A two-thresholds policy to interrupt transmission of West Nile Virus to birds. (21st February 2019) - Zhou, Weike; Xiao, Yanni; Heffernan, Jane M
Highlights: Two thresholds are introduced in a piece-wise smooth model of WNV transmission to estimate whether and which control strategy (of bird- and mosquito- culling) should be implemented. The global stability of the model and all possible dynamic behaviors of the system are investigated theoretically. Model solutions ultimately approach a real endemic equilibrium or a pseudo-equilibrium, or a pseudo-attractor, depending on the chosen threshold levels of birds and mosquitoes. Both the numbers of infected birds and infected mosquitoes can be maintained simultaneously when threshold values are chosen properly. Abstract: 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 C I m, the dynamical behaviour of the proposed non-smooth system is investigated as the bird threshold level C I b 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.