Ways to test stochastic dynamic programming models empirically

Stochastic dynamic programming (SDP) models are widely used to predict optimal behavioural and life history strategies. We discuss a diversity of ways to test SDP models empirically, taking as our main illustration a model of the daily singing routine of birds. One approach to verification is to quantify model parameters, but most SDP models are schematic. Because predictions are therefore qualitative, testing several predictions is desirable. How state determines behaviour (the policy) is a central prediction that should be examined directly if both state and behaviour are measurable. Complementary predictions concern how behaviour and state change through time, but information is discarded by considering behaviour rather than state, by looking only at average state rather than its distribution, and by not following individuals. We identify the various circumstances in which an individual's state/behaviour at one time is correlated with its state/behaviour at a later time. When there are several state variables the relationships between them may be informative. Often model parameters represent environmental conditions that can also be viewed as state variables. Experimental manipulation of the environment has several advantages as a test, but a problem is uncertainty over how much the organism's policy will adjust. As an example we allow birds to use different assumptions about how well past weather predicts future weather. We advocate mirroring planned empirical investigations on the computer to investigate which manipulations and predictions will best test a model. Copyright 2000 The Association for the Study of Animal Behaviour.     

Prediction and error in multi‐stage models for spread of aquatic non‐indigenous species

Aim Predictions of spread of non‐indigenous species allow for greater efficiency in managing invasions by targeting areas for preventative measures. The invasion sequence is a useful concept in predictions of spread, as it allows us to test hypotheses about the transport and establishment of propagules in novel habitats. Our aims are twofold: (1) to develop and validate multi‐stage invasion models for the introduced fishhook waterflea, Cercopagis pengoi, and (2) to assess how variability in the transport patterns of the propagules influences the accuracy and spatial extent for predictions of spread. Location New York State, USA. Methods We developed a two‐stage model for the spread of C. pengoi. First, we developed a stochastic gravity model for dispersal based on surveys of recreational boat traffic in New York State as a proxy for propagule pressure. We then modelled the probability of establishment based on predicted levels of propagule pressure and measures of lakes’ physicochemistry. In addition, we used Monte Carlo simulations based on the gravity model to propagate variability in boater traffic through the establishment model to assess how uncertainty in dispersal influenced predictions of spread. Results The amount recreationalists were willing to spend, lake area and population size of the city nearest to the destination lake were significant factors affecting boater traffic. In turn, boater traffic, lake area, specific conductance and turbidity were significant predictors of establishment. The inclusion of stochastic dispersal reduced the rate of false positives (i.e. incorrect prediction of an invasion) in detecting invasions at the upper 95% prediction interval for the probability of establishment. Main conclusions Combinations of measures of propagule pressure, habitat suitability and stochastic dispersal allow for the most accurate predictions of spread. Further, multi‐stage spread models may overestimate the extent of spread if stochasticity in early stages of the models is not considered.     

Predicting distributions of Mexican birds using ecological niche modelling methods

We tested the utility of the modelling program Genetic Algorithm for Rule-set Prediction (GARP) for modelling ecological niches to make accurate predictions of geographical distributions for 25 bird species across Mexico. Specimen-based point-occurrence data were entered into the algorithm in the form of geographical coordinates, and related to digitized maps of environmental variables, including mean annual precipitation, elevation, mean annual temperature, and potential vegetation. Two Mexican states were used as test areas by withholding their points from model construction; these points were later overlaid on predictions to measure model performance. Statistically, most models (7890%) were significantly more powerful than random models in predicting occurrences in test states; model failures were most often due to low sample size for testing, rather than an inability to model distributions of particular species. The success of this test indicates that ecological niche modelling approaches such as GARP provide a promising tool for exploring a broad range of questions in ecology, biogeography and conservation.     

Validation of species–climate impact models under climate change

Increasing concern over the implications of climate change for biodiversity has led to the use of species–climate envelope models to project species extinction risk under climate‐change scenarios. However, recent studies have demonstrated significant variability in model predictions and there remains a pressing need to validate models and to reduce uncertainties. Model validation is problematic as predictions are made for events that have not yet occurred. Resubstituition and data partitioning of present‐day data sets are, therefore, commonly used to test the predictive performance of models. However, these approaches suffer from the problems of spatial and temporal autocorrelation in the calibration and validation sets. Using observed distribution shifts among 116 British breeding‐bird species over the past ～20 years, we are able to provide a first independent validation of four envelope modelling techniques under climate change. Results showed good to fair predictive performance on independent validation, although rules used to assess model performance are difficult to interpret in a decision‐planning context. We also showed that measures of performance on nonindependent data provided optimistic estimates of models' predictive ability on independent data. Artificial neural networks and generalized additive models provided generally more accurate predictions of species range shifts than generalized linear models or classification tree analysis. Data for independent model validation and replication of this study are rare and we argue that perfect validation may not in fact be conceptually possible. We also note that usefulness of models is contingent on both the questions being asked and the techniques used. Implementations of species–climate envelope models for testing hypotheses and predicting future events may prove wrong, while being potentially useful if put into appropriate context.     

