Figure 2.

Intuitive derivation of type-reproduction numbers: Eqs. (24), (26) and (28). (a) There are four basic processes of infection spread starting from an infected adult woman and two starting from an infected adult man. The top two processes (yellow and green) represent vertical transmission starting from a horizontally infected adult woman, where the average production numbers are calculated as in Fig. 1b,c. Note that these processes contain vertical transmission starting from a vertically infected adult woman. The middle two processes represent horizontal transmission from a horizontally (blue) or vertically (brown) infected woman. The bottom two processes represent horizontal transmission from a horizontally (purple) or vertically (red) infected man. The average production numbers are given by the product of the transmissibility (β_f→m or β_m→f), contact rate (c_f, k_f, c_m or k_m), and duration (1/(${\mu }_{\mathrm{f}}^{\text{'}}$ + η_f) or 1/(${\mu }_{\mathrm{m}}^{\text{'}}$ + η_m)). (b) There are three paths starting from a horizontally infected adult woman and ending with a horizontally infected adult woman: two horizontal transmissions (HH), one vertical and two horizontal transmissions (VHH), and one vertical and one horizontal transmission (VH). Because the HH case is made up of the processes shown in blue and purple in (a), the reproduction rate R_HH for the HH case is given by the product of their average production numbers. We can calculate the reproduction rates R_VHH and R_VH for the VHH and VH cases in the same way. Thus, the type-reproduction number for adult women is given by summing these rates: R_HH + R_VHH + R_VH; thus, Eq. (24) is obtained. (c) The paths starting with a horizontally infected adult man and ending with a horizontally infected adult man are more complicated. The HH case (purple and blue) and the VHH case (purple, yellow, and brown) are similar to (b) (see the top two parts). The average production numbers of these cases are R_HH and R_VHH. In addition to these two cases, secondary infected women may vertically infect their sons, who may then infect other women (i.e., we can insert the green and red parts in the way; see the middle two parts). The average production numbers of these cases are R_HHR_VH and R_VHHR_VH. This procedure can be repeated (see the bottom two parts). Thus, the type-reproduction number for adult men is derived by a geometric series with geometric ratio R_VH, and Eq. (26) is obtained. (d) The paths starting with an infected juvenile and ending with an infected juvenile are also complicated. The case through one woman is simple, and it is obvious that its reproduction rate is α_eff. There are two cases through two women: One includes men infected horizontally (brown and purple), and the other includes those infected vertically (red). Inserting the blue and purple part in the way, we obtain two cases through three women. This procedure can be repeated. Considering two geometric series with geometric ratio R_HH, we obtain the type-reproduction number for juveniles—Eq. (28).