Branching Process Model Research Articles

Early detection of cholera epidemics to support control in fragile states: estimation of delays and potential epidemic sizes

BackgroundCholera epidemics continue to challenge disease control, particularly in fragile and conflict-affected states. Rapid detection and response to small cholera clusters is key for efficient control before an epidemic propagates. To understand the capacity for early response in fragile states, we investigated delays in outbreak detection, investigation, response, and laboratory confirmation, and we estimated epidemic sizes. We assessed predictors of delays, and annual changes in response time.MethodsWe compiled a list of cholera outbreaks in fragile and conflict-affected states from 2008 to 2019. We searched for peer-reviewed articles and epidemiological reports. We evaluated delays from the dates of symptom onset of the primary case, and the earliest dates of outbreak detection, investigation, response, and confirmation. Information on how the outbreak was alerted was summarized. A branching process model was used to estimate epidemic size at each delay. Regression models were used to investigate the association between predictors and delays to response.ResultsSeventy-six outbreaks from 34 countries were included. Median delays spanned 1–2 weeks: from symptom onset of the primary case to presentation at the health facility (5 days, IQR 5–5), detection (5 days, IQR 5–6), investigation (7 days, IQR 5.8–13.3), response (10 days, IQR 7–18), and confirmation (11 days, IQR 7–16). In the model simulation, the median delay to response (10 days) with 3 seed cases led to a median epidemic size of 12 cases (upper range, 47) and 8% of outbreaks ≥ 20 cases (increasing to 32% with a 30-day delay to response). Increased outbreak size at detection (10 seed cases) and a 10-day median delay to response resulted in an epidemic size of 34 cases (upper range 67 cases) and < 1% of outbreaks < 20 cases. We estimated an annual global decrease in delay to response of 5.2% (95% CI 0.5–9.6, p = 0.03). Outbreaks signaled by immediate alerts were associated with a reduction in delay to response of 39.3% (95% CI 5.7–61.0, p = 0.03).ConclusionsFrom 2008 to 2019, median delays from symptom onset of the primary case to case presentation and to response were 5 days and 10 days, respectively. Our model simulations suggest that depending on the outbreak size (3 versus 10 seed cases), in 8 to 99% of scenarios, a 10-day delay to response would result in large clusters that would be difficult to contain. Improving the delay to response involves rethinking the integration at local levels of event-based detection, rapid diagnostic testing for cluster validation, and integrated alert, investigation, and response.

Coupled differentiation and division of embryonic stem cells inferred from clonal snapshots

The deluge of single-cell data obtained by sequencing, imaging and epigenetic markers has led to an increasingly detailed description of cell state. However, it remains challenging to identify how cells transition between different states, in part because data are typically limited to snapshots in time. A prerequisite for inferring cell state transitions from such snapshots is to distinguish whether transitions are coupled to cell divisions. To address this, we present two minimal branching process models of cell division and differentiation in a well-mixed population. These models describe dynamics where differentiation and division are coupled or uncoupled. For each model, we derive analytic expressions for each subpopulation’s mean and variance and for the likelihood, allowing exact Bayesian parameter inference and model selection in the idealised case of fully observed trajectories of differentiation and division events. In the case of snapshots, we present a sample path algorithm and use this to predict optimal temporal spacing of measurements for experimental design. We then apply this methodology to an in vitro dataset assaying the clonal growth of epiblast stem cells in culture conditions promoting self-renewal or differentiation. Here, the larger number of cell states necessitates approximate Bayesian computation. For both culture conditions, our inference supports the model where cell state transitions are coupled to division. For culture conditions promoting differentiation, our analysis indicates a possible shift in dynamics, with these processes becoming more coupled over time.

Implication of backward contact tracing in the presence of overdispersed transmission in COVID-19 outbreaks.

