scholarly journals Qualitative analysis of a diffusive SEIR epidemic model with linear external source and asymptomatic infection in heterogeneous environment

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Tian ◽  
Shangjiang Guo ◽  
Zhisu Liu
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
External Source ◽  
Asymptomatic Infection ◽  
Endemic Equilibrium ◽  
Heterogeneous Environment ◽  
Variable Recruitment ◽  
And Control ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>This paper is devoted to an SEIR epidemic model with variable recruitment and both exposed and infected populations having infectious in a spatially heterogeneous environment. The basic reproduction number is defined and the existence of endemic equilibrium is obtained, and the relationship between the basic reproduction number and diffusion coefficients is established. Then the global stability of the endemic equilibrium in a homogeneous environment is investigated. Finally, the asymptotic profiles of endemic equilibrium are discussed, when the diffusion rates of susceptible, exposed and infected individuals tend to zero or infinity. The theoretical results show that limiting the movement of exposed, infected and recovered individuals can eliminate the disease in low-risk sites, while the disease is still persistent in high-risk sites. Therefore, the presence of exposed individuals with infectious greatly increases the difficulty of disease prevention and control.</p>

scholarly journals Spatial dynamics of a diffusive SIRI model with distinct dispersal rates and heterogeneous environment

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lian Duan ◽  
Lihong Huang ◽  
Chuangxia Huang
Keyword(s):  
Steady State ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Spatial Dynamics ◽  
Heterogeneous Environment ◽  
Threshold Parameter ◽  
Dispersal Rate ◽  
Siri Model ◽  
The Basic Reproduction Number

<p style='text-indent:20px;'>In this paper, we are concerned with the dynamics of a diffusive SIRI epidemic model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected individuals. We first establish the basic properties of solutions to the model, and then identify the basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{R}_{0} $\end{document}</tex-math></inline-formula> which serves as a threshold parameter that predicts whether epidemics will persist or become globally extinct. Moreover, we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected individuals approaches zero. Our analytical results reveal that the epidemics can be extinct by limiting the movement of the susceptible individuals, and the infected individuals concentrate on certain points in some circumstances when limiting their mobility.</p>

scholarly journals APLIKASI MODEL EPIDEMIK SEIAR-SEI PADA PENYEBARAN PENYAKIT MALARIA DENGAN INFEKSI ASIMTOMATIK DAN SUPER INFEKSI

2018 ◽  
Vol 15 (2) ◽  
pp. 67
Author(s):  
Stella Maryana Belwawin
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Asymptomatic Infection ◽  
Endemic Equilibrium ◽  
Asymptomatic Infections ◽  
The Stability ◽  
Super Infection ◽  
The Basic Reproduction Number

AbstractThis aim of this study is to determine the point of equilibrium and analyze the stability of SEIAR-SEI model on malaria disease with asymptomatic infection, super infection and the effect of the mosquito's life cycle. This study also aim is to measure the sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes through the magnitude of . The method in this research is deductively, through several stage, such as  determination of disease-free equilibrium point and endemic equilibrium point, determination of basic reproduction number (), analyze of the basic reproduction number sensitivity of the spread of malaria to the parameters of asymptomatic infections, the rate of treatment, and the rate of birth of mosquitoes. The endemic equilibrium point was obtained using rule of Descartes. The result show that the change in the value of parameter , , and  has effect on the basic reproduction number (). Treatment factors in the human population influence the elimination of malaria in a population. Whereas asymptomatic infection factors and the birth rate of adult mosquitoes influence the increase in malaria infection. Keywords:  Malaria, asymptomatic infection, super infection, basic reproduction number, rule of descrates. AbstrakPenelitian ini bertujuan menentukan titik keseimbangan dan menganalisis kestabilan dari model SEIAR_SEI pada penyakit malaria dengan pengaruh infeksi asimtomatik, super infeksi, dan siklus hidup nyamuk. Penelitian ini juga bertujuan mengukur tingkat sensitivitas penyebaran penyakit malaria terhadap parameter infeksi asimtomatik, laju pengobatan, serta laju kelahiran nyamuk.melalu besaran .  Metode yang digunakan dalam penelitian ini adalah metode deduktif dengan langkah-langkah : menentukan titik keseimbangan bebas penyakit dan endemik dan menentukan bilangan reproduksi dasar ). Analisis sensitivitas bilangan reproduksi dasar dilakukan terhadap parameter infeksi asimtomatik, pengobatan, dan laju kelahiran nyamuk. Tititk keseimbangan endemik diperoleh dengan aturan descrates. Hasil yang diperoleh menunjukkan parameter , , dan  berpengaruh terhadap bilangan reproduksi dasar (). Faktor pengobatan berpengaruh terhadap eliminasi penyakit malaria. Sedangkan faktor infeksi asimtomatik dan laju kelahiran nyamuk dewasa berpengaruh terhadap peningkatan infeksi penyakit malaria. Kata kunci: Malaria, Infeksi Asimtomatik, Super Infeksi, Bilangan Reproduksi Dasar, Aturan Descrates . 

