Estimating the stage-specific numbers of HIV infection using a Markov model and back-calculation

Estimating seroconversion rates accounting for repeated infections by approximate Bayesian computation

This study presents a novel approach for inferring the incidence of infections by employing a quantitative model of the serum antibody response. Current methodologies often overlook the cumulative effect of an individual's infection history, making it challenging to obtain a marginal distribution for antibody concentrations. Our proposed approach leverages approximate Bayesian computation to simulate cross-sectional antibody responses and compare these to observed data, factoring in the impact of repeated infections. We then assess the empirical distribution functions of the simulated and observed antibody data utilizing Kolmogorov deviance, thereby incorporating a goodness-of-fit check. This new method not only matches the computational efficiency of preceding likelihood-based analyses but also facilitates the joint estimation of antibody noise parameters. The results affirm that the predictions generated by our within-host model closely align with the observed distributions from cross-sectional samples of a well-characterized population. Our findings mirror those of likelihood-based methodologies in scenarios of low infection pressure, such as the transmission of pertussis in Europe. However, our simulations reveal that in settings of higher infection pressure, likelihood-based approaches tend to underestimate the force of infection. Thus, our novel methodology presents significant advancements in estimating infection incidence, thereby enhancing our understanding of disease dynamics in the field of epidemiology.

Modeling methods for estimating HIV incidence: a mathematical review

Estimating HIV incidence is crucial for monitoring the epidemiology of this infection, planning screening and intervention campaigns, and evaluating the effectiveness of control measures. However, owing to the long and variable period from HIV infection to the development of AIDS and the introduction of highly active antiretroviral therapy, accurate incidence estimation remains a major challenge. Numerous estimation methods have been proposed in epidemiological modeling studies, and here we review commonly-used methods for estimation of HIV incidence. We review the essential data required for estimation along with the advantages and disadvantages, mathematical structures and likelihood derivations of these methods. The methods include the classical back-calculation method, the method based on CD4+ T-cell depletion, the use of HIV case reporting data, the use of cohort study data, the use of serial or cross-sectional prevalence data, and biomarker approach. By outlining the mechanistic features of each method, we provide guidance for planning incidence estimation efforts, which may depend on national or regional factors as well as the availability of epidemiological or laboratory datasets.

Extending Bayesian back-calculation to estimate age and time specific HIV incidence

CD4-based multi-state back-calculation methods are key for monitoring the HIV epidemic, providing estimates of HIV incidence and diagnosis rates by disentangling their inter-related contribution to the observed surveillance data. This paper, extends existing approaches to age-specific settings, permitting the joint estimation of age- and time-specific incidence and diagnosis rates and the derivation of other epidemiological quantities of interest. This allows the identification of specific age-groups at higher risk of infection, which is crucial in directing public health interventions. We investigate, through simulation studies, the suitability of various bivariate splines for the non-parametric modelling of the latent age- and time-specific incidence and illustrate our method on routinely collected data from the HIV epidemic among gay and bisexual men in England and Wales.

Estimation of the burden of people living with Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome (HIV/AIDS) in Kerala state, India

Background

Worldwide, 36.7 million people were infected with Acquired Immunodeficiency Syndrome (AIDS) by the end of 2015. Over the period 2007 to 2015, there was a declining trend in the prevalence of adult Human Immunodeficiency Virus (HIV) in the state of Kerala, India. The current study aims to find a suitable statistical modelling technique for the distribution of HIV incubation time and predict the cumulative number of AIDS cases.

Materials and Methods

The requisite data were obtained from the Kerala State AIDS Control Society (KSACS) for the years 2007 to 2015. To assess the distribution of HIV incubation time, the data of 22 HIV-infected Keralite patients were retrieved from the medical records of a teaching hospital. Data included age, gender, and incubation time. The back-calculation method was utilized to predict the cumulative HIV/AIDS cases.

Results

The estimated total cumulative AIDS cases in Kerala for the years 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 and 2015 were found to be 35,777, 48,944, 62,039, 45,669, 45,668, and 43,605, 42,377, 39,362, 37,617, 39,583, 25,414 respectively using back-calculation method with Weibull (2) incubation time distribution. The mean incubation time of the total HIV cases (male and female) was 4.4 years which indicates a rapid progression of the disease in the state of Kerala.

