Numerous African studies^{1-11} have demonstrated that children of HIV-infected mothers have higher mortality than children of uninfected mothers, and many have tried to measure the proportion of children infected through mother-to-child transmission of the HIV virus.^{3,12-15} To project the overall effect that the HIV epidemic will have on child mortality at the population level in countries with generalized epidemics, it is necessary to estimate a realistic schedule of “net” age-specific mortality rates that would be observed if the only causes of death affecting children were HIV-related. Such a schedule could then be applied to age-specific mortality patterns experienced by uninfected children to obtain the expected child mortality patterns of infected children in populations with varying background mortality. Surprisingly, the data for estimating such net mortality schedules are not widely available. This article explains the mathematics behind the demographic techniques that can be used to make such estimates and explores ways in which widely available data on the mortality of children of infected and uninfected mothers can be manipulated to compensate for missing information on the HIV status of children. A critical literature review is used to identify African studies that measured the survival of infected children or children of infected mothers. Data from studies meeting *a priori* quality criteria and satisfying certain internal consistency checks are used to derive a net mortality schedule for HIV infection that can then be used for estimating the impact of HIV on child mortality.

#### DATA NEEDS AND AVAILABILITY

The complete information needed to derive the net mortality schedule that could be used to estimate current and future overall child mortality levels in populations affected by HIV is a set of 3 schedules of single-year age-specific mortality rates (or proportions surviving from birth to exact ages), ideally up to the age of 15 years, for the following:

* HIV-infected children

* Uninfected children of HIV-infected mothers

* Children of uninfected mothers

In Table 1, we present available data on child survival from studies in which children have been followed continuously from birth, and their HIV status (or that of their mothers) has been reliably measured. Not all studies provide data in the form that is needed. Some do not present information on the survival of uninfected children, and some present data on the HIV status of mothers but are unable to provide information on the HIV status of the child. The following 5 categories of information about children are distinguished:

* M^{−}: Children of HIV-negative mothers

* M^{+}: Children of HIV-positive mothers, status of child not known

* M^{+}C^{−}: HIV-negative children of HIV-positive mothers

* M^{+}C^{+}: HIV-positive children

* C^{−}: HIV-negative children, status of mother not known

Apart from the sources listed in Table 1, other published studies were reviewed but rejected for various reasons. Rejected studies included those in which survival proportions were calculated from some time after birth^{16,17}; antiretroviral drugs were used,^{18} which may have changed the natural course of the disease; follow-up lasted less than 1 year^{19,20}; losses to follow-up were unreasonably large (30% of the children of HIV-positive mothers, nearly all occurring before 12 months)^{21}; atypically high numbers of AIDS cases (>20%) were included in the sample of infected mothers (hospital A) in the Kinshasa study^{6,22}; patients were selected on the basis of some other illness (1 study site is a tuberculosis [TB] clinic)^{23}; follow-up time was not given^{24}; and information was based on incompatible ages.^{25}

Four of the studies in Table 1 (Rakai, Masaka, Kisesa, and Karonga) were community-based,^{1,9-11} 1 was hospital-based,^{14} and all the other studies were based on antenatal clinics (ANCs) or mother and child clinics. The table shows that the Rakai and Kampala studies (group 1) are the only studies that provide survival data on all 3 categories of interest. Of the other 4 studies that are able to identify HIV-infected children, the reports on the Côte d'Ivoire and Burkina Faso studies and the Nairobi study (group 2) provide information on the survival of uninfected children of infected mothers, whereas the Kigali study (group 2) provides information on the survival of children of HIV-negative mothers, although not of the same ages as the HIV-positive children. The South African study (group 2) gives no information on the survival of HIV-negative children. The remaining 8 studies (group 3) listed in the table only have data broken down by HIV status of the mother.

