Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China.
Math Biosci Eng. 2013 Apr;10(2):445-61. doi: 10.3934/mbe.2013.10.445.
This paper proposes and analyzes a mathematical model on an infectious disease system with a piecewise smooth incidence rate concerning media/psychological effect. The proposed models extend the classic models with media coverage by including a piecewise smooth incidence rate to represent that the reduction factor because of media coverage depends on both the number of cases and the rate of changes in case number. On the basis of properties of Lambert W function the implicitly defined model has been converted into a piecewise smooth system with explicit definition, and the global dynamic behavior is theoretically examined. The disease-free is globally asymptotically stable when a certain threshold is less than unity, while the endemic equilibrium is globally asymptotically stable for otherwise. The media/psychological impact although does not affect the epidemic threshold, delays the epidemic peak and results in a lower size of outbreak (or equilibrium level of infected individuals).
本文提出并分析了一个具有分段光滑发生率的传染病系统的数学模型，该发生率与媒体/心理效应有关。所提出的模型通过包含分段光滑发生率来扩展经典模型，以表示由于媒体报道而减少的因素取决于病例数和病例数变化率。基于 Lambert W 函数的性质，隐式定义的模型已转换为具有显式定义的分段光滑系统，并从理论上检验了全局动态行为。当某个阈值小于 1 时，无病平衡点全局渐近稳定，否则地方病平衡点全局渐近稳定。尽管媒体/心理影响不会影响传染病的阈值，但它会延迟传染病的高峰期，并导致疫情规模较小（或感染个体的平衡水平）。
