Q:

Were death rates greater than birth rates in premodern urban populations?

In the book by Peter Turchin, War and Peace and War. The rise and fall of empires, on page 151, I found the following sentence:

Until modern times, death rates of urban populations always outweighed their birth
  rates

This is said as a well-established fact, with no comments and no references.
This sounds very surprising to me, and I wonder, even if true, how could this be
determined. 
Can someone shed any light on this?

A:

While Peter Turchin has overstated the case, it is most likely that this was usually true. Infant mortality rates were very high until they started to decline in the early decades of the 20th century (especially in Europe and the US). To this can be added the generally high death rate from lack of effective medical care and frequent epidemics. These factors meant that the birth rate had to be very high in order for it to exceed the overall death rate.
For ancient Rome, it is impossible to be certain but there is some fairly strong evidence that Turchin's statement can be fairly applied to the late Republic and early Empire (pdf).

There is considerable evidence to show that Roman society in the late
  Republic and early Empire was afflicted by a low birthrate. Augustus
  in 18 B.C. found it necessary to pass the lex Iulia de maritandis
  ordinibus in the hope of raising the birth-rate by penalizing the
  unmarried and the childless. In 9 A.D. he attempted to supplement this
  law with the lex Papia Poppaea. The very existence of this legislation
  indicates that the problem of childlessness was widespread and
  long-Iasting, a view which is further supported by references to this
  subject in Latin literature).

The causes of this low birth rate are disputed by scholars with reasons ranging from too many hot baths adversely affecting male fertility to the use of contraception and abortion. What other evidence we have comes from funerary monuments, and sources commenting on factors such as epidemics, seasonal deaths and birth control. None of these, though, allow us produce any firm statistics and for other periods in Rome's history we have much less information.
In the Princeton/Stanford Working Paper in Classics Disease and death in the ancient city of Rome (pdf), Walter Scheidel states that the demographic 'balance sheet' is "profoundly unclear", adding that

A very rough guesstimate for the early modern city of London
  envisioning an annual excess of deaths over births equivalent to 1% of
  the total size of the population has repeatedly been acknowledged by
  Roman historians, yet its relevance is doubtful: it might equally well
  be too pessimistic (because imperial Rome enjoyed better
  infrastructure and welfare provisions) or too optimistic (because
  London was free from falciparian malaria).

Thus, as Scheidel notes:

It is important to remember that not all premodern metropolises were equally deadly,...

The reference to early modern London leads to more concrete evidence outlined in The Population History of England 1541-1871 by E. A. Wrigley, R. S. Schofield and Roger Schofield. By looking at baptism and burial records for London from 1550 to 1824, the authors tentatively conclude that

it would appear that the latter [burials] regularly exceeded the former [baptisms] until 1802

Although the pattern was more pronounced in London than elsewhere, they also found that deficits in urban parishes and market towns were more common than surpluses. Nonetheless, there were urban areas which did have surpluses of births over deaths for several decades at a time.

