scholarly journals Stability, Bifurcation, and a Pair of Conserved Quantities in a Simple Epidemic System with Reinfection for the Spread of Diseases Caused by Coronaviruses

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Jorge Fernando Camacho ◽  
Cruz Vargas-De-León
Keyword(s):  
Numerical Solutions ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Nonlinear Ordinary Differential Equations ◽  
Conserved Quantities ◽  
Asymptotically Stable ◽  
Infinite Line ◽  
Disease Free ◽  
Significant Mortality

In this paper, we study a modified SIRI model without vital dynamics, based on a system of nonlinear ordinary differential equations, for epidemics that exhibit partial immunity after infection, reinfection, and disease-induced death. This model can be applied to study epidemics caused by SARS-CoV, MERS-CoV, and SARS-CoV-2 coronaviruses, since there is the possibility that, in diseases caused by these pathogens, individuals recovered from the infection have a decrease in their immunity and can be reinfected. On the other hand, it is known that, in populations infected by these coronaviruses, individuals with comorbidities or older people have significant mortality rates or deaths induced by the disease. By means of qualitative methods, we prove that such system has an endemic equilibrium and an infinite line of nonhyperbolic disease-free equilibria, we determine the local and global stability of these equilibria, and we also show that it has no periodic orbits. Furthermore, we calculate the basic reproductive number R 0 and find that the system exhibits a forward bifurcation: disease-free equilibria are stable when R 0 < 1 / σ and unstable when R 0 > 1 / σ , while the endemic equilibrium consist of an asymptotically stable upper branch that appears from R 0 > 1 / σ , σ being the rate that quantifies reinfection. We also show that this system has two conserved quantities. Additionally, we show some of the most representative numerical solutions of this system.

scholarly journals Dynamical Behavior of a New Epidemiological Model

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zizi Wang ◽  
Zhiming Guo
Keyword(s):  
Time Delay ◽  
Dynamical Behavior ◽  
Endemic Equilibrium ◽  
Epidemiological Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

A new epidemiological model is introduced with nonlinear incidence, in which the infected disease may lose infectiousness and then evolves to a chronic noninfectious disease when the infected disease has not been cured for a certain timeτ. The existence, uniqueness, and stability of the disease-free equilibrium and endemic equilibrium are discussed. The basic reproductive numberR0is given. The model is studied in two cases: with and without time delay. For the model without time delay, the disease-free equilibrium is globally asymptotically stable provided thatR0≤1; ifR0>1, then there exists a unique endemic equilibrium, and it is globally asymptotically stable. For the model with time delay, a sufficient condition is given to ensure that the disease-free equilibrium is locally asymptotically stable. Hopf bifurcation in endemic equilibrium with respect to the timeτis also addressed.

GLOBAL STABILITY OF AN HIV/AIDS MODEL WITH STOCHASTIC PERTURBATION

2011 ◽  
Vol 04 (02) ◽  
pp. 349-358 ◽  
Author(s):  
Junyuan Yang ◽  
Xiaoyan Wang ◽  
Xuezhi Li
Keyword(s):  
Stochastic Perturbation ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Ode Model ◽  
Globally Asymptotically Stable ◽  
Stochastic Perturbations ◽  
Hiv Model ◽  
Disease Free

In this paper, we investigate the dynamic behavior of an HIV model with stochastic perturbation. Firstly, in ODE model, the disease-free equilibrium E0 is globally asymptotically stable if the basic reproductive number R0 < 1. When R0 > 1, the endemic equilibrium E* is globally asymptotically stable. Secondly, the criterion for robustness of the system is established under stochastic perturbations. The conditions of stochastic stability of the endemic equilibrium E* are obtained. Finally, we simulate our analytical results.

scholarly journals Modelling and Simulating a Transmission of COVID-19 Disease: Niger Republic Case

2020 ◽  
Vol 13 (3) ◽  
pp. 549-566
Author(s):  
Abba Mahamane Oumarou ◽  
Saley Bisso
Keyword(s):  
Matrix Method ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Next Generation Matrix ◽  
The Impact ◽  
Disease Free

This paper focuses on the dynamics of spreads of a coronavirus disease (Covid-19).Through this paper, we study the impact of a contact rate in the transmission of the disease. We determine the basic reproductive number R0, by using the next generation matrix method. We also determine the Disease Free Equilibrium and Endemic Equilibrium points of our model. We prove that the Disease Free Equilibrium is asymptotically stable if R0 < 1 and unstable if R0 > 1. The asymptotical stability of Endemic Equilibrium is also establish. Numerical simulations are made to show the impact of contact rate in the spread of disease.

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.

