具有Logistic增长和治疗的SIRS传染病模型的后向分支

利用传染病动力学理论,建立一类具有Logistic增长和治疗的SIRS传染病模型,并假设模型中总人口增长满足Logistic方程,考虑治疗项为非线性的可微函数,以描述有限的医疗资源对延迟患者治疗的影响.综合运用微分方程稳定性理论中的Routh-Hurwitz判据、LaSalle不变集原理并构造适当的Lyapunov函数方法,获得了该模型的无病平衡点和地方病平衡点全局渐近稳定的代数判据及后项分支存在条件.结果表明,当延迟患者治疗的影响较严重时,该模型会出现后向分支现象,基本再生数已不再是疾病是否消亡的阀值,需要加强医疗资源建设,通过提供及时优质的医疗服务,才能有效控制或根除疾病.

Backward Bifurcation of a SIRS Epidemic Model with Logistic Growth and Treatment

Based on the epidemic dynamic theory, a SIRS epidemic model with logistic growth and treatment was established. On the basis of the growth of the total population satisfying Logistic equation and nolinear differentiable function being the treatment term, the model describes the effect of limited medical resources to delaying treatment of the sick. Based on the Routh Hurwitz criterion, LaSalle invariant set principle and structured proper Lyapunov function method of stability theory in the differential equation, global asymptoticstable algebraic criterion of disease free equilibrium and endemic equilibrium
and backward bifurcation conditions of the model were obtained. The research results show when the delayed effect is serious, backward bifurcation will take place and the basic reproduction number is not the threshold value of the eradicating of the disease. In order to eradicating the disease, medical resources construction should be strengthened, and prompt and superior medical service should be provided.