Inferring the dynamics of a spatial epidemic from time-series data

Spatial interactions are key determinants in the dynamics of many epidemiological and ecological systems; therefore it is important to use spatio-temporal models to estimate essential parameters. However, spatially-explicit data sets are rarely available; moreover, fitting spatially-explicit models to such data can be technically demanding and computationally intensive. Thus non-spatial models are often used to estimate parameters from temporal data. We introduce a method for fitting models to temporal data in order to estimate parameters which characterise spatial epidemics. The method uses semi-spatial models and pair approximation to take explicit account of spatial clustering of disease without requiring spatial data. The approach is demonstrated for data from experiments with plant populations invaded by a common soilborne fungus, Rhizoctonia solani. Model inferences concerning the number of sources of disease and primary and secondary infections are tested against independent measures from spatio-temporal data. The applicability of the method to a wide range of host-pathogen systems is discussed.     

Effects of dispersal mechanisms on spatio-temporal development of epidemics

The nature of pathogen transport mechanisms strongly determines the spatial pattern of disease and, through this, the dynamics and persistence of epidemics in plant populations. Up to recently, the range of possible mechanisms or interactions assumed by epidemic models has been limited: either independent of the location of individuals (mean-field models) or restricted to local contacts (between nearest neighbours or decaying exponentially with distance). Real dispersal processes are likely to lie between these two extremes, and many are well described by long-tailed contact kernels such as power laws. We investigate the effect of different spatial dispersal mechanisms on the spatio-temporal spread of disease epidemics by simulating a stochastic Susceptible-infective model motivated by previous data analyses. Both long-term stationary behaviour (in the presence of a control or recovery process) and transient behaviour (which varies widely within and between epidemics) are examined. We demonstrate the relationship between epidemic size and disease pattern (characterized by spatial autocorrelation), and its dependence on dispersal and infectivity parameters. Special attention is given to boundary effects, which can decrease disease levels significantly relative to standard, periodic geometries in cases of long-distance dispersal. We propose and test a definition of transient duration which captures the dependence of transients on dispersal mechanisms. We outline an analytical approach that represents the behaviour of the spatially-explicit model, and use it to prove that the epidemic size is predicted exactly by the mean-field model (in the limit of an infinite system) when dispersal is sufficiently long ranged (i.e. when the power-law exponent a相似文献   

Modeling dengue outbreaks

We introduce a dengue model (SEIR) where the human individuals are treated on an individual basis (IBM) while the mosquito population, produced by an independent model, is treated by compartments (SEI). We study the spread of epidemics by the sole action of the mosquito. Exponential, deterministic and experimental distributions for the (human) exposed period are considered in two weather scenarios, one corresponding to temperate climate and the other to tropical climate. Virus circulation, final epidemic size and duration of outbreaks are considered showing that the results present little sensitivity to the statistics followed by the exposed period provided the median of the distributions are in coincidence. Only the time between an introduced (imported) case and the appearance of the first symptomatic secondary case is sensitive to this distribution. We finally show that the IBM model introduced is precisely a realization of a compartmental model, and that at least in this case, the choice between compartmental models or IBM is only a matter of convenience.     

Effects of spatial scale and foraging efficiency on the predictions made by spatially-explicit models of fish growth rate potential

Synopsis Spatially-explicit modeling of fish growth rate potential is a relatively new approach that uses physical and biological properties of aquatic habitats to map spatial patterns of fish growth rate potential. Recent applications of spatially-explicit models have used an arbitrary spatial scale and have assumed a fixed foraging efficiency. We evaluated the effects of spatial scale, predator foraging efficiency (combined probabilities of prey recognition, attack, capture, and ingestion), and predator spatial distribution on estimates of mean growth rate potential of chinook salmon,Oncorhynchus tshawytscha. We used actual data on prey densities and water temperatures taken from Lake Ontario during the summer, as well as, simulated data assuming binomial distribution of prey. Results show that a predator can compensate for low foraging efficiency by inhabiting the most profitable environments (regions of high growth rate potential). Differences exist in predictions of growth rate potential across spatial scales of observation and a single scale may not be adequate for interpreting model results across seasons. Continued refinements of this modeling approach must focus on the assumptions of stationary distributions of predator and prey populations and predator foraging tactics.     

