Elementary proof of convergence to the mean-field model for the SIR process.

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi: 10.1093/imamat/hxw010 , 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.

Population and dyadic-based seroincidence of herpes simplex virus-2 and syphilis in southern India.

OBJECTIVES: Herpes simplex virus-2 (HSV-2) and syphilis are associated with increased risk of HIV, highlighting the importance of understanding their transmission dynamics. In India, most studies of HSV-2 and syphilis incidence are in high-risk populations and may not accurately reflect infectious activity. In this study, we aim to define HSV-2/syphilis incidence and risk factors in a population sample. METHODS: We conducted a longitudinal population-based survey in Andhra Pradesh, India, in two rounds: 2004-2005 and 2010-2011. Sociodemographic and behavioural data were collected, and dried blood spots tested for HSV-2 and Treponema pallidum IgG. After calculating sexually transmitted infection (STI) incidence, associated factors were assessed using modified Poisson regression and within-couple transmission rates modelled using seroconcordance/discordance data. RESULTS: 12,617 adults participated at baseline with 8494 at follow-up. Incidence of HSV-2 and syphilis per 1000 person-years was 25.6 (95% CI 24.1 to 27.2) and 3.00 (95% CI 2.52 to 3.54). Incidence of HSV-2 was higher in women vs. men (31.1 vs. 20.2) and in rural vs urban residents (31.1 vs 19.0) (p<0.05 for both). STI seroincidence increased in a step-wise fashion with age and was associated with spousal seropositivity for both sexes (incidence rate ratio (IRR) 2.59 to 6.78). Within couples the rate of transmission per 1000 couple-years from men to women vs. women to men was higher for HSV-2 (193.3 vs. 119.0) compared with syphilis (27.6 vs. 198.8), p<0.05 for both. CONCLUSIONS: HSV-2 has higher incidence among subpopulations such as women, rural residents and older-aged individuals, suggesting a need for more generalised STI prevention approaches among populations traditionally considered low risk.

Effectiveness of a no-sex or safe-sex month in reducing HIV transmission.

OBJECTIVE: To build a deterministic compartmental model for exploring the effects on the transmission of human immunodeficiency virus (HIV) of a population abstaining from sex or practising only "safe" sex for one month each year. METHODS: A model of HIV transmission was built to simulate the effects of the intervention (i.e. an annual no-sex or safe-sex month in which no transmission occurred) in three countries, under several optimistic assumptions. The reduction in the modelled annual incidence of transmission that was attributable to this "test" intervention was compared with that seen with an alternative intervention. In the latter, monthly incidences of transmission were each reduced by one twelfth, so that, essentially, the month-long interruption was spread evenly across a full year. FINDINGS: Over the first modelled year, the test intervention averted only 2.5% (Kenya), 3.3% (South Africa) and 1.6% (Swaziland) more HIV infections than the alternative interruption. According to the model, if the test intervention were repeated each January, it would avert only 2% (Kenya), 2% (South Africa) and 1% (Swaziland) more HIV infections over 5 years than the alternative intervention. CONCLUSION: Although it did not appear markedly more effective than the alternative intervention, the test intervention may still be more feasible and therefore worthwhile. Before the test intervention can be recommended, the cost-effectiveness and feasibility of such an annual month-long break in HIV transmission need to be assessed and compared with those of other interventions that may reduce new HIV infections, such as circumcision and concurrency-reduction campaigns.

Formulating adaptive radiation therapy (ART) treatment planning into a closed-loop control framework.

While ART has been studied for years, the specific quantitative implementation details have not. In order for this new scheme of radiation therapy (RT) to reach its potential, an effective ART treatment planning strategy capable of taking into account the dose delivery history and the patient's on-treatment geometric model must be in place. This paper performs a theoretical study of dynamic closed-loop control algorithms for ART and compares their utility with data from phantom and clinical cases. We developed two classes of algorithms: those Adapting to Changing Geometry and those Adapting to Geometry and Delivered Dose. The former class takes into account organ deformations found just before treatment. The latter class optimizes the dose distribution accumulated over the entire course of treatment by adapting at each fraction, not only to the information just before treatment about organ deformations but also to the dose delivery history. We showcase two algorithms in the class of those Adapting to Geometry and Delivered Dose. A comparison of the approaches indicates that certain closed-loop ART algorithms may significantly improve the current practice. We anticipate that improvements in imaging, dose verification and reporting will further increase the importance of adaptive algorithms.

