HMisiri_Estimating HIV incidence from grouped cross-sectional data in settings where anti-retroviral therapy is provided

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  Abstract Prevalence and incidence are measures that are used for monitoring the occurrence of a disease. Prevalence can be computed from readily available cross-sectional data but incidence is traditionally computed from longitudinal data from longitudinal studies. Longitudinal studies are characterised by financial and logistical problems where as cross-sectional studies are easy to conduct. This paper introduces a new method for estimating HIV incidence from grouped cross-sectional sero-prevalence data from settings where antiretroviral therapy is provided to those who are eligible according to recommended criteria for the administration of such drugs.
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  • 1. Estimating HIV incidence from grouped cross-sectional data in settings where anti-retroviral therapy is provided Humphrey Misiri hmisiri@gmail.com Public Health Department, College of Medicine, University of Malawi, Blantyre, Malawi Key words: incidence, antiretroviral 1
  • 2. Abstract Prevalence and incidence are measures that are used for monitoring the occurrence of a disease. Prevalence can be computed from readily available cross-sectional data but incidence is traditionally computed from longitudinal data from longitudinal studies. Longitudinal studies are characterised by financial and logistical problems where as cross-sectional studies are easy to conduct. This paper introduces a new method for estimating HIV incidence from grouped cross-sectional sero-prevalence data from settings where antiretroviral therapy is provided to those who are eligible according to recommended criteria for the administration of such drugs. Introduction Antiretroviral therapy (ART) has helped to alleviate the suffering of AIDS patients in the world. In many countries, patients have access to ART. In Malawi, ART is also available for free but not all HIV positive persons have access to ART. By 2011, over 30% of HIV positive persons were on ART [1]. Incidence is a very important measure of disease occurrence. If the incidence of HIV is known, it is easy to monitor its spread. On the other hand, prevalence alone does not give complete information about the magnitude of the spread of HIV or any disease in general. Consider a virulent disease like Ebola which kills after just a few days from infection. Individuals who are infected with the Ebola virus die after a very short illness if no meaningful therapeutic intervention is available. In that case, prevalence can never give a true picture of the extent of an Ebola epidemic since those who die from the disease are never counted. As a result, a low prevalence of Ebola does not mean Ebola is about to be non- existent or is almost eradicated from a community. On the other hand, the incidence of Ebola 2
  • 3. is the best measure which can be used to monitor the disease since Ebola deaths are included in its computation. Consequently, incidence gives a true picture of an Ebola epidemic. In the same vein, HIV incidence gives a true picture of the spread of HIV in a community. Traditionally, incidence is computed from data from longitudinal studies. Unfortunately, there are many financial and logistical problems associated with conducting longitudinal studies. To avoid these drawbacks, a viable alternative is to estimate incidence from data from cross-sectional studies. Two good examples of methods for achieving this are models by Podgor and Leske (1986) and Misiri et al (2012)[2, 3]. These models produce estimates of incidence which are adjusted for differential mortality. Both approaches are for estimating HIV incidence where ART is not properly rolled out in the community. It is possible to estimate the incidence of HIV from cross-sectional data from a population where ART is provided. The aim of this paper is to introduce a new method