The Spatial Resolution of Epidemic Peaks

The emergence of novel respiratory pathogens can challenge the capacity of key health care resources, such as intensive care units, that are constrained to serve only specific geographical populations. An ability to predict the magnitude and timing of peak incidence at the scale of a single large population would help to accurately assess the value of interventions designed to reduce that peak. However, current disease-dynamic theory does not provide a clear understanding of the relationship between: epidemic trajectories at the scale of interest (e.g. city); population mobility; and higher resolution spatial effects (e.g. transmission within small neighbourhoods). Here, we used a spatially-explicit stochastic meta-population model of arbitrary spatial resolution to determine the effect of resolution on model-derived epidemic trajectories. We simulated an influenza-like pathogen spreading across theoretical and actual population densities and varied our assumptions about mobility using Latin-Hypercube sampling. Even though, by design, cumulative attack rates were the same for all resolutions and mobilities, peak incidences were different. Clear thresholds existed for all tested populations, such that models with resolutions lower than the threshold substantially overestimated population-wide peak incidence. The effect of resolution was most important in populations which were of lower density and lower mobility. With the expectation of accurate spatial incidence datasets in the near future, our objective was to provide a framework for how to use these data correctly in a spatial meta-population model. Our results suggest that there is a fundamental spatial resolution for any pathogen-population pair. If underlying interactions between pathogens and spatially heterogeneous populations are represented at this resolution or higher, accurate predictions of peak incidence for city-scale epidemics are feasible.     

Inferring the Structure of Social Contacts from Demographic Data in the Analysis of Infectious Diseases Spread

Social contact patterns among individuals encode the transmission route of infectious diseases and are a key ingredient in the realistic characterization and modeling of epidemics. Unfortunately, the gathering of high quality experimental data on contact patterns in human populations is a very difficult task even at the coarse level of mixing patterns among age groups. Here we propose an alternative route to the estimation of mixing patterns that relies on the construction of virtual populations parametrized with highly detailed census and demographic data. We present the modeling of the population of 26 European countries and the generation of the corresponding synthetic contact matrices among the population age groups. The method is validated by a detailed comparison with the matrices obtained in six European countries by the most extensive survey study on mixing patterns. The methodology presented here allows a large scale comparison of mixing patterns in Europe, highlighting general common features as well as country-specific differences. We find clear relations between epidemiologically relevant quantities (reproduction number and attack rate) and socio-demographic characteristics of the populations, such as the average age of the population and the duration of primary school cycle. This study provides a numerical approach for the generation of human mixing patterns that can be used to improve the accuracy of mathematical models in the absence of specific experimental data.     

Individual-based modeling of phytoplankton: evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.     

Testing,tracing and isolation in compartmental models

Existing compartmental mathematical modelling methods for epidemics, such as SEIR models, cannot accurately represent effects of contact tracing. This makes them inappropriate for evaluating testing and contact tracing strategies to contain an outbreak. An alternative used in practice is the application of agent- or individual-based models (ABM). However ABMs are complex, less well-understood and much more computationally expensive. This paper presents a new method for accurately including the effects of Testing, contact-Tracing and Isolation (TTI) strategies in standard compartmental models. We derive our method using a careful probabilistic argument to show how contact tracing at the individual level is reflected in aggregate on the population level. We show that the resultant SEIR-TTI model accurately approximates the behaviour of a mechanistic agent-based model at far less computational cost. The computational efficiency is such that it can be easily and cheaply used for exploratory modelling to quantify the required levels of testing and tracing, alone and with other interventions, to assist adaptive planning for managing disease outbreaks.     

Steady state analysis of influx and transbilayer distribution of ergosterol in the yeast plasma membrane

Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.     

Modeling and artificial intelligence approaches to enzyme systems

Modeling is a means of formulating and testing complex hypotheses. Useful modeling is now possible with biological laboratory microcomputers with which experimenters feel comfortable. Artificial intelligence (AI) is sufficiently similar to modeling that AI techniques, now becoming usable on microcomputers, are applicable to modeling. Microcomputer and AI applications to physiological system studies with multienzyme models and with kinetic models of isolated enzymes are described. Using an IBM PC microcomputer, we have been able to fit kinetic enzyme models; to extend this process to design kinetic experiments by determining the optimal conditions; and to construct an enzyme (hexokinase) kinetics data base. We have also used a PC to do most of the constructing of complex multienzyme models, initially with small simple BASIC programs; alternative methods with standard spreadsheet or data base programs have been defined. Formulating and solving differential equations in appropriate representational languages, and sensitivity analysis, are soon likely to be feasible with PCs. Much of the modeling process can be stated in terms of AI expert systems, using sets of rules for fitting and evaluating models and designing further experiments. AI techniques also permit critiquing and evaluating the data, experiments, and hypotheses being modeled, and can be extended to supervise the calculations involved.     