Optimal mix of screening and contact tracing for endemic diseases.

Two common means of controlling infectious diseases are screening and contact tracing. Which should be used, and when? We consider the problem of determining the cheapest mix of screening and contact tracing necessary to achieve a desired endemic prevalence of a disease or to identify a specified number of cases. We perform a partial equilibrium analysis of small-scale interventions, assuming that prevalence is unaffected by the intervention; we develop a full equilibrium analysis where we compare the long-term cost of various combinations of screening and contact tracing needed to achieve a given equilibrium prevalence; and we solve the problem of minimizing the total costs of identifying and treating disease cases plus the cost of untreated disease cases. Our analysis provides several insights. First, contact tracing is only cost effective when prevalence is below a threshold value. This threshold depends on the relative cost per case found by screening versus contact tracing. Second, for a given contact tracing policy, the screening rate needed to achieve a given prevalence or identify a specified number of cases is a decreasing function of disease prevalence. As prevalence increases above the threshold (and contact tracing is discontinued), the screening rate jumps discontinuously to a higher level. Third, these qualitative results hold when we consider unchanged or changed prevalence, and short-term or long-term costs.

Forward reachable sets: Analytically derived properties of connected components for dynamic networks.

Formal analysis of the emergent structural properties of dynamic networks is largely uncharted territory. We focus here on the properties of forward reachable sets (FRS) as a function of the underlying degree distribution and edge duration. FRS are defined as the set of nodes that can be reached from an initial seed via a path of temporally ordered edges; a natural extension of connected component measures to dynamic networks. Working in a stochastic framework, we derive closed-form expressions for the mean and variance of the exponential growth rate of the FRS for temporal networks with both edge and node dynamics. For networks with node dynamics, we calculate thresholds for the growth of the FRS. The effects of finite population size are explored via simulation and approximation. We examine how these properties vary by edge duration and different cross-sectional degree distributions that characterize a range of scientifically interesting normative outcomes (Poisson and Bernoulli). The size of the forward reachable set gives an upper bound for the epidemic size in disease transmission network models, relating this work to epidemic modeling (Ferguson, 2000; Eames, 2004).

A Data-Driven Simulation of HIV Spread Among Young Men Who Have Sex With Men: Role of Age and Race Mixing and STIs.

OBJECTIVE: Young men who have sex with men (YMSM) in the United States have a high HIV incidence with substantial racial disparities that are poorly understood. We use a data-driven simulation model to understand the impact of network-level mechanisms and sexually transmitted infections on the spread of HIV among YMSM. METHODS: We designed and parameterized a stochastic agent-based network simulation model using results of a longitudinal cohort study of YMSM in Chicago. Within this model, YMSM formed and dissolved partnerships over time, and partnership types were stratified by length of partnership, sex, and age of the partner. In each partnership, HIV, gonorrhea, and chlamydia could be transmitted. Counterfactual scenarios were run to examine drivers of HIV. RESULTS: Over a 15-year simulation, the HIV epidemic among YMSM continued to rise, with Latino/white YMSM facing a steeper increase in the HIV burden compared with black YMSM. YMSM in partnerships with older MSM, in particular black YMSM with older black MSM, were at highest risk for HIV, and 1 infection prevented with an older partner would prevent 0.8 additional infections among YMSM. Additionally, racial disparities in HIV were driven by differences in the HIV prevalence of YMSM partners. Finally, of all HIV infections among YMSM, 14.6% were attributable to NG and CT infections. CONCLUSION: Network-level mechanisms and sexually transmitted infections play a significant role in the spread of HIV and in racial disparities among YMSM. HIV prevention efforts should target YMSM across race, and interventions focusing on YMSM partnerships with older MSM might be highly effective.