of estimating HIV incidence in settings where ART is provided to HIV positive people who need it regardless of the extent of coverage of such services. This method also adjusts for differential mortality. Materials and Methods Motivation Podgor and Leske(1986) proposed a method for estimating incidence from grouped cross- sectional data [3]. In the spirit of Podgor and Leske(1986), we proceed to motivate our approach. Let be the rate of natural mortality, be the HIV incidence, be the rate of HIV mortality in the absence of ART, be the rate of recruitment to ARV therapy, be the rate of mortality among ART recipients. 3 1λ2λ3λ4λ5λ
  • 4. Let X1, X2, X3, X4 and X5 be independent random variables where X1 is the time to death from natural causes, X2 is the time to HIV infection, X3 is the time to death whilst HIV positive, X4 is the time to ART registration and X5 is the time to death whilst on ART. It follows from the above description that X1, X2, ... , X5 have exponential distributions with parameters and respectively. We will proceed by dividing the population into three strata namely: HIV negative persons, HIV positives on ART and HIV positives who are not on ART. Denote the total proportion of HIV positives by P0, the proportion of positives who are not on ART by P01 and the proportion of positives who are on ART by P02. Both P01 and P02 are proportions of the population. Consider an interval [x, x+t]. The number of HIV positives at the end of the interval is N1P1=N0P0S1+ N0(1-P0)S2 (1) where S1 is the probability of surviving the interval given that one entered the interval already infected. S2 is the probability of being infected in the interval given that one was HIV negative at the beginning of the interval. Furthermore, the number of HIV negatives at the end of the interval is N1(1-P1)=N0(1-P0)S3 (2) where S3 is the probability of surviving the interval without contracting HIV According to the relationship among these exponential random variables [3, 4] 4 4321 ,,, λλλλ 5λ
  • 5. , S2= and [3] In the interval [x, x+t], some people may have just been registered to receive ART but some were already registered prior to entering the interval. Therefore the formula in (1) above does not capture the number of infected people in [x, x+t] in a setting where ART is provided. If ART is provided, at the end of the interval there are two groups of HIV positive individuals namely those who are not on ART and those who are on ART. Not every infected person is eligible for ART. For example, an individual who gets infected with HIV in a 5-year interval can never be eligible for ART as the therapy is for HIV positives who are in a reasonably advanced stage of infection. Therefore, the number of HIV positive individuals who are on ART at the end of the interval is the sum of old HIV positives who entered the interval already on ART and those HIV positives who are newly registered to receive ART. This can be denoted by N0P02S4 + N0P01S5 (4) where S4 is the probability of surviving to the end of the interval whilst on ART given than one was already on ART at the beginning of the interval S5 is the probability of surviving to the end of the interval having been newly recruited to receive ART given than one was not on ART at the beginning of the interval Using the relationship between independent exponential random variables as described in Lagakos(1976) on pages 553 through 555[4], these probabilities are defined as follows: 5 ( ) ( )4343 1 0 431 )(1 λλλλ λλ +−+− =+−= ∫ edteS ( ) ( ) ( ) ( )321 2 1 0 2 213 3321 1 λλλ λ λ λλλ λλλλ −+ − =− +−− −−+− ∫ ee dte t ( ) ( )2121 1 0 213 )(1 λλλλ λλ +−+− =+−= ∫ edteS t
