Estimating the basic reproductive number during the early stages of an emerging epidemic.

A novel outbreak will generally not be detected until such a time that it has become established. When such an outbreak is detected, public health officials must determine the potential of the outbreak, for which the basic reproductive numberR0 is an important factor. However, it is often the case that the resulting estimate of R0 is positively-biased for a number of reasons. One commonly overlooked reason is that the outbreak was not detected until such a time that it had become established, and therefore did not experience initial fade out. We propose a method which accounts for this bias by conditioning the underlying epidemic model on becoming established and demonstrate that this approach leads to a less-biased estimate of R0 during the early stages of an outbreak. We also present a computationally-efficient approximation scheme which is suitable for large data sets in which the number of notified cases is large. This methodology is applied to an outbreak of pandemic influenza in Western Australia, recorded in 2009.

Intervention to maximise the probability of epidemic fade-out.

The emergence of a new strain of a disease, or the introduction of an existing strain to a naive population, can give rise to an epidemic. We consider how to maximise the probability of epidemic fade-out - that is, disease elimination in the trough between the first and second waves of infection - in the Markovian SIR-with-demography epidemic model. We assume we have an intervention at our disposal that results in a lowering of the transmission rate parameter, ß, and that an epidemic has commenced. We determine the optimal stage during the epidemic in which to implement this intervention. This may be determined using Markov decision theory, but this is not always practical, in particular if the population size is large. Hence, we also derive a formula that gives an almost optimal solution, based upon the approximate deterministic behaviour of the model. This formula is explicit, simple, and, perhaps surprisingly, independent of ß and the effectiveness of the intervention. We demonstrate that this policy can give a substantial increase in the probability of epidemic fade-out, and we also show that it is relatively robust to a less than ideal implementation.

Hybrid Markov chain models of S-I-R disease dynamics.

Deterministic epidemic models are attractive due to their compact nature, allowing substantial complexity with computational efficiency. This partly explains their dominance in epidemic modelling. However, the small numbers of infectious individuals at early and late stages of an epidemic, in combination with the stochastic nature of transmission and recovery events, are critically important to understanding disease dynamics. This motivates the use of a stochastic model, with continuous-time Markov chains being a popular choice. Unfortunately, even the simplest Markovian S-I-R model-the so-called general stochastic epidemic-has a state space of order [Formula: see text], where N is the number of individuals in the population, and hence computational limits are quickly reached. Here we introduce a hybrid Markov chain epidemic model, which maintains the stochastic and discrete dynamics of the Markov chain in regions of the state space where they are of most importance, and uses an approximate model-namely a deterministic or a diffusion model-in the remainder of the state space. We discuss the evaluation, efficiency and accuracy of this hybrid model when approximating the distribution of the duration of the epidemic and the distribution of the final size of the epidemic. We demonstrate that the computational complexity is [Formula: see text] and that under suitable conditions our approximations are highly accurate.

The probability of epidemic fade-out is non-monotonic in transmission rate for the Markovian SIR model with demography.

Epidemic fade-out refers to infection elimination in the trough between the first and second waves of an outbreak. The number of infectious individuals drops to a relatively low level between these waves of infection, and if elimination does not occur at this stage, then the disease is likely to become endemic. For this reason, it appears to be an ideal target for control efforts. Despite this obvious public health importance, the probability of epidemic fade-out is not well understood. Here we present new algorithms for approximating the probability of epidemic fade-out for the Markovian SIR model with demography. These algorithms are more accurate than previously published formulae, and one of them scales well to large population sizes. This method allows us to investigate the probability of epidemic fade-out as a function of the effective transmission rate, recovery rate, population turnover rate, and population size. We identify an interesting feature: the probability of epidemic fade-out is very often greatest when the basic reproduction number, R0, is approximately 2 (restricting consideration to cases where a major outbreak is possible, i.e., R0>1). The public health implication is that there may be instances where a non-lethal infection should be allowed to spread, or antiviral usage should be moderated, to maximise the chance of the infection being eliminated before it becomes endemic.