Elimination of HIV in South Africa through Expanded Access to Antiretroviral Therapy: A Model Comparison Study

Background

Expanded access to antiretroviral therapy (ART) using universal test and treat (UTT) has been suggested as a strategy to eliminate HIV in South Africa within 7 y based on an influential mathematical modeling study. However, the underlying deterministic model was criticized widely, and other modeling studies did not always confirm the study''s finding. The objective of our study is to better understand the implications of different model structures and assumptions, so as to arrive at the best possible predictions of the long-term impact of UTT and the possibility of elimination of HIV.

Methods and Findings

We developed nine structurally different mathematical models of the South African HIV epidemic in a stepwise approach of increasing complexity and realism. The simplest model resembles the initial deterministic model, while the most comprehensive model is the stochastic microsimulation model STDSIM, which includes sexual networks and HIV stages with different degrees of infectiousness. We defined UTT as annual screening and immediate ART for all HIV-infected adults, starting at 13% in January 2012 and scaled up to 90% coverage by January 2019. All models predict elimination, yet those that capture more processes underlying the HIV transmission dynamics predict elimination at a later point in time, after 20 to 25 y. Importantly, the most comprehensive model predicts that the current strategy of ART at CD4 count ≤350 cells/µl will also lead to elimination, albeit 10 y later compared to UTT. Still, UTT remains cost-effective, as many additional life-years would be saved. The study''s major limitations are that elimination was defined as incidence below 1/1,000 person-years rather than 0% prevalence, and drug resistance was not modeled.

Conclusions

Our results confirm previous predictions that the HIV epidemic in South Africa can be eliminated through universal testing and immediate treatment at 90% coverage. However, more realistic models show that elimination is likely to occur at a much later point in time than the initial model suggested. Also, UTT is a cost-effective intervention, but less cost-effective than previously predicted because the current South African ART treatment policy alone could already drive HIV into elimination. Please see later in the article for the Editors'' Summary     

Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents

Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.     

Predicting the process of extinction in experimental microcosms and accounting for interspecific interactions in single‐species time series

Predicting population extinction risk is a fundamental application of ecological theory to the practice of conservation biology. Here, we compared the prediction performance of a wide array of stochastic, population dynamics models against direct observations of the extinction process from an extensive experimental data set. By varying a series of biological and statistical assumptions in the proposed models, we were able to identify the assumptions that affected predictions about population extinction. We also show how certain autocorrelation structures can emerge due to interspecific interactions, and that accounting for the stochastic effect of these interactions can improve predictions of the extinction process. We conclude that it is possible to account for the stochastic effects of community interactions on extinction when using single‐species time series.     

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton–Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.     

The dual klepsydra model of internal time representation and time reproduction

We present a model of the internal representation and reproduction of temporal durations, the 'dual klepsydra' model (DKM). Unlike most contemporary models operating on a 'pacemaker-counter' scheme, the DKM does not assume an oscillatory process as the internal time-base. It is based on irreversible, dissipative processes in inflow/outflow systems (leaky klepsydrae), whose states are continuously compared; if their states are equal, durations are subjectively perceived as equal. Model-based predictions fit experimental time reproduction data with good accuracy, and show qualitative features not accounted for by other models. The deterministic model is characterized by two parameters, kappa (outflow rate coefficient) and eta (ratio of inflow rates). A stochastic version of the model (SDKM) assumes randomly fluctuating inflows, involves two more parameters, and accounts for intra-individual variance of reproduced durations. Analysis of the SDKM leads to non-trivial problems in the stochastic theory, briefly sketched here. Methods of parameter estimation for both deterministic and stochastic versions are given. Applying the DKM to the subjective experience of time passage, we show how subjective measure of elapsed time is constituted. Finally, essential features of the model and its possible neurophysiological interpretation are discussed.     

How accurate are population models? Lessons from landscape‐scale tests in a fragmented system

There is a growing debate about the ability of Population Viability Analysis (PVA) to predict the risk of extinction. Previously, the debate has focused largely on models where spatial variation and species movement are ignored. We present a synthesis of the key results for an array of different species for which detailed tests of the accuracy of PVA models were completed. These models included spatial variation in habitat quality and the movement of individuals across a landscape. The models were good approximations for some species, but poor for others. Predictive ability was limited by complex processes typically overlooked in spatial population models, these being interactions between landscape structure and life history attributes. Accuracy of models could not be determined a priori, although model tests indicated how they might be improved. Importantly, model predictions were poor for some species that are among the best‐studied vertebrates in Australia. This indicated that although the availability of good life history data is a key part of PVA other factors also influence model accuracy. We were also able to draw broad conclusions about the sorts of populations and life history characteristics where model predictions are likely to be less accurate. Predictions of extinction risk are often essential for real‐world population management. Therefore, we believe that although PVA has been shown to be less than perfect, it remains a useful tool particularly in the absence of alternative approaches. Hence, tests of PVA models should be motivated by the cycle of testing and improvement.