Spectrum reconstruction from dose measurements as a linear inverse problem.

There are three ways to determine the spectrum of a clinical photon beam: direct measurement, modelling the source and reconstruction from ion-chamber measurements. We focus on reconstruction because the necessary equipment is readily available and it provides independent confirmation of source models for a given machine. Reconstruction methods involve measuring the dose in an ion chamber after the beam passes through an attenuator. We gain information about the spectrum from measurements using attenuators of differing compositions and thicknesses since materials have energy dependent attenuation. Unlike the procedures used in other papers, we do not discretize or parametrize the spectrum. With either of these two approximations, reconstruction is a least squares problem. The forward problem of going from a spectrum to a series of dose measurements is a linear operator, with the composition and thickness of the attenuators as parameters. Hence the singular value decomposition (SVD) characterizes this operator. The right singular vectors form a basis for the spectrum, and, at first approximation, only those corresponding to singular values above a threshold are measurable. A more rigorous error analysis shows with what confidence different components of the spectrum can be measured. We illustrate this theory with simulations and an example utilizing six sets of dose measurements with water and lead as attenuators.

The importance of extended high viremics in models of HIV spread in South Africa.

Recent studies found a substantial fraction of 'extended high viremics' among HIV-1 subtype C, the most common subtype in southern Africa. Extended high viremics are HIV infected individuals who maintain a high viral load for a longer time period than usual after the initial infection. They are more infectious during this period, and their infection progresses to full-blown AIDS and death much faster than usual. This study investigates the impact of extended high viremics on the spread of the HIV epidemic in South Africa. We develop a simple deterministic compartmental model for HIV infection that includes extended high viremics. As the available data on extended high viremics are limited, we parameterize this model using only the fraction of extended high viremics among new infections and the reduced life-span of extended high viremics. We find that without extended high viremics, the HIV prevalence in South Africa would have remained close to its 1990 level, instead of increasing to the current epidemic levels. We also find that the greater the fraction of extended high viremics among susceptibles, the greater the steady-state HIV prevalence and the more sensitive the steady-state prevalence is to the HIV transmission probability. These results suggest that extended high viremics have an impact on the HIV epidemic in South Africa; justify the need for comprehensive epidemiological studies since the current data is limited; and suggest that future models of HIV for southern Africa should explicitly model extended high viremics.

The cost-effectiveness of expanded HIV screening in the United States.

OBJECTIVE: The current Centers of Disease Control and Prevention (CDC) guidelines from 2006 recommend a one-time test for low-risk individuals and annual testing for those at high risk. These guidelines may not be aggressive enough, even for those at low risk of infection, due to the earlier initiation of HAART and a movement towards a test-and-treat environment. We evaluated the optimal testing frequencies for various risk groups in comparison to the CDC recommendations. METHODS: We build a deterministic mathematical model optimizing the tradeoff between the societal cost of testing and the benefits over a patient's lifetime of earlier diagnosis. RESULTS: Under a test-and-treat scenario with immediate initiation of HAART, the optimal testing frequency is every 2.4 years for low-risk (0.01% annual incidence) individuals; every 9 months for moderate risk (0.1% incidence) individuals; and every 3 months for high-risk (1.0% incidence) individuals. The incremental cost-effectiveness of the optimal policy is $ 36 ,342/quality-adjusted life-years (QALY) for low-risk individuals and $ 45 ,074/QALY for high-risk individuals compared with 20-year and annual testing, respectively. CONCLUSION: The current CDC guidelines for HIV testing are too conservative, and more frequent testing is cost-effective for all risk groups.

Sex role segregation and mixing among men who have sex with men: implications for biomedical HIV prevention interventions.

