Vol.8, no.6. 1992 Pages 579-581
CABIOS
AIDS in Ireland: the reporting delay distribution and the implementation of integral equation models C.M.Comiskey and H.J.Ruskin Abstract
provides a prediction of future numbers of both HTV and ADDS cases.
Faculty of Computing and Mathematical Sciences, Dublin G/v University, Dublin 9, Ireland
© Oxford University Press
a(t) = \'h(t-
u
(1)
579
Downloaded from http://bioinformatics.oxfordjournals.org/ at East Carolina University on July 8, 2015
This paper deals with two basic aspects concerning the modelling of AIDS incidence in the context of Irish data. We Adjustment for reporting delays describe initially the adjustment of the number of AIDS cases (Xjj) to allow for reporting delays, where a simple form of the We used the method proposed by Brookmeyer and Damiano likelihood function for the Xjj is supported by GLIM. (1989) to estimate reporting delays in Ireland. AIDS cases are Subsequently, we consider the accessibility of numerical solution typically cross-classified by length of reporting delay and date (through a NAG routine) of the integral equation models of diagnosis so that models of log-linear form, appropriate for generated by the back-projection method for the adjusted AIDS contingency table data are considered (with cells equal to the number of cases). Dividing calendar time into intervals s.t. cases. Results for the Irish data are summarized for various [Tj _ |, Tj),j = 1, . . .,R, with TR the most recent report date, choices of the incidence distribution. we let Xy represent the number of AIDS cases diagnosed in the yth calendar interval which had a reporting delay in the Introduction interval [d, _ ,, d,), i = 1, . . . , / . The analysis for reporting Fundamental to the modelling of AIDS incidence is the need delays presented in Table I was based on cases reported to the both to obtain numerical solutions to the appropriate integral Irish Department of Health up to the end of March 1990. Delays equation models and to ensure the relevance of the data set. of all cases diagnosed up to 31 December 1989 were examined, In the latter case, incidence figures for new AIDS cases can a total of 125 correctly documented cases were observed. As typically be disorted by reporting delays (i.e. delays between the matrix of Irish data was sparse, reporting delay and date the diagnosis and the notification of the case) of anything up of diagnosis were grouped into 3 month intervals and the model to three or four years. Clearly, accurate representation of the fitted to the quarterly data. current status of diagnosed AIDS cases requires an adjustment Adjusted figures were fitted as below. Table I shows the of the reported figures. Brookmeyer and Damiano (1989) and reporting delay probabilities P, when the Brookmeyer and Downs el al. (1987) describe a method, based on the conditional Damiano model was fitted to the adjusted figures. This model likelihood, for the estimation of the reporting delay distribufitted well with a scaled deviance (x2) of 99.36 with 455 tion, and in the following we evaluate the results obtained using degrees of freedom. a generalized linear model approach. It is evident from Table I that almost 85% of all cases are Early figures for the number of AIDS cases in Europe reported within 6 months of diagnosis. (The maximum delay predicted exponential growth of the epidemic (Downs et al., of 42 months represents a case that was diagnosed early in the 1987) but while there is now reason to believe that early epidemic.) As numbers of cases in Ireland are relatively few projections were unduly pessimistic, the poor prognosis (WHO, 1989) and delays short, little adjustment was needed associated with the disease serves to emphasize the critical for the present annual AIDS incidence figures employed in the importance of accurate predictions for the future. The integral following analysis. equation model of back-projection has proved extremely useful both in terms of evaluating current data and in terms of The method of back-projection providing estimates of future cases (Brookmeyer and Gail, 1986, 1987, 1988; Isham, 1988; Anderson, 1989; Brookmeyer and Brookmeyer and Gail (1986, 1987, 1988), Isham (1988), Damiano, 1989). The method involves back-calculation from Anderson (1989) and Brookmeyer and Damiano (1989) have the number of diseased cases (suitably adjusted for reporting discussed the back-projection approach to investigating the delays), via the incubation period distribution to the number relationship between the incidence of AIDS cases, the incidence of previously infected cases. Alternatively, working forward of HFV infection and the distribution of the incubation period. from the current figures under a similar set of assumptions This relationship may be expressed in the form,
C.M.Comiskey and H.Ruskin
where a(t) is the rate of appearance of newly diagnosed ADDS cases, h(t) is the incidence of HTV cases and/fr) is the incubation period distribution. Clearly for any two of a(t), h(t) or fit) known, then the third can be derived. To describe, then, the rate of appearance of new AIDS cases we adopt three forms for a(t) that reflect the empirical data and are similar to those suggested by Isham (1988). We have, a{t) = Ooexp(a,/ - a-/)
(3)
a(t) = c0 + cxt + c-2?-
(4)
The incubation period distribution has been studied by Blythe and Anderson (1988), Medley et al. (1987) and by Anderson and Medley (1988). To model rapid progression rates to the disease the latter suggest a Weibull distribution with mean fi = 7.3 years and having the general form,
fit) =
-(
CABIOS
AIDS in Ireland: the reporting delay distribution and the implementation of integral equation models C.M.Comiskey and H.J.Ruskin Abstract
provides a prediction of future numbers of both HTV and ADDS cases.
