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In this paper we investigate the within-host dynamics of the foot-and-mouth disease virus (FMDV) in cattle using previously published data for 8 experimentally infected cows. An 8-compartment, 14-parameter differential equation model was fitted to data collected from each cow every 24 h over the course of an infection on: (i) the concentration of FMDV genomes in the blood, (ii) the concentration of infectious virus in the blood, (iii) antibody levels, and (iv) interferon levels. Model parameters were estimated using maximum-likelihood methods. The likelihood surface was sampled using Markov chain Monte Carlo methods giving credible intervals for each of the model parameters. The model was able to capture the within-host dynamics well for 6 of the infections, with both the innate (type 1 interferon) and antibody responses playing key roles in determining the height and duration of peak levels of virus. There was considerable variation between virus dynamics in individual cattle which was only partly accounted for by inferred differences in the dose of virus received. A better understanding of the within-host dynamics also provides insights into the dynamics of infectiousness and the transmission of virus to new hosts.

Parasite strategies for exploiting host resources are key determinants of disease severity (i.e., virulence) and infectiousness (i.e., transmission between hosts). By iterating the development of theory and empirical tests, we investigated whether variation in parasite traits across two genetically distinct clones of the rodent malaria parasite, Plasmodium chabaudi, explains differences in within-host infection dynamics and virulence. First, we experimentally tested key predictions of our earlier modeling work. As predicted, the more virulent genotype produced more progeny parasites per infected cell (burst size), but in contrast to predictions, invasion rates of red blood cells (RBCs) did not differ between the genotypes studied. Second, we further developed theory by confronting our earlier model with these new data, testing a new set of models that incorporate more biological realism, and developing novel theoretical tools for identifying differences between parasite genotypes. Overall, we found robust evidence that differences in burst sizes contribute to variation in dynamics and that differential interactions between parasites and host immune responses also play a role. In contrast to previous work, our model predicts that RBC age structure is not important for explaining dynamics. Integrating theory and empirical tests is a potentially powerful way of progressing understanding of disease biology.

Mathematical model-based statistical inference applied to within-host dynamics of infectious diseases can help dissect complex interactions between hosts and microbes. This work has applied advances in model-based inference to understand colonization of cattle by enterohaemorrhagic Escherichia coli O157 : H7 at the terminal rectum. A mathematical model was developed based on niche replication and transition rates at this site. A nested-model comparison, applied to excretion curves from 25 calves, was used to reduce complexity while maintaining integrity. We conclude that, 5–9 days post inoculation, the innate immune response negates bacterial replication on the epithelium and either reduces attachment to or increases detachment from the epithelium of the terminal rectum. Thus, we provide a broadly applicable model that gives novel insights into bacterial replication rates in vivo and the timing and impact of host responses.

Identifying risk factors for the presence of Escherichia coli O157 infection on cattle farms is important for understanding the epidemiology of this zoonotic infection in its main reservoir and for informing the design of interventions to reduce the public health risk. Here, we use data from a large-scale field study carried out in Scotland to fit both "SIS"-type dynamical models and statistical risk factor models. By comparing the fit (assessed using maximum likelihood) of different dynamical models we are able to identify the most parsimonious model (using the AIC statistic) and compare it with the model suggested by risk factor analysis. Both approaches identify 2 key risk factors: the movement of cattle onto the farm and the number of cattle on the farm. There was no evidence for a role of other livestock species or seasonality, or of significant risk of local spread. However, the most parsimonious dynamical model does predict that farms can infect other farms through routes other than cattle movement, and that there is a nonlinear relationship between the force of infection and the number of infected farms. An important prediction from the most parsimonious model is that although only approximately 20% farms may harbour E. coli O157 infection at any given time approximately 80% may harbour infection at some point during the course of a year.

Background: An understanding of the factors that determine the risk of members of a susceptible population becoming infected is essential for estimating the potential for disease spread, as opposed to just focusing on transmission from an infected population. Furthermore, analysis of the risk factors can reveal important characteristics of an epidemic and further develop understanding of the processes operating. Methods: This paper describes the development of a mixed effects logistic regression model of susceptibility of holdings to foot and mouth disease (FMD) during the 2001 epidemic in Great Britain following the imposition of a national ban on the movements of susceptible animals (NMB). Results: The principal risk factors identified in the model were shorter distances to the nearest infectious seed (a holding infected before the NMB) and the county of the holding (principally Cumbria). Additional risk factors included holdings that are mixed species rather than single species, the surface area of the holding, and the number of cattle within 10 km (all p 0.1). The fit of the model was evaluated using the area under the receiver operator characteristic curve (ROC) and the Hosmer and Lemeshow Chi-squared statistic; the fit was good with both tests (area under the ROC = 0.962 and Hosmer and Lemeshow Chi-squared statistic = 49.98 (p > 0.1)). Conclusions: Holdings at greatest risk of infection can be identified using simple readily available risk factors; this information could be employed in the control of future FMD epidemics. (C) 2009 International Society for Infectious Diseases. Published by Elsevier Ltd. All rights reserved.

