Effect of Changes in Breast-feeding on the Age Distribution of HTLV-I Carriers Using a Mathematical Model

Based on a mathematical model of human T-cell leukemia virus type I (HTLV-I) transmission, the non-vanishing condition of HTLV-I was derived, and the effect of changes in breastfeeding patterns on the prevalence of HTLV-I was evaluated. The non-vanishing condition was shown to be a>1/(1+b), where a is the probability of mother-to-child transmission and b is the probability of husband-to-wife transmission. This condition implies that if a is under 0.5, HTLV-I cannot be sustained in populations over a long term. The age-specific prevalences in 1985 were calculated under this model and the assumption of changes in breast-feeding over the period 1925-1985, based on information from many surveys on breast-feeding in Japan. The prevalence increased with age, like the prevalences observed in many populations in Japan. These findings suggest that the probability of mother-to-child transmission had been over 0.5 and was reduced by the spread of compound milk and the shorter duration of breast-feeding in recent years in Japan, and that it has resulted in a decrease in HTLV-I carriers among younger age groups. J Epidemiol, 1991 ; 1 : 1-5.

Adult T-cell leukemia (ATL) has been shown to be caused by infection of human T-cell leukemia virus type I (HTLV-I)1,2).
Since HTLV-I is transmitted in a cell-to-cell mode and its target is the T-cell3), the transmission of HTLV-I is likely to follow the liver transfer of T-cells4).
The main routes of HTLV-I transmission seem to be mother-to-child through breast-feeding, husband-to-wife and blood-borne transmission4-6). The probability of transmission via breast-feeding has been reported to be about 0.37-9). However, if this probability is about 0.3 or less, it has been shown that the prevalence of HTLV-I could not have been sustained in populations until today10). In many populations in Japan, the prevalence of HTLV-I has been observed to increase with age""', and no explanation for this has as yet been given13). This paper aims to derive a non-vanishing condition of HTLV-I, and to show that the increase of the prevalence with age can be caused by changes in breast-feeding. Pn and P(n+1) are prevalences in mothers of the nth generation and in those of the (n+ 1 )th generation, respectively. a and b are probabilities of mother-to-child and husband-to-wife transmissions, respectively.

MATERIALS AND METHODS
2. Non-vanishing condition of HTLV-I In order to derive a non-vanishing condition of HTLV-I, we consider that the prevalence is stationary, that is, Pn never changes through generations.
Let P be the stationary prevalence in mothers. From (1), the relationship among P, a and b can be obtained.
3. Effect of changes in breast-feeding patterns on prevalence The age-specific prevalences of HTLV-I in 1985 were calculated under the mathematical model in Figure 1 and the following assumptions.
First, consider the relation between the probability of mother-to-child transmission, a, and the total amount of breast milk given to an infant during the period of breast-feeding.
The solid and broken lines in Figure 2 represent the amount of breast milk per day in two patterns of breast-feeding.
The first pattern (the solid line in Figure  2) is modified from the observed data on breast-feeding in recent years in Japan, and the second pattern (the broken line in Figure 2) is modified from those of several decades ago14). The ratio of the total amount of breast milk in the first pattern to that in the second pattern is calculated to be about 3 : 7. If a=0 for no breast-feeding, a=0.3 for the first pattern, and a is proportional to the total amount of breast milk, a for the second pattern becomes 0.7. It is, therefore, assumed that a=0.15 when the total amount of breast milk is less than that in the first pattern, as its value is the average of a for no breast-feeding and a for the first pattern, that a= 0.5 when the total amount of breast milk is between that in the first pattern and that in the second pattern, and that a=0.7 when the total amount of breast milk is more than that in the second pattern.
Infants are categorized by their total amount of breast milk into 3 groups (the total amount of breast milk is less than that in the first pattern, between that in the first pattern and that in the second pattern, and more than that in the second pattern). Proportions of each of 3 groups of infants to all infants are assumed to be shown in Figure 3 from 1925 to 1985, based on information from many surveys on breast-feeding in Japan14-18), and are assumed to have been constant before 1925.
Other assumptions are as follows : Males and females get married at the age of 25 years, and a carrier husband quickly transmits HTLV-I to his wife. Children are borne to 25 year old mothers, and a carrier  mother quickly transmits HTLV-I to her children. The prevalence of HTLV-I in mothers was 25% before 1925, since the prevalence in females older than 60 in a population was reported to be about 25% in 1984-8513). The probability of husband-to-wife transmission, b=0.68, as this value was determined by the following non-vanishing condition.

RESULTS
1. Non-vanishing condition of HTLV-I For example, taking the prevalence in mothers of the baseline generation, P0=0.5, and assuming a=0.3 and b=0.7, we obtain a prevalence trend through generations in Figure 4. The prevalence in mothers of the first generation, P1=0.24, P2=0.12, and P6 is less than 0.01. In this case, HTLV-I can not be sustained.
Noticing that P(n+1)=Pn=P in (1) The relation among P, a and b in (2) are illustrated in Figure 5. If a given point (a, b) is on the curve with the value of P, Pn converges to P as n increases. For example, putting a=0.6 and b=0.8, Pn converges to about 30%.
This is a non-vanishing condition. Furthermore, since b is 1 or less, a>0.5.
This condition implies that when a is 0.5 or less, the prevalence decreases towards zero, even if b=1, that is, a carrier husband always transmits HTLV-I to his wife.  Figure   6). This increase with age was like prevalences observed, for example, in Figure  713). In Figure  6, the differences between prevalences with changes in   Female. with breast-feeding pattern change Male. with breast-feeding pattern change Female. without breast-feeding pattern change Male. without breast-feeding pattern change Figure 6. Prevalences calculated under the model. In the present study, it was shown that if the probability of mother-to-child transmission was under 0.5, HTLV-I could not be sustained. However, from recent studies7-9), this probability has been reported to be about 0.3. These findings suggest that this probability had been over 0.5 and was reduced to the above value by some factors in recent years.
If not, other unknown important routes of HTLV-I transmission would exist.

Effect of changes
in breast-feeding patterns on prevalence The increase of prevalence was partially caused by blood-borne transmission4). Differences between prevalences in males and those in females were associated with husband-to-wife transmissions). However, the increase of prevalence with age in both sexes could not be caused by only these routes13). The changes in breast-feeding ( Figure  3) are not only an impractical assumption, since they are based on information from many surveys on breast-feeding in Japan14-18).
Furthermore, it was reported that the duration of breast-feeding has become shorter by the education of proper duration of breast-feeding for mothers in recent years, and that the use of compound milk has been widely spread from 1950s in Japan14).
Therefore, the changes in breast-feeding of our assumption may reflect on those in most of the Japanese Effect of Breast-feeding on HTLV- I 5 population. Based on our model, the increase of prevalence with age, which was observed in prevalences in many populations in Japan, was induced by these changes in breast-feeding patterns.
The curve of the age-specific prevalences could not be greatly distorted by small changes in other assumptions.
The above discussion suggests that the probability of mother-to-child transmission had been over 0.5 and was reduced by the spread of compound milk and the shorter duration of breast-feeding in recent years as compared to previous years in Japan, and that these resulted in the increasing prevalence with age. Furthermore, without great changes in breast-feeding patterns tending to produce an increase in the probability of mother-to-child transmission, the prevalence of HTLV-I will converge to zero in the future.