scholarly journals Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chengxia Lei ◽  
Yi Shen ◽  
Guanghui Zhang ◽  
Yuxiang Zhang
Keyword(s):  
Epidemic Model ◽  
Disease Transmission ◽  
Lyapunov Functions ◽  
Migration Rate ◽  
Global Attractivity ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Transmission Risk ◽  
Threshold Behaviour ◽  
Homogeneous Environment

<p style='text-indent:20px;'>In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.</p>

Dynamics of an infection age-space structured cholera model with Neumann boundary condition

2021 ◽  
pp. 1-30
Author(s):  
WEIWEI LIU ◽  
JINLIANG WANG ◽  
RAN ZHANG
Keyword(s):  
Vibrio Cholerae ◽  
Disease Transmission ◽  
Global Attractivity ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Human Populations ◽  
Global Dynamics ◽  
Infection Age ◽  
Reaction Diffusion Model ◽  
Disease Persistence

This paper investigates global dynamics of an infection age-space structured cholera model. The model describes the vibrio cholerae transmission in human population, where infection-age structure of vibrio cholerae and infectious individuals are incorporated to measure the infectivity during the different stage of disease transmission. The model is described by reaction–diffusion models involving the spatial dispersal of vibrios and the mobility of human populations in the same domain Ω ⊂ ℝ n . We first give the well-posedness of the model by converting the model to a reaction–diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggest that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability of equilibria, disease persistence and global attractivity of equilibria.

scholarly journals Analysis of a diffusive cholera model incorporating latency and bacterial hyperinfectivity

2021 ◽  
Vol 20 (11) ◽  
pp. 3921
Author(s):  
Wei Yang ◽  
Jinliang Wang
Keyword(s):  
Lyapunov Functional ◽  
Global Attractivity ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Threshold Dynamics ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Nonlocal Reaction ◽  
Theoretical Results ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the threshold dynamics of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Our model takes the form of spatially nonlocal reaction-diffusion system associated with zero-flux boundary condition and time delay. By studying the associated eigenvalue problem, we establish the threshold dynamics that determines whether or not cholera will spread. We also confirm that the threshold dynamics can be determined by the basic reproduction number. By constructing Lyapunov functional, we address the global attractivity of the unique positive equilibrium whenever it exists. The theoretical results are still hold for the case when the constant parameters are replaced by strictly positive and spatial dependent functions.</p>

scholarly journals On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

2020 ◽  
Vol 10 (22) ◽  
pp. 8296 ◽  
Author(s):  
Malen Etxeberria-Etxaniz ◽  
Santiago Alonso-Quesada ◽  
Manuel De la Sen
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Impulsive Vaccination ◽  
Disease Free ◽  
The Basic Reproduction Number

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

scholarly journals Analysis of a Diffusive Heroin Epidemic Model in a Heterogeneous Environment

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jinliang Wang ◽  
Hongquan Sun
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Threshold Dynamics ◽  
Heterogeneous Environment ◽  
Threshold Parameter ◽  
Simulation Results ◽  
Heroin Model

This paper is concerned with a reaction-diffusion heroin model in a bound domain. The objective of this paper is to explore the threshold dynamics based on threshold parameter and basic reproduction number (BRN) ℜ0, and it is proved that if ℜ0<1, heroin spread will be extinct, while if ℜ0>1, heroin spread is uniformly persistent and there exists a positive heroin-spread steady state. We also obtain that the explicit formula of ℜ0 and global attractiveness of constant positive steady state (PSS) when all parameters are positive constants. Our simulation results reveal that compared to the homogeneous setting, the spatial heterogeneity has essential impacts on increasing the risk of heroin spread.

scholarly journals Analysis of COVID-19 based on SEIR epidemic models in a multi-patch environment

2021 ◽  
Author(s):  
Lan Meng ◽  
Wei Zhu
Keyword(s):  
Numerical Simulations ◽  
Dynamic Behavior ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Migration Rate ◽  
Reproduction Number ◽  
Population Migration ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

