Extinction in host-vector infection models and the role of heterogeneity.

For infections that become endemic in a population, the process may appear stable over a long time scale, but stochastic fluctuations can lead to eventual disease extinction. We consider the effects of model parameters and of population heterogeneities upon the expected time to extinction for host-vector disease systems. We find that non-homogeneous host selection by vectors increases persistence times relative to the homogeneous case, and that the effect becomes even more marked when there are strong associations between particular groups of vectors and hosts. Heterogeneity in vector lifespans, in contrast, is found to decrease persistence times relative to the homogeneous case. Neither the basic reproduction number R0, nor the endemic prevalence level in the corresponding deterministic model, is found to be sufficient to predict (for a given population size) time to extinction. The endemic level, in particular, proves a very unreliable guide to the duration of long-term persistence.

Precise Estimates of Persistence Time for SIS Infections in Heterogeneous Populations.

For a susceptible-infectious-susceptible infection model in a heterogeneous population, we derive simple and precise estimates of mean persistence time, from a quasi-stationary endemic state to extinction of infection. Heterogeneity may be in either individuals' levels of infectiousness or of susceptibility, as well as in individuals' infectious period distributions. Infectious periods are allowed to follow arbitrary non-negative distributions. We also obtain a new and accurate approximation to the quasi-stationary distribution of the process, as well as demonstrating the use of our estimates to investigate the effects of different forms of heterogeneity. Our model may alternatively be interpreted as describing an infection spreading through a heterogeneous directed network, under the annealed network approximation.

Persistence time of SIS infections in heterogeneous populations and networks.

For a susceptible-infectious-susceptible infection model in a heterogeneous population, we present simple formulae giving the leading-order asymptotic (large population) behaviour of the mean persistence time, from an endemic state to extinction of infection. Our model may be interpreted as describing an infection spreading through either (1) a population with heterogeneity in individuals' susceptibility and/or infectiousness; or (2) a heterogeneous directed network. Using our asymptotic formulae, we show that such heterogeneity can only reduce (to leading order) the mean persistence time compared to a corresponding homogeneous population, and that the greater the degree of heterogeneity, the more quickly infection will die out.

Among-site variability in the stochastic dynamics of East African coral reefs.

Coral reefs are dynamic systems whose composition is highly influenced by unpredictable biotic and abiotic factors. Understanding the spatial scale at which long-term predictions of reef composition can be made will be crucial for guiding conservation efforts. Using a 22-year time series of benthic composition data from 20 reefs on the Kenyan and Tanzanian coast, we developed Bayesian vector autoregressive state-space models for reef dynamics, incorporating among-site variability, and quantified their long-term behaviour. We estimated that if there were no among-site variability, the total long-term variability would be approximately one-third of its current value. Thus, our results showed that among-site variability contributes more to long-term variability in reef composition than does temporal variability. Individual sites were more predictable than previously thought, and predictions based on current snapshots are informative about long-term properties. Our approach allowed us to identify a subset of possible climate refugia sites with high conservation value, where the long-term probability of coral cover ≤0.1 (as a proportion of benthic cover of hard substrate) was very low. Analytical results show that this probability is most strongly influenced by among-site variability and by interactions among benthic components within sites. These findings suggest that conservation initiatives might be successful at the site scale as well as the regional scale.

Generality of endemic prevalence formulae.

In simple infection models, the susceptible proportion s(*) in endemic equilibrium is related to the basic reproduction number R0 by s(*)=1/R0. We investigate the extent to which this relationship remains valid under more realistic modelling assumptions. In particular, we relax the biologically implausible assumptions that individuals' lifetimes and infectious periods follow exponential distributions; allow a general recruitment process; allow for multiple stages of infection; and consider extension to a multigroup model in which the groups may represent, for instance, spatial heterogeneity, or the existence of super-spreaders. For a homogeneous population, we find that: (i) the susceptible proportion is s(*)=1/R0(e), where R0(e) is a modified reproduction number, equal to R0 only in certain circumstances; (ii) the proportions of the population in each stage of infection are proportional to the expected time spent by an infected individual in that stage before recovery or death. We demonstrate robustness of the formula s(*)=1/R0 for many human infections by noting conditions under which R0(e) is approximately equal to R0, while pointing out other circumstances under which this approximation fails. For heterogeneous populations, the formula s(*)=1/R0 does not hold in general, but we are able to exhibit symmetry conditions under which it is valid.

Effectiveness of simulated interventions in reducing the estimated prevalence of Salmonella in UK pig herds.

