◽  
The Right ◽  
The Impact ◽  
Disease Free

The spread of coronavirus across the world has become a major pandemic following the Spanishflu of 1918. A mathematical model of the spread of the coronavirus with social distancing effect is studied. Amathematical model of the spread of the virus form Wuhan in China to the rest of the world is suggested andanalyzed. Another mathematical model with quarantine and social distancing factors is proposed and analyzed.Stability analysis for both models were carried out and data fitting was performed to predict the possible extinctionof the disease. The disease free equilibria of both models were locally and globally asymptotically stable. Themodels suggest that with interventions such as lock downs and social distancing the extinction of the coronaviruscan be achieved. Increasing social distancing could reduce the number of new cases by up to 30%. The paperpresents a unique style of considering both theoretical and data analysis which is rarely studied in the literature.Questions arising from this study for further research include the right time to apply interventions and the state ofpreparedness in case of similar pandemics.

scholarly journals Global Stability of a Variation Epidemic Spreading Model on Complex Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
De-gang Xu ◽  
Xi-yang Xu ◽  
Chun-hua Yang ◽  
Wei-hua Gui
Keyword(s):  
Complex Networks ◽  
Stability Criterion ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Epidemic Spreading ◽  
Hurwitz Stability ◽  
Spreading Model ◽  
Disease Free

Epidemic spreading on networks becomes a hot issue of nonlinear systems, which has attracted many researchers’ attention in recent years. A novel epidemic spreading model with variant factors in complex networks is proposed and investigated in this paper. One main feature of this model is that virus variation is investigated in the process of epidemic dynamical spreading. The global dynamics of this model involving an endemic equilibrium and a disease-free equilibrium are, respectively, discussed. Some sufficient conditions are given for the existence of the endemic equilibrium. In addition, the global asymptotic stability problems of the disease-free equilibrium and the endemic equilibrium are also investigated by the Routh-Hurwitz stability criterion and Lyapunov stability criterion. And the uniform persistence condition of the new system is studied. Finally, numerical simulations are provided to illustrate obtained theoretical results.

scholarly journals A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Bakary Traoré ◽  
Boureima Sangaré ◽  
Sado Traoré
Keyword(s):  
Mathematical Model ◽  
Malaria Transmission ◽  
Climatic Factors ◽  
Positive Periodic Solution ◽  
Uniform Persistence ◽  
Basic Reproduction Ratio ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Vector Population ◽  
Biting Rate

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.

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