Getting the most out of maths: How to coordinate mathematical modelling research to support a pandemic, lessons learnt from three initiatives that were part of the COVID-19 response in the UK

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Commentary on the use of the reproduction number R during the COVID-19 pandemic

Since the beginning of the COVID-19 pandemic, the reproduction number [Formula: see text] has become a popular epidemiological metric used to communicate the state of the epidemic. At its most basic, [Formula: see text] is defined as the average number of secondary infections caused by one primary infected individual. [Formula: see text] seems convenient, because the epidemic is expanding if [Formula: see text] and contracting if [Formula: see text]. The magnitude of [Formula: see text] indicates by how much transmission needs to be reduced to control the epidemic. Using [Formula: see text] in a naïve way can cause new problems. The reasons for this are threefold: (1) There is not just one definition of [Formula: see text] but many, and the precise definition of [Formula: see text] affects both its estimated value and how it should be interpreted. (2) Even with a particular clearly defined [Formula: see text], there may be different statistical methods used to estimate its value, and the choice of method will affect the estimate. (3) The availability and type of data used to estimate [Formula: see text] vary, and it is not always clear what data should be included in the estimation. In this review, we discuss when [Formula: see text] is useful, when it may be of use but needs to be interpreted with care, and when it may be an inappropriate indicator of the progress of the epidemic. We also argue that careful definition of [Formula: see text], and the data and methods used to estimate it, can make [Formula: see text] a more useful metric for future management of the epidemic.

Modelling: Understanding pandemics and how to control them

New disease challenges, societal demands and better or novel types of data, drive innovations in the structure, formulation and analysis of epidemic models. Innovations in modelling can lead to new insights into epidemic processes and better use of available data, yielding improved disease control and stimulating collection of better data and new data types. Here we identify key challenges for the structure, formulation, analysis and use of mathematical models of pathogen transmission relevant to current and future pandemics.

Challenges on the interaction of models and policy for pandemic control

The COVID-19 pandemic has seen infectious disease modelling at the forefront of government decision-making. Models have been widely used throughout the pandemic to estimate pathogen spread and explore the potential impact of different intervention strategies. Infectious disease modellers and policymakers have worked effectively together, but there are many avenues for progress on this interface. In this paper, we identify and discuss seven broad challenges on the interaction of models and policy for pandemic control. We then conclude with suggestions and recommendations for the future.

Key questions for modelling COVID-19 exit strategies

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

Analysing Interrupted Time Series with a Control

Interrupted time series are increasingly being used to evaluate the population-wide implementation of public health interventions. However, the resulting estimates of intervention impact can be severely biased if underlying disease trends are not adequately accounted for. Control series offer a potential solution to this problem, but there is little guidance on how to use them to produce trend-adjusted estimates. To address this lack of guidance, we show how interrupted time series can be analysed when the control and intervention series share confounders, i. e. when they share a common trend. We show that the intervention effect can be estimated by subtracting the control series from the intervention series and analysing the difference using linear regression or, if a log-linear model is assumed, by including the control series as an offset in a Poisson regression with robust standard errors. The methods are illustrated with two examples.

Gaussian process approximations for fast inference from infectious disease data

We present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. We develop this for the SIR and SEIR models, and we show how it can be used to perform quick maximum likelihood inference for the underlying parameters given population estimates of the number of infecteds or cases at given time points. We also show how the unobserved processes can be inferred at the same time as the underlying parameters.

Response to the Letter to the Editor by Eberhard et al

In a Letter to the Editor, Eberhard et al. question the validity of our model of skin snip sensitivity and argue against the use of skin snips to evaluate onchocerciasis elimination by mass drug administration. Here we discuss their arguments and compare model predictions with observed data to assess the validity of our model.

Series expansions for the properties of a birth process of controlled variability

A birth process is studied in which the birth rate at any time is a function of the difference between the current population size and a target corresponding to unit growth rate. If this controlling function is a decreasing function of its argument a stabilizing effect is to be expected. By supposing that the controlling function varies very slowly, series expansions for the properties of the process are obtained, the leading term corresponding to a diffusion approximation. The sequence of births considered as a point process of controlled variability is examined and approximations to the interval distribution and covariance density obtained.

