  Concentrations of pathogen genomes measured in wastewater have recently
become available as a new data source to use when modeling the spread of
infectious diseases. One promising use for this data source is inference of the
effective reproduction number, the average number of individuals a newly
infected person will infect. We propose a model where new infections arrive
according to a time-varying immigration rate which can be interpreted as an
average number of secondary infections produced by one infectious individual
per unit time. This model allows us to estimate the effective reproduction
number from concentrations of pathogen genomes while avoiding difficult to
verify assumptions about the dynamics of the susceptible population. As a
byproduct of our primary goal, we also produce a new model for estimating the
effective reproduction number from case data using the same framework. We test
this modeling framework in an agent-based simulation study with a realistic
data generating mechanism which accounts for the time-varying dynamics of
pathogen shedding. Finally, we apply our new model to estimating the effective
reproduction number of SARS-CoV-2 in Los Angeles, California, using pathogen
RNA concentrations collected from a large wastewater treatment facility.
