Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

Viral load monitoring of antiretroviral therapy, cohort viral load and HIV transmission in Southern Africa: A mathematical modelling analysis

In low-income settings, treatment failure is often identified using CD4 cell count monitoring. Consequently, patients remain on a failing regimen, resulting in a higher risk of transmission. We investigated the benefit of routine viral load monitoring for reducing HIV transmission.

Timing of progression from Chlamydia trachomatis infection to pelvic inflammatory disease: a mathematical modelling study

Background Pelvic inflammatory disease (PID) results from the ascending spread of microorganisms from the vagina and endocervix to the upper genital tract. PID can lead to infertility, ectopic pregnancy and chronic pelvic pain. The timing of development of PID after the sexually transmitted bacterial infection Chlamydia trachomatis (chlamydia) might affect the impact of screening interventions, but is currently unknown. This study investigates three hypothetical processes for the timing of progression: at the start, at the end, or throughout the duration of chlamydia infection. Methods We develop a compartmental model that describes the trial structure of a published randomised controlled trial (RCT) and allows each of the three processes to be examined using the same model structure. The RCT estimated the effect of a single chlamydia screening test on the cumulative incidence of PID up to one year later. The fraction of chlamydia infected women who progress to PID is obtained for each hypothetical process by the maximum likelihood method using the results of the RCT. Results The predicted cumulative incidence of PID cases from all causes after one year depends on the fraction of chlamydia infected women that progresses to PID and on the type of progression. Progression at a constant rate from a chlamydia infection to PID or at the end of the infection was compatible with the findings of the RCT. The corresponding estimated fraction of chlamydia infected women that develops PID is 10% (95% confidence interval 7-13%) in both processes. Conclusions The findings of this study suggest that clinical PID can occur throughout the course of a chlamydia infection, which will leave a window of opportunity for screening to prevent PID.

Describing the progression from Chlamydia trachomatis and Neisseria gonorrhoeae to pelvic inflammatory disease: systematic review of mathematical modeling studies

Chlamydia screening is recommended to prevent pelvic inflammatory disease (PID). A systematic review was conducted to determine how the natural history of Chlamydia trachomatis or Neisseria gonorrhoeae infection and progression to PID have been described in mathematical modeling studies.

Resurgence of HIV infection among men who have sex with men in Switzerland: mathematical modelling study

Background New HIV infections in men who have sex with men (MSM) have increased in Switzerland since 2000 despite combination antiretroviral therapy (cART). The objectives of this mathematical modelling study were: to describe the dynamics of the HIV epidemic in MSM in Switzerland using national data; to explore the effects of hypothetical prevention scenarios; and to conduct a multivariate sensitivity analysis. Methodology/Principal Findings The model describes HIV transmission, progression and the effects of cART using differential equations. The model was fitted to Swiss HIV and AIDS surveillance data and twelve unknown parameters were estimated. Predicted numbers of diagnosed HIV infections and AIDS cases fitted the observed data well. By the end of 2010, an estimated 13.5% (95% CI 12.5, 14.6%) of all HIV-infected MSM were undiagnosed and accounted for 81.8% (95% CI 81.1, 82.4%) of new HIV infections. The transmission rate was at its lowest from 1995–1999, with a nadir of 46 incident HIV infections in 1999, but increased from 2000. The estimated number of new infections continued to increase to more than 250 in 2010, although the reproduction number was still below the epidemic threshold. Prevention scenarios included temporary reductions in risk behaviour, annual test and treat, and reduction in risk behaviour to levels observed earlier in the epidemic. These led to predicted reductions in new infections from 2 to 26% by 2020. Parameters related to disease progression and relative infectiousness at different HIV stages had the greatest influence on estimates of the net transmission rate. Conclusions/Significance The model outputs suggest that the increase in HIV transmission amongst MSM in Switzerland is the result of continuing risky sexual behaviour, particularly by those unaware of their infection status. Long term reductions in the incidence of HIV infection in MSM in Switzerland will require increased and sustained uptake of effective interventions.

