Department of Applied Mathematics and Physics, Air Force Engineering University, Xi'an 710051, PR China.
J Theor Biol. 2011 Jul 7;280(1):108-16. doi: 10.1016/j.jtbi.2011.03.013. Epub 2011 Apr 6.
Vaccination is important for the control of some infectious diseases. This paper considers two SIR-SVS epidemic models with vaccination, where it is assumed that the vaccination for the newborns is continuous in the two models, and that the vaccination for the susceptible individuals is continuous and impulsive, respectively. The basic reproduction numbers of two models, determining whether the disease dies out or persists eventually, are all obtained. For the model with continuous vaccination for the susceptibles, the global stability is proved by using the Lyapunov function. Especially for the endemic equilibrium, to prove the negative definiteness of the derivative of the Lyapunov function for all the feasible values of parameters, it is expressed in three different forms for all the feasible values of parameters. For the model with pulse vaccination for the susceptibles, the global stability of the disease free periodic solution is proved by the comparison theorem of impulsive differential equations. At last, the effect of vaccination strategies on the control of the disease transmission is discussed, and two types of vaccination strategies for the susceptible individuals are also compared.
接种疫苗对于控制某些传染病非常重要。本文考虑了两种具有接种疫苗的 SIR-SVS 传染病模型，其中假设两种模型中的新生儿接种疫苗是连续的，而易感个体的接种疫苗是连续和脉冲的。两个模型的基本再生数，决定疾病是否最终消亡或持续存在，都得到了确定。对于具有连续接种疫苗的易感个体的模型，通过使用 Lyapunov 函数证明了全局稳定性。特别是对于地方病平衡点，为了证明 Lyapunov 函数导数对于所有可行参数值的负定性，对于所有可行参数值，将其表示为三种不同的形式。对于具有脉冲接种疫苗的易感个体的模型，通过脉冲微分方程的比较定理证明了无病周期解的全局稳定性。最后，讨论了接种策略对疾病传播控制的影响，并比较了两种易感个体的接种策略。