To maximize the use of these data in estimating a net schedule of age-specific HIV mortality, the following methodologic problems need to be addressed:

* How to estimate the mortality of uninfected children in studies in which the only data available are for HIV-positive children

* How to estimate the mortality of infected children in studies in which we only have information on the maternal status

#### THEORETICAL DEVELOPMENT

Notational conventions have been developed to explain the relations between the subpopulations of interest. The subscript “*M*” is used to denote the mortality of children with HIV-positive mothers, and the subscript “*O*” is used for mortality from all causes other than those related to HIV-this is the same as the (observed) mortality of children of HIV-negative mothers. The subscript “*A*” is used to denote the (theoretical) net mortality from AIDS and other HIV-related causes when all other causes of death are not operating; the subscript “*A*+*O*” is used to denote the gross (observed) mortality of HIV-infected children who are also subject to risks of dying from other causes. The subscript “*S*” is used to denote a standard mortality schedule appropriate for non-HIV mortality tabulated for all childhood ages.

##### Estimating Net Mortality as a Result of HIV

The net survival probability, *lA*(*x*), if HIV-related mortality is the only operative cause of death, can be calculated from the proportions of HIV-positive children surviving to age *x*, ι_{A+O}(*x*), and the proportion of uninfected children surviving to age *x*, *lO*(*x*), using the usual relation for cause-deleted life tables:

In the case of complete direct data (group 1), the *lA*(*x*) ratios are based on the empiric values shown in Table 1; for incompleted data (group 2), estimates of *lO*(*x*) are derived as set out below. Also derived are the estimates for the indirect data (group 3) of *lO+A*(*x*).

It has been suggested^{3,26-28} that infected infants experience different rates of progression through the disease stages leading to AIDS and death, with those who acquired the infection *in utero* experiencing a more rapid progression than those acquiring the infection at parturition or during breast feeding. The theoretical form of the survival curve for net HIV mortality in pediatric infections embodying this frailty assumption has been described as taking the form of a double Weibull curve^{29} (incubation period of vertically acquired AIDS):^{30,31}

where λ_{1}, μ_{1}, λ_{2}, and μ_{2} are the parameters of the component Weibull curves describing mortality among rapid and slow progressors, respectively, and π is the proportion of children in the rapid progression group. In theory, these 5 parameters can be fitted to empiric estimates of *lA*(*x*) using general function minimization techniques. In practice, only π and the parameters governing the rapid progression component are fitted, because little evidence is available about survival after 5 years of age. By fitting π to obtain a smooth curve through available data points, we may be moving away from the literal interpretation of its link to mode of transmission (ie, there may be some fast progressors who acquire the infection at or immediately after birth).

Two external constraints are introduced to fit a realistic curve beyond the age of 5 years. First, it is assumed that, overall, less than 1% of infected children survive to age 15. This assumption effectively forces the second Weibull component to produce a pattern of mortality risks that increase with duration since infection, as is observed for infected adults. Second, we assume that net HIV mortality rates of infected children are at least as high as those of adults at an equivalent duration of infection. In effect, this constraint is mainly important for death rates of children aged between 4 and 6 years; in this age range, most fast progressors will have died, but high mortality in slow progressors may not yet have begun to take a serious toll. After reviewing the adult mortality data for the Masaka and Kisesa cohorts,^{32-34} the second constraint was expressed as a requirement that the annual net risk of dying from HIV-related causes in children should not fall below 0.07, which is the approximate net risk of dying for adults infected for 4 or more years.

To obtain plausible upper and lower bounds for child mortality, confidence intervals were constructed based on the proportion surviving to each age in the best-fitting curve, coupled with the overall number of uncensored observations at that age from all the studies that contributed data at that age point. Double Weibull curves were also fitted through these upper and lower bounds.

##### Estimating Mortality of Uninfected Children (Groups 1 and 2)

There is some evidence to suggest that there is a difference in the mortality of HIV-negative children born to HIV-positive mothers compared with HIV-negative mothers. In the case of studies that have incomplete information on the mortality of uninfected children (eg, only providing information on the uninfected children of infected mothers), this hazard ratio needs to be determined to obtain the mortality of the HIV-negative children of negative mothers. This hazard ratio for uninfected children aged *x* of HIV-infected mothers compared to children of uninfected mothers is denoted by *C*(*x*), where *q*(*x*), the probability of dying before age *x* is given by:

The Rakai study^{1} found that, overall, for *x* less than 2 years of age, *C*(*x* < 2) = 1.3, there was some evidence of variation of *C*(*x*) with age, with *C*(*x* < 1) = 1.1 and *C*(1 ≤ *x* < 2) = 1.8. The study in Kampala^{2} showed a similar pattern with the same overall value for *C*(*x* < 2) = 1.3 and a similar increase with age on subdivision of the interval. We have assumed that *C*(*x*) takes the values observed in Rakai and that in children older than 2 years of age, it remains at a constant level of 1.8.

