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Three borehole gravity (BHG) surveys were performed in 2013, 2016, and 2018 to monitor the changes in gravity/density as a result of the injection and withdrawal of carbon dioxide ([Formula: see text]) into and out of the Dover 33 carbonate reservoir reef in Northern Michigan. The observed gravity changes and inferred density changes have been modeled to determine the flow and storage zones of the injected [Formula: see text] in the reef. The high quality and low level of uncertainty of the data collected make them useful for delineating the [Formula: see text] plume position over time and for identifying the oil sweeping extent and mechanisms in the Dover 33 reef. The time-lapse gravity results indicate the effects of the changing [Formula: see text] mass within the reservoir, consistent with increasing mass from 2013 to 2016 (following [Formula: see text] injection) and a decreasing mass from 2016 to 2018 (after [Formula: see text] withdrawal). Three-dimensional imaging of fluid migrations in the reef has been obtained by coupling the time-lapse BHG results to a 3D porosity and permeability model. This coupled approach allows the evaluation of the volume of the reef affected by the injection of [Formula: see text] between 2013 and 2016, the efficiency of the oil sweeping between 2016 and 2018, and the location of the residual [Formula: see text] plume in the reef after 2018.

The ability of health systems to cope with coronavirus disease (COVID-19) cases is of major concern. In preparation, we used clinical pathway models to estimate healthcare requirements for COVID-19 patients in the context of broader public health measures in Australia. An age- and risk-stratified transmission model of COVID-19 demonstrated that an unmitigated epidemic would dramatically exceed the capacity of the health system of Australia over a prolonged period. Case isolation and contact quarantine alone are insufficient to constrain healthcare needs within feasible levels of expansion of health sector capacity. Overlaid social restrictions must be applied over the course of the epidemic to ensure systems do not become overwhelmed and essential health sector functions, including care of COVID-19 patients, can be maintained. Attention to the full pathway of clinical care is needed, along with ongoing strengthening of capacity.

Previous work has shown how a minimal ecological structure consisting of patchily distributed resources and recurrent dispersal between patches can scaffold Darwinian properties onto collections of cells. When the timescale of dispersal is long compared with the time to consume resources, patches evolve such that their size increases, but at the expense of cells whose growth rate decreases within patches. This creates the conditions that initiate evolutionary transitions in individuality. A key assumption of this scaffolding is that a bottleneck is created during dispersal, so patches are founded by single cells. The bottleneck decreases competition within patches and hence creates a strong hereditary link at the level of patches. Here we construct a fully stochastic model of nested Darwinian populations and investigate how larger bottlenecks affect the evolutionary dynamics at both cell and collective levels. It is shown that, up to a point, larger bottlenecks simply slow the dynamics, but at some point, which depends on the parameters of the within-patch model, the direction of evolution toward the equilibrium is reversed. Introducing random bottleneck sizes with some positive probability of smaller sizes can counteract this, even if the probability of smaller bottlenecks is small.Competing Interest StatementThe authors have declared no competing interest.

While the use of pesticides continues to rise worldwide, our understanding of the pervasiveness of associated contamination and the health risks humans may be exposed to remain limited to small samples size, and based on small geographic scales, the exposed population, or the pesticide types. Using our recent mapping of global pesticide use, we quantify three complementary health risk metrics for 92 active ingredients: (i) the pesticide hazard load (PHL); (ii) the population exposure (PE); and (iii) the human intake relative to the acceptable dose (INTR). We integrated these metrics into the pesticide health risk index (PHRI) to assess the standing of 133 nations against the global averages of PHL and PE and the acceptable levels of INTR using data of 2015 (PHRI > 1 indicates a concern). We found that some low-toxicity ingredients have PHL values equivalent to high-toxicity ones, and hence neglecting low-toxicity ingredients may cause biases in risk assessments. The geography of PHL, PE, and INTR show hotspots across the Americas, East and South Asia, and Europe, but with the EU27 countries generally showing lower PHL than other countries possibly due to strict governance on pesticide use. By our measure, about 1.7 billion people (24% of the world population) reside in close proximity to where pesticide applications are greater than 100 kg-a.i. km−2 year−1; about 2.3 billion people (32% of the world population) may exceed the acceptable pesticide intake and about 1.1 billion (15% of the world population) may exceed this by 10 fold. We identified 36 countries with PHRI > 1 and 6 countries with PHRI > 5; of these countries, 10 belong to lower-middle and low income economies. Our analyses show that proximity exposure to pesticides may be more widespread than revealed in occupational studies, and therefore assessments of potential health effects over wider scales may be needed.