Conclusion

The back-calculation method is a powerful tool to estimate the cumulative frequency of AIDS cases; which predicted a declining HIV trend among Keralites. Moreover, the Weibull distribution is the best fitted distribution for HIV incubation time in our population.

Modeling of HIV/AIDS dynamic evolution using non-homogeneous semi-markov process

The purpose of this study is to model the progression of HIV/AIDS disease of an individual patient under ART follow-up using non-homogeneous semi-Markov processes. The model focuses on the patient's age as a relevant factor to forecast the transitions among the different levels of seriousness of the disease. A sample of 1456 patients was taken from a hospital record at Amhara Referral Hospitals, Amhara Region, Ethiopia, who were under ART follow up from June 2006 to August 2013. The states of disease progression adopted in the model were defined based on of the following CD4 cell counts: >500 cells/mm(3) (SI); 349 to 500 cells/mm(3) (SII); 199 to 350 cells/mm(3)(SIII); ≤200 cells/mm(3) (SIV); and death (D). The first four states are referred as living states. The probability that an HIV/AIDS patient with any one of the living states will transition to the death state is greater with increasing age, irrespective of the current state and age of the patient. More generally, the probability of dying decreases with increasing CD4 counts over time. For an HIV/AIDS patient in a specific state of the disease, the probability of remaining in the same state decreases with increasing age. Within the living states, the results show that the probability of being in a better state is non-zero, but less than the probability of being in a worse state for all ages. A reliability analysis also revealed that the survival probabilities are all declining over time. Computed conditional probabilities show differential subject response that depends on the age of the patient. The dynamic nature of AIDS progression is confirmed with particular findings that patients are more likely to be in a worse state than a better one unless interventions are made. Our findings suggest that ongoing ART treatment services could be provided more effectively with careful consideration of the recent disease status of patients.

Probability of a false-negative HIV antibody test result during the window period: a tool for pre- and post-test counselling

Failure to understand the risk of false-negative HIV test results during the window period results in anxiety. Patients typically want accurate test results as soon as possible while clinicians prefer to wait until the probability of a false-negative is virtually nil. This review summarizes the median window periods for third-generation antibody and fourth-generation HIV tests and provides the probability of a false-negative result for various days post-exposure. Data were extracted from published seroconversion panels. A 10-day eclipse period was used to estimate days from infection to first detection of HIV RNA. Median (interquartile range) days to seroconversion were calculated and probabilities of a false-negative result at various time periods post-exposure are reported. The median (interquartile range) window period for third-generation tests was 22 days (19-25) and 18 days (16-24) for fourth-generation tests. The probability of a false-negative result is 0.01 at 80 days' post-exposure for third-generation tests and at 42 days for fourth-generation tests. The table of probabilities of falsely-negative HIV test results may be useful during pre- and post-test HIV counselling to inform co-decision making regarding the ideal time to test for HIV.

A multi-stage compartmental model for HIV-infected individuals: I--waiting time approach

Traditionally, epidemic processes have focused on establishing systems of differential-difference equations governing the number of individuals at each stage of the epidemic. Except for simple situations such as when transition rates are linear, these equations are notoriously intractable mathematically. In this work, the process is described as a compartmental model. The model also allows for individuals to go directly from any prior compartment directly to a final stage corresponding to death. This allows for the possibility that individuals can die earlier due to some non-disease related cause. Then, the model is based on waiting times in each compartment. Survival probabilities of moving from a given compartment to another compartment are established. While our approach can be used for general epidemic processes, our framework is for the HIV/AIDS process. It is then possible to establish the impact of the HIV/AIDS epidemic process on, e.g., insurance premiums and payouts and health-care costs. The effect of changing model parameter values on these entities is investigated.