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 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.

Global stability of an SEIR epidemic model with vaccination

2016 ◽  
Vol 09 (06) ◽  
pp. 1650082 ◽  
Author(s):  
Lili Wang ◽  
Rui Xu
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Globally Asymptotically Stable ◽  
Compound Matrices ◽  
Comparison Arguments ◽  
The Basic Reproduction Number

In this paper, an SEIR epidemic model with vaccination is formulated. The results of our mathematical analysis indicate that the basic reproduction number plays an important role in studying the dynamics of the system. If the basic reproduction number is less than unity, it is shown that the disease-free equilibrium is globally asymptotically stable by comparison arguments. If it is greater than unity, the system is permanent and there is a unique endemic equilibrium. In this case, sufficient conditions are established to guarantee the global stability of the endemic equilibrium by the theory of the compound matrices. Numerical simulations are presented to illustrate the main results.

scholarly journals Analysis of an SQEIAR epidemic model with media coverage and asymptomatic infection

2021 ◽  
Vol 6 (11) ◽  
pp. 12298-12320
Author(s):  
Xiangyun Shi ◽  
◽  
Xiwen Gao ◽  
Xueyong Zhou ◽  
Yongfeng Li ◽  
...  
Keyword(s):  
Maximum Principle ◽  
Cost Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Media Coverage ◽  
Reproduction Number ◽  
Asymptomatic Infection ◽  
Quadratic Cost ◽  
Quadratic Cost Function ◽  
The Basic Reproduction Number

<abstract><p>An SQEIAR model with media coverage and asymptomatic infection is proposed for populations with a certain level of immunity. Firstly, we discuss the extinction and persistence for the diseases of the model by using basic reproduction number $ \mathcal{R}_C $. Then the parameter threshold is analyzed and the effect of parameters on the basic reproduction number is discussed. Furthermore, the optimal media coverage strategy and quarantine strategy for optimal problems under quadratic cost function are derived by applying Pontryagin's Maximum Principle.</p></abstract>

Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate

2016 ◽  
Vol 09 (05) ◽  
pp. 1650068 ◽  
Author(s):  
Muhammad Altaf Khan ◽  
Yasir Khan ◽  
Sehra Khan ◽  
Saeed Islam
Keyword(s):  
Global Stability ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free ◽  
The Basic Reproduction Number

This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.

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.

Analysis of a delayed epidemic model with non-monotonic incidence rate and vertical transmission

2014 ◽  
Vol 07 (04) ◽  
pp. 1450041
Author(s):  
Jinhu Xu ◽  
Wenxiong Xu ◽  
Yicang Zhou
Keyword(s):  
Vertical Transmission ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sufficient Conditions ◽  
Endemic Equilibrium ◽  
Globally Asymptotically Stable ◽  
Comparison Arguments ◽  
Disease Free ◽  
The Basic Reproduction Number

A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R0 < 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R0 > 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R0 > 1.

scholarly journals On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. De la Sen ◽  
A. Ibeas
Keyword(s):  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Antiviral Treatment ◽  
Stability Properties ◽  
Local Asymptotic Stability ◽  
Proposed Model ◽  
And Control ◽  
The Basic Reproduction Number

AbstractIn this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

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