OBJECTIVE: Men who have sex with men (MSM) practice role segregation - insertive or receptive only sex positions instead of a versatile role - in several international settings where candidate biomedical HIV prevention interventions (e.g., circumcision, anal microbicide) will be tested. The effects of these position-specific interventions on HIV incidence are modeled. MATERIALS AND METHODS: We developed a deterministic compartmental model to predict HIV incidence among Indian MSM using data from 2003-2010. The model's sex mixing matrix was derived from network data of Indian MSM (n=4604). Our model captures changing distribution of sex roles over time. We modeled microbicide and circumcision efficacy on trials with heterosexuals. RESULTS: Increasing numbers of versatile MSM resulted in little change in HIV incidence over 20 years. Anal microbicides and circumcision would decrease the HIV prevalence at 10 years from 15.6% to 12.9% and 12.7% respectively. Anal microbicides would provide similar protection to circumcision at the population level despite lower modeled efficacy (54% and 60% risk reduction, respectively). Combination of the interventions were additive: in 5 years, the reduction in HIV prevalence of the combination (-3.2%) is almost the sum of their individual reductions in HIV prevalence (-1.8% and -1.7%). CONCLUSIONS: MSM sex role segregation and mixing, unlike changes in the sex role distribution, may be important for evaluating HIV prevention interventions in international settings. Synergies between some position-specific prevention interventions such as circumcision and anal microbicides warrant further study.

HIV status awareness, partnership dissolution and HIV transmission in generalized epidemics.

OBJECTIVES: HIV status aware couples with at least one HIV positive partner are characterized by high separation and divorce rates. This phenomenon is often described as a corollary of couples HIV Testing and Counseling (HTC) that ought to be minimized. In this contribution, we demonstrate the implications of partnership dissolution in serodiscordant couples for the propagation of HIV. METHODS: We develop a compartmental model to study epidemic outcomes of elevated partnership dissolution rates in serodiscordant couples and parameterize it with estimates from population-based data (Rakai, Uganda). RESULTS: Via its effect on partnership dissolution, every percentage point increase in HIV status awareness reduces HIV incidence in monogamous populations by 0.27 percent for women and 0.63 percent for men. These effects are even larger when the assumption of monogamy can be relaxed, but are moderated by other behavior changes (e.g., increased condom use) in HIV status aware serodiscordant partnerships. When these behavior changes are taken into account, each percentage point increase in HIV status awareness reduces HIV incidence by 0.13 and 0.32 percent for women and men, respectively (assuming monogamy). The partnership dissolution effect exists because it decreases the fraction of serodiscordant couples in the population and prolongs the time that individuals spend outside partnerships. CONCLUSION: Our model predicts that elevated partnership dissolution rates in HIV status aware serodiscordant couples reduce the spread of HIV. As a consequence, the full impact of couples HTC for HIV prevention is probably larger than recognized to date. Particularly high partnership dissolution rates in female positive serodiscordant couples contribute to the gender imbalance in HIV infections.

Exploring the relative costs of contact tracing for increasing HIV case finding in sub-Saharan countries.

BACKGROUND: Contact tracing (CT) has rarely been used to improve HIV case finding in sub-Saharan Africa because of concerns regarding privacy protection and possibly high costs. METHODS: We estimate the relative cost of identifying an undiagnosed HIV infection through CT compared with client-initiated voluntary counseling and testing (VCT) and door-to-door provider-initiated testing (PIT). We used data from the Likoma Network Study, a study of sexual networks and HIV infection conducted on the small island of Likoma in northern Malawi, to inform these calculations. RESULTS: In Likoma, the probability that partners of HIV index cases could be traced and that they would consent to HIV testing and counseling was high and varied by partner type. HIV prevalence ranged from 48.1% to 66.7% among the partners who were tested. Per newly diagnosed infection, CT is rarely a cheaper case finding approach than VCT in populations with HIV prevalence > 5%. In populations with HIV prevalence < 5%, CT is an attractive case-finding approach relative to VCT when few HIV-infected individuals are aware of their status. Compared with door-to-door PIT, CT is almost always preferred when the population prevalence is below 10%, unless CT costs are prohibitively high. When HIV prevalence is >10%, providing CT for current spouses of index cases remains an attractive approach to HIV case finding. CONCLUSIONS: CT could complement client-initiated VCT or door-to-door PIT in a large number of sub-Saharan populations affected by generalized epidemics of varying magnitudes.