For studies providing information on the mortality of un-infected children in a limited age range, the probability of dying outside this range can be derived if the probability of dying is known at just 2 age points, because it has been shown^{35} that mortality schedules are linearly related on the logit scale. The probability of surviving to age *x* is denoted by:

and the logit transformation is defined by:

Comparing observed values with those tabulated in an appropriate reference standard schedule of mortality, the parameters of the linear relation of the observed mortality schedule with respect to the standard are given by:

allowing the calculation of survival probabilities for other childhood ages, *x*, using the relation:

##### Estimating Mortality of Infected Children (Group 3)

We denote the proportion of children surviving to age *x* among HIV-infected mothers as *lM*(*x*) and define *V*(*x*) to be the proportion of HIV-infected children alive at age *x* among all children surviving to age *x* born to infected mothers. The observed hazard ratio at age *x* between the mortality of children of HIV-infected mothers and the children of uninfected mothers is denoted by *K*(*x*), and, as before, the unobserved hazard ratio for *x*-year-old uninfected children of HIV-infected mothers compared with children of uninfected mothers by *C*(*x*). The mortality hazard at age *x* is denoted by λ(*x*).

By definition, the probability of surviving from age *x* to age *x* + 1 among children of HIV-infected mothers is as follows:

where λ_{O}(*x*) denotes the mortality hazard experienced by children of uninfected mothers at age *x*.

This survival probability can also be expressed in terms of the survival of HIV-infected and -uninfected children of HIV-infected mothers as follows:

Equating these 2 expressions for survival yields an expression for the mortality hazard affecting HIV-infected children at age *x* as follows:

*V*(*x*) can be found recursively, if λ_{O}(*x*), λ_{A+O}(*x*), and *C*(*x*) are known. If we denote the total number of children born to HIV-positive mothers who are still alive at age *x* by *T*(*x*), the numbers of HIV-positive and HIV-negative children in this group are given by *T*(*x*) · *V*(*x*) and *T*(*x*) · [1 − *V*(*x*)], respectively. The numbers surviving to age *x* + 1 in these 2 groups are as follows:

for HIV infected children, and

for uninfected children.

*V*(*x* + 1) is the proportion of these children who are infected, therefore

By definition, *V*(0) is the mother-to-child vertical transmission probability, a quantity that has been estimated in various studies to be between 25% and 45% in breast-feeding populations in the absence of antiretroviral treatment,^{36} so that the limits between which it varies in the absence of antiretroviral treatment (ART) are reasonably well known. This makes it possible to estimate the gross mortality of HIV-positive children, λ_{A+O}(*x*), for all childhood ages for an assumed value of *V*(0) if the mortality of HIV-negative children, λ_{O}(*x*), and the hazard ratio, *K*(*x*), faced by children of HIV-positive mothers can be measured and a reasonable value assumed for *C*(*x*), the hazard ratio of HIV-negative children of HIV-positive mothers.

A first approximation to a reference curve for survival is obtained, assuming that vertical transmission *V*(0) is 35%, using general function minimization techniques. Allowance would be made for variable vertical transmission rates between sites using an iterative process. The best-fitting survival curve is used as a reference curve to obtain site-specific values of *V*(0) that give the closest alignment of age-specific data points to each study using general minimization techniques.

With these assumptions, we can estimate λ_{A+O}(*x*), for all childhood ages and use these values to estimate survival proportions for infected children even for those studies in which only maternal HIV status is known.