Faculty of Computing and Mathematical Sciences, Dublin G/v University, Dublin 9, Ireland
© Oxford University Press
a(t) = \'h(t-
u
(1)
579
Downloaded from http://bioinformatics.oxfordjournals.org/ at East Carolina University on July 8, 2015
This paper deals with two basic aspects concerning the modelling of AIDS incidence in the context of Irish data. We Adjustment for reporting delays describe initially the adjustment of the number of AIDS cases (Xjj) to allow for reporting delays, where a simple form of the We used the method proposed by Brookmeyer and Damiano likelihood function for the Xjj is supported by GLIM. (1989) to estimate reporting delays in Ireland. AIDS cases are Subsequently, we consider the accessibility of numerical solution typically cross-classified by length of reporting delay and date (through a NAG routine) of the integral equation models of diagnosis so that models of log-linear form, appropriate for generated by the back-projection method for the adjusted AIDS contingency table data are considered (with cells equal to the number of cases). Dividing calendar time into intervals s.t. cases. Results for the Irish data are summarized for various [Tj _ |, Tj),j = 1, . . .,R, with TR the most recent report date, choices of the incidence distribution. we let Xy represent the number of AIDS cases diagnosed in the yth calendar interval which had a reporting delay in the Introduction interval [d, _ ,, d,), i = 1, . . . , / . The analysis for reporting Fundamental to the modelling of AIDS incidence is the need delays presented in Table I was based on cases reported to the both to obtain numerical solutions to the appropriate integral Irish Department of Health up to the end of March 1990. Delays equation models and to ensure the relevance of the data set. of all cases diagnosed up to 31 December 1989 were examined, In the latter case, incidence figures for new AIDS cases can a total of 125 correctly documented cases were observed. As typically be disorted by reporting delays (i.e. delays between the matrix of Irish data was sparse, reporting delay and date the diagnosis and the notification of the case) of anything up of diagnosis were grouped into 3 month intervals and the model to three or four years. Clearly, accurate representation of the fitted to the quarterly data. current status of diagnosed AIDS cases requires an adjustment Adjusted figures were fitted as below. Table I shows the of the reported figures. Brookmeyer and Damiano (1989) and reporting delay probabilities P, when the Brookmeyer and Downs el al. (1987) describe a method, based on the conditional Damiano model was fitted to the adjusted figures. This model likelihood, for the estimation of the reporting delay distribufitted well with a scaled deviance (x2) of 99.36 with 455 tion, and in the following we evaluate the results obtained using degrees of freedom. a generalized linear model approach. It is evident from Table I that almost 85% of all cases are Early figures for the number of AIDS cases in Europe reported within 6 months of diagnosis. (The maximum delay predicted exponential growth of the epidemic (Downs et al., of 42 months represents a case that was diagnosed early in the 1987) but while there is now reason to believe that early epidemic.) As numbers of cases in Ireland are relatively few projections were unduly pessimistic, the poor prognosis (WHO, 1989) and delays short, little adjustment was needed associated with the disease serves to emphasize the critical for the present annual AIDS incidence figures employed in the importance of accurate predictions for the future. The integral following analysis. equation model of back-projection has proved extremely useful both in terms of evaluating current data and in terms of The method of back-projection providing estimates of future cases (Brookmeyer and Gail, 1986, 1987, 1988; Isham, 1988; Anderson, 1989; Brookmeyer and Brookmeyer and Gail (1986, 1987, 1988), Isham (1988), Damiano, 1989). The method involves back-calculation from Anderson (1989) and Brookmeyer and Damiano (1989) have the number of diseased cases (suitably adjusted for reporting discussed the back-projection approach to investigating the delays), via the incubation period distribution to the number relationship between the incidence of AIDS cases, the incidence of previously infected cases. Alternatively, working forward of HFV infection and the distribution of the incubation period. from the current figures under a similar set of assumptions This relationship may be expressed in the form,
C.M.Comiskey and H.Ruskin
where a(t) is the rate of appearance of newly diagnosed ADDS cases, h(t) is the incidence of HTV cases and/fr) is the incubation period distribution. Clearly for any two of a(t), h(t) or fit) known, then the third can be derived. To describe, then, the rate of appearance of new AIDS cases we adopt three forms for a(t) that reflect the empirical data and are similar to those suggested by Isham (1988). We have, a{t) = Ooexp(a,/ - a-/)
(3)
a(t) = c0 + cxt + c-2?-
(4)
The incubation period distribution has been studied by Blythe and Anderson (1988), Medley et al. (1987) and by Anderson and Medley (1988). To model rapid progression rates to the disease the latter suggest a Weibull distribution with mean fi = 7.3 years and having the general form,
fit) =
-(
