Global stability of an SEIR epidemic model with vaccination

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.

EXISTENCE AND UNIQUENESS OF ENDEMIC STATES FOR THE AGE-STRUCTURED SEIR EPIDEMIC MODEL

An age-structured SEIR epidemic model of a vertically as well as horizontally transmitted disease is investigated. Threshold results for the existence of endemic states are established for most cases. Under certain conditions, uniqueness is also shown. Threshold used are explicitly computable in term of demographic and epiderniological parameters of the model.

Backward bifurcation,basic reinfection number and robustness of an SEIRE epidemic model with reinfection

Recent evidences show that individuals who recovered from COVID-19 can be reinfected.However,this phenomenon has rarely been studied using mathematical models.In this paper,we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection.We obtain the important thresholds R_(0)(the basic reproduction number)and R_(c)(a threshold less than one).Our investigations show that when R_(0)>1,the system has an endemic equilibrium,which is globally asymptotically stable.When R_(c)<R_(0)<1,the epidemic system exhibits bistable dynamics.That is,the system has backward bifurcation and the disease cannot be eradicated.In order to eradicate the disease,we must ensure that the basic reproduction number R_(0) is less than R_(c).The basic reinfection number is obtained to measure the reinfection force,which turns out to be a new tipping point for disease dynamics.We also give definition of robustness,a new concept to measure the dificulty of completely eliminating the disease for a bistable epidemic system.Numerical simulations are carried out to verify the conclusions.

AN SEIRS EPIDEMIC MODEL WITH TWO DELAYS AND PULSE VACCINATION

Pulse vaccination is an effective and important strategy to eradicate an infectious disease. The authors investigate an SEIRS epidemic model with two delays and pulse vaccination. By using the discrete dynamical system determined by stroboscopic map, the authors obtain that the infectious population dies out if R△ 〈 1, and the infectious population is uniformly persistent if R^△ 〉 1. The results indicate that a short period of pulse vaccination or a large pulse vaccination rate is a sufficient condition to eradicate the disease.

Analysis of an Age Structured SEIRS Epidemic Model with Varying Total Population Size and Vaccination

This article focuses on the study of an age structured SEIRS epidemic model with a vaccination program when the total population size is not kept at constant. We first give the explicit expression of the reproduction number in the presence of vaccine ( is the exponent of growth of total population), and show that the infection-free steady state is linearly stable if and unstable if , then we apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.

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

A delayed SEIR epidemic model with vertical transmission and non-inonotonic 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 equilib- rium and endemic equilibrium is established by analyzing the corresponding character- istic equations. By comparison arguments, it is proved that, if Ro 〈 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 repro- duction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R0 〉 1.

Evaluation of the effect of the state of emergency for the first wave of COVID-19 in Japan

In this paper,we evaluate the effect of the state of emergency for the first wave of COVID-19 in Japan,2020 from the viewpoint of mathematical modelling.In Japan,it was announced during the period of the state of emergency from April 7 to May 25,2020 that the 80%reduction of the contact rate is needed to control the outbreak.By numerical simulation,we show that the reduction rate seems to have reached up to 86%.Moreover,we estimate the control reproduction number R c during the period of the state of emergency as R c?0:36(95%CI,0.34e0.39),and show that the effective reproduction number R e after the lifting of the state of emergency could be greater than 1.This result suggests us that the second wave of COVID-19 in Japan could possibly occur if any effective intervention will not be taken again.