Viral infections are one of the leading source of mortality worldwide. The great majority of them circulate and persist in wild reservoirs and periodically spill over into humans or domestic animals. In the wild reservoirs, the progression of disease is frequently quite different from that in spillover hosts. We propose a mathematical treatment of the dynamics of viral infections in wild mammals using models with alternative outcomes. We develop and analyze compartmental epizootic models assuming permanent or temporary immunity of the individuals surviving infections and apply them to rabies in bats. We identify parameter relations that support the existing patterns in the viral ecology and estimate those parameters that are unattainable through direct measurement. We also investigate how the duration of the acquired immunity affects the disease and population dynamics.