Control strategies and prevention of infectious illnesses are challenges faced by the healthcare systems. Infectious diseases are caused by organisms and can be transmitted through different ways, such as contact between individuals of the same species or interspecies (zoonotic diseases). One contagious viral disease is the avian influenza, known as bird flu, that infects wild birds, as well as domestic poultry. It can also contaminate some mammalian species and was detected in humans. Epidemiologic models have been used to study the transmission dynamics of bird flu. In this work, we analyse a model that describes the spread of a avian influenza from birds to humans. We consider a compartmental model in which the bird system is divided into susceptible and infected birds (SI), whilst the human system is separated into susceptible, infected and recovered (SIR). In addition, we substantially extend this model by including seasonal effects in the transmission rate of the system and investigate the behaviour of the model through the waves of infection. We uncover that depending on the parameters and initial conditions, the number of disease cases can be kept under control, as well as it is possible to observe even disease-free state. Our results show that the initial conditions play a crucial role in the number of avian influenza waves. Furthermore, we demonstrate that the seasonal forcing increases the infection numbers and induces a more complex dynamical behaviour.