Global Dynamics and Rich Sliding Motion in an Avian-Only Filippov System in Combating Avian Influenza

Culling birds has always been an effective method to control the spread of avian influenza. Here, we introduce a Filippov avian-only model with culling of both susceptible and infected birds. The Filippov-type model is formulated by considering that no control strategy is taken if the number of infected birds is less than an infected threshold level [Formula: see text]; further, we cull infected birds once the number of infected birds exceeds [Formula: see text]; meanwhile, we cull susceptible birds if the number of susceptible birds exceeds a susceptible threshold level [Formula: see text]. The global dynamical behavior of the Filippov system, including the existence and stability of various types of equilibria, the existence of the sliding mode and its dynamics, together with bifurcation analyses with regard to local sliding bifurcations, is investigated. It is shown that model solutions ultimately converge to the positive equilibrium that lies in the region above [Formula: see text], or below [Formula: see text], or on [Formula: see text], as we vary the susceptible and infected threshold values [Formula: see text] and [Formula: see text]. Our results indicate that proper combinations of the susceptible and infected threshold values based on the threshold policy can maintain the number of infected birds either below a certain threshold level or at a previously given level.