Predicting the population impact of chlamydia screening programmes: comparative mathematical modelling study

BACKGROUND: Published individual-based, dynamic sexual network modelling studies reach different conclusions about the population impact of screening for Chlamydia trachomatis. The objective of this study was to conduct a direct comparison of the effect of organised chlamydia screening in different models. METHODS: Three models simulating population-level sexual behaviour, chlamydia transmission, screening and partner notification were used. Parameters describing a hypothetical annual opportunistic screening program in 16-24 year olds were standardised, whereas other parameters from the three original studies were retained. Model predictions of the change in chlamydia prevalence were compared under a range of scenarios. RESULTS: Initial overall chlamydia prevalence rates were similar in women but not men and there were age and sex-specific differences between models. The number of screening tests carried out was comparable in all models but there were large differences in the predicted impact of screening. After 10 years of screening, the predicted reduction in chlamydia prevalence in women aged 16-44 years ranged from 4% to 85%. Screening men and women had a greater impact than screening women alone in all models. There were marked differences between models in assumptions about treatment seeking and sexual behaviour before the start of the screening intervention. CONCLUSIONS: Future models of chlamydia transmission should be fitted to both incidence and prevalence data. This meta-modelling study provides essential information for explaining differences between published studies and increasing the utility of individual-based chlamydia transmission models for policy making.

Tracing of patients lost to follow-up and HIV transmission: Mathematical modelling study based on two large ART programmes in Malawi

OBJECTIVES: Treatment as prevention depends on retaining HIV-infected patients in care. We investigated the effect on HIV transmission of bringing patients lost to follow up (LTFU) back into care. DESIGN: Mathematical model. METHODS: Stochastic mathematical model of cohorts of 1000 HIV-infected patients on antiretroviral therapy (ART), based on data from two clinics in Lilongwe, Malawi. We calculated cohort viral load (CVL; sum of individual mean viral loads each year) and used a mathematical relationship between viral load and transmission probability to estimate the number of new HIV infections. We simulated four scenarios: 'no LTFU' (all patients stay in care); 'no tracing' (patients LTFU are not traced); 'immediate tracing' (after missed clinic appointment); and, 'delayed tracing' (after six months). RESULTS: About 440 of 1000 patients were LTFU over five years. CVL (million copies/ml per 1000 patients) were 3.7 (95% prediction interval [PrI] 2.9-4.9) for no LTFU, 8.6 (95% PrI 7.3-10.0) for no tracing, 7.7 (95% PrI 6.2-9.1) for immediate, and 8.0 (95% PrI 6.7-9.5) for delayed tracing. Comparing no LTFU with no tracing the number of new infections increased from 33 (95% PrI 29-38) to 54 (95% PrI 47-60) per 1000 patients. Immediate tracing prevented 3.6 (95% PrI -3.3-12.8) and delayed tracing 2.5 (95% PrI -5.8-11.1) new infections per 1000. Immediate tracing was more efficient than delayed tracing: 116 and to 142 tracing efforts, respectively, were needed to prevent one new infection. CONCLUSION: Tracing of patients LTFU enhances the preventive effect of ART, but the number of transmissions prevented is small.

Effectiveness and cost-effectiveness of traditional and new partner notification technologies for curable sexually transmitted infections: observational study, systematic reviews and mathematical modelling