On a point process with independent locations

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

A Markov construction for a multidimensional point process

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

Dependent thinning of point processes

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

The virtual waiting-time and related processes

The virtual waiting-time process of Takács is one of the simplest examples of a stochastic process with a continuous state space in continuous time in which jump transitions interrupt periods of deterministic decay. Properties of the process are reviewed, and the transient behaviour examined in detail. Several generalizations of the process are studied. These include two-sided jumps, periodically varying ‘arrival’ rate and the presence of a state-dependent decay rate; the last case is motivated by the properties of soil moisture in hydrology. Throughout, the emphasis is on the derivation of simple interpretable results.

Modelling Neglected Tropical Diseases diagnostics: the sensitivity of skin snips for Onchocerca volvulus in near elimination and surveillance settings

BACKGROUND

The African Programme for Onchocerciasis Control has proposed provisional thresholds for the prevalence of microfilariae in humans and of L3 larvae in blackflies, below which mass drug administration (MDA) with ivermectin can be stopped and surveillance started. Skin snips are currently the gold standard test for detecting patent Onchocerca volvulus infection, and the World Health Organization recommends their use to monitor progress of treatment programmes (but not to verify elimination). However, if they are used (in transition and in parallel to Ov-16 serology), sampling protocols should be designed to demonstrate that programmatic goals have been reached. The sensitivity of skin snips is key to the design of such protocols.

METHODS

We develop a mathematical model for the number of microfilariae in a skin snip and parameterise it using data from Guatemala, Venezuela, Ghana and Cameroon collected before the start of ivermectin treatment programmes. We use the model to estimate sensitivity as a function of time since last treatment, number of snips taken, microfilarial aggregation and female worm fertility after exposure to 10 annual rounds of ivermectin treatment.

RESULTS

The sensitivity of the skin snip method increases with time after treatment, with most of the increase occurring between 0 and 5 years. One year after the last treatment, the sensitivity of two skin snips taken from an individual infected with a single fertile female worm is 31 % if there is no permanent effect of multiple ivermectin treatments on fertility; 18 % if there is a 7 % reduction per treatment, and 0.6 % if there is a 35 % reduction. At 5 years, the corresponding sensitivities are 76 %, 62 % and 4.7 %. The sensitivity improves significantly if 4 skin snips are taken: in the absence of a permanent effect of ivermectin, the sensitivity of 4 skin snips is 53 % 1 year and 94 % 5 years after the last treatment.

CONCLUSIONS

Our model supports the timelines proposed by APOC for post-MDA follow-up and surveillance surveys every 3-5 years. Two skin snips from the iliac region have reasonable sensitivity to detect residual infection, but the sensitivity can be significantly improved by taking 4 snips. The costs and benefits of using four versus two snips should be evaluated.

Modeling infectious disease dynamics in the complex landscape of global health

Despite some notable successes in the control of infectious diseases, transmissible pathogens still pose an enormous threat to human and animal health. The ecological and evolutionary dynamics of infections play out on a wide range of interconnected temporal, organizational, and spatial scales, which span hours to months, cells to ecosystems, and local to global spread. Moreover, some pathogens are directly transmitted between individuals of a single species, whereas others circulate among multiple hosts, need arthropod vectors, or can survive in environmental reservoirs. Many factors, including increasing antimicrobial resistance, increased human connectivity and changeable human behavior, elevate prevention and control from matters of national policy to international challenge. In the face of this complexity, mathematical models offer valuable tools for synthesizing information to understand epidemiological patterns, and for developing quantitative evidence for decision-making in global health.

Five challenges for spatial epidemic models

Infectious disease incidence data are increasingly available at the level of the individual and include high-resolution spatial components. Therefore, we are now better able to challenge models that explicitly represent space. Here, we consider five topics within spatial disease dynamics: the construction of network models; characterising threshold behaviour; modelling long-distance interactions; the appropriate scale for interventions; and the representation of population heterogeneity.