Modern data and computational resources, coupled with algorithmic and theoretical advances to exploit these, allow disease dynamic models to be parameterised with increasing detail and accuracy. While this enhances models' usefulness in prediction and policy, major challenges remain. In particular, lack of identifiability of a model's parameters may limit the usefulness of the model. While lack of parameter identifiability may be resolved through incorporation into an inference procedure of prior knowledge, formulating such knowledge is often difficult. Furthermore, there are practical challenges associated with acquiring data of sufficient quantity and quality. Here, we discuss recent progress on these issues.

BackgroundThe ability of global health systems to cope with increasing numbers of COVID-19 cases is of major concern. In readiness for this challenge, Australia has drawn on clinical pathway models developed over many years in preparation for influenza pandemics. These models have been used to estimate health care requirements for COVID-19 patients, in the context of broader public health measures.MethodsAn age and risk stratified transmission model of COVID-19 infection was used to simulate an unmitigated epidemic with parameter ranges reflecting uncertainty in current estimates of transmissibility and severity. Overlaid public health measures included case isolation and quarantine of contacts, and broadly applied social distancing. Clinical presentations and patient flows through the Australian health care system were simulated, including expansion of available intensive care capacity and alternative clinical assessment pathways.FindingsAn unmitigated COVID-19 epidemic would dramatically exceed the capacity of the Australian health system, over a prolonged period. Case isolation and contact quarantine alone will be insufficient to constrain case presentations within a feasible level of expansion of health sector capacity. Overlaid social restrictions will need to be applied at some level over the course of the epidemic to ensure that systems do not become overwhelmed, and that essential health sector functions, including care of COVID-19 patients, can be maintained. Attention to the full pathway of clinical care is needed to ensure access to critical care.InterpretationReducing COVID-19 morbidity and mortality will rely on a combination of measures to strengthen and extend public health and clinical capacity, along with reduction of overall infection transmission in the community. Ongoing attention to maintaining and strengthening the capacity of health care systems and workers to manage cases is needed.FundingAustralian Government Department of Health Office of Health Protection, Australian Government National Health and Medical Research Council

Evolutionary transitions in individuality are central to the emergence of biological complexity. Recent experiments provide glimpses of processes underpinning the transition from single cells to multicellular life and draw attention to the critical role of ecology. Here we emphasise this ecological dimension and argue that its current absence from theoretical frameworks hampers development of general explanatory solutions. Using mechanistic mathematical models, we show how a minimal ecological structure comprised of patchily distributed resources and between patch dispersal can scaffold Darwinian-like properties on collectives of cells. This scaffolding causes cells to participate directly in the process of evolution by natural selection as if they were members of multicellular collectives, with collectives participating in a death-birth process arising from the interplay between the timing of dispersal events and the rate of resource utilisation by cells. When this timescale is sufficiently long and new collectives are founded by single cells, collectives experience conditions that favour evolution of a reproductive division of labour. Together our simple model makes explicit key events in the major evolutionary transition to multicellularity. It also makes predictions concerning the life history of certain pathogens and serves as an ecological recipe for experimental realisation of evolutionary transitions.

Background: The increasing use of pesticides is raising attention on the pervasiveness of contamination far from emission points and the pathways to human exposure. Recent progress in mapping global pesticide use can substantiate an assessment of human exposure and intake worldwide and the risks of attenuated life expectancy. Methods: We quantified the human intake ratio (INTR) relative to the acceptable level for 92 active substances. We intersected INTR with the Global Burden of Diseases (GBD) dataset, and we analysed the disability-adjusted life years (DALY) of both aggregated and individual non-communicable diseases (NCDs) against the INTR of major pesticide chemical classes. Using a stochastic two-stage regression analysis, we estimated the associations between pesticide INTR and DALYs, and we determined whether these associations were positive (i.e., adverse effect on life expectancy), uncertain, or negative. Findings: About 5.7 billion people exceed the acceptable pesticide intake; about 4.6 billion exceed this by 100 times. Pesticides are associated with DALYs of NCDs in 47% of countries; organophosphates, inorganics, triazines, and triazoles are associated with NCDs in 100%, 68%, 66% and 66% of countries, respectively. Prevalent NCDs are musculoskeletal disorders, neoplasms, and mental disorders in 72%, 62%, and 54% of countries. In average, an increase by one order in INTR is associated with a decrease of 1.35 days/person in neoplasms (95% CI, 0.27-2.48 days/person), and 1.09 days/person in musculoskeletal disorders (95% CI, 0.45-1.74 days/person). In spite of geographic variability, high income countries are more affected than low income countries due to raised health risk factors. Interpretation: Although the biological mechanisms determining health effects in human subjects remain largely undiscovered, statistically significant associations of chronic exposure to pesticide mixtures on various NCDs show the relative impact of different pesticide classes on exposed populations and indicate geographic regions, socioeconomic, and health indicators associated to risks of attenuated life expectancy. Funding Statement: The University of Sydney, SREI2020 EnviroSphere Program, SOAR Fellowship (FM). Declaration of Interests: The authors declare no competing interests.

