An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza

Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level $$S_b$$Sb, we derive the phase portrait for the dynamical system as we vary the infected threshold level $$I_b$$Ib, focusing on the existence of endemic states; the endemic states are represented by real equilibria, pseudoequilibria and pseudo-attractors. We show generically that all solutions of this model will approach one of the endemic states. Our results suggest that the spread of avian influenza in bird populations is tolerable if the trajectories converge to the equilibrium point that lies in the region below the threshold level $$I_b$$Ib or if they converge to one of the pseudoequilibria or a pseudo-attractor on the surface of discontinuity. However, we have to cull birds whenever the solution of this model converges to an equilibrium point that lies in the region above the threshold level $$I_b$$Ib in order to control the outbreak. Hence a good threshold policy is required to combat bird flu successfully and to prevent overkilling birds.