Seven challenges for modelling indirect transmission: vector-borne diseases, macroparasites and neglected tropical diseases

Many of the challenges which face modellers of directly transmitted pathogens also arise when modelling the epidemiology of pathogens with indirect transmission – whether through environmental stages, vectors, intermediate hosts or multiple hosts. In particular, understanding the roles of different hosts, how to measure contact and infection patterns, heterogeneities in contact rates, and the dynamics close to elimination are all relevant challenges, regardless of the mode of transmission. However, there remain a number of challenges that are specific and unique to modelling vector-borne diseases and macroparasites. Moreover, many of the neglected tropical diseases which are currently targeted for control and elimination are vector-borne, macroparasitic, or both, and so this article includes challenges which will assist in accelerating the control of these high-burden diseases. Here, we discuss the challenges of indirect measures of infection in humans, whether through vectors or transmission life stages and in estimating the contribution of different host groups to transmission. We also discuss the issues of “evolution-proof” interventions against vector-borne disease.

Seven challenges for metapopulation models of epidemics, including households models

This paper considers metapopulation models in the general sense, i.e. where the population is partitioned into sub-populations (groups, patches,...), irrespective of the biological interpretation they have, e.g. spatially segregated large sub-populations, small households or hosts themselves modelled as populations of pathogens. This framework has traditionally provided an attractive approach to incorporating more realistic contact structure into epidemic models, since it often preserves analytic tractability (in stochastic as well as deterministic models) but also captures the most salient structural inhomogeneity in contact patterns in many applied contexts. Despite the progress that has been made in both the theory and application of such metapopulation models, we present here several major challenges that remain for future work, focusing on models that, in contrast to agent-based ones, are amenable to mathematical analysis. The challenges range from clarifying the usefulness of systems of weakly-coupled large sub-populations in modelling the spread of specific diseases to developing a theory for endemic models with household structure. They include also developing inferential methods for data on the emerging phase of epidemics, extending metapopulation models to more complex forms of human social structure, developing metapopulation models to reflect spatial population structure, developing computationally efficient methods for calculating key epidemiological model quantities, and integrating within- and between-host dynamics in models.

Eight challenges for network epidemic models

Networks offer a fertile framework for studying the spread of infection in human and animal populations. However, owing to the inherent high-dimensionality of networks themselves, modelling transmission through networks is mathematically and computationally challenging. Even the simplest network epidemic models present unanswered questions. Attempts to improve the practical usefulness of network models by including realistic features of contact networks and of host-pathogen biology (e.g. waning immunity) have made some progress, but robust analytical results remain scarce. A more general theory is needed to understand the impact of network structure on the dynamics and control of infection. Here we identify a set of challenges that provide scope for active research in the field of network epidemic models.

A mathematical model of serotype replacement in pneumococcal carriage following vaccination

A number of childhood vaccination programmes have recently introduced vaccination against Streptococcus pneumoniae, the pneumococcus, a major cause of pneumonia and meningitis. The pneumococcal conjugate vaccines (PCVs) that are currently in use only protect against some serotypes of the bacterium, and there is now strong evidence that those serotypes not included in the vaccine increase in prevalence among most vaccinated populations. We present a mathematical model for the dynamics of nasopharyngeal carriage of S. pneumoniae that allows for carriage with multiple serotypes. The model is used to predict the prevalence of vaccine type (VT) and non-VT (NVT) serotypes following the introduction of PCV. Parameter estimates for the model are obtained by maximum likelihood using pre-vaccination data from The Gambia. The model predicts that low (1, 6A and 9V) and medium (4, 5, 7F, 14, 18C, 19A and 19F) prevalence serotypes can be eliminated through vaccination, but that the overall prevalence of carriage will be reduced only slightly because of an increase in the prevalence of NVT serotypes. Serotype replacement will be sequential, with high and medium prevalence NVT serotypes dominating initially, followed by an increase of serotypes of low prevalence. We examine the impact of a hypothetical vaccine that provides partial protection against all serotypes, and find that this reduces overall carriage, but is unable to eliminate low or medium prevalence serotypes.