The Limits of Mean-Field Heterozygosity Estimates under Spatial Extension in Simulated Plant Populations

Computational models of evolutionary processes are increasingly required to incorporate multiple and diverse sources of data. A popular feature to include in population genetics models is spatial extension, which reflects more accurately natural populations than does a mean field approach. However, such models necessarily violate the mean field assumptions of classical population genetics, as do natural populations in the real world. Recently, it has been questioned whether classical approaches are truly applicable to the real world. Individual based models (IBM) are a powerful and versatile approach to achieve integration in models. In this study an IBM was used to examine how populations of plants deviate from classical expectations under spatial extension. Populations of plants that used three different mating strategies were placed in a range of arena sizes giving crowded to sparse occupation densities. Using a measure of population density, the pollen communication distance (P(cd)), the deviation exhibited by outbreeding populations differed from classical mean field expectations by less than 5% when P(cd) was less than 1, and over this threshold value the deviation significantly increased. Populations with an intermediate mating strategy did not have such a threshold and deviated directly with increasing isolation between individuals. Populations with a selfing strategy were influenced more by the mating strategy than by increased isolation. In all cases pollen dispersal was more influential than seed dispersal. The IBM model showed that mean field calculations can be reasonably applied to natural outbreeding plant populations that occur at a density in which individuals are less than the average pollen dispersal distance from their neighbors.     

Edge-Based Compartmental Modelling of an SIR Epidemic on a Dual-Layer Static–Dynamic Multiplex Network with Tunable Clustering

The duration, type and structure of connections between individuals in real-world populations play a crucial role in how diseases invade and spread. Here, we incorporate the aforementioned heterogeneities into a model by considering a dual-layer static–dynamic multiplex network. The static network layer affords tunable clustering and describes an individual’s permanent community structure. The dynamic network layer describes the transient connections an individual makes with members of the wider population by imposing constant edge rewiring. We follow the edge-based compartmental modelling approach to derive equations describing the evolution of a susceptible–infected–recovered epidemic spreading through this multiplex network of individuals. We derive the basic reproduction number, measuring the expected number of new infectious cases caused by a single infectious individual in an otherwise susceptible population. We validate model equations by showing convergence to pre-existing edge-based compartmental model equations in limiting cases and by comparison with stochastically simulated epidemics. We explore the effects of altering model parameters and multiplex network attributes on resultant epidemic dynamics. We validate the basic reproduction number by plotting its value against associated final epidemic sizes measured from simulation and predicted by model equations for a number of set-ups. Further, we explore the effect of varying individual model parameters on the basic reproduction number. We conclude with a discussion of the significance and interpretation of the model and its relation to existing research literature. We highlight intrinsic limitations and potential extensions of the present model and outline future research considerations, both experimental and theoretical.     

Optimal control of the spatial allocation of COVID-19 vaccines: Italy as a case study

While campaigns of vaccination against SARS-CoV-2 are underway across the world, communities face the challenge of a fair and effective distribution of a limited supply of doses. Current vaccine allocation strategies are based on criteria such as age or risk. In the light of strong spatial heterogeneities in disease history and transmission, we explore spatial allocation strategies as a complement to existing approaches. Given the practical constraints and complex epidemiological dynamics, designing effective vaccination strategies at a country scale is an intricate task. We propose a novel optimal control framework to derive the best possible vaccine allocation for given disease transmission projections and constraints on vaccine supply and distribution logistics. As a proof-of-concept, we couple our framework with an existing spatially explicit compartmental COVID-19 model tailored to the Italian geographic and epidemiological context. We optimize the vaccine allocation on scenarios of unfolding disease transmission across the 107 provinces of Italy, from January to April 2021. For each scenario, the optimal solution significantly outperforms alternative strategies that prioritize provinces based on incidence, population distribution, or prevalence of susceptibles. Our results suggest that the complex interplay between the mobility network and the spatial heterogeneities implies highly non-trivial prioritization strategies for effective vaccination campaigns. Our work demonstrates the potential of optimal control for complex and heterogeneous epidemiological landscapes at country, and possibly global, scales.     