BACKGROUND Partner notification is essential to the comprehensive case management of sexually transmitted infections. Systematic reviews and mathematical modelling can be used to synthesise information about the effects of new interventions to enhance the outcomes of partner notification. OBJECTIVE To study the effectiveness and cost-effectiveness of traditional and new partner notification technologies for curable sexually transmitted infections (STIs). DESIGN Secondary data analysis of clinical audit data; systematic reviews of randomised controlled trials (MEDLINE, EMBASE and Cochrane Central Register of Controlled Trials) published from 1 January 1966 to 31 August 2012 and of studies of health-related quality of life (HRQL) [MEDLINE, EMBASE, ISI Web of Knowledge, NHS Economic Evaluation Database (NHS EED), Database of Abstracts of Reviews of Effects (DARE) and Health Technology Assessment (HTA)] published from 1 January 1980 to 31 December 2011; static models of clinical effectiveness and cost-effectiveness; and dynamic modelling studies to improve parameter estimation and examine effectiveness. SETTING General population and genitourinary medicine clinic attenders. PARTICIPANTS Heterosexual women and men. INTERVENTIONS Traditional partner notification by patient or provider referral, and new partner notification by expedited partner therapy (EPT) or its UK equivalent, accelerated partner therapy (APT). MAIN OUTCOME MEASURES Population prevalence; index case reinfection; and partners treated per index case. RESULTS Enhanced partner therapy reduced reinfection in index cases with curable STIs more than simple patient referral [risk ratio (RR) 0.71; 95% confidence interval (CI) 0.56 to 0.89]. There are no randomised trials of APT. The median number of partners treated for chlamydia per index case in UK clinics was 0.60. The number of partners needed to treat to interrupt transmission of chlamydia was lower for casual than for regular partners. In dynamic model simulations, > 10% of partners are chlamydia positive with look-back periods of up to 18 months. In the presence of a chlamydia screening programme that reduces population prevalence, treatment of current partners achieves most of the additional reduction in prevalence attributable to partner notification. Dynamic model simulations show that cotesting and treatment for chlamydia and gonorrhoea reduce the prevalence of both STIs. APT has a limited additional effect on prevalence but reduces the rate of index case reinfection. Published quality-adjusted life-year (QALY) weights were of insufficient quality to be used in a cost-effectiveness study of partner notification in this project. Using an intermediate outcome of cost per infection diagnosed, doubling the efficacy of partner notification from 0.4 to 0.8 partners treated per index case was more cost-effective than increasing chlamydia screening coverage. CONCLUSIONS There is evidence to support the improved clinical effectiveness of EPT in reducing index case reinfection. In a general heterosexual population, partner notification identifies new infected cases but the impact on chlamydia prevalence is limited. Partner notification to notify casual partners might have a greater impact than for regular partners in genitourinary clinic populations. Recommendations for future research are (1) to conduct randomised controlled trials using biological outcomes of the effectiveness of APT and of methods to increase testing for human immunodeficiency virus (HIV) and STIs after APT; (2) collection of HRQL data should be a priority to determine QALYs associated with the sequelae of curable STIs; and (3) standardised parameter sets for curable STIs should be developed for mathematical models of STI transmission that are used for policy-making. FUNDING The National Institute for Health Research Health Technology Assessment programme.

Monitoring of antiretroviral therapy and mortality in HIV programmes in Malawi, South Africa and Zambia: mathematical modelling study

OBJECTIVES Mortality in patients starting antiretroviral therapy (ART) is higher in Malawi and Zambia than in South Africa. We examined whether different monitoring of ART (viral load [VL] in South Africa and CD4 count in Malawi and Zambia) could explain this mortality difference. DESIGN Mathematical modelling study based on data from ART programmes. METHODS We used a stochastic simulation model to study the effect of VL monitoring on mortality over 5 years. In baseline scenario A all parameters were identical between strategies except for more timely and complete detection of treatment failure with VL monitoring. Additional scenarios introduced delays in switching to second-line ART (scenario B) or higher virologic failure rates (due to worse adherence) when monitoring was based on CD4 counts only (scenario C). Results are presented as relative risks (RR) with 95% prediction intervals and percent of observed mortality difference explained. RESULTS RRs comparing VL with CD4 cell count monitoring were 0.94 (0.74-1.03) in scenario A, 0.94 (0.77-1.02) with delayed switching (scenario B) and 0.80 (0.44-1.07) when assuming a 3-times higher rate of failure (scenario C). The observed mortality at 3 years was 10.9% in Malawi and Zambia and 8.6% in South Africa (absolute difference 2.3%). The percentage of the mortality difference explained by VL monitoring ranged from 4% (scenario A) to 32% (scenarios B and C combined, assuming a 3-times higher failure rate). Eleven percent was explained by non-HIV related mortality. CONCLUSIONS VL monitoring reduces mortality moderately when assuming improved adherence and decreased failure rates.