Optimal prophylactic vaccination in segregated populations: When can we improve on the equalising strategy?

One of the fundamental problems in public health is how to allocate a limited set of resources to have the greatest benefit on the health of the population. This often leads to difficult value judgements about budget allocations. However, one scenario that is directly amenable to mathematical analysis is the optimal allocation of a finite stockpile of vaccine when the population is partitioned into many relatively small cliques, often conceptualised as households. For the case of SIR (susceptible-infectious-recovered) dynamics, analysis and numerics have supported the conjecture that an equalising strategy (which leaves equal numbers of susceptible individuals in each household) is optimal under certain conditions. However, there exists evidence that some of these conditions may be invalid or unsuitable in many situations. Here we consider how well the equalising strategy performs in a range of other scenarios that deviate from the idealised household model. We find that in general the equalising strategy often performs optimally, even far from the idealised case. However, when considering large subpopulation sizes, frequency-dependent transmission and intermediate levels of vaccination, optimality is often achieved through more heterogeneous vaccination strategies.

Computation of epidemic final size distributions.

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.

On the derivation of approximations to cellular automata models and the assumption of independence.

Cellular automata are discrete agent-based models, generally used in cell-based applications. There is much interest in obtaining continuum models that describe the mean behaviour of the agents in these models. Previously, continuum models have been derived for agents undergoing motility and proliferation processes, however, these models only hold under restricted conditions. In order to narrow down the reason for these restrictions, we explore three possible sources of error in deriving the model. These sources are the choice of limiting arguments, the use of a discrete-time model as opposed to a continuous-time model and the assumption of independence between the state of sites. We present a rigorous analysis in order to gain a greater understanding of the significance of these three issues. By finding a limiting regime that accurately approximates the conservation equation for the cellular automata, we are able to conclude that the inaccuracy between our approximation and the cellular automata is completely based on the assumption of independence.

Epidemiological consequences of household-based antiviral prophylaxis for pandemic influenza.

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.

On parameter estimation in population models III: time-inhomogeneous processes and observation error.

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the population size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Continuing previous work in this series on the use of diffusion approximations for efficient calibration of continuous-time Markov chains, I present efficient techniques for time-inhomogeneous chains and accounting for observation error. Observation error (partial observability) is accounted for by joint estimation using a scaled unscented Kalman filter for state-space models. The methodology will be illustrated with respect to models of disease dynamics incorporating seasonal transmission rate and in the presence of observation error, including application to two influenza outbreaks and measles in London in the pre-vaccination era.

Invasion of infectious diseases in finite homogeneous populations.

We consider the initial invasion of an infectious disease in a finite, homogeneous population. Methodology for evaluating the basic reproduction number, R(0), and the probability mass function of secondary infections is presented. The impact of finite population size, and infectious period distribution (between exponential, two-phase gamma, and constant), is assessed. Implications for infectious disease invasion and estimation of infectious disease model and parameters from data of secondary infections by initially infected individuals in naive, finite, homogeneous populations are reported. As any individual interacts with a finite number of contacts during their infectious period, these results are important to the study of infectious disease dynamics.

Measuring social networks in British primary schools through scientific engagement.

Primary schools constitute a key risk group for the transmission of infectious diseases, concentrating great numbers of immunologically naive individuals at high densities. Despite this, very little is known about the social patterns of mixing within a school, which are likely to contribute to disease transmission. In this study, we present a novel approach where scientific engagement was used as a tool to access school populations and measure social networks between young (4-11 years) children. By embedding our research project within enrichment activities to older secondary school (13-15) children, we could exploit the existing links between schools to achieve a high response rate for our study population (around 90% in most schools). Social contacts of primary school children were measured through self-reporting based on a questionnaire design, and analysed using the techniques of social network analysis. We find evidence of marked social structure and gender assortativity within and between classrooms in the same school. These patterns have been previously reported in smaller studies, but to our knowledge no study has attempted to exhaustively sample entire school populations. Our innovative approach facilitates access to a vitally important (but difficult to sample) epidemiological sub-group. It provides a model whereby scientific communication can be used to enhance, rather than merely complement, the outcomes of research.