Spread of information and infection on finite random networks

The modeling of epidemic-like processes on random networks has received considerable attention in recent years. While these processes are inherently stochastic, most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit. In a previous paper, for a class of epidemic and rumor processes, we derived approximate models for the full probability distribution of the final size of the epidemic, as opposed to only mean values. In this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies: homogeneous networks, Erdös-Rényi (ER) random graphs, Barabasi-Albert scale-free networks, and random geometric graphs. We find that the approximate model is reasonably accurate in predicting the probability of spread. However, the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks. We attribute this failure to the presence of other structural properties beyond degree-degree correlations, and in particular clustering, which are present in any finite network but are not incorporated in the approximate model. In order to test this "hypothesis"  we perform additional simulations on a set of ER random graphs where degree-degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature. Our results show that even strong degree-degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size. On the other hand, the introduction of clustering greatly affects both the position of the threshold and the conditional mean. Similar analysis for the Barabasi-Albert scale-free network confirms the significance of clustering on the dynamics of rumor spread. For this network, though, with its highly skewed degree distribution, the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph.

Designs for clinical trials with time-to-event outcomes based on stopping guidelines for lack of benefit

background

The pace of novel medical treatments and approaches to therapy has accelerated in recent years. Unfortunately, many potential therapeutic advances do not fulfil their promise when subjected to randomized controlled trials. It is therefore highly desirable to speed up the process of evaluating new treatment options, particularly in phase II and phase III trials. To help realize such an aim, in 2003, Royston and colleagues proposed a class of multi-arm, two-stage trial designs intended to eliminate poorly performing contenders at a first stage (point in time). Only treatments showing a predefined degree of advantage against a control treatment were allowed through to a second stage. Arms that survived the first-stage comparison on an intermediate outcome measure entered a second stage of patient accrual, culminating in comparisons against control on the definitive outcome measure. The intermediate outcome is typically on the causal pathway to the definitive outcome (i.e. the features that cause an intermediate event also tend to cause a definitive event), an example in cancer being progression-free and overall survival. Although the 2003 paper alluded to multi-arm trials, most of the essential design features concerned only two-arm trials. Here, we extend the two-arm designs to allow an arbitrary number of stages, thereby increasing flexibility by building in several 'looks' at the accumulating data. Such trials can terminate at any of the intermediate stages or the final stage.

Methods

We describe the trial design and the mathematics required to obtain the timing of the 'looks' and the overall significance level and power of the design. We support our results by extensive simulation studies. As an example, we discuss the design of the STAMPEDE trial in prostate cancer.

Results

The mathematical results on significance level and power are confirmed by the computer simulations. Our approach compares favourably with methodology based on beta spending functions and on monitoring only a primary outcome measure for lack of benefit of the new treatment.

Conclusions

The new designs are practical and are supported by theory. They hold considerable promise for speeding up the evaluation of new treatments in phase II and III trials.

Point process models of rainfall: developments for fine-scale structure

A conceptual stochastic model of rainfall is proposed in which storm origins occur in a Poisson process, where each storm has a random lifetime during which rain cell origins occur in a secondary Poisson process. In addition, each cell has a random lifetime during which instantaneous random depths (or ‘pulses’) of rain occur in a further Poisson process. A key motivation behind the model formulation is to account for the variability in rainfall data over small (e.g. 5 min) and larger time intervals. Time-series properties are derived to enable the model to be fitted to aggregated rain gauge data. These properties include moments up to third order, the probability that an interval is dry, and the autocovariance function. To allow for distinct storm types (e.g. convective and stratiform), several processes may be superposed. Using the derived properties, a model consisting of two storm types is fitted to 60 years of 5 min rainfall data taken from a site near Wellington, New Zealand, using sample estimates taken at 5 min, 1 hour, 6 hours and daily levels of aggregation. The model is found to fit moments of the depth distribution up to third order very well at these time scales. Using the fitted model, 5 min series are simulated, and annual maxima are extracted and compared with equivalent values taken from the historical record. A good fit in the extremes is found at both 1 and 24 hour levels of aggregation, although at the 5 min level there is some underestimation of the historical values. Proportions of time intervals with depths below various low thresholds are extracted from the simulated and historical series and compared. A tendency for underestimation of the historical values is evident at some time scales, with a close fit being obtained as the threshold is increased.

