Viet Chi Tran

HIV with contact tracing: a case study in approximate Bayesian computation.

Missing data is a recurrent issue in epidemiology where the infection process may be partially observed. Approximate Bayesian computation (ABC), an alternative to data imputation methods such as Markov chain Monte Carlo (MCMC) integration, is proposed for making inference in epidemiological models. It is a likelihood-free method that relies exclusively on numerical simulations. ABC consists in computing a distance between simulated and observed summary statistics and weighting the simulations according to this distance. We propose an original extension of ABC to path-valued summary statistics, corresponding to the cumulated number of detections as a function of time. For a standard compartmental model with Suceptible, Infectious and Recovered individuals (SIR), we show that the posterior distributions obtained with ABC and MCMC are similar. In a refined SIR model well suited to the HIV contact-tracing data in Cuba, we perform a comparison between ABC with full and binned detection times. For the Cuban data, we evaluate the efficiency of the detection system and predict the evolution of the HIV-AIDS disease. In particular, the percentage of undetected infectious individuals is found to be of the order of 40%.

Hepatitis C treatment as prevention of viral transmission and liver‐related morbidity in persons who inject drugs

Hepatitis C virus (HCV) seroprevalence remains high in people who inject drug (PWID) populations, often above 60%. Highly effective direct‐acting antiviral (DAA) regimens (90% efficacy) are becoming available for HCV treatment. This therapeutic revolution raises the possibility of eliminating HCV from this population. However, for this, an effective cascade of care is required. In the context of the available DAA therapies, we used a dynamic individual‐based model including a model of the PWID social network to simulate the impact of improved testing, linkage to care, and adherence to treatment, and of modified treatment recommendation on the transmission and on the morbidity of HCV in PWID in France. Under the current incidence and cascade of care, with treatment initiated at fibrosis stage ≥F2, HCV prevalence decreased from 42.8% to 24.9% (95% confidence interval: 24.8‐24.9) after 10 years. Changing treatment initiation criteria to treat from F0 was the only intervention leading to a substantial additional decrease in prevalence, which fell to 11.6% (95% CI: 11.6‐11.7) at 10 years. Combining this change with improved testing, linkage to care, and adherence to treatment decreased HCV prevalence to 7.0% (95% CI: 7.0‐7.1) at 10 years and avoided 15% (95% CI: 14‐17) and 29% (95% CI: 28‐30) of cirrhosis complications over 10 and 40 years, respectively. Conclusions: Major decreases in prevalent HCV infections occur only when treatment is initiated at early stages of fibrosis, suggesting that systematic treatment in PWID, where incidence remains high, would be beneficial. However, elimination within the 10 next years will be difficult to achieve using treatment alone, even with a highly improved cascade of care. (Hepatology 2016;63:1090–1101)

Large graph limit for an SIR process in random network with heterogeneous connectivity

We consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by Volz (2008).

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.

The 2D-directed spanning forest is almost surely a tree

We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF.

Dynamic modelling of hepatitis C virus transmission among people who inject drugs: a methodological review.

Equipment sharing among people who inject drugs (PWID) is a key risk factor in infection by hepatitis C virus (HCV). Both the effectiveness and cost–effectiveness of interventions aimed at reducing HCV transmission in this population (such as opioid substitution therapy, needle exchange programmes or improved treatment) are difficult to evaluate using field surveys. Ethical issues and complicated access to the PWID population make it difficult to gather epidemiological data. In this context, mathematical modelling of HCV transmission is a useful alternative for comparing the cost and effectiveness of various interventions. Several models have been developed in the past few years. They are often based on strong hypotheses concerning the population structure. This review presents compartmental and individual‐based models to underline their strengths and limits in the context of HCV infection among PWID. The final section discusses the main results of the papers.

Slow and fast scales for superprocess limits of age-structured populations

Abstract A superprocess limit for an interacting birth–death particle system modeling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence.

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.

Level sets estimation and Vorob'ev expectation of random compact sets

Abstract The issue of a “mean shape” of a random set X often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob’ev expectation E V ( X ) , which is closely linked to level or quantile sets. In this paper, we propose a consistent and ready to use estimator of E V ( X ) built from independent copies of X with spatial discretisation. The control of discretisation errors is handled with a mild regularity assumption on the boundary of X . Several examples are developed and an application to cosmological data is presented.

Predicting data saturation in qualitative surveys with mathematical models from ecological research

OBJECTIVE Sample size in surveys with open-ended questions relies on the principle of data saturation. Determining the point of data saturation is complex because researchers have information on only what they have found. The decision to stop data collection is solely dictated by the judgment and experience of researchers. In this article, we present how mathematical modeling may be used to describe and extrapolate the accumulation of themes during a study to help researchers determine the point of data saturation. STUDY DESIGN AND SETTING The model considers a latent distribution of the probability of elicitation of all themes and infers the accumulation of themes as arising from a mixture of zero-truncated binomial distributions. We illustrate how the model could be used with data from a survey with open-ended questions on the burden of treatment involving 1,053 participants from 34 different countries and with various conditions. The performance of the model in predicting the number of themes to be found with the inclusion of new participants was investigated by Monte Carlo simulations. Then, we tested how the slope of the expected theme accumulation curve could be used as a stopping criterion for data collection in surveys with open-ended questions. RESULTS By doubling the sample size after the inclusion of initial samples of 25 to 200 participants, the model reliably predicted the number of themes to be found. Mean estimation error ranged from 3% to 1% with simulated data and was <2% with data from the study of the burden of treatment. Sequentially calculating the slope of the expected theme accumulation curve for every five new participants included was a feasible approach to balance the benefits of including these new participants in the study. In our simulations, a stopping criterion based on a value of 0.05 for this slope allowed for identifying 97.5% of the themes while limiting the inclusion of participants eliciting nothing new in the study. CONCLUSION Mathematical models adapted from ecological research can accurately predict the point of data saturation in surveys with open-ended questions.