Modelling population processes with random initial conditions.

Population dynamics are almost inevitably associated with two predominant sources of variation: the first, demographic variability, a consequence of chance in progenitive and deleterious events; the second, initial state uncertainty, a consequence of partial observability and reporting delays and errors. Here we outline a general method for incorporating random initial conditions in population models where a deterministic model is sufficient to describe the dynamics of the population. Additionally, we show that for a large class of stochastic models the overall variation is the sum of variation due to random initial conditions and variation due to random dynamics, and thus we are able to quantify the variation not accounted for when random dynamics are ignored. Our results are illustrated with reference to both simulated and real data.

Computationally exact methods for stochastic periodic dynamics: Spatiotemporal dispersal and temporally forced transmission.

The dynamics of many diseases and populations possess distinct recurring phases. For example, many species breed only during a subset of the year and the infection dynamics of many pathogens have transmission rates that vary with season. Here I investigate computational methods for studying transient and long-term behaviour of stochastic models which have periodic phases-several different potential techniques for studying long-term behaviour will be contrasted. I illustrate the results with two studies: The first is of a spatially realistic metapopulation model of malleefowl (Leipoa ocellata), a species which disperses only during a quarter of the year; this model is used to highlight the advantages and disadvantages of the particular methods presented. The second study is of a model for disease dynamics which incorporates seasonality in both the rate of within-population transmission and also in the rate of transmission effected via aerosol importation. This model has applications to studying disease invasion and persistence in captive-breeding populations. We demonstrate, via comparison to appropriately matched models with constant transmission rates and also no aerosol transmission, that seasonality and aerosol importation may alter control choices, with possibly an increase in the threshold population size for local control surveillance, transfer of importance to limiting aerosol transmission, and the use of temporally targetted surveillance. The methodology presented is the gold-standard for dealing with many phased processes in ecology and epidemiology, but its application is limited to systems of small size.

Efficient methods for studying stochastic disease and population dynamics.

Stochastic ecological and epidemiological models are now routinely used to inform management and decision making throughout conservation and public-health. A difficulty with the use of such models is the need to resort to simulation methods when the population size (and hence the size of the state space) becomes large, resulting in the need for a large amount of computation to achieve statistical confidence in results. Here we present two methods that allow evaluation of all quantities associated with one- (and higher) dimensional Markov processes with large state spaces. We illustrate these methods using SIS disease dynamics and studying species that are affected by catastrophic events. The methods allow the possibility of extending exact Markov methods to real-world problems, providing techniques for efficient parameterisation and subsequent analysis.

On parameter estimation in population models II: multi-dimensional processes and transient dynamics.

Recently, a computationally-efficient method was presented for calibrating a wide-class of Markov processes from discrete-sampled abundance data. The method was illustrated with respect to one-dimensional processes and required the assumption of stationarity. Here we demonstrate that the approach may be directly extended to multi-dimensional processes, and two analogous computationally-efficient methods for non-stationary processes are developed. These methods are illustrated with respect to disease and population models, including application to infectious count data from an outbreak of "Russian influenza" (A/USSR/1977 H1N1) in an educational institution. The methodology is also shown to provide an efficient, simple and yet rigorous approach to calibrating disease processes with gamma-distributed infectious period.

Integrating stochasticity and network structure into an epidemic model.