Sexually transmitted disease epidemics in a natural insect population

1. The epidemiology of sexually transmitted diseases (STDs) in human and domesticated populations is well documented. However, there has been less study of STDs in natural populations. 2. We investigated STD dynamics in the model system involving a host from the most speciose group of animals: the insects. We investigated temporal variation in the prevalence of the sexually transmitted mite Coccipolipus hippodamiae on its ladybird host, Adalia bipunctata. 3. Field surveys over two seasons showed a repeated pattern of a profound epidemic in the overwintered cohort and a later prevalence decline. 4. In order to understand the key factors in the dynamics of this system we studied the phenology of the host and simulated parasite dynamics in the overwintered cohort using a model with within-sex homogeneity in mating rate and field-measured parameter values. The similarity of natural and simulation prevalence levels allowed us to carry out sensitivity analysis and hence to identify the key determinants of the dynamics. 5. The observed pattern of periodic extreme prevalence combined with system persistence probably results from time lags in host recruitment and widespread promiscuity. 6. Our findings improve our understanding of STDs in natural populations and illustrate the importance of examining seasonality and time delays in population dynamics in order to fully understand the characteristics of natural populations and their parasites.

Representation of space–time variability of soil moisture

A simplified spatial-temporal soil moisture model driven by stochastic spatial rainfall forcing is proposed. The model is mathematically tractable, and allows the spatial and temporal structure of soil moisture fields, induced by the spatial-temporal variability of rainfall and the spatial variability of vegetation, to be explored analytically. The influence of the main model parameters, reflecting the spatial scale of rain cells, the soil storage capacity, the rainfall interception and the soil water loss rate (representing evaporation and deep infiltration) is investigated. The variabilities of the spatially averaged soil moisture process, and that averaged in both space and time, are derived. The present analysis focuses on spatially uniform vegetation conditions; a follow-up paper will incorporate stochastically heterogeneous vegetation.

Population biology of multispecies helminth infection: interspecific interactions and parasite distribution

Despite evidence for the existence of interspecific interactions between helminth species, there has been no theoretical exploration of their effect on the distribution of the parasite species in a host population. We use a deterministic model for the accumulation and loss of adult worms of 2 interacting helminth species to motivate an individual-based stochastic model. The mean worm burden and variance[ratio ]mean ratio (VMR) of each species, and the correlation between the two species are used to describe the distribution within different host age classes. We find that interspecific interactions can produce convex age-intensity profiles and will impact the level of aggregation (as measured by the VMR). In the absence of correlated exposure, the correlation in older age classes may be close to zero when either intra- or interspecific synergistic effects are strong. We therefore suggest examining the correlation between species in young hosts as a possible means of identifying interspecific interaction. The presence of correlation between the rates of exposure makes the interpretation of correlations between species more difficult. Finally we show that in the absence of interaction, strong positive correlations are generated by averaging across most age classes.

Surveillance of antimicrobial resistance--what, how and whither?

OBJECTIVE

To express the views of a working party held to consider antibiotic resistance surveillance systems, their strengths and weaknesses, and their current and future applications.

METHODS

The participants, all of whom were experienced in this field, discussed the development of surveillance systems in relation to the increasing prevalence of resistance to antibacterial agents and the current interest in surveillance systems shown by many official bodies, in both the human and veterinary fields. The problems inherent in surveillance systems were considered together with the applications of different systems.

RESULTS

The properties of good antibiotic resistance surveillance systems were defined. Surveillance systems vary widely from those with a narrow base, focusing on few organisms in one disease area, to those covering many diseases, many organisms (including normal flora) and many compounds. Whatever their design, they should be able to detect significant differences and shifts in susceptibility to various antibacterial agents, and the information derived from them should reach as many interested parties as possible in a timely manner. In using this information to decide strategies, criteria for action need to be determined by pragmatic consensus. Funding remains a major problem, with few large studies being supported by official bodies in spite of their professed enthusiasm for surveillance. In consequence, many current systems are funded by the pharmaceutical industry and are of necessity restricted in their focus.

CONCLUSIONS

Antibiotic resistance surveillance studies should and can be well planned and well executed. Many current systems suffer from well-recognized but uncorrected biases. Consortium funding will be necessary for large schemes to be successful. There is no "ideal" surveillance system.

A study of the role of the transmission mechanism in macroparasite aggregation

The dynamics of host-macroparasite infections pose considerable challenges for stochastic modelling because of the need to take into account a large number of relevant factors and many nonlinear interactions between them. This paper focuses attention on the infection transmission process and the effects of specific modelling assumptions about the mechanisms involved. Some dramatically simplified linear models are considered; they are based on multidimensional linear birth and death processes, and are designed to illuminate qualitative effects of interest. Both single and compound infections are allowed. It is shown that such simple models can generate and increase dispersion of parasite counts, even among homogeneous hosts.