The reliability of sexual partnership histories: implications for the measurement of partnership concurrency during surveys.

OBJECTIVES: To measure the reliability of sexual partnership histories collected during survey interviews and to assess the impact of measurement error on survey estimates of partnership concurrency. METHODS: We used sociocentric data collected on Likoma Island (Malawi). Up to five of the respondents' most recent sexual partners were identified in population rosters. We assessed interpartner agreement (IPA) in reports of sexual partnerships (i.e. whether partners concordantly report that they have had sexual relations with each other) and its association with respondent and partnership characteristics. We estimated the extent of bias in the point prevalence of concurrency and the duration of overlap between concurrent partnerships according to two scenarios: one in which only partnerships reported by both partners were considered as 'true' ('concordant scenario'), and one in which partnerships reported by either partner were included ('complete scenario'). FINDINGS: IPA was low in nonmarital relations, but was significantly higher in ongoing than in dissolved nonmarital relations. IPA was further associated with the number of other partners the respondents or their partner(s) had, as well as with the duration of ongoing partnerships. Biases in measurements of the prevalence of concurrent partnerships were large: concurrent partnerships were rare in the concordant scenario, but common in the complete scenario. This was particularly true among never married women. Estimates of the average duration of overlap between concurrent partnerships derived from self-reported survey data were also biased, particularly among married respondents. CONCLUSION: Future empirical tests of the 'concurrency hypothesis' and interventions targeting concurrent partnerships should take reporting biases into account.

Cost-effective control of chronic viral diseases: finding the optimal level of screening and contact tracing.

Chronic viral diseases such as human immunodeficiency virus (HIV) and hepatitis B virus (HBV) afflict millions of people worldwide. A key public health challenge in managing such diseases is identifying infected, asymptomatic individuals so that they can receive antiviral treatment. Such treatment can benefit both the treated individual (by improving quality and length of life) and the population as a whole (through reduced transmission). We develop a compartmental model of a chronic, treatable infectious disease and use it to evaluate the cost and effectiveness of different levels of screening and contact tracing. We show that: (1) the optimal strategy is to get infected individuals into treatment at the maximal rate until the incremental health benefits balance the incremental cost of controlling the disease; (2) as one reduces the disease prevalence by moving people into treatment (which decreases the chance that they will infect others), one should increase the level of contact tracing to compensate for the decreased effectiveness of screening; (3) as the disease becomes less prevalent, it is optimal to spend more per case identified; and (4) the relative mix of screening and contact tracing at any level of disease prevalence is such that the marginal efficiency of contact tracing (cost per infected person found) equals that of screening if possible (e.g., when capacity limitations are not binding). We also show how to determine the cost-effective equilibrium level of disease prevalence (among untreated individuals), and we develop an approximation of the path of the optimal prevalence over time. Using this, one can obtain a close approximation of the optimal solution without having to solve an optimal control problem. We apply our methods to an example of hepatitis B virus.

Contact tracing to control infectious disease: when enough is enough.

Contact tracing (also known as partner notification) is a primary means of controlling infectious diseases such as tuberculosis (TB), human immunodeficiency virus (HIV), and sexually transmitted diseases (STDs). However, little work has been done to determine the optimal level of investment in contact tracing. In this paper, we present a methodology for evaluating the appropriate level of investment in contact tracing. We develop and apply a simulation model of contact tracing and the spread of an infectious disease among a network of individuals in order to evaluate the cost and effectiveness of different levels of contact tracing. We show that contact tracing is likely to have diminishing returns to scale in investment: incremental investments in contact tracing yield diminishing reductions in disease prevalence. In conjunction with a cost-effectiveness threshold, we then determine the optimal amount that should be invested in contact tracing. We first assume that the only incremental disease control is contact tracing. We then extend the analysis to consider the optimal allocation of a budget between contact tracing and screening for exogenous infection, and between contact tracing and screening for endogenous infection. We discuss how a simulation model of this type, appropriately tailored, could be used as a policy tool for determining the appropriate level of investment in contact tracing for a specific disease in a specific population. We present an example application to contact tracing for chlamydia control.