Introduction: Contact tracing has the potential to control outbreaks without the need for stringent physical distancing policies, e.g. civil lockdowns. Unlike forward contact tracing, backward contact tracing identifies the source of newly detected cases. This approach is particularly valuable when there is high individual-level variation in the number of secondary transmissions (overdispersion). Methods: By using a simple branching process model, we explored the potential of combining backward contact tracing with more conventional forward contact tracing for control of COVID-19. We estimated the typical size of clusters that can be reached by backward tracing and simulated the incremental effectiveness of combining backward tracing with conventional forward tracing. Results: Across ranges of parameter values consistent with dynamics of SARS-CoV-2, backward tracing is expected to identify a primary case generating 3-10 times more infections than average, typically increasing the proportion of subsequent cases averted by a factor of 2-3. The estimated number of cases averted by backward tracing became greater with a higher degree of overdispersion. Conclusion: Backward contact tracing can be an effective tool for outbreak control, especially in the presence of overdispersion as was observed with SARS-CoV-2.

Containment, Contact Tracing and Asymptomatic Transmission of Novel Coronavirus Disease (COVID-19): A Modelling Study.

When a novel infectious disease emerges, enhanced contact tracing and isolation are implemented to prevent a major epidemic, and indeed, they have been successful for the control of severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS), which have been greatly reduced without causing a global pandemic. Considering that asymptomatic and pre-symptomatic infections are substantial for the novel coronavirus disease (COVID-19), the feasibility of preventing the major epidemic has been questioned. Using a two-type branching process model, the present study assesses the feasibility of containing COVID-19 by computing the probability of a major epidemic. We show that if there is a substantial number of asymptomatic transmissions, cutting chains of transmission by means of contact tracing and case isolation would be very challenging without additional interventions, and in particular, untraced cases contribute to lowering the feasibility of containment. Even if isolation of symptomatic cases is conducted swiftly after symptom onset, only secondary transmissions after the symptom onset can be prevented.

Using contact data to model the impact of contact tracing and physical distancing to control the SARS-CoV-2 outbreak in Kenya

Background: Across the African continent, other than South Africa, COVID-19 cases have remained relatively low. Nevertheless, in Kenya, despite early implementation of containment measures and restrictions, cases have consistently been increasing. Contact tracing forms one of the key strategies in Kenya, but may become infeasible as the caseload grows. Here we explore different contact tracing strategies by distinguishing between household and non-household contacts and how these may be combined with other non-pharmaceutical interventions. Methods: We extend a previously developed branching process model for contact tracing to include realistic contact data from Kenya. Using the contact data, we generate a synthetic population of individuals and their contacts categorised by age and household membership. We simulate the initial spread of SARS-CoV-2 through this population and look at the effectiveness of a number of non-pharmaceutical interventions with a particular focus on different contact tracing strategies and the potential effort involved in these. Results: General physical distancing and avoiding large group gatherings combined with contact tracing, where all contacts are isolated immediately, can be effective in slowing down the outbreak, but were, under our base assumptions, not enough to control it without implementing extreme stay at home policies. Under optimistic assumptions with a highly overdispersed R0 and a short delay from symptom onset to isolation, control was possible with less stringent physical distancing and by isolating household contacts only. Conclusions: Without strong physical distancing measures, controlling the spread of SARS-CoV-2 is difficult. With limited resources, physical distancing combined with the isolation of households of detected cases can form a moderately effective strategy, and control is possible under optimistic assumptions. More data are needed to understand transmission in Kenya, in particular by studying the settings that lead to larger transmission events, which may allow for more targeted responses, and collection of representative age-related contact data.

Exploration of Superspreading Events in 2015 MERS-CoV Outbreak in Korea by Branching Process Models.

