A fully coupled, mechanistic model for infectious disease dynamics in a metapopulation: movement and epidemic duration

The drive to understand the invasion, spread and fade out of infectious disease in structured populations has produced a variety of mathematical models for pathogen dynamics in metapopulations. Very rarely are these models fully coupled, by which we mean that the spread of an infection within a subpopulation affects the transmission between subpopulations and vice versa. It is also rare that these models are accessible to biologists, in the sense that all parameters have a clear biological meaning and the biological assumptions are explained. Here we present an accessible model that is fully coupled without being an individual-based model. We use the model to show that the duration of an epidemic has a highly non-linear relationship with the movement rate between subpopulations, with a peak in epidemic duration appearing at small movement rates and a global maximum at large movement rates. Intuitively, the first peak is due to asynchrony in the dynamics of infection between subpopulations; we confirm this intuition and also show the peak coincides with successful invasion of the infection into most subpopulations. The global maximum at relatively large movement rates occurs because then the infectious agent perceives the metapopulation as if it is a single well-mixed population wherein the effective population size is greater than the critical community size.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

Estimating Temporal Transmission Parameters from Infectious Disease Household Data, with Application to Taiwan SARS Data

Taking households having at least one infective as standard units and considering both a within-household infection rate and a global infection rate, we propose a Bayesian two level mixing S-I-R (susceptible-infective-removed) counting process model in which the transmission parameters may change over time and the parameters of interest are the within-household infection rate and the removal rate. Customized Markov chain Monte Carlo methods are developed for generating samples from the posterior distribution for inference purpose, based only on the removal times. The numerical performance of this method is examined in a simulation study. Applying this method to 2003 Taiwan SARS data, we find that the within-household infection rate decreases, the removal rate increases and their ratio is less than one and decreases significantly during the epidemic. This method allows the estimation of these parameters during the epidemic. For a rapidly transmitted disease, it provides a method to nearly real-time tracking of infection measures.     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

Modeling epidemic spread with awareness and heterogeneous transmission rates in networks

During an epidemic outbreak in a human population, susceptibility to infection can be reduced by raising awareness of the disease. In this paper, we investigate the effects of three forms of awareness (i.e., contact, local, and global) on the spread of a disease in a random network. Connectivity-correlated transmission rates are assumed. By using the mean-field theory and numerical simulation, we show that both local and contact awareness can raise the epidemic thresholds while the global awareness cannot, which mirrors the recent results of Wu et al. The obtained results point out that individual behaviors in the presence of an infectious disease has a great influence on the epidemic dynamics. Our method enriches mean-field analysis in epidemic models.     

Modelling pathogen transmission: the interrelationship between local and global approaches

We describe two spatial (cellular automaton) host-pathogen models with contrasting types of transmission, where the biologically realistic transmission mechanisms are based entirely on 'local' interactions. The two models, fixed contact area (FCA) and fixed contact number (FCN), may be viewed as local 'equivalents' of commonly used global density- (and frequency-) dependent models. Their outputs are compared with each other and with the patterns generated by these global terms. In the FCN model, unoccupied cells are bypassed, but in the FCA model these impede pathogen spread, extending the period of the epidemic and reducing the prevalence of infection when the pathogen persists. Crucially, generalized linear modelling reveals that the global transmission terms betaSI and beta'SI/N are equally good at describing transmission in both the FCA and FCN models when infected individuals are homogeneously distributed and N is approximately constant, as at the quasi-equilibrium. However, when N varies, the global frequency-dependent term beta'SI/N is better than the density-dependent one, betaSI, at describing transmission in both the FCA and FCN models. Our approach may be used more generally to compare different local contact structures and select the most appropriate global transmission term.     

Evolution of migration under kin selection and local adaptation

We present here a stochastic two-locus, two-habitat model for the evolution of migration with local adaptation and kin selection. One locus determines the migration rate while the other causes local adaptation. We show that the opposing forces of kin competition and local adaptation can lead to the existence of one or two convergence stable migration rates, notably depending on the recombination rate between the two loci. We show that linkage between migration and local adaptation loci has two antagonist effects: when linkage is tight, cost of local adaptation increases, leading to smaller equilibrium migration rates. However, when linkage is tighter, the population structure at the migration locus tends to be very high because of the indirect selection, and thus equilibrium migration rates increases. This result, qualitatively different from results obtained with other models of migration evolution, indicates that ignoring drift or the detail of the genetic architecture may lead to incorrect conclusions.     

