Modeling avian influenza using Filippov systems to determine culling of infected birds and quarantine

The growing number of reported avian influenza cases has prompted awareness of the effectiveness of pharmaceutical or/and non-pharmaceutical interventions that aim to suppress the transmission rate. We propose two Filippov models with threshold policy: the avian-only model with culling of infected birds and the SIIR (Susceptible–Infected–Infected–Recovered) model with quarantine. The dynamical systems of these two models are governed by nonlinear ordinary differential equations with discontinuous right-hand sides. The solutions of these two models will converge to either one of the two endemic equilibria or the sliding equilibrium on the discontinuous surface. We prove that the avian-only model achieves global stability. Moreover, by choosing an appropriate quarantine threshold level IcIc in the SIIR model, this model converges to an equilibrium in the region below IcIc or a sliding equilibrium, suggesting the outbreak can be controlled. Therefore a well-defined threshold policy is important for us to combat the influenza outbreak efficiently.