While the foundations of modern epidemiology are based upon deterministic models with homogeneous mixing, it is being increasingly realized that both spatial structure and stochasticity play major roles in shaping epidemic dynamics. The integration of these two confounding elements is generally ascertained through numerical simulation. Here, for the first time, we develop a more rigorous analytical understanding based on pairwise approximations to incorporate localized spatial structure and diffusion approximations to capture the impact of stochasticity. Our results allow us to quantify, analytically, the impact of network structure on the variability of an epidemic. Using the susceptible-infectious-susceptible framework for the infection dynamics, the pairwise stochastic model is compared with the stochastic homogeneous-mixing (mean-field) model--although to enable a fair comparison the homogeneous-mixing parameters are scaled to give agreement with the pairwise dynamics. At equilibrium, we show that the pairwise model always displays greater variation about the mean, although the differences are generally small unless the prevalence of infection is low. By contrast, during the early epidemic growth phase when the level of infection is increasing exponentially, the pairwise model generally shows less variation.

On methods for studying stochastic disease dynamics.

Models that deal with the individual level of populations have shown the importance of stochasticity in ecology, epidemiology and evolution. An increasingly common approach to studying these models is through stochastic (event-driven) simulation. One striking disadvantage of this approach is the need for a large number of replicates to determine the range of expected behaviour. Here, for a class of stochastic models called Markov processes, we present results that overcome this difficulty and provide valuable insights, but which have been largely ignored by applied researchers. For these models, the so-called Kolmogorov forward equation (also called the ensemble or master equation) allows one to simultaneously consider the probability of each possible state occurring. Irrespective of the complexities and nonlinearities of population dynamics, this equation is linear and has a natural matrix formulation that provides many analytical insights into the behaviour of stochastic populations and allows rapid evaluation of process dynamics. Here, using epidemiological models as a template, these ensemble equations are explored and results are compared with traditional stochastic simulations. In addition, we describe further advantages of the matrix formulation of dynamics, providing simple exact methods for evaluating expected eradication (extinction) times of diseases, for comparing expected total costs of possible control programmes and for estimation of disease parameters.

On parameter estimation in population models.

We describe methods for estimating the parameters of Markovian population processes in continuous time, thus increasing their utility in modelling real biological systems. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to a class of models called density-dependent Markov population processes. We illustrate the versatility of both approaches by estimating the parameters of the stochastic SIS logistic model from simulated data. This model is also fitted to data from a population of Bay checkerspot butterfly (Euphydryas editha bayensis), allowing us to assess the viability of this population.

Stochastic models for mainland-island metapopulations in static and dynamic landscapes.

This paper has three primary aims: to establish an effective means for modelling mainland-island metapopulations inhabiting a dynamic landscape; to investigate the effect of immigration and dynamic changes in habitat on metapopulation patch occupancy dynamics; and to illustrate the implications of our results for decision-making and population management. We first extend the mainland-island metapopulation model of Alonso and McKane [Bull. Math. Biol. 64:913-958, 2002] to incorporate a dynamic landscape. It is shown, for both the static and the dynamic landscape models, that a suitably scaled version of the process converges to a unique deterministic model as the size of the system becomes large. We also establish that, under quite general conditions, the density of occupied patches, and the densities of suitable and occupied patches, for the respective models, have approximate normal distributions. Our results not only provide us with estimates for the means and variances that are valid at all stages in the evolution of the population, but also provide a tool for fitting the models to real metapopulations. We discuss the effect of immigration and habitat dynamics on metapopulations, showing that mainland-like patches heavily influence metapopulation persistence, and we argue for adopting measures to increase connectivity between this large patch and the other island-like patches. We illustrate our results with specific reference to examples of populations of butterfly and the grasshopper Bryodema tuberculata.

Extinction times for a birth-death process with two phases.

Many populations have a negative impact on their habitat or upon other species in the environment if their numbers become too large. For this reason they are often subjected to some form of control. One common control regime is the reduction regime: when the population reaches a certain threshold it is controlled (for example culled) until it falls below a lower predefined level. The natural model for such a controlled population is a birth-death process with two phases, the phase determining which of two distinct sets of birth and death rates governs the process. We present formulae for the probability of extinction and the expected time to extinction, and discuss several applications.