Bayesian back-calculation using a multi-state model with application to HIV

Back-calculation is a method of obtaining estimates of the number of infections of a disease over time. Data on an endpoint of the disease, together with knowledge of the time from infection to endpoint, allows reconstruction of the incidence of infection. The technique has had much success when applied to the HIV epidemic, using incidence of AIDS diagnoses to inform past HIV infections. In recent years, the period from infection to AIDS has changed considerably due to new regimes of anti-viral therapies. This has led to attempts to use incidence of first positive HIV test as an alternative basis for back-calculation. Developing on earlier work, this paper explores the feasibility of a multi-state formulation of the back-calculation method that models the disease and diagnosis processes and uses HIV diagnoses as an endpoint. Estimation is carried out in a Bayesian framework, which naturally allows incorporation of external information to inform the diagnosis probabilities. The idea is illustrated on data from the HIV epidemic in homosexuals in England and Wales.

Estimating HIV Incidence and Detection Rates From Surveillance Data

BACKGROUND

Markov models that incorporate HIV test information can increase precision in estimates of new infections and permit the estimation of detection rates. The purpose of this study was to assess the functioning of a Markov model for estimating new HIV infections and HIV detection rates in Louisiana using surveillance data.

METHODS

We expanded a discrete-time Markov model by accounting for the change in AIDS case definition made by the Centers for Disease Control and Prevention in 1993. The model was applied to quarterly HIV/AIDS surveillance data reported in Louisiana from 1981 to 1996 for various exposure and demographic subgroups. When modeling subgroups defined by exposure categories, we adjusted for the high proportion of missing exposure information among recent cases. We ascertained sensitivity to changes in various model assumptions.

RESULTS

The model was able to produce results consistent with other sources of information in the state. Estimates of new infections indicated a transition of the HIV epidemic in Louisiana from (1) predominantly white men and men who have sex with men to (2) women, blacks, and high-risk heterosexuals. The model estimated that 61% of all HIV/AIDS cases were detected and reported by 1996, yet half of all HIV/non-AIDS cases were yet to be detected. Sensitivity analyses demonstrated that the model was robust to several uncertainties.

CONCLUSIONS

In general, the methodology provided a useful and flexible alternative for estimating infection and detection trends using data from a U.S. surveillance program. Its use for estimating current infection will need further exploration to address assumptions related to newer treatments.

Timing of human immunodeficiency virus type 1 (HIV-1) transmission from mother to child: bayesian estimation using a mixture

The timing of mother-to-child HIV transmission is not directly observable but influences the infected child's viral and immune status in the neonatal period. A hierarchical model was developed in a Bayesian framework to 'back-calculate' the timing of HIV-1 transmission from mother to child from the virological and immunological kinetics in the infected infant. Joint evolution of viral markers and immune response was modelled as a continuous time Markov process. The modelling of the period from infection to birth was based on a mixture of three distributions taking into account the various mother-to-child transmission pathways: In utero (early or late in gestation) and intrapartum (during the delivery process), integrating the fact that transmission is a continuum during the pregnancy. Gibbs sampling was used to estimate the marginal posterior distributions of the transition intensities between stages of HIV infection and those of the individual times from infection to birth. We applied our model to data on 135 perinatally HIV-1-infected children included in the French Prospective Study on Pediatric HIV infection. The model suggested that transmission occurred late in utero during the last month of pregnancy and that the day of delivery was a particularly critical time in HIV-1 transmission from mother to child. The paper ends with a discussion of model assumptions and a comparison with results obtained using a non-parametric method.

A partial ranking method for identifying repeated inclusion of individuals in anonymized HIV infection reports

Diagnoses of HIV infection are reported to the Public Health Laboratory Service (PHLS) AIDS Centre under a voluntary surveillance scheme. Names are not held in the data set, but the date of birth of the individual concerned is usually available. This paper describes a statistical method for identifying whether there are likely to be individuals repeatedly represented in the resulting data set, which is considered by birth year. A partial ordering method is used that is especially useful for years where the number of birth years in the sample is too small for chi2 tests to be used. At the 5% level, one of the five birth years tested in the data supplied to us by the PHLS shows evidence of more replication than would be expected from independent random sampling from the population. The results are compared with an alternative maximum-likelihood-based test that reaches the same conclusions. Maximum likelihood methods are further used to estimate the percentage of overcounting of individuals in the sample at 2.7%.

