This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model,denoted by SVIQR,with vaccination-age.Moreover,the considered model contains a relapse term and a gener...This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model,denoted by SVIQR,with vaccination-age.Moreover,the considered model contains a relapse term and a general form of the incidence function.This model is formulated to show the effect of sanitary isolation and vaccination on the disease prevalence when the relapse phenomena occurs.First,we show that the model is mathematically well posed by using the integrated semigroup theory.Second,we prove the existence of equilibrium points based on the basic reproduction number R_(0)^(ω,ξ).Moreover,by developing appropriate Lyapunov functionals and applying the LaSalles invariance principle,we established the global dynamics of the model,especially,we show that if R_(0)^(ω,ξ)≤1 the the disease-free equilibrium is globally asymptotically stable and,if Rω,ξ0>1 the endemic equilibrium is globally asymptotically stable.Finally,using the finite difference method,numerical simulations are made to reinforce the theoretical findings and to prove the contribution of the vaccination and the isolation on the disease disappearing.展开更多
Seasonality is repetitive in the ecological,biological and human systems.Seasonal factors affect both pathogen survival in the environment and host behavior.In this study,we considered a five-dimensional system of ord...Seasonality is repetitive in the ecological,biological and human systems.Seasonal factors affect both pathogen survival in the environment and host behavior.In this study,we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate.We started by studying the autonomous system by investigating the global stability of steady states.Later,we proved the existence,uniqueness,positivity and boundedness of a periodic orbit in a non-autonomous system.We demonstrate that the global dynamics are determined using the basic reproduction number R_(o) which is defined by the spectral radius of a linear integral operator.We showed that if R_(o)<1,then the disease-free periodic solution is globally asymptotically stable and if R_(o)>1,then the trajectories converge to a limit cycle reflecting the persistence of the disease.Finally,we present a numerical investigation that support our results.展开更多
In the paper, we study a kind of time-delayed novel coronavirus pneumonia dynamical model with vaccination. This model considers that people are vaccinated, and the human immune system has a series of processes, which...In the paper, we study a kind of time-delayed novel coronavirus pneumonia dynamical model with vaccination. This model considers that people are vaccinated, and the human immune system has a series of processes, which need a certain time. We first obtain the disease-free equilibrium and the basic reproduction number R<sub>0</sub>, and the system has a unique endemic equilibrium when R<sub>0</sub> > 1. Then we discuss the stability of the disease-free equilibrium and the endemic equilibrium with different delays τ. For τ = 0, using the Lyapunov function approach, we obtained the stability of disease-free equilibrium and the endemic equilibrium, respectively. For any delay τ ≠ 0, using the Routh-Hurwitz Criteria, we obtained that the disease-free equilibrium is locally asymptotically stable. We also find the critical value τ<sub>0</sub> at the endemic equilibrium, and obtain the condition that the system has a Hopf bifurcation at the endemic equilibrium. Finally, with the suitable choices of the parameters, some numerical simulations are presented in order to verify the effectiveness of the obtained theoretical results.展开更多
文摘This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model,denoted by SVIQR,with vaccination-age.Moreover,the considered model contains a relapse term and a general form of the incidence function.This model is formulated to show the effect of sanitary isolation and vaccination on the disease prevalence when the relapse phenomena occurs.First,we show that the model is mathematically well posed by using the integrated semigroup theory.Second,we prove the existence of equilibrium points based on the basic reproduction number R_(0)^(ω,ξ).Moreover,by developing appropriate Lyapunov functionals and applying the LaSalles invariance principle,we established the global dynamics of the model,especially,we show that if R_(0)^(ω,ξ)≤1 the the disease-free equilibrium is globally asymptotically stable and,if Rω,ξ0>1 the endemic equilibrium is globally asymptotically stable.Finally,using the finite difference method,numerical simulations are made to reinforce the theoretical findings and to prove the contribution of the vaccination and the isolation on the disease disappearing.
基金funded by the University of Jeddah,Jeddah,Saudi Arabia,under Grant No.(UJ-23-DR-279)。
文摘Seasonality is repetitive in the ecological,biological and human systems.Seasonal factors affect both pathogen survival in the environment and host behavior.In this study,we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate.We started by studying the autonomous system by investigating the global stability of steady states.Later,we proved the existence,uniqueness,positivity and boundedness of a periodic orbit in a non-autonomous system.We demonstrate that the global dynamics are determined using the basic reproduction number R_(o) which is defined by the spectral radius of a linear integral operator.We showed that if R_(o)<1,then the disease-free periodic solution is globally asymptotically stable and if R_(o)>1,then the trajectories converge to a limit cycle reflecting the persistence of the disease.Finally,we present a numerical investigation that support our results.
文摘In the paper, we study a kind of time-delayed novel coronavirus pneumonia dynamical model with vaccination. This model considers that people are vaccinated, and the human immune system has a series of processes, which need a certain time. We first obtain the disease-free equilibrium and the basic reproduction number R<sub>0</sub>, and the system has a unique endemic equilibrium when R<sub>0</sub> > 1. Then we discuss the stability of the disease-free equilibrium and the endemic equilibrium with different delays τ. For τ = 0, using the Lyapunov function approach, we obtained the stability of disease-free equilibrium and the endemic equilibrium, respectively. For any delay τ ≠ 0, using the Routh-Hurwitz Criteria, we obtained that the disease-free equilibrium is locally asymptotically stable. We also find the critical value τ<sub>0</sub> at the endemic equilibrium, and obtain the condition that the system has a Hopf bifurcation at the endemic equilibrium. Finally, with the suitable choices of the parameters, some numerical simulations are presented in order to verify the effectiveness of the obtained theoretical results.
