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Modelers typically use detailed simulation models and vary the fraction vaccinated to study outbreak control. However, there is currently no guidance for modelers on how much detail (i.e., heterogeneity) is necessary and how large a population to simulate. We provide theoretical and numerical guidance for those decisions and also analyze the benefit of a faster public health response through a stochastic simulation model in the case of measles in the United States. Theoretically, we prove that the outbreak size converges as the simulation population increases and that the outbreaks are slightly larger with a heterogeneous community structure. We find that the simulated outbreak size is not sensitive to the size of the simulated population beyond a certain size. We also observe that in case of an outbreak, a faster public health response provides benefits similar to increased vaccination. Insights from this study can inform the control and elimination measures of the ongoing coronavirus disease (COVID-19) as measles has shown to have a similar structure to COVID-19.

Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given [Formula: see text] red points and [Formula: see text] blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless [Formula: see text]. We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points.

A recent study showed that progestogen-only injectable hormonal contraception (POIHC) doubles the risk of HIV transmission. This may affect contraceptive use and HIV-related outcomes, if women switch away from POIHC. A deterministic compartmental model of individuals aged 15–49 distinguishing gender and HIV status was used to simulate various contraceptive use scenarios. We specifically tracked HIV prevalence, new infections, HIV-related deaths, vertical transmission, and births over a 15-year period for five African countries. Stopping POIHC use will result in a large increase in births and vertical transmission. Switching from POIHC to other contraceptives limits these increases while still improving HIV outcomes.

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

We consider the problem of optimal decision making under uncertainty but assume that the decision maker’s utility function is not completely known. Instead, we consider all the utilities that meet some criteria, such as preferring certain lotteries over other lotteries and being risk averse, S-shaped, or prudent. These criteria extend the ones used in the first- and second-order stochastic dominance framework. We then give tractable formulations for such decision-making problems. We formulate them as robust utility maximization problems, as optimization problems with stochastic dominance constraints, and as robust certainty equivalent maximization problems. We use a portfolio allocation problem to illustrate our results. This paper was accepted by Dimitris Bertsimas, optimization.

The use of a stochastic dominance constraint to specify risk preferences in a stochastic program has been recently proposed in the literature. Such a constraint requires the random outcome resulting from one’s decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. This article proposes a constraint using a different version of multivariate stochastic dominance. This version is natural due to its connection to expected utility maximization theory and relatively tractable. In particular, it is shown that such a constraint can be formulated with linear constraints for the second-order dominance relation and with mixed-integer constraints for the first-order relation. This is in contrast with a constraint on second-order positive linear dominance, for which no efficient algorithms are known. The proposed formulations are tested in the context of two applications: budget a...

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).

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.

Concurrent partnerships (CPs) can accelerate the transmission of HIV (and other sexually transmitted infections) in a population [1–4]. CPs have been described as the “key driver” of generalized HIV epidemics [5]. Long-term CPs (i.e., partnerships that overlap for months, possibly years) in particular, may play a crucial role in connecting the sexual networks that transmit HIV [6]. Despite a recent surge of interest in targeting CPs for HIV prevention [7], the evidence that they are an important risk factor of HIV transmission remains limited [8–12]. The lack of association between partnership concurrency and HIV infection in empirical investigations could in part be due to the poor quality of survey data on CPs. To estimate the extent of CPs, survey respondents are typically asked questions about their 3–5 most recent sexual partners, including the dates at first and last sex, as well as whether the relationship with a given partner is still ongoing. These questions are used to check whether relationship intervals overlap [13, 14]. Such data are potentially affected by large biases [15–19], but few studies have evaluated the impact of measurement errors on survey estimates of partnership concurrency [14, 20], particularly in sub-Saharan settings. Only Morris and O'Gorman [15] argued that “measurement errors introduce a slight positive bias in estimates of the prevalence of CPs, and a slight negative bias in the length of the overlap [between CPs]”. This pioneering study suffers from two major limitations however: first, it only considered the impact of date heaping and recall errors on survey reports of partnership dates, but it did not consider the likely hypothesis that partnership histories (on which estimates are based) may not be reliable; second, it hypothesized that reporting errors occurred at random among population members, in ways that are unrelated to parameters of partnership concurrency. If patterns of misreporting are associated with partnership duration (for example), biases in concurrency measurements are likely to be significantly larger than previously thought. We use sociocentric network data collected on Likoma (an island located in the northern region of Lake Malawi) to (i) assess the inter-partner reliability of partnership histories collected during sexual behaviors surveys, and (ii) test whether reliability varies with partnership duration and timing. In doing so, and contrary to Morris and O'Gorman's previous assessment, we show that biases in survey measurements of CPs are large and of unknown direction.

Prediction markets with buyers and sellers of contracts on multiple outcomes are shown to have unique equilibrium prices, which can be computed in polynomial time.