  • 6. . Therefore (4) becomes . (5) The number of HIV positives at the end of the interval is therefore . (6) The number of HIV negative persons at the end of the interval is N1(1-P1) =N0(1-P0)S6 (7) where S6 is the probability of remaining HIV negative having survived the interval Now, . Therefore the equation in (7) becomes (8) From (8) we have that . Therefore the left hand side of equation (6) becomes . Consequently, equation (6) becomes § From this expression we define a function as follows : 6 5 4 λ− = eS ( ) ( ) ( ) ( )543 4 1 0 45 435 5543 λλλ λ λ λλλ λλλλ −+ − == +−− −−+− ∫ ee dteS t ( ) ( ) ( )543 4 010020 435 5 λλλ λ λλλ λ −+ − + +−− − ee PNePN ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       −+ − ++      −+ − −+= +−− − +−− +− 543 4 010020 321 2 0001011 435 5 213 43 1 λλλ λ λλλ λ λλλ λ λλλ λλ ee PNePN ee PNePNPN ( ) ( )2121 1 0 216 )( λλλλ λλ +−+− ∫ =+= edteS ( ) ( ) ( )21 0011 11 λλ +− −=− ePNPN ( ) ( ) ( )21 1 0 01 1 1 λλ +− − − = e P P NN ( ) ( ) ( )21 1 1 0 0 1 1 λλ +− − − eP P P N ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       −+ − ++      −+ − += − − +−− − +−− +− +− 543 4 010020 321 2 010010 1 00 435 5 213 43 21 1 1 λλλ λ λλλ λ λλλ λ λλλ λλ λλ ee PNePN ee PNePN P ePP )( 2λf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )       −+ − −−      −+ − −−− − − = +−−+−− −+− +− 321 20 543 4010201 1 10 2 213435 543 21 1 1 1 λλλ λ λλλ λλ λλλλλλ λλλ λλ ee P ee PePeP P ePP f
  • 7. where 1- P1 > 0, and . Using the Newton-Raphson method, the value of can be estimated given appropriate data. According to the Newton-Raphson method: . The derivative of is Note that the graph of has an asymptote at . Because of this, it is possible forto have more than 1 root on either side of the asymptote. Nevertheless, we will retain the roots of which are to the right of the asymptote because these are the only values which satisfy the condition that = 0 given . The standard error of was estimated using the delta transformation. An explanation of how the formula for the standard error was derived is given in the Annex. Application of the method to population-based data from the Malawi Demographic Survey 2010 Description of the data The estimated population of Malawi in 2011 was 14,388,550[5]. The national prevalence of HIV was 10% in 2010[6]. The provision of ARV therapy in Malawi is overseen by the HIV Unit in the Ministry of Health and Population. By 2011, 382,953 people were on ARV therapy[1]. The remaining 1,055,902 were not on ARV therapy. In the same year, the number of deaths due to HIV was 43,000 [1]. 7 0321 >−+ λλλ 0543 >−+ λλλ 2λ ( ) ( ) ( ) ( )2 2 2 1 2 ' λ λ λλ f fnn −=+ ( )2λf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( )321 02 321 0 321 02 1 10 2 2132132121 111 1 1 ' λλλ λ λλλλλλ λ λ λλλλλλλλλλ −+ −− + −+ −− − −+ − − − − −= +−−+−−+−+− eePeePeP P ePP f ( )2λf 132 λλλ −=( )2λf ( )2λf ( )2λf 0321 >−+ λλλ 2λ
  • 8. From the ARV Supervision database for 2004-2009 which was maintained by the HIV Unit, in 2004 there were 3,262 ART registrations [7]. By the end of 2008, a total of 20,393 HIV positive persons were recruited to receive ARV therapy. This gives a recruitment rate () of 3,426 people per year on average. Studies [8, 9] conducted in Malawi found that ART reduces mortality by 10%[8]. Therefore given HIV mortality rates, the rate of mortality among those on ART is . The age-specific HIV sero-prevalence data analysed for this paper are extracted from the database of the Malawi Demographic and Health Survey (MDHS2010) which was conducted in 2010. The data are in Table 1 below. Table 1:Nationally representative HIV sero-prevalence data for Malawi, 2010 HIV- % HIV + Not on ARV % On ARV % Agegroup Number (p03) Number % Number (p01) Number (p02) 15-19 3208 0.022 71 0.02 2 63 0.02 0 8 0.002 20-24 2370 0.051 122 0.05 1 114 0.04 8 8 0.003 25-29 2141 0.108 232 0.10 8 197 0.09 2 35 0.016 30-34 1560 0.181 283 0.18 1 227 0.14 6 56 0.036 35-39 1224 0.246 301 0.24 6 232 0.19 0 69 0.056 40-44 870 0.247 215 0.24 7 155 0.17 8 60 0.069 45-49 817 0.193 158 0.19 3 95 0.11 6 63 0.077 50-54 295 0.129 38 0.12 9 25 0.08 5 13 0.044 In 1992, HIV was not endemic as it is today. Mortality, in general, was mainly due to causes other than HIV. As HIV spread throughout Malawi, HIV became the leading cause of 8 4λ 35 *9.0 λλ =