South Korea has learned a valuable lesson from the Middle East respiratory syndrome (MERS) coronavirus outbreak in 2015. The 2015 MERS-CoV outbreak in Korea was the largest outbreak outside the Middle Eastern countries and was characterized as a nosocomial infection and a superspreading event. To assess the characteristics of a super spreading event, we specifically analyze the behaviors and epidemiological features of superspreaders. Furthermore, we employ a branching process model to understand a significantly high level of heterogeneity in generating secondary cases. The existing model of the branching process (Lloyd-Smith model) is used to incorporate individual heterogeneity into the model, and the key epidemiological components (the reproduction number and the dispersive parameter) are estimated through the empirical transmission tree of the MERS-CoV data. We also investigate the impact of control intervention strategies on the MERS-CoV dynamics of the Lloyd-Smith model. Our results highlight the roles of superspreaders in a high level of heterogeneity. This indicates that the conditions within hospitals as well as multiple hospital visits were the crucial factors for superspreading events of the 2015 MERS-CoV outbreak.

Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China.

Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R 0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from theWorld Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R 0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R 0 and k (95% CrIs: R 0 1.4-12; k 0.04-0.2); however, the upper bound of R 0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.

Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China

Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.

Using information theory to optimise epidemic models for real-time prediction and estimation.

The effective reproduction number, Rt, is a key time-varying prognostic for the growth rate of any infectious disease epidemic. Significant changes in Rt can forewarn about new transmissions within a population or predict the efficacy of interventions. Inferring Rt reliably and in real-time from observed time-series of infected (demographic) data is an important problem in population dynamics. The renewal or branching process model is a popular solution that has been applied to Ebola and Zika virus disease outbreaks, among others, and is currently being used to investigate the ongoing COVID-19 pandemic. This model estimates Rt using a heuristically chosen piecewise function. While this facilitates real-time detection of statistically significant Rt changes, inference is highly sensitive to the function choice. Improperly chosen piecewise models might ignore meaningful changes or over-interpret noise-induced ones, yet produce visually reasonable estimates. No principled piecewise selection scheme exists. We develop a practical yet rigorous scheme using the accumulated prediction error (APE) metric from information theory, which deems the model capable of describing the observed data using the fewest bits as most justified. We derive exact posterior prediction distributions for infected population size and integrate these within an APE framework to obtain an exact and reliable method for identifying the piecewise function best supported by available epidemic data. We find that this choice optimises short-term prediction accuracy and can rapidly detect salient fluctuations in Rt, and hence the infected population growth rate, in real-time over the course of an unfolding epidemic. Moreover, we emphasise the need for formal selection by exposing how common heuristic choices, which seem sensible, can be misleading. Our APE-based method is easily computed and broadly applicable to statistically similar models found in phylogenetics and macroevolution, for example. Our results explore the relationships among estimate precision, forecast reliability and model complexity.

Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China

Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R 0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R 0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R 0 and k (95% CrIs: R 0 1.4-12; k 0.04-0.2); however, the upper bound of R 0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.

How Mathematical Approaches Could Help Decision-Making to Epidemic Control? The Successful Experience against COVID-19 in Cuba

Background. There is a gap for the effective use of mathematical models for real-time decision-making. We aimed to illustrate with the Cuban experience to control the COVID-19, how mathematical models can be put in place to answer key decision-makers´ questions.Methods. A science-policy partnership was created to mutually define questions, communicate results and facilitate the translation of modeling advice into actions. For forecasting and planning at national level mechanistic models and machine learning based on the epidemic patterns in other countries were used. Statistical models to explain the variability of transmission was used to stratify control actions. The effect of interventions was assessed using branching process models, time varying reproduction number (Rt) and social mixing patterns by location, and by age group.Findings. The mathematical approach implemented contribute to successful control of the COVID-19 in Cuba. The urbanization, living conditions and the economic index explain the 73% of the variability of the transmission at provincial level. Increased risk of transmission were identified in 33 municipalities mostly in densely populated urban areas with high aging index. Control intervention reduced the transmission from R0=2.84 (95% CI: 1.52 - 4.76) to Rt=0.6 (95% CI:0.2-2.38 ). The highest transmission was detected among adolescents and from people older than 60 years.Conclusions. Understanding the key questions for decision-making at all times, translating problems into a mathematical language, integrating different approaches to their solution and being able to present the results in an easy-to-understand way is vital to have a timely impact on controlling the epidemic.