Evolutionary transitions in individuality are central to the emergence of biological complexity. Recent experiments provide glimpses of processes underpinning the transition from single cells to multicellular life and draw attention to the critical role of ecology. Here we emphasise this ecological dimension and argue that its current absence from theoretical frameworks hampers development of general explanatory solutions. Using mechanistic mathematical models, we show how a minimal ecological structure comprised of patchily distributed resources and between patch dispersal can scaffold Darwinian-like properties on collectives of cells. This scaffolding causes cells to participate directly in the process of evolution by natural selection as if they were members of multicellular collectives, with collectives participating in a death-birth process arising from the interplay between the timing of dispersal events and the rate of resource utilisation by cells. When this timescale is sufficiently long and new collectives are founded by single cells, collectives experience conditions that favour evolution of a reproductive division of labour. Together our simple model makes explicit key events in the major evolutionary transition to multicellularity. It also makes predictions concerning the life history of certain pathogens and serves as an ecological recipe for experimental realisation of evolutionary transitions.

An efficient method for Bayesian model selection is presented for a broad class of continuous-time Markov chain models and is subsequently applied to two important problems in epidemiology. The first problem is to identify the shape of the infectious period distribution; the second problem is to determine whether individuals display symptoms before, at the same time, or after they become infectious. In both cases we show that the correct model can be identified, in the majority of cases, from symptom onset data generated from multiple outbreaks in small populations. The method works by evaluating the likelihood using a particle filter that incorporates a novel importance sampling algorithm designed for partially-observed continuous-time Markov chains. This is combined with another importance sampling method to unbiasedly estimate the model evidence. These come with estimates of precision, which allow for stopping criterion to be employed. Our method is general and can be applied to a wide range of model selection problems in biological and epidemiological systems with intractable likelihood functions.

Assessing the risk of disease spread between communities is important in our highly connected modern world. However, the impact of disease- and population-specific factors on the time taken for an epidemic to spread between communities, as well as the impact of stochastic disease dynamics on this spreading time, are not well understood. In this study, we model the spread of an acute infection between two communities ('patches') using a susceptible-infectious-removed (SIR) metapopulation model. We develop approximations to efficiently evaluate the probability of a major outbreak in a second patch given disease introduction in a source patch, and the distribution of the time taken for this to occur. We use these approximations to assess how interventions, which either control disease spread within a patch or decrease the travel rate between patches, change the spreading probability and median spreading time. We find that decreasing the basic reproduction number in the source patch is the most effective way of both decreasing the spreading probability, and delaying epidemic spread to the second patch should this occur. Moreover, we show that the qualitative effects of interventions are the same regardless of the approximations used to evaluate the spreading time distribution, but for some regions in parameter space, quantitative findings depend upon the approximations used. Importantly, if we neglect the possibility that an intervention prevents a large outbreak in the source patch altogether, then intervention effectiveness is not estimated accurately.

Recently, pandemic response has involved the use of antivirals. These antivirals are often allocated to households dynamically throughout the pandemic with the aim being to retard the spread of infection. A drawback of this is that there is a delay until infection is confirmed and antivirals are delivered. Here an alternative allocation scheme is considered, where antivirals are instead preallocated to households at the start of a pandemic, thus reducing this delay. To compare these two schemes, a deterministic approximation to a novel stochastic household model is derived, which allows efficient computation of key quantities such as the expected epidemic final size, expected early growth rate, expected peak size and expected peak time of the epidemic. It is found that the theoretical best choice of allocation scheme depends on strain transmissibility, the delay in delivering antivirals under a dynamic allocation scheme and the stockpile size. A broad summary is that for realistic stockpile sizes, a dynamic allocation scheme is superior with the important exception of the epidemic final size under a severe pandemic scenario. Our results, viewed in conjunction with the practical considerations of implementing a preallocation scheme, support a focus on attempting to reduce the delay in delivering antivirals under a dynamic allocation scheme during a future pandemic.

