Modelling the impact of temperature and bird migration on the spread of West Nile virus

West Nile virus (WNV) is a climate-sensitive mosquito-borne arbovirus that circulates between mosquitoes of the genus Culex and birds, with potential spillover to humans and other mammals. Recent trends in climatic change, characterised by early and/or prolonged summer seasons, increased temperatures, and above-average rainfall, are likely to have facilitated the spread of WNV in Europe. In this work, we formulate a spatial WNV model as a system of parabolic partial differential equations (PDEs), incorporating diffusion, advection, and temperature-dependent parameters, namely the mosquito birth rate, mosquito biting rate, extrinsic incubation rate, and mortality rate. Diffusion represents the random movement of both mosquitoes and birds across space, while advection captures the directed movement of migratory birds. The model is first studied mathematically, and we show that it has non-negative, unique, and bounded solutions in time and space. Numerical simulations of the PDE model are performed using temperature data for Germany (2019-2025). The simulation results showed strong agreement with the reported WNV cases among birds and equids in Germany. The observed spread patterns throughout the years were mainly driven by the combination of temperature, diffusion processes of hosts and vectors, and the biting preference of mosquitoes between resident and migratory birds. The model better explained the observed WNV spreading pattern, including distant, isolated cases in Germany, when more bites were allocated to migratory birds than to resident birds.