Extinction probabilities as a function of temperature for populations of tsetse (Glossina spp.).

Significant reductions in populations of tsetse (Glossina spp) in parts of Zimbabwe have been attributed to increases in temperature over recent decades. Sustained increases in temperature might lead to local extinctions of tsetse populations. Extinction probabilities for tsetse populations have not so far been estimated as a function of temperature. We develop a time-homogeneous branching process model for situations where tsetse live at different levels of fixed temperature. We derive a probability distribution pk(T) for the number of female offspring an adult female tsetse is expected to produce in her lifetime, as a function of the fixed temperature at which she is living. We show that pk(T) can be expressed as a geometric series: its generating function is therefore a fractional linear type. We obtain expressions for the extinction probability, reproduction number, time to extinction and growth rates. The results are valid for all tsetse, but detailed effects of temperature will vary between species. No G. m. morsitans population can escape extinction if subjected, for extended periods, to temperatures outside the range 16°C–32°C. Extinction probability increases more rapidly as temperatures approach and exceed the upper and lower limits. If the number of females is large enough, the population can still survive even at high temperatures (28°C–31°C). Small decreases or increases in constant temperature in the neighbourhoods of 16°C and 31°C, respectively, can drive tsetse populations to extinction. Further study is needed to estimate extinction probabilities for tsetse populations in field situations where temperatures vary continuously.

Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China

Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.

A model of tuberculosis clustering in low incidence countries reveals more transmission in the United Kingdom than the Netherlands between 2010 and 2015.

Tuberculosis (TB) remains a public health threat in low TB incidence countries, through a combination of reactivated disease and onward transmission. Using surveillance data from the United Kingdom (UK) and the Netherlands (NL), we demonstrate a simple and predictable relationship between the probability of observing a cluster and its size (the number of cases with a single genotype). We demonstrate that the full range of observed cluster sizes can be described using a modified branching process model with the individual reproduction number following a Poisson lognormal distribution. We estimate that, on average, between 2010 and 2015, a TB case generated 0.41 (95% CrI 0.30,0.60) secondary cases in the UK, and 0.24 (0.14,0.48) secondary cases in the NL. A majority of cases did not generate any secondary cases. Recent transmission accounted for 39% (26%,60%) of UK cases and 23%(13%,37%) of NL cases. We predict that reducing UK transmission rates to those observed in the NL would result in 538(266,818) fewer cases annually in the UK. In conclusion, while TB in low incidence countries is strongly associated with reactivated infections, we demonstrate that recent transmission remains sufficient to warrant policies aimed at limiting local TB spread.

The transmissibility of novel Coronavirus in the early stages of the 2019-20 outbreak in Wuhan: Exploring initial point-source exposure sizes and durations using scenario analysis

Background: The current novel coronavirus outbreak appears to have originated from a point-source exposure event at Huanan seafood wholesale market in Wuhan, China. There is still uncertainty around the scale and duration of this exposure event. This has implications for the estimated transmissibility of the coronavirus and as such, these potential scenarios should be explored. Methods: We used a stochastic branching process model, parameterised with available data where possible and otherwise informed by the 2002-2003 Severe Acute Respiratory Syndrome (SARS) outbreak, to simulate the Wuhan outbreak. We evaluated scenarios for the following parameters: the size, and duration of the initial transmission event, the serial interval, and the reproduction number (R0). We restricted model simulations based on the number of observed cases on the 25th of January, accepting samples that were within a 5% interval on either side of this estimate. Results: Using a pre-intervention SARS-like serial interval suggested a larger initial transmission event and a higher R0 estimate. Using a SARs-like serial interval we found that the most likely scenario produced an R0 estimate between 2-2.7 (90% credible interval (CrI)). A pre-intervention SARS-like serial interval resulted in an R0 estimate between 2-3 (90% CrI). There were other plausible scenarios with smaller events sizes and longer duration that had comparable R0 estimates. There were very few simulations that were able to reproduce the observed data when R0 was less than 1. Conclusions: Our results indicate that an R0 of less than 1 was highly unlikely unless the size of the initial exposure event was much greater than currently reported. We found that R0 estimates were comparable across scenarios with decreasing event size and increasing duration. Scenarios with a pre-intervention SARS-like serial interval resulted in a higher R0 and were equally plausible to scenarios with SARs-like serial intervals.