Previous work has developed algorithms for finding an equitable convex partition that partitions the plane into n convex pieces each containing an equal number of red and blue points. Motivated by a vehicle routing heuristic, we look at a related problem where each piece must contain one point and an equal fraction of the area of some convex polygon. We first show how algorithms for solving the older problem lead to approximate solutions for this new equitable convex partition problem. Then we demonstrate a new algorithm that finds an exact solution to our problem in O ( N n log N ) time or operations, where n is the number of points, m the number of vertices or edges of the polygon, and N := n + m the sum.

Ye [?Competitive Communication Spectrum Economy and Equilibrium,? 2008, working paper] and Lin [?Budget Allocation in a Competitive Communication Spectrum Economy,? EURASIP J. Adv. Signal Process., Article ID: 963717, vol. 2009, p. 12, Sep. 2009] have shown that dynamic spectrum management (DSM) using the market competitive equilibrium (CE), which sets a price for transmission power on each channel, leads to better system performance in terms of the total data transmission rate (by reducing cross talk), than using the Nash equilibrium (NE). But how to achieve such a CE is an open problem. We show that the CE is the solution of a linear complementarity problem (LCP) and can be computed efficiently. We propose a decentralized tatonnement process for adjusting the prices to achieve a CE. We show that under reasonable conditions, any tatonnement process converges to the CE. The conditions are that users of a channel experience the same noise levels and that the crosstalk effects between users are low-rank and weak.

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 (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.

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.

We analyze a problem in computer network security, wherein packet filters are deployed to defend a network against spoofed denial of service attacks. Information on the Internet is transmitted by the exchange of IP packets, which must declare their origin and destination addresses. A route-based packet filter verifies whether the purported origin of a packet is correct with respect to the current route map. We examine the optimization problem of finding a minimum cardinality set of nodes to filter in the network such that no spoofed packet can reach its destination. We prove that this problem is NP-hard, and derive properties that explicitly relate the filter placement problem to the vertex cover problem. We identify topologies and routing policies for which a polynomial-time solution to the minimum filter placement problem exists, and prove that under certain routing conditions a greedy heuristic for the filter placement problem yields an optimal solution.

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.

It is common to use a compartmental, fluid model described by a system of ordinary differential equations (ODEs) to model disease spread. In addition to their simplicity, these models are also the mean-field approximations of more accurate stochastic models of disease spread on contact networks. For the simplest case of a stochastic susceptible-infected-susceptible (SIS) process (infection with recovery) on a complete network, it has been shown that the fraction of infected nodes converges to the mean-field ODE as the number of nodes increases. However the proofs are not simple, requiring sophisticated probability, partial differential equations (PDE), or infinite systems of ODEs. We provide a short proof in this case for convergence in mean-square on finite time-intervals using a system of two ODEs and a moment inequality and also obtain the first lower bound on the expected fraction of infected nodes.

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 (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., Comment: 18 pages, 2 figures

Background: Empirical estimates of the female-to-male ratio of infections in generalized HIV epidemics in sub-Saharan Africa range from 1.31 in Zambia to 2.21 in Ivory Coast. Inequalities in the gender ratio of infections can arise because of differences in exposure (to HIV-positive partners), susceptibility (given exposure), and survival (once infected). Differences in susceptibility have to date received most attention, but neither the relatively high gender ratio of infections nor the heterogeneity in empirical estimates is fully understood. Objective: Demonstrate the relevance of partnership network attributes and sexual mixing patterns to gender differences in the exposure to HIV-positive partners and the gender ratio of infections. Methods: Agent-based simulation model built in NetLogo. Results: The female-to-male ratio of infections predicted by our model ranges from 1.13 to 1.75. Gender-asymmetric partnership concurrency, rapid partnership turnover, elevated partnership dissolution in female-positive serodiscordant couples, and lower partnership re-entry rates among HIV-positive women can produce (substantial) differences in the gender ratio of infections. Coital dilution and serosorting have modest moderating effects. Conclusions: Partnership network attributes and sexual mixing patterns can have a considerable effect on the gender ratio of HIV infections. We need to look beyond individual behavior and gender differences in biological susceptibility if we are to fully understand, and remedy, gender inequalities in HIV infection in generalized epidemics.

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.

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.

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.

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 testand-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 $36342/quality-adjusted life-years (QALY) for low-risk individuals and $45074/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. 2013 Wolters Kluwer Health | Lippincott Williams & Wilkins AIDS 2013, 27:795‐801

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.

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.

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.