  • 9. mortality. The provision of ART to HIV positives has reversed this trend in mortality. Therefore, the mortality estimates for 1992 represent true natural mortality rates for Malawi which are not contaminated by HIV mortality. The source of HIV mortalities is a study by Crampin et al (2002). This study reports mortality rates for HIV persons not on ARV therapy from a study conducted in a typical rural setting representative of an average rural area in Malawi [10. These estimates represent HIV mortality rates in rural Malawi in the absence of ARV therapy. Table 2 below contains the natural and HIV mortality rates. Table 2: Natural and AIDS Mortality rates for Malawi Age group index Natural mortality rates() AIDS mortality rates() (j) Age group Men Women for men and women 1 15-19 0.0038 0.0053 0.0471 2 20-24 0.0041 0.0036 0.0593 3 25-29 0.0068 0.0068 0.0675 4 30-34 0.0084 0.0072 0.1354 5 35-39 0.0076 0.009 0.1354 6 40-44 0.0101 0.0089 0.1427 7 45-49 0.0097 0.0096 0.1427 8 50+ 0.0097 0.0096 0.2339 Results HIV incidence estimates for 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54 age groups are in Table 3. The 95% confidence interval for each estimate is also presented. The incidence estimates were obtained by using the Newton-Raphson method. The initial values of plucked into the Newton-Raphson algorithm were obtained by a combination of methods which include inspection , use of the R function uniroot and numerical search procedures. Table 3: Nationally representative HIV incidence estimates for Malawi 9 1λ 3λ 2λ
  • 10. Incidenc e 95% CI for Incidence Agegroup FOI SE per 5years Lower limit Upper Limit 15-19 0.0607 0.000358 61 60 61 20-24 0.0858 0.000779 86 84 87 25-29 0.1171 0.001806 117 114 121 30-34 0.1628 0.00157 163 160 166 35-39 0.1428 0.00108 143 141 145 40-44 0.1446 0.000897 145 143 146 45-49 0.1447 0.000755 145 143 146 50-54 - - - - The age group with the highest incidence is the 30-34 year age group. The smallest incidence is for the 15-19 year age group. Although 40-44 and 45-49 age groups have the same incidence estimate the two estimates are different correct to 6 decimal places. All the standard errors of the FOI estimates are very small. Furthermore, the 95 % confidence intervals for the 15-19 through 45-49 age groups are very narrow. Discussion This method is a very good way of estimating incidence from cross-sectional data. It is impossible to estimate the HIV incidence for the age group 50-54 years because the structure of the model does not permit it. Our new method relies heavily on the existence of the roots of. For the 50-54 year age group, no estimate is possible because of the nature of the model used. We tested the sensitivity of the method to the size of P01 and P02. According to our findings, big values of P01 and P02 resulted in whose roots were hard to estimate. In order to have reasonable smaller P01 and P02 for the Newton-Raphson method to converge efficiently, both parameters (P01 and P02) must be defined as proportions of the sample for each age 10 ( )2λf ( )2λf
  • 11. group. In any case, the number of people on ART is bound to be small, therefore as a fraction of the sample for each age group, this produces proportions which make it easy to achieve convergence when using the Newton - Raphson algorithm. The objective of the method is to produce incidence estimates. Therefore, defining P01 and P02 as proposed above does not make the results of the current method unusable. The reader who wants the proportions P01 and P02 to be defined otherwise can do so and can compute the proportions based on his own definitions from data [3]. The fact that all the confidence intervals were narrow can be explained by the size of the samples for each age group. All sample sizes were very big. In such cases, standard errors are very small. These affect the size of the margin of error. Eventually, confidence intervals computed from such standard errors are likewise narrow. Besides, the narrow confidence intervals are indicative of high precision in the estimation of FOI. Conclusion The novel method introduced in this paper is a very good approach for estimating HIV incidence from aggregated data collected from settings where ART is provided to HIV infected individuals. This method is timely as it comes at a time when provision of ART is rampant in many countries of the world. Competing interests There are no competing interests. Acknowledgments I am very grateful to ORC Macro International for allowing me to analyse the MDHS2010 data. Authors' contributions HM conceived the study, conceived the method, obtained the data, analyzed the data, drafted 11