Predicting colorectal cancer risk from adenoma detection via a two-type branching process model.

Despite advances in the modeling and understanding of colorectal cancer development, the dynamics of the progression from benign adenomatous polyp to colorectal carcinoma are still not fully resolved. To take advantage of adenoma size and prevalence data in the National Endoscopic Database of the Clinical Outcomes Research Initiative (CORI) as well as colorectal cancer incidence and size data from the Surveillance Epidemiology and End Results (SEER) database, we construct a two-type branching process model with compartments representing adenoma and carcinoma cells. To perform parameter inference we present a new large-size approximation to the size distribution of the cancer compartment and validate our approach on simulated data. By fitting the model to the CORI and SEER data, we learn biologically relevant parameters, including the transition rate from adenoma to cancer. The inferred parameters allow us to predict the individualized risk of the presence of cancer cells for each screened patient. We provide a web application which allows the user to calculate these individual probabilities at https://ccrc-eth.shinyapps.io/CCRC/. For example, we find a 1 in 100 chance of cancer given the presence of an adenoma between 10 and 20mm size in an average risk patient at age 50. We show that our two-type branching process model recapitulates the early growth dynamics of colon adenomas and cancers and can recover epidemiological trends such as adenoma prevalence and cancer incidence while remaining mathematically and computationally tractable.

The transmissibility of novel Coronavirus in the early stages of the 2019-20 outbreak in Wuhan: Exploring initial point-source exposure sizes and durations using scenario analysis.

Background: The current novel coronavirus outbreak appears to have originated from a point-source exposure event at Huanan seafood wholesale market in Wuhan, China. There is still uncertainty around the scale and duration of this exposure event. This has implications for the estimated transmissibility of the coronavirus and as such, these potential scenarios should be explored. Methods: We used a stochastic branching process model, parameterised with available data where possible and otherwise informed by the 2002-2003 Severe Acute Respiratory Syndrome (SARS) outbreak, to simulate the Wuhan outbreak. We evaluated scenarios for the following parameters: the size, and duration of the initial transmission event, the serial interval, and the reproduction number (R0). We restricted model simulations based on the number of observed cases on the 25th of January, accepting samples that were within a 5% interval on either side of this estimate. Results: Using a pre-intervention SARS-like serial interval suggested a larger initial transmission event and a higher R0 estimate. Using a SARs-like serial interval we found that the most likely scenario produced an R0 estimate between 2-2.7 (90% credible interval (CrI)). A pre-intervention SARS-like serial interval resulted in an R0 estimate between 2-3 (90% CrI). There were other plausible scenarios with smaller events sizes and longer duration that had comparable R0 estimates. There were very few simulations that were able to reproduce the observed data when R0 was less than 1. Conclusions: Our results indicate that an R0 of less than 1 was highly unlikely unless the size of the initial exposure event was much greater than currently reported. We found that R0 estimates were comparable across scenarios with decreasing event size and increasing duration. Scenarios with a pre-intervention SARS-like serial interval resulted in a higher R0 and were equally plausible to scenarios with SARs-like serial intervals.

Branching Out to Speciation in a Model of Fractionation: The Malvaceae.

Fractionation is the genome-wide process of losing one gene per duplicate pair following whole genome doubling (WGD). An important type of evidence for duplicate gene loss is the frequency distribution of similarities between paralogous gene pairs in a genome or orthologous gene pairs in two species. We extend a previous branching process model for fractionation, originally accounting for paralog similarities, to encompass the distribution of ortholog similarities, after multiple rounds of whole genome doubling and fractionation, with the speciation event occurring at any point. We estimate the fractionation rates during all the inter-event periods in each lineage of the plant family Malvaceae. We suggest a major correction of the phylogenetic position of the durian sub-family, and discover a new triplication event in this lineage.

