Change in fertility centers on the current value of the total fertility rate (TFR).  IFs determines the TFR and then imposes that on the cohort-specific fertility distribution (FERDST) of the region/country.

IFs uses three key variables to drive TFR forecasts over time.  One of those accounts for the change that typically accompanies long-term development and social evolution. The two principal candidates to represent such change in the long term across all IFs models are GDP per capita at purchasing power parity (GDPPCP) and the years of formal education attained by adults (EDYRSAG15).  Our own analysis and that by Angeles (2010) both suggest that the latter is the stronger predictor for TFR.  Frequently in IFs we find (Hughes 2001) that the relationship between one of those two deep distal drivers and any specific element of social change is logarithmic (that is, social change happens especially rapidly at lower levels of income and education and then saturates) and this is the case in this instance also.  You can see the approximate form of that relationship by examining a scattergram of TFR as a function of EDYRSAG15 in the initial model year, or you can look at the multivariate relationship that IFs actually uses (in Scenario Analysis/Change Selected Functions ).

In addition to long-term development and the deep or distal variables associated with it, societies are subject to short-term factors, most of which are in turn influenced heavily over time by the distal variables.  These more proximate variables do, however, exhibit patterns of change that are at least somewhat independent of the distal drivers and more dependent on societal choices and policies.  In the case of fertility change, two such variables often identified to be important are the rate of mortality, often infant mortality in particular (INFMORT) and the rate of use of modern contraception (CONTRUSE).

In the equation above we have used a lagged form of infant mortality.  The lag uses 10 percent of the new value for infant mortality and 90 percent of lagged (therefore actually moving average) value; the proportions are subject to change, but were chosen to capture roughly the 10-year lag to peak effect identified by Angeles (2010). When such a moving average is initiated with the value of the first year of the model run, rather than with a value computed over an historical period preceding that first year, it gives rise to a pattern of slow change in initial years (values of early years tend to be very close to those of the initial value) and then accelerating change over time up to the about the 10th year.  We therefore phase in the effect of the moving average, also over 10 years.

The additional term involving the parameter ttfrr is used to represent time change that is independent of the relationship estimated via cross-sectional analysis with recent data. There has been a global ideational change with respect to fertility that the term can represent; in addition, it can be a tuning parameter and normally the value is very low in IFs.  Finally in the equation above, the user can adjust a multiplier parameter (tfrm ) from its default value of 1 so as to force higher or lower fertility.

There are also, however, three important algorithmic elements that wrap this equation in more extensive model code.  First, we compute in the model preprocessor the historical growth rate of TFR (TFRgr) and use that to help drive year-to-year change in TFR.  In fact, in the first year the change in TFR is fully driven by that internal variable, but attention to it is phased out over 10 years.  Second, we have captured in the first year of the model forecast the difference between TFR from the function and TFR from the data.  This difference or shift could be viewed as a country-specific fixed effect dependent on variables such as historical paths and cultural factors. We choose, however, to phase it out over a fairly long period of time specified by the parameter tfrconv .  Often in IFs the reduction of such shift factors is done over a half century or more, and, at the time of this writing, the parameter's value was 100.  Third, total fertility rate is unlikely to shift indefinitely toward zero. In fact, it requires a value of about 2.1 simply to maintain a steady population (unless life expectancies are growing). TFR is therefore bound by a minimum that responds to a global parameter (tfrmin). The equation below represents that long-tem bound which is again phased in over a very long period of time and algorithmically raises the fertility of countries below the minimum. 

The use of modern contraceptives (CONTRUSE) is itself a function of a key distal driver, in this case GDP per capita at purchasing power parity (GDPPCP).The reader may wish to use the model to look also at a scattergram of CONTRUSE against GDPPCP in the initial year. The "actual" level of contraception use depends not only on GDPPCP, but on an exogenous multiplier ( contrusm ), and on a temporal (t) upward drift in contraception use related to ideational change again, as well as related technological innovation and diffusion (controlled by tconr ).

Once we have computed the total fertility rate (TFR), the number of births in a given year is a simple function of the fertility distribution and the TFR.

If advances in health very substantially affect life expectancy, they may also affect fertility patterns.  Parameters in IFs allow control of the onset age of fertility ( hltfrageinit ), the peak age of it ( hltfragepeak ), the age of menopause ( hltfragestop ), and the rate of decline from peak to menopause ( hltfragehalflife ).  If child-bearing age were greatly extended, it would necessarily lead at some point to a change not only in the peak age of child-bearing, but the rate of child-bearing at that age ( hltfrpeaklevel ), changed from current patterns at a rate controlled in the model by a final fertility parameter ( hltfrconv ).