Abstract In this paper, an n-patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. Some numerical simulations with three patches are given to validate the effectiveness of the theoretical results. The influence of quarantined rate and population migration rate on the basic reproduction number is also discussed by simulation.

scholarly journals Stability Analysis of a Reaction-Diffusion Heroin Epidemic Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Liang Zhang ◽  
Yifan Xing
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Globally Asymptotically Stable ◽  
D Model ◽  
Drug Free ◽  
The Basic Reproduction Number

A reaction-diffusion (R-D) heroin epidemic model with relapse and permanent immunization is formulated. We use the basic reproduction number R0 to determine the global dynamics of the models. For both the ordinary differential equation (ODE) model and the R-D model, it is shown that the drug-free equilibrium is globally asymptotically stable if R0≤1, and if R0>1, the drug-addiction equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.

scholarly journals The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yanan Zhao ◽  
Daqing Jiang
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Stochastic Perturbation ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Time Average ◽  
The Difference ◽  
Disease Free

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction numberR0. We deduce the globally asymptotic stability of the disease-free equilibrium whenR0≤ 1and the perturbation is small, which means that the disease will die out. WhenR0>1, we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

scholarly journals The impact of active case finding among high-risk populations on the decline of tuberculosis incidence

2021 ◽  
Author(s):  
Aleksandra Tomczak ◽  
Dominika Warmjak ◽  
Aneta Wiśniewska
Keyword(s):  
High Risk ◽  
Disease Transmission ◽  
Reproduction Number ◽  
Case Finding ◽  
Transmission Risk ◽  
Active Case Finding ◽  
Disease Dynamics ◽  
Active Screening ◽  
Active Case ◽  
The Impact

Introduction: Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis. In 2019 the WHO reported approximately 10 million TB cases and 1.4 million deaths worldwide. TB still remains one of the leading causes of death in humans. Brazil is one of 30 countries with the highest TB burden with 96,000 new cases and 6,700 deaths reported in 2019. From 2015 the TB incidence is increasing by 2%–3% annually. It means that TB control programs need to be improved. Aim: Our aim is to show the impact of active case finding of TB cases among a high-risk subpopulation on decline of the incidence in the general population. Material and methods: We use a SIS-type compartmental mathematical model to describe the disease dynamics. We consider the population as a heterogeneous population which differ in disease transmission risk. Using best-fit techniques we compare the actual data with the model. For the fitted parameters we calculate the basic reproduction number and estimate the TB trends for the next few years applying several preventative protocols. Results and discussion: Using numerical simulations we examine the impact of ACF on the disease dynamics. We show that active screening among high risk subpopulations can help to reduce TB spread. We show how the reproduction number and estimated incidence decline depend on the detection rate. Conclusions: Active screening is one of the most effective ways for reducing the spread of disease. However, due to financial constraints, it can only be used to a limited extent. Properly applied detection can limit the spread of the disease while minimizing costs.

scholarly journals Partial Differential Equations of an Epidemic Model with Spatial Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
El Mehdi Lotfi ◽  
Mehdi Maziane ◽  
Khalid Hattaf ◽  
Noura Yousfi
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Reaction Diffusion ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

Dynamical Behavior of an Epidemic Model in a Fuzzy Transmission

2015 ◽  
Vol 23 (05) ◽  
pp. 651-665 ◽  
Author(s):  
Prasanta Kumar Mondal ◽  
Soovoojeet Jana ◽  
Palash Haldar ◽  
T. K. Kar
Keyword(s):  
Epidemic Model ◽  
Disease Transmission ◽  
Fuzzy Numbers ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Dynamical Behavior ◽  
Threshold Condition ◽  
Treatment Control ◽  
Fuzzy Expected Value ◽  
Treatment Function

In this paper, we have formulated a simple SIS type epidemic model in the presence of treatment control, and we have discussed the dynamical behavior of the system. The system is modified by considering both the disease transmission rate and the treatment function as fuzzy numbers, and also the fuzzy expected value of the infected individuals is calculated. Furthermore, the fuzzy basic reproduction number is investigated and a threshold condition of pathogen is obtained at which the system undergoes a transcritical bifurcation.

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