Table 1 Calculations of remaining lifespan at age 105 for Italian data and projections of population sizes required to reach selected age maxima.

Each model begins with the 463 males and 3373 females in the Barbi et al. 1904 birth cohort; P represents initial population size. In the “radioactive decay” model, N/P = exp(–mt), where m is the annual mortality rate; for age of the last survivor, N is set at 1, yielding Tmax = –(1/m) ln(1/P). In the Gompertz model as calculated in Barbi et al., m(x) = a × exp(bx) × exp(β1C + β2M) → S(x) = exp{a/b × [1 – exp(bx)] × exp(β1C + β2M)}, where a and b are the Gompertz parameters; C is cohort birth year minus 1904; M = 1 for males and 0 for females; β1 and β2 are coefficient estimates for cohort and sex, respectively; and S(x) is the survival function at age x. Then, the expression 1/P = exp{a/b × [1 – exp(bx)] × exp(β1C + β2M)} yields Tmax = ln{1 + [b × ln(P)]/[a × exp(β1C + β2M)]}/b.

MaleFemale
PTmaxTmax agePTmaxTmax age
Radioactive decay model
4639.43114.433,37312.92117.92
4,63012.97117.9733,73016.58121.58
46,30016.51121.51337,30020.24125.24
463,00020.05125.053,373,00023.90128.90
4,630,000,000,00044.82149.823,373,000,000,00045.86150.86
Gompertz model as calculated in Barbi et al. (1)
4638.90113.93,37311.94116.9
4,63012.00117.033,73015.01120.0
46,30015.00120.0337,30017.97123.0
463,00017.81122.83,373,00020.81125.8
46,300,000,000,000,00043.77148.833,730,000,000,000,00044.64149.6