The probability to surviving to age *x* + 1 is simply:

#### RESULTS

The theoretical relations derived above are applied to the incomplete data shown in Table 1 to obtain estimates of net mortality caused by HIV alone from each of the sources quoted.

##### Estimating Net Mortality From HIV in Studies With Direct Data Available (Groups 1 and 2)

The mortality of the HIV-negative children of HIV-negative mothers was estimated using Equation 3 with values of *C*(*x* < 1) = 1.1 and *C*(1 < *x* < 2) = 1.8 (the excess risk faced by uninfected children of infected mothers) for the Côte d'Ivoire and Burkina Faso studies and the Kenyan study (Table 2).^{12,13,37}

Equation 3 Image Tools |
Table 2 Image Tools |

For the time of the Durban study,^{14} we estimated infant mortality, *q*(1), to be 49 per 1000 and child mortality, *q*(5), to be 59 per 1000 for uninfected children (Ian Timaeus, pers. commun., 2003). To find the mortality at other childhood ages (see Table 2) for the Durban and Kigali studies,^{3,14} where survival for HIV-negative children is given at different points from that of HIV-positive children, logit transformations were used and compared with the Brass general standard.^{35}

A double Weibull curve was fitted to studies with direct data on the survival of HIV-positive children by minimizing the difference between the fitted and observed survivorship. A general function minimization routine was used^{38} with the sum of squares of differences as the objective function, varying the parameters λ_{1}, μ_{1}, λ_{2}, μ_{2}, and π subject to the constraint that the single-year age-specific mortality rate did not fall below 70 per 1000. This resulted in the curve shown in the graph in Figure 1, generating parameters of π = 0.66, λ_{1} = 0.65, μ_{1} = 0.96, λ_{2} = 0.11, and μ_{2} = 7.5.

##### Estimating Mortality of Infected Children in Studies in Which Only the Mother's Status Is Known (Group 3)

##### Estimating Mortality of Infected Children

With estimates for *C*(*x* < 1), *C*(1 ≤ *x* < 2), and all values greater than *C*(*x* ≥ 2) as 1.1, 1.8, and 1.8, respectively, and an estimate for the unknown vertical transmission, *V*(0), of 0.35 (the midpoint of the estimated range of *V*(0) of 25%-45% in breast-feeding populations in the absence of ART),^{36} it is possible to obtain estimates of mortality rates of HIV-positive children from data giving only the status of the mothers using the relations embodied in equations 10 and 13. The resulting estimates are shown in Table 3. As with the direct data, a Weibull survival curve was derived using general function minimization procedures to minimize the sum of squares of differences between observed data and the fitted curve, with the constraint that the annual mortality rate must be at least 70 per 1000 to ensure that child mortality was not lower than that of adults with the same duration since infection. The parameters were π = 0.62, λ_{1} = 0.77, μ_{1} = 1.18, λ_{2} = 0.10, and μ_{2} = 3.6.

Equation 10 Image Tools |
Equation 13 Image Tools |
Table 3 Image Tools |

Using the iterative process described in the background generated a substantial improvement in the goodness of fit (the mean sum of squares of differences for all data points from the curve) from 0.23 to 0.03, but further iterations did not produce improvements in fit. Estimates for the mortality of HIV-negative children are shown in Table 3 as well as the site-specific values for vertical transmission. The fitted vertical transmission rates ranged from 0.24 in The Gambia to 0.45 in Blantyre, which is within the range of observed vertical transmission rates in studies in which it could be measured directly.

Shown in Figure 2 is a Weibull survival curve with generating parameters of π = 0.64, λ_{1}= 0.74, μ_{1} = 1.04, λ_{2}= 0.1, and μ_{2} = 4.4. This is the best fit obtained to the indirect data after allowing for variation in the vertical transmission rate between these studies, as shown in Table 3.