We develop a new methodology for the efficient computation of epidemic final size distributions for a broad class of Markovian models. We exploit a particular representation of the stochastic epidemic process to derive a method which is both computationally efficient and numerically stable. The algorithms we present are also physically transparent and so allow us to extend this method from the basic SIR model to a model with a phase-type infectious period and another with waning immunity. The underlying theory is applicable to many Markovian models where we wish to efficiently calculate hitting probabilities., final published version

Gravity can now be measured practically anywhere a drill hole can go. This extends our ability to perform 3D and 4D gravity mapping of the sub-surface with applications in the energy and mineral sectors. New tools have been developed to perform these measurements. The tools address issues of size, measurement accuracy, stability and ruggedness. The data handling has also changed to address the 3 dimensional nature of the measurements. Whereas previous borehole gravity measurements in near vertical wells could ignore latitude corrections and use simpler terrain correction, surveys in highly deviated wells involve calculating and using 3 dimensional coordinates for each gravity station in the correction process. This all needs to be done preserving the survey accuracy requirements that are often only a few microGals. Tool location is extremely important, more in the vertical direction due to the vertical free air and Bouguer gradients. Tool location control tends to be more difficult in more horizontal holes. A new high sampling resolution CCL tool has proven very useful for repeat location of the tool for drift calculations and time lapse measurements. For oil and gas, most applications in highly deviated or horizontal wells are expected to be of a time lapse mode such as monitoring of traditional and SAGD type steam floods. For mining the tool significantly extends the volume of investigation for deep exploration and development holes.

Early estimation of the probable impact of a pandemic influenza outbreak can assist public health authorities to ensure that response measures are proportionate to the scale of the threat. Recently, frameworks based on transmissibility and severity have been proposed for initial characterization of pandemic impact. Data requirements to inform this assessment may be provided by “First Few Hundred” (FF100) studies, which involve surveillance—possibly in person, or via telephone—of household members of confirmed cases. This process of enhanced case finding enables detection of cases across the full spectrum of clinical severity, including the date of symptom onset. Such surveillance is continued until data for a few hundred cases, or satisfactory characterization of the pandemic strain, has been achieved. We present a method for analysing these data, at the household level, to provide a posterior distribution for the parameters of a model that can be interpreted in terms of severity and transmissibility of a pandemic strain. We account for imperfect case detection, where individuals are only observed with some probability that can increase after a first case is detected. Furthermore, we test this methodology using simulated data generated by an independent model, developed for a different purpose and incorporating more complex disease and social dynamics. Our method recovers transmissibility and severity parameters to a high degree of accuracy and provides a computationally efficient approach to estimating the impact of an outbreak in its early stages.

Cost effectiveness for consumers of solar electric systems has been achieved in the California residential market due to Time-of-Use Net Metering, a tiered electric rate structure with high rates, and subsidies by the state government. Three different approaches test and validate this hypothesis for a large fraction of consumers. Solar electricity can generate rates of return of 11–20% exceeding most other common investments, can increase property value by more than the cost to install the system, and generate a positive cash flow if financed. This paper suggests further study in the commercial sector and a survey to document evidence of increased home resale value.

Processes that spread through local contact, including outbreaks of infectious diseases, are inherently noisy, and are frequently observed to be far noisier than predicted by standard stochastic models that assume homogeneous mixing. One way to reproduce the observed levels of noise is to introduce significant individual-level heterogeneity with respect to infection processes, such that some individuals are expected to generate more secondary cases than others. Here we consider a population where individuals can be naturally aggregated into clumps (subpopulations) with stronger interaction within clumps than between them. This clumped structure induces significant increases in the noisiness of a spreading process, such as the transmission of infection, despite complete homogeneity at the individual level. Given the ubiquity of such clumped aggregations (such as homes, schools and workplaces for humans or farms for livestock) we suggest this as a plausible explanation for noisiness of many epidemic time series.

The fabrication of two layer structures of electrically isolated wire — crossed wire structures and a surface coil inductor — is described. The fabrication process utilizes the tools of soft lithography and incorporates two levels of self-assembly. The use of microcontact printing and patterned self-assembly of liquid polymers removes the need for registration of the insulating layer with the underlying layer as required in conventional lithography techniques. The performance characteristics of the surface coil inductor are measured and closely resemble those predicted by theory.