Cost-effectiveness of point-of-care viral load monitoring of antiretroviral therapy in resource-limited settings: mathematical modelling study

BACKGROUND Monitoring of HIV viral load in patients on combination antiretroviral therapy (ART) is not generally available in resource-limited settings. We examined the cost-effectiveness of qualitative point-of-care viral load tests (POC-VL) in sub-Saharan Africa. DESIGN Mathematical model based on longitudinal data from the Gugulethu and Khayelitsha township ART programmes in Cape Town, South Africa. METHODS Cohorts of patients on ART monitored by POC-VL, CD4 cell count or clinically were simulated. Scenario A considered the more accurate detection of treatment failure with POC-VL only, and scenario B also considered the effect on HIV transmission. Scenario C further assumed that the risk of virologic failure is halved with POC-VL due to improved adherence. We estimated the change in costs per quality-adjusted life-year gained (incremental cost-effectiveness ratios, ICERs) of POC-VL compared with CD4 and clinical monitoring. RESULTS POC-VL tests with detection limits less than 1000 copies/ml increased costs due to unnecessary switches to second-line ART, without improving survival. Assuming POC-VL unit costs between US$5 and US$20 and detection limits between 1000 and 10,000 copies/ml, the ICER of POC-VL was US$4010-US$9230 compared with clinical and US$5960-US$25540 compared with CD4 cell count monitoring. In Scenario B, the corresponding ICERs were US$2450-US$5830 and US$2230-US$10380. In Scenario C, the ICER ranged between US$960 and US$2500 compared with clinical monitoring and between cost-saving and US$2460 compared with CD4 monitoring. CONCLUSION The cost-effectiveness of POC-VL for monitoring ART is improved by a higher detection limit, by taking the reduction in new HIV infections into account and assuming that failure of first-line ART is reduced due to targeted adherence counselling.

Reinfection by untreated partners of people treated for Chlamydia trachomatis and Neisseria gonorrhoeae: mathematical modelling study

OBJECTIVES Reinfection after treatment for Chlamydia trachomatis or Neisseria gonorrhoeae reduces the effect of control interventions. We explored the impact of delays in treatment of current partners on the expected probability of reinfection of index cases using a mathematical model. METHODS We used previously reported parameter distributions to calculate the probability that index cases would be reinfected by their untreated partners. We then assumed different delays between index case and partner treatment to calculate the probabilities of reinfection. RESULTS In the absence of partner treatment, the medians of the expected reinfection probabilities are 19.4% (IQR 9.2-31.6%) for C trachomatis and 12.5% (IQR 5.6-22.2%) for N gonorrhoeae. If all current partners receive treatment 3 days after the index case, the expected reinfection probabilities are 4.2% (IQR 2.1-6.9%) for C trachomatis and 5.5% (IQR 2.6-9.5%) for N gonorrhoeae. CONCLUSIONS Quicker partner referral and treatment can substantially reduce reinfection rates for C trachomatis and N gonorrhoeae by untreated partners. The formula we used to calculate reinfection rates can be used to inform the design of randomised controlled trials of novel partner notification technologies like accelerated partner therapy.

Direct and indirect effects of screening for Chlamydia trachomatis on the prevention of pelvic inflammatory disease: a mathematical modeling study

BACKGROUND Pelvic inflammatory disease (PID) results from the ascending spread of microorganisms, including Chlamydia trachomatis, to the upper genital tract. Screening could improve outcomes by identifying and treating chlamydial infections before they progress to PID (direct effect) or by reducing chlamydia transmission (indirect effect). METHODS We developed a compartmental model that represents a hypothetical heterosexual population and explicitly incorporates progression from chlamydia to clinical PID. Chlamydia screening was introduced, with coverage increasing each year for 10 years. We estimated the separate contributions of the direct and indirect effects of screening on PID cases prevented per 100,000 women. We explored the influence of varying the time point at which clinical PID could occur and of increasing the risk of PID after repeated chlamydial infections. RESULTS The probability of PID at baseline was 3.1% by age 25 years. After 5 years, the intervention scenario had prevented 187 PID cases per 100,000 women and after 10 years 956 PID cases per 100,000 women. At the start of screening, most PID cases were prevented by the direct effect. The indirect effect produced a small net increase in PID cases, which was outweighed by the effect of reduced chlamydia transmission after 2.2 years. The later that progression to PID occurs, the greater the contribution of the direct effect. Increasing the risk of PID with repeated chlamydial infection increases the number of PID cases prevented by screening. CONCLUSIONS This study shows the separate roles of direct and indirect PID prevention and potential harms, which cannot be demonstrated in observational studies.