scholarly journals A Two-Strain Deterministic Dengue Model With Seasonality Effect

2019 ◽  
Vol 1 (2) ◽  
pp. 13-15
Author(s):  
Afeez Abidemi ◽  
Mohd Ismail Abd Aziz ◽  
Rohanin Ahmad
Keyword(s):  
Disease Transmission ◽  
Birth Rate ◽  
Climatic Factors ◽  
Secondary Infection ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Dengue Disease ◽  
Reproduction Numbers ◽  
Disease Free

A compartmental model is proposed to describe the dynamics of vector-host interations for dengue disease transmission with coexistence of two virus serotypes. The model is modified to incorporate  seasonal-dependent mosquito birth rate in order to examine the influence of climatic factors such as rainfall and temperature on the dynamics of mosquito population and dengue disease transmission. The Next Generation Matrix method is used to obtain the basic reproduction number associated with the model without seasonality effect. The global dynamics of the model is analysed using the Comparison Theorem. The model is simulated in MATLAB with ode45 routine for two cases, namely: the less aggressive case (Case ) and the more aggressive case (Case ).  Analysis of the model shows that the Disease-Free Equilibrium (DFE) is locally asymptotically stable whenever both the basic reproduction numbers  R01 (associated with strain  only) and  R0j (associated with strain  only) are below unity. It is shown that the DFE is globally asymptotically stable when the susceptibility indices for secondary infection in strain  1 ( sigma1) and strain j ( sigmaj), and  are all less than 1.

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.

scholarly journals Global Stability of an SEIS Epidemic Model with General Saturation Incidence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hui Zhang ◽  
Li Yingqi ◽  
Wenxiong Xu
Keyword(s):  
Latent Period ◽  
Epidemic Model ◽  
Convergence Theorem ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T0, and the disease spreads to be endemic.

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.

INTRINSIC TRANSMISSION GLOBAL DYNAMICS OF TUBERCULOSIS WITH AGE STRUCTURE

2011 ◽  
Vol 04 (03) ◽  
pp. 329-346 ◽  
Author(s):  
JUN-YUAN YANG ◽  
XIAO-YAN WANG ◽  
XUE-ZHI LI ◽  
FENG-QIN ZHANG
Keyword(s):  
Steady State ◽  
Disease Transmission ◽  
Death Rate ◽  
Transmission Dynamics ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Age Structured

An age-structured epidemiological model for the disease transmission dynamics of TB is studied. We show that the infection-free steady state is locally and globally asymptotically stable if the basic reproductive number is below one, and in this case, the disease always dies out. We prove that the endemic steady state exists when the basic reproductive number is above one. In addition, the endemic steady state is globally asymptotically stable if the basic reproductive number is above one and death rate due to TB is zero.

Lyapunov function and global stability for a two-strain SEIR model with bilinear and non-monotone incidence

2019 ◽  
Vol 12 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Dounia Bentaleb ◽  
Saida Amine
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Reproduction Numbers ◽  
The Status ◽  
Incidence Functions ◽  
Disease Free ◽  
Disease Parameters

In this paper, we study a multi-strain SEIR epidemic model with both bilinear and non-monotone incidence functions. Under biologically motivated assumptions, we show that the model has two basic reproduction numbers that we noted [Formula: see text] and [Formula: see text]; and four equilibrium points. Using the Lyapunov method, we prove that if [Formula: see text] and [Formula: see text] are less than one then the disease-free equilibrium is Globally Asymptotically Stable, thus the disease will be eradicated. However, if one of the two basic reproduction numbers is greater than one, then the strain that persists is that with the larger basic reproduction number. And finally if both of the two basic reproduction numbers are equal or greater than one then the total endemic equilibrium is globally asymptotically stable. A numerical simulation is also presented to illustrate the influence of the psychological effect, of people to infection, on the spread of the disease in the population. This simulation can be used to determine the status of different diseases in a region using the corresponding data and infectious disease parameters.

scholarly journals Global Stability Analysis of Two-Stage Quarantine-Isolation Model with Holling Type II Incidence Function

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 350 ◽  
Author(s):  
Mohammad A. Safi
Keyword(s):  
Disease Transmission ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Control Measures ◽  
Asymptotically Stable ◽  
Two Stage ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Special Case ◽  
Disease Free

A new two-stage model for assessing the effect of basic control measures, quarantine and isolation, on a general disease transmission dynamic in a population is designed and rigorously analyzed. The model uses the Holling II incidence function for the infection rate. First, the basic reproduction number ( R 0 ) is determined. The model has both locally and globally asymptotically stable disease-free equilibrium whenever R 0 < 1 . If R 0 > 1 , then the disease is shown to be uniformly persistent. The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used in conjunction with LaSalle Invariance Principle to show that the endemic equilibrium is globally asymptotically stable for a special case.

scholarly journals Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hai-Feng Huo ◽  
Li-Xiang Feng
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Incomplete Treatment ◽  
Disease Free ◽  
The Basic Reproduction Number

An epidemic model with incomplete treatment and vaccination for the newborns and susceptibles is constructed. We establish that the global dynamics are completely determined by the basic reproduction numberR0. IfR0≤1, then the disease-free equilibrium is globally asymptotically stable. IfR0>1, the endemic equilibrium is globally asymptotically stable. Some numerical simulations are also given to explain our conclusions.

scholarly journals Global Dynamic Behavior of a Multigroup Cholera Model with Indirect Transmission

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ming-Tao Li ◽  
Gui-Quan Sun ◽  
Juan Zhang ◽  
Zhen Jin
Keyword(s):  
Classical Method ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Theoretic Approach ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Graph Theoretic ◽  
Indirect Transmission ◽  
Global Dynamic ◽  
Disease Free

For a multigroup cholera model with indirect transmission, the infection for a susceptible person is almost invariably transmitted by drinking contaminated water in which pathogens,V. cholerae, are present. The basic reproduction numberℛ0is identified and global dynamics are completely determined byℛ0. It shows thatℛ0is a globally threshold parameter in the sense that if it is less than one, the disease-free equilibrium is globally asymptotically stable; whereas if it is larger than one, there is a unique endemic equilibrium which is global asymptotically stable. For the proof of global stability with the disease-free equilibrium, we use the comparison principle; and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach.

A TWO-STRAIN EPIDEMIC MODEL ON COMPLEX NETWORKS WITH DEMOGRAPHICS

2016 ◽  
Vol 24 (04) ◽  
pp. 577-609 ◽  
Author(s):  
YANRU YAO ◽  
JUPING ZHANG
Keyword(s):  
Basic Reproduction Number ◽  
Disease Transmission ◽  
Degree Distribution ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sis Model ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, we develop a two-strain SIS model on heterogeneous networks with demographics for disease transmission. We calculate the basic reproduction number [Formula: see text] of infection for the model. We prove that if [Formula: see text], the disease-free equilibrium is globally asymptotically stable. If [Formula: see text], the conditions of the existence and global asymptotical stability of two boundary equilibria and the existence of endemic equilibria are established, respectively. Numerical simulations illustrate that the degree distribution of population varies with time before it reaches the stationary state. What is more, the basic reproduction number [Formula: see text] does not depend on the degree distribution like in the static network but depend on the demographic factors.

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.

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