Research On A Class Of Epidemic Model With The Generalized Nonlinear Vaccination Rate And The Suboptimal Immunity

The density of infected persons has an effect on the vaccination rate of susceptible persons.Motivated by this,a generalized nonlinear vaccination function is established based on the continuous vaccination model,and furthermore,two type of epidemic models are proposed:(1)an epidemic model with a generalized nonlinear vaccination rate and the temporary immunity,(2)an epidemic model with a generalized nonlinear vaccination rate and the partial immunity.Firstly,the dynamic behaviors of the epidemic model with the temporary immunity is studied,and the corresponding theoretical analysis results are obtained,such as(1)the well-posedness of solutions of the considered model is obtained by verifying the nonnegativity and boundedness of solutions of the corresponding model under the initial conditions.(2)the effective reproduction number Rv is calculated.Moreover,the threshold dynamics of the considered model with the effective reproduction number Rv as the controlling condition are obtained.The results showed that the disease-free equilibrium uniquely exists and the disease-free equilibrium of the considered model is locally asymptotically stable when the effective reproduction number Rv<1.Furthermore,the globally asymptotic stability of the disease-free equilibrium is proved by constructing the suitable Lyapunov function.Otherwise,the endemic equilibrium is locally asymptotically stable when the effective reproduction number Rv>1.Moreover,the globally asymptotic stability of the endemic equilibrium is proved by constructing the suitable Lyapunov function.(3)The numerical simulations are conducted to verify the reliability of the corresponding theoretical conclusions.Meanwhile,based on the local sensitivity analysis of Rv to the model parameters and global sensitivity analysis of Rv,the impacts of the critical parameters,such as the generalized nonlinear continuous vaccination rate coefficient ? and the immune loss rate ?,on the disease prevention and control are revealed,and furthermore the measures for the disease prevention and control are proposed.Secondly,the dynamic behaviors of the epidemic model with the partial immunity is investigated,and the existence and well-posedness of the model solutions are obtained under the initial conditions.The effective reproduction number Rv is computed.It is shown that disease-free equilibrium is locally asymptotically stable when Rv<1.Under some certain conditions,the global asymptotic stability of disease-free equilibrium is obtained by applying the upper and lower limit method and comparison principle.Otherwise,if Rv>1,the endemic equilibrium is locally asymptotically stable.Finally,the theoretical results are verified by numerical simulations.The global asymptotic stability of the endemic equilibrium point E*is observed from the sequence diagram.Furthermore,based on the local sensitivity analysis of Rv to the model parameters and global sensitivity analysis of Rv,the sequence of the importance of some parameters which impact the dynamical behaviors is defined,and then the effective measures of epidemic prevention and control are proposed.