Figure 2 Image Tools |
Table 4 Image Tools |

##### Pooled Estimate of Net Mortality as a Result of HIV

A comparison of the curves shown in Figures 1 and 2 suggests that the indirect estimates, community- and clinic-based, generate a net mortality pattern that closely matches the pattern obtained by fitting to the direct estimates of net HIV mortality. The parameter values and curves are compared in Table 4 and Figure 3. The choice of an independent value for vertical transmission in each indirect data set, although affecting the scatter of the data points relative to the best-fitting curve, has little effect on the overall shape of the curve. Figure 4 shows the final survival curve fitting to the pooled data from indirect and direct sources. The generating parameters are π = 0.58, λ_{1} = 0.89, μ_{1} = 1.13, λ_{2} = 0.11, and μ_{2} = 3.8. The 95% confidence intervals (CIs) for the model are also shown as well as the double Weibull survival curves that fit these maximum and minimum levels.

Figure 3 Image Tools |
Figure 4 Image Tools |

#### DISCUSSION

There seems to be a common “net” mortality pattern, which can be described well by the double Weibull curve at early ages. The overall curve predicts 67% survival at 1 year and 39% at 5 years. The differences between the results of the indirect and direct techniques are small: 67% compared with 68% for 1-year survival and 36% to 37% for 5-year survival. Survival patterns derived from community data and hospital/ANC data were similar after excluding those hospitals with high proportions of AIDS cases in the sample.

##### Estimating Excess Risk Faced by Uninfected Children of Infected Mothers

In the Rakai study,^{1} the hazard ratio, *C*(*x*), describing the excess risk faced by uninfected children of infected mothers increased with age from 1.1 between birth and the age of 1 year to 1.8 between the ages of 1 and 2 years. They study in Kampala^{2} suggested a similar overall excess risk: 1.3 between birth and the age of 2 years, which is the same as in the Rakai study had we ignored the available age breakdown. Others studies evaluated *C*(*x*) between the ages of 1 and 2 years as 1.3 (0.49-3.32) in Malawi^{17} or from birth to 5 years of age in Rwanda as 0.4 (0.1-1.6)^{3}; neither of these values is significantly different from 1.0. We suspect that the value of *C*(*x*) may vary widely between populations and that many factors (social and biologic) might affect its level and direction.

One of the factors that cause uninfected children of infected mothers to experience higher mortality is the disruption to care and nutrition caused by the death of their mothers. HIV is associated with higher mortality among mothers.^{22,39} Those HIV-infected mothers who give birth are less likely to have developed AIDS,^{40} however, so the ratio of mortality in infected and uninfected mothers would be expected in increase with time as the epidemic matures. The impact of the mother's death is greater on early childhood mortality,^{41,42} although it is expected that more infected mothers will die when their children are older. It is uncertain whether to expect the net effect of mortality among mothers would be stronger at younger or older childhood ages.

Socioeconomic factors also affect values of *C*(*x*), and these characteristics may differ between HIV-positive and HIV-negative mothers in relation to their setting. In many hospital/ANC studies, controls were matched for age and parity,^{3,6,7} and other studies reported differences between HIV-positive mothers and HIV-negative mothers, with HIV-negative mothers having a higher level of education, more likely to be married, and expecting their first child.^{4} In general, HIV-positive mothers are more likely to be younger, because most are infected in their mid-20s and do not survive past their mid-30s.^{43} There is also some inconclusive evidence to suggest that HIV-positive mothers give to birth to more low-weight and premature babies,^{44,45} thereby possibly increasing the risk of an infant's death even if the baby is not infected.^{46} Another possible explanation for the increased mortality of uninfected children of infected mothers is low maternal weight.^{47}

##### Estimation of Vertical Transmission

Indirect estimates of the vertical transmission probability for those studies with only the status of the mothers recorded yielded values between 24% and 45%. The same estimation technique applied to a study in the Democratic Republic of Congo by Ryder et al,^{22} where 23% of the mothers had symptomatic AIDS, yielded a value of 57%, which is consistent with the observation that vertical transmission increases when the viral load is high.^{48,49}

Vertical transmission is thought to be around 35% on average in the absence of ART,^{36} with around one third to one half occurring during breast-feeding.^{50} The indirect method uses the assumption that all vertical transmission occurs during infancy, although not necessarily at birth. This results in a slight overestimation of the survival of infected infants. Direct estimates based on HIV status of the child are equally affected, because those children who seroconverted after birth are generally classified with those who are HIV-positive from birth, thereby introducing an upward bias to survival from birth estimates in the breast-feeding period. This bias decreases with time and does not affect mortality after breast-feeding (approximately 2 years). It does, however, remain as a component of the survival curve, because survival is cumulative. This bias would be complicated to eliminate in the indirect data without introducing further assumptions about the difference in survival of children according to time of infection.