Purpose: While ART has been studied for years, the specific quantitative implementation details have not. In order for this new scheme of radiation therapy to reach its potential, an effective ART planning strategy capable of taking into account the dosedelivery history and patient's on‐treatment geometric model must be in place. This work performs a study of dynamic closed‐loop control algorithms for ART and demonstrates their utilities with data from phantom and clinical cases with on‐treatment cone‐beam CTimages.Method: In closed‐loop control, the controller is not run just once but repeatedly, each time receiving the current state of the system as its input. To meet the requirements of different clinical applications, two classes of algorithms are developed: those Adapting to Changing Geometry (ACG) and those Adapting to Geometry and DeliveredDose (AGDD). The former takes into account organ deformations found just before treatment. The latter optimizes the dose distribution accumulated over the entire course of treatment by adapting at each fraction not only to the anatomic information just before treatment but also to the dosedelivery history. The closed‐loop algorithms are showcased by phantom and clinical cases. Results: A comparison of the approaches with conventional open‐loop IMRT without adaptively incorporating feedback information indicates that closed‐loop ART may significantly improve the current practice. In both phantom and clinical studies, AGDD outperforms ACG algorithms in three aspects: target dose coverage, sensitive‐structure sparing, and steeper gradients around the tumor. Within the AGDD formalism, it is beneficial not to correct all the previous dosimetric errors at once right after the information is available but over a number of fractions until next set of feedback data is available. Conclusion:ART with closed‐loop dynamic algorithms substantially improve the dose distribution. In addition, the differing performance of the specific implementations shows that the algorithmic details matter.

Purpose: Current radiation treatment plans do not adapt to errors (and other deviations from the target dose distribution) in already delivered fractions. We assume that the dose actually delivered (including any errors) is known after its delivery. We devise a dynamic control algorithm compensating for these errors by adapting the plan for the current fraction. Method and Materials: We wrote MATLAB code that models treatment of a 2D phantom using the beamlet model. We used SNOPT (a commercial SQP optimization code) to select beamlet weights that minimize the weighted quadratic deviation from some desired dose. We also apply our algorithm to a head and neck case. Suppose the prescription dose is D* over N fractions; Di * is the dose planned for fraction i; and Di is the dose actually delivered in fraction i. Our baseline (algorithm) chooses for fraction i the feasible plan, Di *, that minimizes the weighted quadratic deviation from D*/N. Our adaptive algorithm selects for fraction k the feasible plan Dk * that minimizes the weighted quadratic deviation from D*(k/N)−D 1−…−Dk−1. Simulation determines the delivereddoseDi from the anticipated doseDi * by adding noise and incorporating setup error (translation and rotation of the patient) and tissue distortion caused by small organ motion. We compare the adaptive algorithm to the baseline and an algorithm with perfect foresight. For the algorithms, we compare the DVH of the cumulative doseD 1+…+DN and the margin needed to achieve a satisfactory cumulative delivereddose.Results: We achieved significant improvement in the objective function while delivering more dose to the tumor and less to the sensitive structures on our test case with setup errors of ±1cm and ±2°. Conclusion: This new paradigm of adaptive radiation therapy (ART) holds significant promise for us to improve the current radiation therapy.

Purpose: Current radiation treatment practice neither detects nor adapts to intrafraction organ motion beyond gating. We develop a simple optimization scheme for 4D IMRT which doesn't rely on gating and simulate its performance in the presence of the anticipated organ motion and unanticipated setup errors and tissue motion. Method and Materials: We wrote MATLAB code modeling treatment of a 2D phantom using the beamlet model. We also used geometry from a lung case. SNOPT (a commercial SQP optimization code) selects beamlet weights minimizing the weighted quadratic deviation from some desired dose. Suppose the beam‐on time is divided into N phases and the prescription dose is D*. For location r in phase i, let Di *(r) be the planned dose;Di (r) the actually delivered dose; and r=Ai (v) the anticipated location of voxel v. Our two baseline algorithms use static plans (Di *=Dj * for any phases i,j) and gating (Di *=0 for phases i≠1). In both cases we choose feasible D 1*+…+DN * minimizing the weighted quadratic deviation from D*. Our 4D algorithm selects (D 1 *,…,D N *)∈ arg min (D1∈ F,…, DN ∈ F) , Σ v α(v)(D1(A 1 (v))+…+DN(A N (v)) − D*(v)) 2 Simulation determines the delivered doseDi from the anticipated doseDi * by adding noise and incorporating setup error (translation and rotation of the patient) and tissue distortion caused by unanticipated small organ motion. For our algorithm and the baseline, we compare the DVH of the cumulative doseD 1+…+DN and the margin needed to achieve a satisfactory cumulative delivered dose.Results: We achieved significant improvement in the objective function (delivering more dose to the tumor and less to the organ) on our test case with 3cm motions. Conclusion: This new paradigm of 4D IMRT holds significant promise for improving the current radiation therapy.

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{0,1,\ldots,N\}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model., Comment: 14 pages, 4 figures, revised