  • 12. the manuscript and revised it. References 1. HIV Unit: 2012 Global AIDS Response Progress Report:Malawi Country Report for 2010 and 2011. Lilongwe,Malawi: Ministry of Health,Malawi Government; 2012. 2. Misiri HE, Edriss A, Aalen OO, Dahl FA: Estimation of HIV incidence in Malawi from cross-sectional population-based sero-prevalence data. Journal of the International AIDS Society 2012, 15(14). 3. Podgor MJ, Leske M: Estimating incidence from age-specific prevalence for irreversible diseases with differential mortality. Statistics in Medicine 1986, 5:573- 578. 4. Lagakos SW: A stochastic model for censored-survival data in the presence of an auxiliary variable. Biometrics 1976, 32(3):551-559. 5. "Population projections for Malawi." [http://www.nso.malawi.net/index.php? option=com_content&view=article&id=134%3Apopulation-projections-for- malawi&catid=8&Itemid=3. ] 6. National Statistical Office (NSO) ORC Macro: Malawi Demographic and Health Survey 2010. Zomba: National Statistical Office (NSO) and O. R. C. Macro; 2010. 7. HIV Unit. ART Supervision Database. Lilongwe: Ministry of Health and 12
  • 13. Population, Malawi Government; 2009. 8. Jahn A, Floyd S, Crampin AC, Mwaungulu F, Mvula H, Munthali F, McGrath N,, JMwalaso J MV, Mangongo B, Fine PEM, Glynn JR: Population-level effect of HIV on adult mortality and early evidence of reversal after introduction of antiretroviral therapy in Malawi. Lancet 2008, 371:1603-1611. 9. Floyd S, Molesworth A, Dube A, Banda E, Jahn A, Mwafulirwa C, Ngwira B, Branson K, Crampin AC, Zaba B, Glynn JR, French N : Population-level reduction in adult mortality after extension of free Anti-Retroviral Therapy provision into rural areas in Northern Malawi. PLoS ONE 2010, 5(10). 10. Crampin AC, Floyd S, Glynn JR, Sibande F, Mulawa D, Nyondo A, Broadbent P, Bliss L, Ngwira B, Fine PE: Long-term follow-up of HIV-positive and HIVnegative individuals in rural Malawi. AIDS 2002, 16:1545–1550. 13
  • 14. ANNEX Derivation of the standard error of Obviously, the force of infection (FOI) is the function of both P0 and P1. That is to say . Therefore to find the variance of we use the delta method of transformation. Using the delta method, . We will define a function y in this way: (1) Therefore. (2) Similarly, (3). It is also true that (4). Now and . The partial derivative is the quotient when the result in (2) is divided by the result in (3) above. Similarly, the partial derivative is the quotient when the result in (4) is divided by the result in (3) above. The variances of P0 is . Similarly the variance of P1 is . 14 2λ 2λ( )102 , PPf=λ 2λ ( ) ( ) ( )1 2 1 2 0 2 0 2 2 PVar P PVar P Var       ∂ ∂ +      ∂ ∂ = λλ λ ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ]213 435 543 21 321 2 5430 4 0201 01 1 11 1 1 λλλ λλλ λλλ λλ λλλ λ λλλ λ +−− +−− −+− +− − −+ +      −+ − − −− − − − = ee ee P ePeP PP eP y ( ) ( ) [ ] ( ) ( ) ( )      −+ − − −− − −= ∂ ∂ +−− −+− 543 2 0 4 02012 00 435 543 11 1 λλλ λ λλλ λλλ ee P ePeP PP y ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 321 2 321321 2 1 1 2 2132132121 . 1 λλλ λ λλλλλλ λ λ λλλλλλλλλλ −+ − + −+ − + −+ + − −= ∂ ∂ +−−+−−+−+− eeeee P ePy ( ) ( )1 2 1 1 1 11 2121 P e P eP P y − + − = ∂ ∂ −−−− λλλλ       ∂ ∂       ∂ ∂ = ∂ ∂ 2 0 0 2 λ λ y P y P       ∂ ∂       ∂ ∂ = ∂ ∂ 2 1 1 2 λ λ y P y P 0 2 P∂ ∂λ 1 2 P∂ ∂λ ( ) ( ) 0 00 0 1 N PP PVar − = ( ) ( ) 1 11 1 1 N PP PVar − =
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