Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, PR China.
Department of Mathematics & Statistics, York University, Toronto, ON, M3J 1P3, Canada.
J Theor Biol. 2019 Feb 21;463:22-46. doi: 10.1016/j.jtbi.2018.12.013. Epub 2018 Dec 12.
This paper proposes a model of West Nile Virus (WNV) including threshold control policies concerning the culling of mosquitoes and birds under different conditions. Two thresholds are introduced to estimate whether and which control strategy should be implemented. For each mosquito threshold level [Formula: see text] the dynamical behaviour of the proposed non-smooth system is investigated as the bird threshold level [Formula: see text] varies, focusing on the existence of sliding domains, the existence of pseudo-equilibria, real or virtual of the endemic equilibria, global stability of these steady states, and the most interesting case of the occurrence of a novel globally asymptotically stable pseudo-attractor. The model solutions ultimately converge to a real equilibrium or a pseudo-equilibrium (if it exists), or a pseudo-attractor if no equilibrium is real and no pseudo-equilibrium exists. Here within, we show that the free system has a single stable endemic equilibrium under biologically reasonable assumptions, and show that when the control system has: (1) a bird-culling threshold that is above the bird equilibrium, culling has no advantage; (2) a bird-culling threshold that is below the bird equilibrium, but a mosquito-culling threshold that lies above the mosquito equilibrium, the infected bird population can be reduced but the infected mosquito population will remain the same; (3) a bird-culling threshold and a mosquito-culling threshold that both lie below their respective equilibrium values of the free system, then both the infected bird and mosquito populations can be reduced to lower levels. The results suggest that preset levels of the number of infected birds and infected mosquitoes can be maintained simultaneously when threshold values are chosen properly, which provides a possible control strategy when an emergent infectious disease cannot be eradicated immediately.
本文提出了一个西尼罗河病毒（WNV）模型，包括在不同条件下针对蚊子和鸟类捕杀的阈值控制策略。引入了两个阈值来估计是否以及应实施哪种控制策略。对于每个蚊子阈值 [Formula: see text]，当鸟类阈值 [Formula: see text] 变化时，研究所提出的非光滑系统的动态行为，重点关注滑动域的存在、伪平衡点的存在、地方平衡点的真实或虚拟、这些稳定状态的全局稳定性，以及发生新颖的全局渐近稳定伪吸引子的最有趣情况。模型解最终收敛到真实平衡或伪平衡（如果存在），或者不存在真实平衡且不存在伪平衡时的伪吸引子。在这里，我们表明，在合理的生物学假设下，自由系统只有一个稳定的地方平衡点，并表明当控制系统具有：（1）高于鸟类平衡点的鸟类捕杀阈值时，捕杀没有优势；（2）低于鸟类平衡点但高于蚊子平衡点的鸟类捕杀阈值时，感染鸟类的数量可以减少，但感染蚊子的数量将保持不变；（3）鸟类捕杀阈值和蚊子捕杀阈值都低于自由系统各自的平衡点时，那么感染鸟类和蚊子的数量都可以减少到较低水平。结果表明，当选择适当的阈值时，可以同时维持感染鸟类和感染蚊子的预设数量，这为无法立即根除突发传染病时提供了一种可能的控制策略。