Salmonella spp are a major foodborne zoonotic cause of human illness. Consumption of pork products is believed to be a major source of human salmonellosis and Salmonella control throughout the food-chain is recommended. A number of on-farm interventions have been proposed, and some have been implemented in order to try to achieve Salmonella control. In this study we utilize previously developed models describing Salmonella dynamics to investigate the potential effects of a range of these on-farm interventions. As the models indicated that the number of bacteria shed in the faeces of an infectious animal was a key factor, interventions applied within a high-shedding scenario were also analysed. From simulation of the model, the probability of infection after Salmonella exposure was found to be a key driver of Salmonella transmission. The model also highlighted that minimising physiological stress can have a large effect but only when shedding levels are not excessive. When shedding was high, weekly cleaning and disinfection was not effective in Salmonella control. However it is possible that cleaning may have an effect if conducted more often. Furthermore, separating infectious animals, shedding bacteria at a high rate, from the rest of the population was found to be able to minimise the spread of Salmonella.

Semi-stochastic models for Salmonella infection within finishing pig units in the UK.

A multi-group semi-stochastic model is formulated to describe Salmonella dynamics on a pig herd within the UK and assess whether farm structure has any effect on the dynamics. The models include both direct transmission and indirect (via free-living infectious units in the environment and airborne infection). The basic reproduction number R0 is also investigated. The models estimate approximately 24.6% and 25.4% of pigs at slaughter weight will be infected with Salmonella within a slatted-floored and solid-floored unit respectively, which corresponds to values found in previous abattoir and farm studies, suggesting that the model has reasonable validity. Analysis of the models identified the shedding rate to be of particular importance in the control of Salmonella spread, a finding also evident in an increase in the R0 value.

The effect of population heterogeneities upon spread of infection.

It has often been observed that population heterogeneities can lead to outbreaks of infection being less frequent and less severe than homogeneous population models would suggest. We address this issue by comparing a model incorporating various forms of heterogeneity with a homogenised model matched according to the value of the basic reproduction number [Formula: see text]. We mainly focus upon heterogeneity in individuals' infectivity and susceptibility, though with some allowance also for heterogeneous patterns of mixing. The measures of infectious spread we consider are (i) the probability of a major outbreak; (ii) the mean outbreak size; (iii) the mean endemic prevalence level; and (iv) the persistence time. For each measure, we establish conditions under which heterogeneity leads to a reduction in infectious spread. We also demonstrate that if such conditions are not satisfied, the reverse may occur. As well as comparison with a homogeneous population, we investigate comparisons between two heterogeneous populations of differing degrees of heterogeneity. All of our results are derived under the assumption that the susceptible population is sufficiently large.

Data-driven models for regional coral-reef dynamics.

Coral reefs have been affected by natural and anthropogenic disturbances. Coral cover has declined on many reefs, and macroalgae have increased on some. The existence of alternative stable states with high or low coral cover has been widely debated, but not clearly established. We evaluate the evidence for alternative stable states in benthic coral-reef dynamics in the Caribbean, Kenya and Great Barrier Reef (GBR), using stochastic semi-parametric models based on large numbers of time series of cover of hard corals, macroalgae and other components. Only the GBR showed a consistent short-term regional decline in coral cover. There was no evidence for regional increases in macroalgae. The equilibrium distributions of our models were close to recently observed distributions, and differed among regions. In all three regions, the equilibrium distributions were unimodal rather than bimodal, and thus did not suggest the existence of alternative stable states on a regional scale, under current conditions.

The effect of waning immunity on long-term behaviour of stochastic models for the spread of infection.

In stochastic modelling of infectious spread, it is often assumed that infection confers permanent immunity, a susceptible-infective-removed (SIR) model. We show how results concerning long-term (endemic) behaviour may be extended to a susceptible-infective-removed-susceptible (SIRS) model, in which immunity is temporary. Since the full SIRS model with demography is rather intractable, we also consider two simpler models: the susceptible-infective-susceptible (SIS) model with demography, in which there is no immunity; and the SIRS model in a closed population. For each model, we first analyse a deterministic model, then approximate the quasi-stationary distribution (equilibrium distribution conditional upon non-extinction of infection) using a moment closure technique. We look in particular at the effect of the immune period upon infection prevalence and upon time to fade-out of infection. Our main findings are that a shorter average immune period leads to higher infection prevalence in quasi-stationarity, and to longer persistence of infection in the population.

Quantifying exposure to vero-cytotoxigenic Escherichia coli O157 in milk sold as pasteurized: a model-based approach.