Cancer recurrence times from a branching process model.

As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.

Predicting Minimal Residual Disease in Acute Myeloid Leukemia through Stochastic Modeling of Clonality

Background. Relapse affects about 50% of AML patients who achieved remission after treatment, and the prognosis of relapsed AML is poor. Current evidence has shown that in many patients, mutations giving rise to relapse are already present at diagnosis and remain in small numbers in remission, defined as the minimal residual disease (MRD) [1]. Chemoresistant clones contributing to relapse of the disease arise from minimal residual disease (MRD) rather than resulting from newly acquired mutations during or after chemotherapy. MRD is the presence of measurable leukemic cells using non-morphologic assays. It is considered a strong predictor of relapse. The dynamics of clones comprising MRD is poorly understood and is considered influenced by a form of Darwinian selection.Methods. We propose a stochastic model based on a multitype (multi-clone) age-dependent Markov branching process to study how random events in MRD contribute to the heterogeneity in response to treatment in a cohort of six patients from The Cancer Genome Atlas database with whole genome sequencing data at two time points. Because human bone marrow cell counts are too large for direct stochastic simulation methods to be effective, we developed a hybrid numerical algorithm combining stochastic Gillespie-type and tau-leaping algorithms, and a deterministic differential equation solver, which uses much less computer time than a "straight Gillespie algorithm". Results. We developed a stochastic model of clonal evolution based on a multitype (multi-clone) age-dependent Markov branching process model of cell proliferation. In brief, we consider the critical time interval between diagnosis and initial relapse of AML that includes cytotoxic chemotherapy, chemotherapy-induced myelosuppression and decrease in leukemic cells, non-leukemic marrow recovery, and growth of the leukemic clones due to refractory or relapsed disease. Underlying our model are assumptions regarding the structure of growth, differentiation, and competition of the normal and leukemic clones. Our model reflects the stochasticity inherent when leukemic clones are near depletion after chemotherapy, which we hypothesize strongly contributes to the interpatient heterogeneity in treatment response. The parameters are estimated by fitting the expected-value model to the patient's clinical data. The available data at diagnosis includes patient's weight, percent cellularity, white blood cell count, percentage of blasts in both peripheral blood and bone marrow, and percentage of normal neutrophils in the peripheral blood. Importantly, the time to relapse and percentage of blasts in bone marrow at relapse are available. The parameters fitted to the expected-value model offer an explanation of how a leukemic clone can escape chemotherapy and promote relapse. These clones have either high proliferation rates or high self-renewal rates. As a result, there is a range of different parameter combinations that can explain their ability to succeed. On the other hand, we also study the clones that have been eradicated by the time of relapse and conclude that these clones might be eliminated either because they are not competitive and therefore surrender to other clones, or they are simply killed by chemotherapy. Also, we checked if the parameters are biologically relevant by using the model to compute the corresponding clonal growth rates for each patient. That these values fit in the clinically observed range independently found in for patients with the NPM1 mutations [1], suggests that the model is consistent with clinical data. Conclusions. Our model offers a more accurate understanding of how relapse arises and which properties allow a leukemic clone to thrive in the Darwinian competition among leukemic and normal hematopoietic clones. The model suggests a quantitative relationship between MRD and time to relapse and therefore may aid clinicians in determining when and how to implement treatment changes to postpone or prevent the time to relapse. [1]Assessment of Minimal Residual Disease in Standard-Risk AML. N Engl J Med. 2016;374:422. Relationship between MRD and time to relapse for all patients. Estimates of MRD and time to relapse are mean values from 1000 stochastic simulations. Each color corresponds to a single patient; left triangles, circles, and right triangles correspond to three parameter sets. Simulated points are fitted with a sigmoidal Hill function. Figure Disclosures No relevant conflicts of interest to declare.