This dissertation includes two parts: the theoretical study of an elliptic equation, and the practical study regarding the seasonality of human influenza.

In the first part, we focus on the study of an elliptic equation with a nonlinear boundary conditions. We establish the non-existence and existence results with respect to the different range of a parameter. To give a sense for the abstract existence result, we provide a solution to this equation for a special parameter.

In the second part, we discuss the seasonal dynamics of human influenza. According to the interactions among climate, influenza virus and human beings, we introduce three ecological based response functions: the influenza virus transmission response to the absolute humidity, the virus survival response to the air temperature and the human susceptibility to the environment temperature. The mathematical epidemiological model (SEIRS) incorporated with these response functions enables us to estimate the seasonal variation and the double peaks pattern in the subtropical pattern, as well as the single winter peak pattern observed in the temperate region. Then, we applied the model to a couple of cities along the latitude gradient and extended our simulation results to the global scale. Our model can be used to predict different flu activity pattern all over the world and help us to explore and understand the possible mechanism of the global influenza circulation.