Prior immunity helps to explain wave-like behaviour of pandemic influenza in 1918-9

Table 4 Median parameter estimates (2.5-97.5% credibility intervals) in model to explain the observations in Table 1.

Contrast	R	R ₀	Z	Φ₁	Φ₂	α₁	α₂	α₃
All 12 populations	2.33 (1.57-3.96)	4.59 (3.10 - 6.74)	0.52 (0.41 - 0.66)	0.57 (0.42 - 0.71)	0.35 (0.23-0.46)	0.39 (0.25 - 0.52)	0.43 (0.30 - 0.59)	0.48 (0.31-0.65)
8 urban populations	1.81 (1.16 - 2.94)	3.80 (2.53 - 5.92)	0.49 (0.34-0.62)	0.50 (0.34 - 0.76)	0.38 (0.24-0.57)	0.38 (0.20 - 0.58)	0.34 (0.21 - 0.49)	0.38 (0.22-0.59)
4 "school" populations	2.84 (1.43-5.36)	4.73 (2.43-8.42)	0.61 (0.41-0.81)	0.74 (0.49 - 0.92)	0.38 (0.20-0.57)	0.49 (0.30-0.68)	0.64 (0.43-0.86)	0.71 (0.41-0.91)

For parameter definitions, see methods text. Estimates for R ₀ , Z, Φ ₁ , Φ ₂ , α ₁ , α ₂ , α ₃ were derived from the joint distributions of the relevant hyperparameters; these parameters can be regarded as the typical values for populations such as those studied. By combining information from the joint distributions of R ₀ and Z, it was also possible to derive the medians and credibility intervals for the effective reproduction number R = R ₀ .Z, at the start of the first wave. Population-specific estimates are given in Tables S.3 & S.4. For detailed methods see the supplementary information in Additional file 1. Results to support the validity of estimation procedures are given in Additional File 1 - see Tables S1 & S2. and Figure S1.