A MODEL FOR VECTOR TRANSMITTED DISEASES WITH SATURATION INCIDENCE

2001 ◽  
Vol 09 (04) ◽  
pp. 235-245 ◽  
Author(s):  
LOURDES ESTEVA ◽  
MARIANO MATIAS
Keyword(s):  
Endemic Equilibrium ◽  
Saturation Effect ◽  
Basic Reproductive Number ◽  
Infected Host ◽  
Reproductive Number ◽  
The Stability ◽  
Stability Results ◽  
Disease Free

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R0.

scholarly journals Mathematical Analysis A Continuous Vaccination Strategy Effect against to spread of Measles by Using the Epidemic SVIR Models

2017 ◽  
Author(s):  
Tonaas Kabul Wangkok Yohanis Marentek
Keyword(s):  
Fixed Point ◽  
Vaccination Strategy ◽  
Fixed Time ◽  
Basic Reproductive Number ◽  
Necessary Condition ◽  
Reproductive Number ◽  
Time Interval ◽  
Asymptotically Stable ◽  
Fixed Time Interval ◽  
Disease Free

Vaccination is one way to minimize the spread of disease. To complete a vaccination, it is usually done several times and there should be a fixed time interval. Considering vaccination in the basic SIR model, SVIR model assumes that individuals are vaccinated do not get immediate immunity means that individuals who are vaccinated still allow infected. So according to the process of vaccination on SVIR model, there are two strategies which continuous vaccination strategy (CVS) and disconnected vaccination strategy (PVS). In this study only addressed continuous vaccination strategy in epidemic model SVIR. Results from the study indicate that the dynamics of the CVS system is fully dependent on the basic reproductive number. If the basic reproductive number is less than one then the fixed point asymptotically stable disease-free will which means that eventually the disease will disappear from the population. Conversely, if more than one fixed point is asymptotically stable endemic would mean that the disease will still exist in the population. Mathematical results show that vaccination helps to minimize the spread of disease by reducing the basic reproductive number. But there is a necessary condition for the disease can be eradicated. If the time for the vaccine recipients to obtain immunity or the possibility of vaccine recipients infected neglected, the condition of the disease will disappear and the disease will always be eradicated. This can lead to over-evaluating the effect of vaccination.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0&lt;1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0&gt;1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

MODELING THE SCHISTOSOMIASIS ON THE ISLETS IN NANJING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250037 ◽  
Author(s):  
LONGXING QI ◽  
JING-AN CUI ◽  
YUAN GAO ◽  
HUAIPING ZHU
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Field Study ◽  
Rattus Norvegicus ◽  
Compartmental Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Parasitic Diseases ◽  
Disease Free

A compartmental model is established for schistosomiasis infection in Qianzhou and Zimuzhou, two islets in the center of Yangtzi River near Nanjing, P. R. China. The model consists of five differential equations about the susceptible and infected subpopulations of mammalian Rattus norvegicus and Oncomelania snails. We calculate the basic reproductive number R0 and discuss the global stability of the disease free equilibrium and the unique endemic equilibrium when it exists. The dynamics of the model can be characterized in terms of the basic reproductive number. The parameters in the model are estimated based on the data from the field study of the Nanjing Institute of Parasitic Diseases. Our analysis shows that in a natural isolated area where schistosomiasis is endemic, killing snails is more effective than killing Rattus norvegicus for the control of schistosomiasis.

Analysis of a sex-structured HIV/AIDS model with the effect of screening of infectives

2014 ◽  
Vol 07 (05) ◽  
pp. 1450054 ◽  
Author(s):  
S. Athithan ◽  
Mini Ghosh
Keyword(s):  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Epidemic Threshold ◽  
Heterosexual Contact ◽  
Globally Asymptotically Stable ◽  
Transmission Of Disease ◽  
Globally Stable ◽  
Disease Free ◽  
Hiv Aids

This paper presents a nonlinear sex-structured mathematical model to study the spread of HIV/AIDS by considering transmission of disease by heterosexual contact. The epidemic threshold and equilibria for the model are determined, local stability and global stability of both the "Disease-Free Equilibrium" (DFE) and "Endemic Equilibrium" (EE) are discussed in detail. The DFE is shown to be locally and globally stable when the basic reproductive number ℛ0 is less than unity. We also prove that the EE is locally and globally asymptotically stable under some conditions. Finally, numerical simulations are reported to support the analytical findings.

scholarly journals Analisis Kestabilan Model SEIR Penyebaran Penyakit Campak dengan Pengaruh Imunisasi dan Vaksin MR

2019 ◽  
Vol 16 (1) ◽  
pp. 107
Author(s):  
Willyam Daniel Sihotang ◽  
Ceria Clara Simbolon ◽  
July Hartiny ◽  
Desrinawati Tindaon ◽  
Lasker Pangarapan Sinaga
Keyword(s):  
Infectious Disease ◽  
Equilibrium Point ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Endemic Equilibrium ◽  
Asymptotically Stable ◽  
Seir Model ◽  
Rate Of Spread ◽  
The Stability ◽  
Disease Free

Measles is a contagious infectious disease caused by a virus and has the potential to cause an outbreak. Immunization and vaccination are carried out as an effort to prevent the spread of measles. This study aims to analyze and determine the stability of the SEIR model on the spread of measles with the influence of immunization and MR vaccines. The results obtained from model analysis, namely there are two disease free and endemic equilibrium points. If the conditions are met, the measles-free equilibrium point will be asymptotically stable and the measles endemic equilibrium point will be stable. Numerical solutions show a decrease in the rate of spread of measles due to the effect of immunization and the addition of MR vaccines.

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