Hospital care for persons with AIDS in the European Union

This study estimates the current and future hospital resources for AIDS patients in the European Union (EU), using multinational scenario analysis (EU Concerted Action BMH1-CT-941723). In collaboration with another EU-project ('Managing the Costs of HIV Infection'), six national European studies on the utilization of hospital care for AIDS have been selected to provide the data for our analysis. The selection criteria involve recentness, quality, comparability, accessibility and representativeness. Baseline hospital resource utilization is estimated for hospital inpatient days and outpatient contracts, using a standardized approach controlling for two severity stages of AIDS (chronic stage and late stage). The epidemiological part of the study is based on standard models for backcalculating HIV incidence and projecting AIDS incidence, prevalence and mortality. In the next step, baseline resource utilization is linked to epidemiological information in a mixed prevalence and mortality-based approach. Several scenarios render different future epidemiological developments and hospital resource needs. For the year 1999, hospital bed needs of 10,000-12,700 in the EU are indicated, representing an increase of 20-60% compared to the estimated current (1995) level. The projected range for 1999 corresponds to a maximum of 0.65% of all hospital beds available in the EU. The growth in the number of outpatient hospital contacts is projected to possibly exceed that of inpatient days up to 1.82 million in 1999. Our methodology illustrates that estimation of current and future hospital care for AIDS has to be controlled for severity stages, to prevent biases. Further application of the multinational approach is demonstrated through a 'what-if' analysis of the potential impact of combination triple therapy for HIV/AIDS. Estimation of the economic impact of other diseases could as well benefit from the severity-stages approach.

Reconstruction and prediction of the HIV/AIDS epidemic among adults in the European Union and in the low prevalence countries of central and eastern Europe

OBJECTIVES

To reconstruct the HIV epidemic and to provide forecasts of AIDS incidence among adults in the European Union (EU) and in a group of low prevalence (LP) countries of central and eastern Europe (including the Asian republics of the former Soviet Union).

METHODS

An empirical Bayesian back-calculation method was applied to AIDS incidence data reported by 31 March 1994. The HIV-infection curve was modelled as a yearly step function and a seven-stage Markov model of disease progression, incorporating effects of pre-AIDS treatment, was used. Estimation was by penalized maximum likelihood with empirical Bayesian smoothing. Data were analysed by transmission group and, within the EU, by country. Predictions of AIDS cases to 1998 were made assuming constant annual HIV incidence from 1993 onwards.

RESULTS

Estimated HIV prevalences per 100,000 population aged 15-59 years were, at 31 December 1993, 198 (n = 447,800) in the EU and 2.7 (n = 6840) in the 22 LP countries, with increases of 41% (EU) and 71% (LP) between 1989 and 1993. Among homo/bisexual men in the EU, prevalence appears to have stabilized since 1989 and AIDS incidence appears to be reaching a peak. Among all prevalent HIV infections in the EU, 42% were estimated to be among injecting drug users, 25% among homo-/bisexual men and 18% among persons infected heterosexually, compared with 29%, 35% and 19%, respectively, in the LP countries. Without allowing for the 1993 revision of the case definition, annual AIDS incidence is predicted to increase, between 1994 and 1998, by 24% in the EU and by 48% in the LP countries, with the largest percentage increases among heterosexually-infected persons.

CONCLUSION

The overall HIV prevalence rate is estimated to have been about 70-fold lower in the LP countries than in the EU in the early 1990s, but to be increasing much more rapidly in the former. Moreover, recent reports of rapidly increasing HIV infection rates suggest that back-calculation may seriously underestimate the size of the epidemic in the LP countries. Implementation of effective preventive measures is urgent if large-scale epidemics are to be avoided in the presently LP countries of the European region.

Computing back-calculation estimates of AIDS epidemic

The method of back-calculation estimates the number of HIV infections from AIDS incidence data and projects future AIDS incidence. We explore a conditional likelihood approach for computing estimates of the number of HIV infections and the parameters in the epidemic density. This method is asymptotically equivalent to the usual likelihood method. The asymptotic normal distribution of the estimates facilitates the computation of confidence intervals. We compute standard deviations for the estimates of HIV incidence and project AIDS incidence from the underlying multinomial distributions. We illustrate the methods with applications to AIDS data in the United States.