Several non-photolithographic methods for growing patterned films of parylene-based polymers prepared by chemical vapor deposition are presented. These patterns were generated using transition meta...

This paper describes the development of a new methodology, “topographically directed etching” or TODE, that relies on patterned regions of disorder in SAMs to generate features specifically at the edges of topographically patterned metal films. The features that are produced have lateral dimensions of

This paper describes the oriented nucleation of calcite controlled by self-assembled monolayers (SAMs) of ω-terminated alkanethiols (HS(CH2)nX) supported on metal films. The effect of the chemistry and geometry of the organic surfaces was surveyed by using different functional groups (X = CO2-, SO3-, PO32-, OH, N(CH3)3+, CH3) and different supporting metals (Au and Ag). Optical microscopy, SEM with image analysis, and XRD were employed to characterize density and orientation of the crystals. Compared to the control surfaces of bare metal films, SAMs terminated in CO2-, SO3-, PO32-, and OH groups were more active in inducing nucleation, whereas SAMs terminated in N(CH3)3+ and CH3 inhibited nucleation. The crystallographic orientations of the crystals were distinct and highly homogeneous for each surface, but different on different surfaces. SAMs of CO2-/Au, CO2-/Ag, OH/Au, OH/Ag, SO3-/Au, and SO3-/Ag induced the face-selective nucleation of calcite from the (015), (012), (104), (103), (1 0 12), and (107) c...

An important requirement in the fabrication of advanced inorganic materials, such as ceramics and semiconductors, is control over crystallization1,2,3,4. In principle, the synthetic growth of crystals can be guided by molecular recognition at interfaces5,6,7,8,9,10,11,12,13,14,15,16. But it remains a practical challenge to control simultaneously the density and pattern of nucleation events, and the sizes and orientations of the growing crystals. Here we report a route to crystal formation, using micropatterned self-assembled monolayers17,18, which affords control over all these parameters. We begin with a metal substrate patterned with a self-assembled monolayer having areas of different nucleating activity—in this case, an array of acid-terminated regions separated by methyl-terminated regions. By immersing the patterned substrates in a calcium chloride solution and exposing them to carbon dioxide, we achieve ordered crystallization of calcite in the polar regions, where the rate of nucleation is fastest; crystallization can be completely suppressed elsewhere by a suitable choice of array spacing, which ensures that the solution is undersaturated in the methyl-terminated regions. The nucleation density (the number of crystals formed per active site) may be controlled by varying the area and distribution of the polar regions, and we can manipulate the crystallographic orientation by using different functional groups and substrates.

Micropatterning is a powerful method for controlling surface properties, with applications from cell biology to electronics1,2,3,4,5,6,7,8. Self-assembled monolayers (SAMs) of alkanethiolates on gold and silver9,10,11—the structures most widely used for preparing organic films with specific surface properties—are usually patterned by partitioning the surface into regions formed from different thiols12,13,14,15. Here we describe a way to pattern SAMs using a single alkanethiol on substrates consisting of regions of different topography: planar islands of one metal on the surface of a second (which may be different from or the same as the first). These topographically patterned SAMs consist of three regions: two planar surfaces and a transition region between the two. The characters of the SAMs on these three regions were inferred from images of three structures that form on them: condensation figures, patterns of crystals of CaCO3 and regions of selective etching. The transition region is more active in the processes generating these structures than either of the two planar regions, and we propose that this activity is due to the relatively high disorder in the organic film there. We believe that this ability to control the local disorder in a SAM with high resolution will be important in controlling processes such as nucleation, wetting, adhesion and etching on scales of below 50 nm to 5 µm.

Antiviral prophylaxis forms a significant component of health management plans for many countries around the world. A number of studies have shown that the delays typically encountered in distributing these antivirals to households, following the first infectious case, can result in their efficacy being severely reduced. Here, we investigate the use of contact tracing as a method to reduce the delays and hence mitigate the reduction in efficacy of antivirals. We assess the usefulness of contact tracing in terms of the probability of a major outbreak. It is found, with parameter distributions appropriate to the 2009 H1N1 pandemic and distributions reflecting commonly experienced delays, that standard contact tracing renders an outbreak impossible approximately one in five times compared with approximately one in ten times in its absence. A contact-tracing efficiency of 50% would see further improvements with an outbreak being impossible approximately one in four times, and a reduction of the median probability of a major outbreak from 0.41 to below 0.27.