Milk sold as pasteurized has historically been implicated in the UK and worldwide as a vehicle for outbreaks of food-borne gastrointestinal disease, with a number of causative pathogenic organisms. One such organism is verocytotoxigenic Escherichia coli, or E. coli, O157 (VTEC O157). We present a quantitative assessment of likely exposure to VTEC O157 via milk sold as pasteurized in the UK. Particular interest in our assessment concerns whether there is any differential risk between milk which is processed in on- and off-farm dairies. We model the milk production chain from the farm through to the point of retail and make a comparison between these two production environments. Our model is an example of the Modular Process Risk Modelling (MPRM) approach and represents uncertainty and variability in input parameters using probability distributions. We conclude that milk processed on farm poses the comparatively greater risk, although that risk is still small.

Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations.

Deterministic and stochastic models motivated by Salmonella transmission in unmanaged/managed populations are studied. The SIRS models incorporate three routes of transmission (direct, vertical and indirect via free-living infectious units in the environment). With deterministic models we are able to understand the effects of different routes of transmission and other epidemiological factors on infection dynamics. In particular, vertical transmission has little influence on this dynamics, whereas the higher the indirect (direct) transmission rate the greater the tendency to persistent oscillation (stable endemic states). We show that the sustained cycles are also prone to demographic effect, i.e., persistent oscillation becomes impossible in the managed case (in the sense of balanced recruitment and death rates) by comparing with results in unmanaged populations (exponential population dynamics). Further, approximations of quasi-stationary distributions are derived for stochastic versions of the proposed models based on a diffusion approximation to the infection process. The effect of transmission parameters on the ratio of mean to standard deviation of the approximating distribution, used to judge the validity of the approximations and the expected time until fade out of infection, is further discussed. We conclude that strengthening any route of transmission may or may not reduce the expected time to fade out of infection, depending on the population dynamics.

Pair approximations and the inclusion of indirect transmission: theory and application to between farm transmission of Salmonella.

The spatial-temporal dynamics of farm animal diseases depend both on disease specific processes and the underlying contact network between farms. Indirect transmission via free-living bacteria in the environment is an important transmission route and contributes significantly to the dynamics. The pair-wise model has been developed to include both direct transmission and indirect transmission via free stages. The model is compared with stochastic simulations of epidemics on contact networks. The network framework is applied to the investigation of the epidemiological dynamics of between-herd transmission of Salmonella spp. The main results help to explain differences in observed epidemiological patterns and to identify possible causes for different strains of Salmonella developing so much variation in their infection dynamics in UK dairy herds. Numerical results show that shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower prevalence of infection and a greater tendency for (damped) oscillation. A possible control strategy is consequently suggested. Furthermore, the effect of network structure on long-term dynamics is examined.

Optimal intervention for an epidemic model under parameter uncertainty.

We will be concerned with optimal intervention policies for a continuous-time stochastic SIR (susceptible-->infective-->removed) model for the spread of infection through a closed population. In previous work on such optimal policies, it is common to assume that model parameter values are known; in reality, uncertainty over parameter values exists. We shall consider the effect upon the optimal policy of changes in parameter estimates, and of explicitly taking into account parameter uncertainty via a Bayesian decision-theoretic framework. We consider policies allowing for (i) the isolation of any number of infectives, or (ii) the immunisation of all susceptibles (total immunisation). Numerical examples are given to illustrate our results.

Understanding the dynamics of Salmonella infections in dairy herds: a modelling approach.

There is evidence of variation in the infection dynamics of different Salmonella serotypes in cattle--ranging from transient epidemics to long term persistence and recurrence. We seek to identify possible causes of these differences. In this study we present mathematical models which describe both managed population dynamics and epidemiology and use these to investigate the effects of demographic and epidemiological factors on epidemic behaviour and threshold for invasion. In particular, when the system is perturbed by higher culling or pathogen-induced mortality we incorporate mechanisms to constrain the lactating herd size to remain constant in the absence of pathogen or to lie within a fairly small interval in the presence of pathogen. A combination of numerical and analytical techniques is used to analyse the models. We find that the epidemiologically dominating management group can change from the dry/lactating cycle to the weaned group with increasing culling rate. Pseudovertical transmission is found to have little effect on the invasion criteria, while indirect transmission has significant influence. Pathogen-induced mortality, recovery, immune response and pathogen removal are found to be factors inducing damped oscillations; variation in these factors between Salmonella serotypes may be responsible for some of the observed differences in within herd dynamics. Specifically, higher pathogen-induced mortality, shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower number of infectives and smaller epidemics but a greater tendency for damped oscillations.