COVID-19 is a public health emergency for human beings and brings some very harmful consequences in social and economic fields. In order to model COVID-19 and develop the effective control measures, this paper proposes an SEIR-type epidemic model with the contacting distance between the healthy individuals and the asymptomatic or symptomatic infected individuals, and the immigration rate of the healthy individuals since the contacting distance and the immigration rate are two critical factors which determine the transmission of COVID-19. Firstly, the threshold values of the contacting distance and the immigration rate are obtained to analyze the presented model. Secondly, based on the data from January 10, 2020, to March 18, 2020, for Wuhan city, all parameters are estimated. Finally, based on the estimated parameters, the sensitivity analysis and the numerical study are conducted. The results show that the contacting distance and the immigration rate play an important role in controlling COVID-19. Meanwhile, the extinct lag decreases as the contacting distance increases and/or the immigration rate decreases. Our study could give some reasonable suggestions for the health officials and the public and provide a theoretical issue for globally controlling the COVID-19 pandemic.
To model the COVID-19 infection and develop effective control measures, this paper proposes an SEIR-type epidemic model considering the impact of face-mask wearing and vaccination. Firstly, the effective reproduction number and the threshold conditions are obtained. Secondly, based on the data of South Korea from January 20, 2022 to March 21, 2022, the model parameters are estimated. Finally, a sensitivity analysis and the numerical study are conducted. The results show that the face-mask wearing is associated with $$83\%$$ 83 % and $$90\%$$ 90 % reductions in the numbers of cumulative cases and newly confirmed cases, respectively, after a period of 60 days, when the face mask wearing rate increases by $$15\%$$ 15 % . Furthermore, the vaccination rate is associated with $$75\%$$ 75 % and $$80\%$$ 80 % reductions in the numbers of cumulative cases and the newly confirmed cases, respectively, after the same period of 60 days when the vaccination rate is increased by $$15\%$$ 15 % . A combined measure involving face-mask wearing and vaccination may be more effective and reasonable in preventing and controlling this infection. It is also suggested that disease control departments should strongly recommended the wearing of face masks s as well as vaccination to prevent the unvaccinated people from becoming infected.
In this paper, we introduce Allee effect and predator competition in the Bazykin’s model with Holling I functional response. Theoretically, we analyze the existence and stability of equilibria, and derive the existence conditions of saddle-node bifurcation and Hopf bifurcation. In addition, in order to determine the stability of limit cycles, we explicitly calculate the first Lyapunov coefficient and prove that the positive equilibrium is not a center, but a weak focus with a multiplicity of at least two. Therefore, the system has Hopf bifurcation and Bautin bifurcation with two limit cycles. Our results indicate that the Allee effect and predator competition lead to a series of complex dynamic phenomena. Finally, numerical simulation verifies the effectiveness of the theoretical results.
<abstract> <p>In this paper, an SIR model with a strong Allee effect and density-dependent transmission is proposed, and its characteristic dynamics are investigated. The elementary mathematical characteristic of the model is studied, including positivity, boundedness and the existence of equilibrium. The local asymptotic stability of the equilibrium points is analyzed using linear stability analysis. Our results indicate that the asymptotic dynamics of the model are not only determined using the basic reproduction number ${R_0}$. If ${R_0} < 1$, there are three disease-free equilibrium points, and a disease-free equilibrium is always stable. At the same time, the conditions for other disease-free equilibrium points to be bistable were determined. If ${R_0} > 1$ and in certain conditions, either an endemic equilibrium emerges and is locally asymptotically stable, or the endemic equilibrium becomes unstable. What must be emphasized is that there is a locally asymptotically stable limit cycle when the latter happens. The Hopf bifurcation of the model is also discussed using topological normal forms. The stable limit cycle can be interpreted in a biological significance as a recurrence of the disease. Numerical simulations are used to verify the theoretical analysis. Taking into account both density-dependent transmission of infectious diseases and the Allee effect, the dynamic behavior becomes more interesting than when considering only one of them in the model. The Allee effect makes the SIR epidemic model bistable, which also makes the disappearance of diseases possible, since the disease-free equilibrium in the model is locally asymptotically stable. At the same time, persistent oscillations due to the synergistic effect of density-dependent transmission and the Allee effect may explain the recurrence and disappearance of disease.</p> </abstract>
Vaccination is an effective way to prevent the spread of infectious diseases. In this study, we formulate a VSEIR mathematical model to explore the effects of vaccination rate, vaccine efficacy, and immune decline on the COVID-19 transmission. The existence and stability criteria of equilibrium states were determined by analyzing the model. Model analysis was performed. One of the interesting phenomena involved in this issue is that diseases may or may not die out when the basic reproduction number falls below unity (i.e., a backward bifurcation may exist and cause multistability). The disease eventually becomes endemic in the population when the basic reproduction number exceeds one. By comparing different vaccination rates, vaccine efficacy, and infection rate factors, the diseases can be eliminated, not only by vaccines but also by strict protective measures. In addition, we used the COVID-19 number of reported cases in Xiamen in September 2021 to fit the model, and the model and the reported data were well matched.
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