Bayesian state-space modeling of metapopulation dynamics in the Glanville fritillary butterfly

The complexity of mathematical models of ecological dynamics varies greatly, and it is often difficult to judge what would be the optimal level of complexity in a particular case. Here we compare the parameter estimates, model fits, and predictive abilities of two models of metapopulation dynamics: a detailed individual-based model (IBM) and a population-based stochastic patch occupancy model (SPOM) derived from the IBM. The two models were fitted to a 17-year time series of data for the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a network of 72 small meadows. The data consisted of biannual counts of larval groups (IBM) and the annual presence or absence of local populations (SPOM). The models were fitted using a Bayesian state-space approach with a hierarchical random effect structure to account for observational, demographic, and environmental stochasticities. The detection probability of larval groups (IBM) and the probability of false zeros of local populations (SPOM) in the observation models were simultaneously estimated from the time-series data and independent control data. Prior distributions for dispersal parameters were obtained from a separate analysis of mark–recapture data. Both models fitted the data about equally, but the results were more precise for the IBM than for the SPOM. The two models yielded similar estimates for a random effect parameter describing habitat quality in each patch, which were correlated with independent empirical measures of habitat quality. The modeling results showed that variation in habitat quality influenced patch occupancy more through the effects on movement behavior at patch edges than on carrying capacity, whereas the latter influenced the mean population size in occupied patches. The IBM and the SPOM explained 63% and 45%, respectively, of the observed variation in the fraction of occupied habitat area among 75 independent patch networks not used in parameter estimation. We conclude that, while carefully constructed, detailed models can have better predictive ability than simple models, this advantage comes with the cost of greatly increased data requirements and computational challenges. Our results illustrate how complex models can be helpful in facilitating the construction of effective simpler models.     

Network theory and SARS: predicting outbreak diversity

Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.     

Multi-locus estimates of population structure and migration in a fence lizard hybrid zone

A hybrid zone between two species of lizards in the genus Sceloporus (S. cowlesi and S. tristichus) on the Mogollon Rim in Arizona provides a unique opportunity to study the processes of lineage divergence and merging. This hybrid zone involves complex interactions between 2 morphologically and ecologically divergent subspecies, 3 chromosomal groups, and 4 mitochondrial DNA (mtDNA) clades. The spatial patterns of divergence between morphology, chromosomes and mtDNA are discordant, and determining which of these character types (if any) reflects the underlying population-level lineages that are of interest has remained impeded by character conflict. The focus of this study is to estimate the number of populations interacting in the hybrid zone using multi-locus nuclear data, and to then estimate the migration rates and divergence time between the inferred populations. Multi-locus estimates of population structure and gene flow were obtained from 12 anonymous nuclear loci sequenced for 93 specimens of Sceloporus. Population structure estimates support two populations, and this result is robust to changes to the prior probability distribution used in the Bayesian analysis and the use of spatially-explicit or non-spatial models. A coalescent analysis of population divergence suggests that gene flow is high between the two populations, and that the timing of divergence is restricted to the Pleistocene. The hybrid zone is more accurately described as involving two populations belonging to S. tristichus, and the presence of S. cowlesi mtDNA haplotypes in the hybrid zone is an anomaly resulting from mitochondrial introgression.     

Mechanistic home range models capture spatial patterns and dynamics of coyote territories in Yellowstone

Patterns of space-use by individuals are fundamental to the ecology of animal populations influencing their social organization, mating systems, demography and the spatial distribution of prey and competitors. To date, the principal method used to analyse the underlying determinants of animal home range patterns has been resource selection analysis (RSA), a spatially implicit approach that examines the relative frequencies of animal relocations in relation to landscape attributes. In this analysis, we adopt an alternative approach, using a series of mechanistic home range models to analyse observed patterns of territorial space-use by coyote packs in the heterogeneous landscape of Yellowstone National Park. Unlike RSAs, mechanistic home range models are derived from underlying correlated random walk models of individual movement behaviour, and yield spatially explicit predictions for patterns of space-use by individuals. As we show here, mechanistic home range models can be used to determine the underlying determinants of animal home range patterns, incorporating both movement responses to underlying landscape heterogeneities and the effects of behavioural interactions between individuals. Our analysis indicates that the spatial arrangement of coyote territories in Yellowstone is determined by the spatial distribution of prey resources and an avoidance response to the presence of neighbouring packs. We then show how the fitted mechanistic home range model can be used to correctly predict observed shifts in the patterns of coyote space-use in response to perturbation.     

Aggregate statistical measures and metapopulation dynamics

There are two main types of metapopulation models. Spatially implicit models are analytically tractable but neglect spatial heterogeneities. Spatially explicit models are more realistic but too complex. In this paper, I build a bridge between both approximations. I derive a new metapopulation model using a well-known technique in population genetics. Spatial heterogeneities are captured by an aggregate statistical measure of spatial correlation. When this correlation is zero, i.e., space is homogeneous, the model becomes the well-known Levins' model. As spatial correlation increases, equilibrium patch occupancy decreases from what would be expected under the spatially homogeneous assumption. I proceed by testing how well spatial complexities from a spatially explicit simulation can be encapsulated by such an aggregate statistical measure.