A series of compounds (1−8) has been prepared in which a 1,4-dimethoxynaphthalene (DMN) group and either two thiol groups or a disulfide moiety are attached to the respective termini of rigid “norbornylogous” hydrocarbon bridges, 3, 5, 7, 9, and 13 bonds in length. Each of the molecules forms an adsorbate monolayer when its CH2Cl2 solutions are exposed to etched gold. Chemically reversible one-electron oxidation of the DMN group is observed in CH2Cl2 or CH3CN at potentials of 0.54−0.67 V vs ferrocene. Repeated potential cycling demonstates less monolayer degradation (a) in CH2Cl2 compared to CH3CN and (b) at subambient temperatures. Fast cyclic voltammetry experiments establish that the electron-transfer rates between the naphthalene group and the electrode surface are very fast (>103 s-1).

Monolayer films of several primary substituted hydrocarbons CH3(CH2)nCH2X (X = CH3, OH, NH2, SH, Cl, Br, I; n = 16−30) have been imaged on graphite at the solution−substrate interface using a scanning tunneling microscope (STM). The straight chain hydrocarbons form well-ordered 2-D films on graphite and physisorb with their molecular axes parallel to the surface. The NH2, SH, Br, and I end groups are observed as bright spots in the STM image corresponding to an enhancement in the tunnel current in the vicinity of the functional group, relative to the remainder of the carbon chain. On the other hand, the OH and Cl substituents were not distinguishable from the alkyl chain in the STM images. Comparison of the relative “brightness” of the functional groups with respect to the carbon chain reveals an empirical relationship between increasing relative brightness and increasing molecular polarizability. A model is proposed to describe the STM imaging mechanism for these insulating, physisorbed films in which th...

Antiviral treatment offers a fast acting alternative to vaccination; as such it is viewed as a first-line of defence against pandemic influenza in protecting families and households once infection has been detected. In clinical trials, antiviral treatments have been shown to be efficacious in preventing infection, limiting disease and reducing transmission, yet their impact at containing the 2009 influenza A(H1N1)pdm outbreak was limited. To understand this seeming discrepancy, we develop a general and computationally efficient model for studying household-based interventions. This allows us to account for uncertainty in quantities relevant to the 2009 pandemic in a principled way, accounting for the heterogeneity and variability in each epidemiological process modelled. We find that the population-level effects of delayed antiviral treatment and prophylaxis mean that their limited overall impact is quantitatively consistent (at current levels of precision) with their reported clinical efficacy under ideal conditions. Hence, effective control of pandemic influenza with antivirals is critically dependent on early detection and delivery ideally within 24 h.

Without mutation and migration, evolutionary dynamics ultimately leads to the extinction of all but one species. Such fixation processes are well understood and can be characterized analytically with methods from statistical physics. However, many biological arguments focus on stationary distributions in a mutation-selection equilibrium. Here, we address the equilibration time required to reach stationarity in the presence of mutation, this is known as the mixing time in the theory of Markov processes. We show that mixing times in evolutionary games have the opposite behaviour from fixation times when the intensity of selection increases: In coordination games with bistabilities, the fixation time decreases, but the mixing time increases. In coexistence games with metastable states, the fixation time increases, but the mixing time decreases. Our results are based on simulations and the WKB approximation of the master equation., Comment: 5 pages, 3 figures

We calculate both the exponential and pre-factor contributions in a WKB approximation of the master equation for a stochastic SIR model with highly oscillatory dynamics. Fixing the basic parameters of the model we investigate how the outbreak distribution changes with the population size. We show that this distribution rapidly becomes highly non-Gaussian, acquiring large tails indicating the presence of rare, but large outbreaks, as the population is made smaller. The analytic results are found to be in excellent agreement with simulations until the systems become so small that the dynamics are dominated by fade-out of the disease., Updated with some minor corrections

The increasing use of computer simulation by theoretical ecologists started a move away from models formulated at the population level towards individual-based models. However, many of the models studied at the individual level are not analysed mathematically and remain defined in terms of a computer algorithm. This is not surprising, given that they are intrinsically stochastic and require tools and techniques for their study that may be unfamiliar to ecologists. Here, we argue that the construction of ecological models at the individual level and their subsequent analysis is, in many cases, straightforward and leads to important insights. We discuss recent work that highlights the importance of stochastic effects for parameter ranges and systems where it was previously thought that such effects would be negligible.

We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle is annual, and after a period-doubling when it is biennial. The comprehensive nature of our approach allows us to give a coherent picture of the dynamics which unifies past work, but which also provides a systematic method for predicting the periods of oscillations seen in whooping cough and measles epidemics., Updated with revisions and two new appendices