Stochastic host-parasite interaction models

We contribute to the discussion of causes and effects of aggregation (overdispersion) of macroparasite counts, focussing particularly upon the effects of clumped infections and parasite-induced host mortality. The simple nonlinear stochastic model for the evolution of the parasite load of a single host, investigated in Isham (1995), is extended to allow three parasite stages (larval, mature and offspring), and to allow durations of these stages to be non-exponentially distributed. As in the earlier work, exact algebraic results are possible, providing insight into the aggregation mechanisms, as long as the only source of interaction between host and parasites is an excess host mortality linearly related to the parasite load. Results are obtained on the distribution of parasite load and on host survival. In particular, although parasite-induced host mortality is usually thought of as a process that reduces parasite aggregation (Anderson and Gordon 1982), it is shown that, for this model, parasite-induced host mortality cannot cause the index of dispersion to fall below unity. Host heterogeneity and disease control are also discussed. An approximation based on moment assumptions appropriate to a specially-constructed multivariate negative binomial distribution is proposed. This approximation, which is applicable to other processes, and an alternative based on the multivariate normal distribution are compared with exact results.

Anthelmintic resistance revisited: under-dosing, chemoprophylactic strategies, and mating probabilities

Deterministic and stochastic models are used to examine the evolution of anthelmintic resistance among trichostrongylid parasites of domestic ruminants. We find that the relative selection pressures exerted by chemoprophylactic (preventive) control strategies, chemotherapeutic (salvage) control strategies, and regimens involving "under-dosing" are critically dependent on a variety of host and parasite parameters (particularly host immunity and grazing behaviour, parasite fecundity, and the survival of the free-living stages on the pasture). Chemoprophylactic strategies are not necessarily more likely to exert a stronger selection pressure than chemotherapeutic strategies. Similarly, as one reduces dosage levels, there is a range of dose levels where under-dosing promotes resistance and a range of dose levels where under-dosing impedes resistance. The most dangerous dose is either that necessary to kill all the susceptible homozygotes, or that necessary to kill all the susceptible homozygotes and all the heterozygotes. Which one prevails depends upon model parameters. The stochastic formulation indicates that spatial heterogeneity in transmission may be a significant force in promoting the spread of resistant genotypes--at least when infection is at low levels.

Epidemics: models and data

"The problems of understanding and controlling disease raise a range of challenging mathematical and statistical research topics, from broad theoretical issues to specific practical ones. In particular, recent interest in acquired immune deficiency syndrome has stimulated much progress in diverse areas of epidemic modelling, particularly with regard to the treatment of heterogeneity, both between individuals and in mixing of subgroups of the population. At the same time better data and data analysis techniques have become available, and there have been exciting developments in relevant theory.... This progress in specific areas is now being matched by interdisciplinary cooperation aimed at elucidating relationships between the widely varying types of model that have been found useful, to determine their strengths and limitations in relation to basic aims such as understanding, prediction, and evaluation and implementation of control strategies."

Assessing the variability of stochastic epidemics

In predicting the course of individual realizations of an epidemic it is important to know the magnitude of the variability of such realizations about their mean. In this paper and in the context of the general stochastic epidemic, some methods of obtaining approximate estimates of this variability are investigated; one is a multivariate normal approximation based on an asymptotic Gaussian diffusion process, and another uses an approximating linear stochastic process. The extension of these methods to the more detailed models used to describe the transmission dynamics of HIV infection and AIDS is discussed.

Estimation of the incidence of HIV infection

The aim of the method of 'back projection' is to provide estimates of the number of new infections with the human immunodeficiency virus (HIV) as a function of time, by using the numbers of diagnoses of the acquired immune deficiency syndrome (AIDS) together with information on the distribution of the incubation period between infection and diagnosis. Here, the method is investigated with particular reference to cases of HIV infection and AIDS in the United Kingdom.

The effect of intermittent exposure to risk on failure time distributions

The relation is investigated between the distributions of the total time until failure and the time of exposure to risk until failure, for individuals who are at risk only intermittently during their lifetimes. A specific example is considered in the case of computer failures.