##### Use of Data Sources

This study maximizes the amount of data that can be used by enabling studies in which only the status of the mother is known to be used, as well as, studies with incomplete information on HIV-negative children. For studies that provide no information at all on uninfected children, external data sources can be used. Published estimates of child mortality at the national level usually include the probabilities of dying by the age of 1 year, *q*(1) and the age of 5 years, *q*(5), generally broken down by urban/rural residence and major administrative area. To make sure that the estimates are for uninfected children, published estimates from before or early on in the epidemic must be used and projected to the date required. Trends in mortality over time can change not only the overall level but the relation between levels of mortality at younger ages compared with older ages; therefore, a more accurate forecast can be made by projecting both of these components.

##### Estimation of Net Mortality: Removing Competing Forces

The estimation of the net survival from HIV assumes that the background characteristics of the HIV-positive and HIV-negative mothers are the same. To avoid this assumption, we would need to subtract the hazard rates for HIV-negative children of positive mothers from the hazard rates of HIV-infected children to obtain a net survival from HIV curve. This would represent the biologic survival from HIV only (eg, disregarding social consequences, maternal death). These data were only available in 3 studies, however.^{1,12,13} For application of the methods set out in this report, it is more realistic to use the HIV-negative children of HIV-negative mothers, because these data are much more widely available. Also, it is much more useful to look at the net survival from AIDS not just from a direct biologic point of view; there are direct and indirect causes of mortality from HIV, and it is their total impact that is needed when making estimates and projections.

##### Shape of Net Mortality: The Double Weibull Curve

Because of the lack of data available for survival of HIV-positive children beyond 5 years, we are unable to demonstrate conclusively that the double Weibull curve is the appropriate model to fit to the net survival pattern. Currently, there is little information on survival in relation to the child's time of infection in utero, intrapartum, or through breast-feeding early on or later. Spira et al^{3} showed that for infection occurring in infants aged less than 3 months, there was a relative hazard of death of 10.3 (CI: 3.0-34.6) compared with those infected after 3 months; however, this does not allow for the fact that those children could not have died aged less than 3 months. A meta-analysis^{51} allows for this potential bias and shows a reduction in mortality risk in children with a postnatal infection rate of 26% compared with those with evidence of infection from birth.

The trials of breast-feeding versus formula-feeding represent a possible source of information about infection time effects.^{15} In theory, infected children in the formula-feeding arm could only have been infected in utero or intrapartum. Compliance rates have been as low as 70%, however, and because there are no children of HIV-negative mothers assigned to formula feeding, there is no real comparison group. This is needed, because we might expect excess mortality caused by exclusive formula feeding^{52-54} unrelated to HIV, which needs to be taken into account when allowing for competing forces of mortality.

#### CONCLUSIONS

This article has discussed a method whereby data from studies on children classified by maternal HIV status can be adjusted to obtain the survival of HIV-positive children. This method produces survival curves for children similar to those found using direct measurements on children of known HIV status. The resulting net survival curve for HIV using all available data sources can be used to estimate and project trends in child mortality.

When more data become available from studies following children beyond 5 years, we will be able to validate our parameterization of the double Weibull curve for describing net HIV mortality of children. Improved knowledge of the time of infection of the child (in utero, intrapartum, or postpartum) would enable us to model the individual components, because we would expect those who are infected postpartum to have a longer survival than those infected in utero, or perhaps intrapartum, because they are born with a potentially healthy immune system and experience disease-free life immediately after birth. Such knowledge would be particularly important when antiretroviral interventions seeking to prevent mother-to-child transmission are introduced.