Avian influenza is a contagious viral disease caused by influenza virus type A. Avian influenza can be disastrous (if it occurs), due to the short incubation period (about 1--4 days). Thus it is important to study this disease so that we are more prepared to manage it in the future. A classical system of differential equations (the half-saturated incidence model) and three Filippov models --- an avian-only model with culling of infected birds, an SIIR (Susceptible-Infected-Infected-Recovered) model with quarantine of infected humans and an avian-only model with culling both susceptible and infected birds --- that are governed by ordinary differential equations with discontinuous right-hand sides (i.e., differential inclusion) are proposed to study the transmission of avian influenza. The effect of half-saturated incidence is investigated, and the outcome of this model is compared with the bilinear incidence model. Both models remain endemic whenever their respective basic reproduction numbers are greater than one. The half-saturated incidence model generates more infected individuals than the bilinear incidence model. This may be because the bilinear incidence model is underestimating the number of infected individuals at the outbreak. For the Filippov models, the number of infected individuals is used as a reference in applying control strategies. If this number is greater than a threshold value, a control measure has to be employed immediately to avoid a more severe outbreak. Otherwise, no action is necessary. We perform dynamical system analysis for all models. The existence of sliding modes and the flow on the discontinuity surfaces are determined. In addition, numerical simulations are conducted to illustrate the dynamics of the models. Our results suggest that if appropriate tolerance thresholds are chosen such that all trajectories of the Filippov models are converging to an equilibrium point that lies in the region below the infected tolerance threshold or on the discontinuity surface, then no control strategy is necessary as we consider the outbreak is tolerable. Otherwise, we have to apply control strategies to contain the outbreak. Hence a well-defined threshold policy is crucial for us to combat avian influenza effectively.