Generation interval contraction and epidemic data analysis

The generation interval is the time between the infection time of an infected person and the infection time of his or her infector. Probability density functions for generation intervals have been an important input for epidemic models and epidemic data analysis. In this paper, we specify a general stochastic SIR epidemic model and prove that the mean generation interval decreases when susceptible persons are at risk of infectious contact from multiple sources. The intuition behind this is that when a susceptible person has multiple potential infectors, there is a "race" to infect him or her in which only the first infectious contact leads to infection. In an epidemic, the mean generation interval contracts as the prevalence of infection increases. We call this global competition among potential infectors. When there is rapid transmission within clusters of contacts, generation interval contraction can be caused by a high local prevalence of infection even when the global prevalence is low. We call this local competition among potential infectors. Using simulations, we illustrate both types of competition. Finally, we show that hazards of infectious contact can be used instead of generation intervals to estimate the time course of the effective reproductive number in an epidemic. This approach leads naturally to partial likelihoods for epidemic data that are very similar to those that arise in survival analysis, opening a promising avenue of methodological research in infectious disease epidemiology.     

An epidemiological framework for modelling fungicide dynamics and control

Defining appropriate policies for controlling the spread of fungal disease in agricultural landscapes requires appropriate theoretical models. Most existing models for the fungicidal control of plant diseases do not explicitly include the dynamics of the fungicide itself, nor do they consider the impact of infection occurring during the host growth phase. We introduce a modelling framework for fungicide application that allows us to consider how "explicit" modelling of fungicide dynamics affects the invasion and persistence of plant pathogens. Specifically, we show that "explicit" models exhibit bistability zones for values of the basic reproductive number ([Formula: see text]) less than one within which the invasion and persistence threshold depends on the initial infection levels. This is in contrast to classical models where invasion and persistence thresholds are solely dependent on [Formula: see text]. In addition if initial infection occurs during the growth phase then an additional "invasion zone" can exist for even smaller values of [Formula: see text]. Within this region the system will experience an epidemic that is not able to persist. We further show that ideal fungicides with high levels of effectiveness, low rates of application and low rates of decay lead to the existence of these bistability zones. The results are robust to the inclusion of demographic stochasticity.     

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible → infected → removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback–Leibler divergence for the two fitted models to these data.     

Assessment of arbovirus vector infection rates using variable size pooling

Pool testing of vector samples for arboviruses is widely used in surveillance programmes. The proportion of infected mosquitoes (Diptera: Culicidae) is often estimated from the minimum infection rate (MIR), based on the assumption of only one infected mosquito per positive pool. This assumption becomes problematic when pool size is large and/or infection rate is high. By relaxing this constraint, maximum likelihood estimation (MLE) is more useful for a wide range of infection levels that may be encountered in the field. We demonstrate the difference between these two estimation approaches using West Nile virus (WNV) surveillance data from vectors collected by gravid traps in Chicago during 2002. MLE of infection rates of Culex mosquitoes was as high as 60 per 1000 at the peak of transmission in August, whereas MIR was less than 30 per 1000. More importantly, we demonstrate roles of various pooling strategies for better estimation of infection rates based on simulation studies with hypothetical mosquito samples of 18 pools. Variable size pooling (with a serial pool sizes of 5, 10, 20, 30, 40 and 50 individuals) performed consistently better than a constant size pooling of 50 individuals. We conclude that variable pool size coupled with MLE is critical for accurate estimates of mosquito infection rates in WNV epidemic seasons.     

Multi-state epidemic processes on complex networks

Infectious diseases are practically represented by models with multiple states and complex transition rules corresponding to, for example, birth, death, infection, recovery, disease progression, and quarantine. In addition, networks underlying infection events are often much more complex than described by meanfield equations or regular lattices. In models with simple transition rules such as the SIS and SIR models, heterogeneous contact rates are known to decrease epidemic thresholds. We analyse steady states of various multi-state disease propagation models with heterogeneous contact rates. In many models, heterogeneity simply decreases epidemic thresholds. However, in models with competing pathogens and mutation, coexistence of different pathogens for small infection rates requires network-independent conditions in addition to heterogeneity in contact rates. Furthermore, models without spontaneous neighbor-independent state transitions, such as cyclically competing species, do not show heterogeneity effects.     

On the importance of risky behavior in the transmission of sexually transmitted diseases

We consider a population with two types of individuals: those who engage in risky sexual behavior (the core) and those who do not (the majority). We are interested in the situation where the majority would not be able to sustain an epidemic without the presence of a core. We show that even if the core is randomly spread out in the general population and is very infectious there will be no epidemic if the proportion of individuals in the core is below a certain threshold (that depends on the infection rates). Risky behavior does put the whole population at risk but only if the behavior is widespread enough. This gives support to the idea of concentrating resources to fight sexually transmitted diseases on the core group in order to lower the proportion of individuals in the core. We show our results for two models: one with total mixing and the other with limited mixing.     

