Testing isn't part of the model at all, it operates only on the true numbers of susceptible/infected/recovered. Testing only gives you an idea of what those might be.
That said, it does have the potential to impact the perception of danger to individuals. And perception of danger should influence the β factor the author mentions. Learning that a lot of people have it or a lot of people are dying for example may cause people to take it more seriously, and thus influence β.