Back-projection of German AIDS data using information on dates of tests

The paper presents an application of back-projection methods to the reported AIDS data which are collected by the AIDS Center in the Federal Office of Health, Berlin. The analysis is based on all data reported up to 31 December 1992. The data are broken down by major risk groups. Correction for reporting delays takes into account temporal changes in their distribution. The paper applies the EMS algorithm for the estimation of HIV incidence. The incubation period is modelled according to a convolution of several exponential distributions which describe time dependent phenomena like change of case definition and therapy effects. For each case it is reported whether an antibody test has been performed before diagnosis, and if yes, at what time this test was performed. The modelling of the incubation distribution takes into account the transition from a state 'not yet tested' into a state 'tested'. Only tested individuals in three pre-AIDS states are eligible for treatment. The model allows us to estimate not only the current total HIV prevalence but also a breakdown into those that are not yet tested, those that are tested but not yet treated and those that are under treatment. The results depend on the assumptions about the effect of treatment and on the degree of smoothing applied in the EMS algorithm.

Estimates of disease incidence in women based on antenatal or neonatal seroprevalence data: HIV in New York City

Piecewise constant incidence models were developed to estimate the force of infection in women from age- and time-specific antenatal or neonatal seroprevalence data. Differential inclusion of infected women in sero-surveys compared to uninfected women was taken into account, with respect to both changes in inclusion rate following infection, and changes in relative inclusion rate over calendar time. These models were applied to anonymous HIV seroprevalence data collected from neonates born to black and Hispanic women in New York City 1988-1992, with incidence and fertility parameters estimated by maximum likelihood. Estimates of inclusion rate parameters accorded well with what is known about the natural history of HIV. The data could not distinguish between additive and multiplicative combination of the effects of age and time on incidence. Incidence was strongly dependent on age with the highest incidence in women aged 20-34 years. There was strong evidence that incidence had been falling in Hispanic women since 1982-1984. The results illustrate the extent to which trends in incidence over time may be confounded by changes in the relative inclusion rate of infected and uninfected women.

Using Markov processes to describe the prognosis of HIV-1 infection

In 1989, Turner and colleagues proposed a new method for automated classification and prediction of outcomes for hospitalized patients with AIDS, based on severity of illness. The authors have confirmed this system, which includes three main stages and 20 substages, in a cohort of HIV-1-infected symptomatic patients admitted to the Miami Veteran's Affairs Medical Center (VAMC) since 1988. They propose a new model of Markov illness-and-death processes and use it to describe the disease progression of the patient cohort. The new Markov model also measures transitions between stages, including risk factors associated with the speed of transitions. Significant differences in survival experiences were found from the different disease stages. The estimated mean (median) survival times were: from stage 1, 21.0 (17.8) months; from stage 2, 10.5 (8.4) months; and from stage 3, 6.7 (4.7) months. In addition, age at HIV-1 diagnosis and homosexual behavior were significant prognostic factors for disease progression and survival. The results indicate that the combination of a severity-of-illness clinical staging system with Markov illness-and-death process modeling is particularly useful for the evaluation of prognostic factors influencing the course of HIV-1 disease progression.

Smoothed nonparametric back-projection of AIDS incidence data with adjustment for therapy

Back-projection is a major tool for assessing the extent of the HIV epidemic. Its application relies on a model for the incubation period of AIDS into which the administration and effect of therapy has been incorporated. We propose a compartmental model with proportional transition rates to describe variation and changes in the duration of the incubation period. The model is easily related to available data and offers fast computation. Its detailed and direct reflection of the administration and effect of therapy makes it relatively easy to ensure that therapy has been accommodated to an appropriate extent. The effect of therapy on back-projection estimates of the HIV epidemic curve is demonstrated with an application to Australian AIDS incidence data.

A review and synthesis of the HIV/AIDS epidemic as a multi-stage process

As our understanding of the dynamics of the transmission of the HIV virus toward diagnosis of AIDS increases, numerous models are being developed to help explain the underlying mechanism. We show that many of the models established so far in the literature are examples of a general class of multi-stage epidemic models that are themselves right-shift processes as developed in 1969 by Severo. We focus attention on those models that can be described as three-stage, four-stage, five-stage, etc. models.