Brauer Fred, van den Driessche P, Wang Lin
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V8N 3R4.
Math Biosci. 2008 Sep;215(1):1-10. doi: 10.1016/j.mbs.2008.05.001. Epub 2008 May 16.
For a single patch SIRS model with a period of immunity of fixed length, recruitment-death demographics, disease related deaths and mass action incidence, the basic reproduction number R(0) is identified. It is shown that the disease-free equilibrium is globally asymptotically stable if R(0)<1. For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation analysis about this equilibrium are carried out. Moreover, a practical numerical approach to locate the bifurcation values for a characteristic equation with delay-dependent coefficients is provided. For a two patch SIRS model with travel, it is shown that there are several threshold quantities determining its dynamic behavior and that travel can reduce oscillations in both patches; travel may enhance oscillations in both patches; or travel can switch oscillations from one patch to another.
对于具有固定长度免疫期、招募-死亡人口统计学特征、疾病相关死亡和质量作用发生率的单斑块SIRS模型，确定了基本再生数R(0)。结果表明，如果R(0)<1，则无病平衡点是全局渐近稳定的。对于R(0)>1的情况，进行了地方病平衡点的局部稳定性分析以及关于该平衡点的霍普夫分岔分析。此外，还提供了一种实用的数值方法来确定具有延迟相关系数的特征方程的分岔值。对于具有迁移的双斑块SIRS模型，结果表明有几个阈值量决定其动态行为，并且迁移可以减少两个斑块中的振荡；迁移可能增强两个斑块中的振荡；或者迁移可以将振荡从一个斑块切换到另一个斑块。