Deterministic epidemic models with explicit household structure

For a wide range of airborne infectious diseases, transmission within the family or household is a key mechanism for the spread and persistence of infection. In general, household-based transmission is relatively strong but only involves a limited number of individuals in contact with each infectious person. In contrast, transmission outside the household can be characterised by many contacts but a lower probability of transmission. Here we develop a relatively simple dynamical model that captures these two transmission regimes. We compare the dynamics of such models for a range of household sizes, whilst constraining all models to have equal early growth rate so that all models fit to the same early incidence observations of an epidemic. Finally we consider the use of prophylactic vaccination, responsive vaccination, or antivirals to combat epidemic spread and focus on whether it is optimal to target controls at entire households or to treat individuals independently.     

Epidemics in Competition: Partial Cross-Immunity

The competition between two pathogen strains during the course of an epidemic represents a fundamental step in the early evolution of emerging diseases as well as in the antigenic drift process of influenza. The outcome of the competition, however, depends not only on the epidemic properties of the two strains but also on the timing and size of the introduction, characteristics that are poorly captured by deterministic mean-field epidemic models. We describe those aspects of the competition that can be determined from the mean-field models giving the range of possible final sizes of susceptible hosts and cumulated attack rates that could be observed after an epidemic with two cross-reacting strains. In the limit where the size of the initial infection goes to zero, the possible outcomes lie on a (one dimensional) curve in the outcome space.     

The size of epidemics in populations with heterogeneous susceptibility

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.     

Effects of heterogeneous and clustered contact patterns on infectious disease dynamics

The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have far-reaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.     

Development of the Dutch elm disease epidemic in southern England, 1971-6

The current epidemic of Dutch elm disease was studied by recording the fate of individual hedgerow elms (Ulmus procera) in five plots in the West Midlands, and by analysing data from successive Forestry Commission surveys of non-woodland elms in 234 plots in southern England. Ninty-five percent of the individual trees died between May 1972 and September 1975. The average infection rate (r) was found to be 1 -35 during the period when the proportion of disease, x, increased from 0–16 to 0–42. In the plots of the main survey the average infection rate was 0–65 and the cumulative loss increased from 6 to 62% between 1971 and 1976, with little evidence that the course of the epidemic was influenced by variations in the weather from year to year. These infection rates are as high as those recorded in Dutch elm disease epidemics elsewhere in the world. The infection rate in English elm was higher than in either the wych elm or the heterogeneous ‘smooth-leaved elm’. The study of English elm in four geographical areas of southern Britain showed that there was an initial drop in infection rate until x = 0–12, when a steady infection rate obtained in all four areas, ranging from 0–56 in the Midlands to 0–76 in the south-east. It is concluded that the epidemic is likely to continue at a high rate until most non-woodland elm have died. Most trees which survive are likely to be smooth-leaved elm in East Anglia. Few communities in southern England have been able to practice vigorous sanitation control programmes, but data from two, in East Sussex and Brighton, are analysed and the effect on disease progress discussed.     

Global behavior of epidemic transmission on heterogeneous networks via two distinct routes

In the study of epidemic spreading two natural questions are: whether the spreading of epidemics on heterogenous networks have multiple routes, and whether the spreading of an epidemic is a local or global behavior? In this paper, we answer the above two questions by studying the SIS model on heterogenous networks, and give the global conditions for the endemic state when two distinct routes with uniform rate of infection are considered. The analytical results are also verified by numerical simulations.     

Modelling the impact of colonisation on genetic diversity and differentiation of forest trees: interaction of life cycle,pollen flow and seed long-distance dispersal

It was shown previously that the long lifespan and juvenile phase of trees strongly attenuate founder effects during colonisation in a diffusive dispersal model. However, this model yielded too slow a colonisation rate in comparison with palynological data for temperate forest trees. Since rare long-distance dispersal events have been shown to increase considerably colonisation rates in population dynamics models, we investigate here the impact of long-distance dispersal on within-population diversity (H(S)) and among-population differentiation (F(ST)) during the colonisation process. We use a stochastic approach and compare several dispersal strategies, ranging from very rare dispersal events of large amplitude to more frequent events of smaller amplitude. Using a simulation approach, which takes into account tree life-history traits, we show that long-distance dispersal events increase colonisation speed, and yield much larger founder effects in comparison with the diffusive model. The two models that include intermediate- and long-distance dispersal events show stronger deviations from experimental F(ST) values during and at the end of the colonisation process than the model with more frequent events of smaller dispersal variance. Furthermore, the introduction of a high level of pollen flow has a much more limited impact on models that include long-distance dispersal than on a diffusive dispersal model. The relatively high H(S) values that were obtained in all models are discussed according to the assumed mutation rate and effective population size. This study is an example of how observed genetic data can provide additional evidence on the best demographic model for a given species or group of species.