Epidemic outbreaks in structured host populations

For a heterogeneous host population, the basic reproduction number of an infectious disease, ${\text{cR}}_0$, is defined as the spectral radius of the next generation operator (NGO). The threshold properties of the basic reproduction number are typically established by imposing conditions that make ${\text{cR}}_0$ an eigenvalue of the NGO associated with a positive eigenvector and a positive eigenfunctional (eigenvector of the dual of the NGO). More general results can be obtained by imposing conditions that associate ${\text{cR}}_0$ just with a positive eigenfunctional. The next generation operator is conveniently expressed by a measure kernel or a Feller kernel which enables the use of analytic rather than functional analytic methods.