Wang Lulu, Jin Zhen, Wang Hao
Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, China.
Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, 030006, Shanxi, China.
J Math Biol. 2018 Oct;77(4):1093-1115. doi: 10.1007/s00285-018-1245-7. Epub 2018 May 9.
To study the effects of an environmental toxin, such as fine particles in hazy weather, on the spread of infectious diseases, we derive a toxin-dependent dynamic model that incorporates the birth rate with the toxin-dependent switching mode, the mortality rate, and infection rate with the toxin-dependent saturation effect. We analyze the model by showing the positive invariance, existence and stability of equilibria, and bifurcations. Numerical simulation is adopted to verify the mathematical results and exhibit transcritical and Hopf bifurcations. Our theoretical results show that there exists a threshold value of the environmental toxin: if the environmental toxin concentration is lower than the threshold, the system has a disease-free equilibrium and an interior equilibrium; if the environmental toxin concentration is higher than the threshold, the system has the extinction equilibrium. For the case where the disease-induced death is ignored, we show the global stability results. Numerical simulations clearly show that the environmental toxin facilitates the spread of infectious diseases. This study provides a theoretical basis for uncovering the impact of toxins on the spread of infectious diseases and for guiding the decision making by disease control agencies and governments.
为研究环境毒素（如雾霾天气中的细颗粒物）对传染病传播的影响，我们推导了一个依赖毒素的动态模型，该模型纳入了具有毒素依赖切换模式的出生率、死亡率以及具有毒素依赖饱和效应的感染率。我们通过展示模型的正不变性、平衡点的存在性与稳定性以及分岔情况来分析该模型。采用数值模拟来验证数学结果并展示跨临界分岔和霍普夫分岔。我们的理论结果表明，存在一个环境毒素的阈值：如果环境毒素浓度低于该阈值，系统存在无病平衡点和内部平衡点；如果环境毒素浓度高于该阈值，系统存在灭绝平衡点。对于忽略疾病导致死亡的情况，我们给出了全局稳定性结果。数值模拟清楚地表明，环境毒素促进了传染病的传播。本研究为揭示毒素对传染病传播的影响以及为疾病控制机构和政府的决策提供理论依据。