Despite considerable success elucidating important immunological and resource-based mechanisms that control the dynamics of infection in some diseases, little is known about how differences in these mechanisms result in strain differences in patterns of pathogenesis. Using a combination of data and theory, we disentangle the role of ecological factors (e.g., resource abundance) in the dynamics of pathogenesis for the malaria species Plasmodium chabaudi in CD4+ T cell-depleted mice. We build a series of nested models to systematically test a number of potential regulatory mechanisms and determine the "best" model using statistical techniques. The best-fit model is further tested using an independent data set from mixed-clone competition experiments. We find that parasites preferentially invade older red blood cells even when they are more fecund in younger reticulocytes and that inoculum size has a strong effect on burst size in reticulocytes. Importantly, the results suggest that strain-specific differences in virulence arise from differences in red blood cell age-specific invasion rates and burst sizes, since these are lower for the less virulent strain, as well as from differences in levels of erythropoesis induced by each strain. Our analyses highlight the importance of model selection and validation for revealing new biological insights.

Foot-and-mouth disease (FMD) in the UK provides an ideal opportunity to explore optimal control measures for an infectious disease. The presence of fine-scale spatio-temporal data for the 2001 epidemic has allowed the development of epidemiological models that are more accurate than those generally created for other epidemics and provide the opportunity to explore a variety of alternative control measures. Vaccination was not used during the 2001 epidemic; however, the recent DEFRA (Department for Environment Food and Rural Affairs) contingency plan details how reactive vaccination would be considered in future. Here, using the data from the 2001 epidemic, we consider the optimal deployment of limited vaccination capacity in a complex heterogeneous environment. We use a model of FMD spread to investigate the optimal deployment of reactive ring vaccination of cattle constrained by logistical resources. The predicted optimal ring size is highly dependent upon logistical constraints but is more robust to epidemiological parameters. Other ways of targeting reactive vaccination can significantly reduce the epidemic size; in particular, ignoring the order in which infections are reported and vaccinating those farms closest to any previously reported case can substantially reduce the epidemic. This strategy has the advantage that it rapidly targets new foci of infection and that determining an optimal ring size is unnecessary.

Recent experiments have shown that in patients with psoriasis, highly elevated levels of nitric oxide (NO) are released at the surface of psoriatic plaques. Nitric oxide is a central biological regulator of many aspects of physiology, and it is a natural possibility that the high nitric oxide levels in psoriasis play a causal role in the onset of the disease. Here, we use mathematical modelling to investigate this possibility. We begin by discussing a simple model consisting of a single equation for nitric oxide concentration, which enables nitric oxide secretion rates in the basal epidermis to be calculated from the observed NO release rates at the skin surface. Using this key parameter value, we then develop an extended model that tests the hypothesis that nitric oxide regulates the formation of the extended rete pegs seen in psoriatic plaques. This occurs via the peroxynitrite-dependent activation of the collagenase MMP-8, which is produced by neutrophils present at high levels in psoriatic plaques. The plausibility of the hypothesis is demonstrated and specific testable quantitative predictions about the roles of the various cell types and signalling molecules are made.

Marek's disease virus (MDV), a commercially important disease of poultry, has become substantially more virulent over the last 60 years. This evolution was presumably a consequence of changes in virus ecology associated with the intensification of the poultry industry. Here, we assess whether vaccination or reduced host life span could have generated natural selection, which favored more virulent strains. Using previously published experimental data, we estimated viral fitness under a range of cohort durations and vaccine treatments on broiler farms. We found that viral fitness maximized at intermediate virulence, as a result of a trade-off between virulence and transmission previously reported. Our results suggest that vaccination, acting on this trade-off, could have led to the evolution of increased virulence. By keeping the host alive, vaccination prolongs infectious periods of virulent strains. Improvements in host genetics and nutrition, which reduced broiler life spans below 50 days, could have also increased the virulence of the circulating MDV strains because shortened cohort duration reduces the impact of host death on viral fitness. These results illustrate the dramatic impact anthropogenic change can potentially have on pathogen virulence.

Background Marek's disease virus (MDV) is an economically important oncogenic herpesvirus of poultry. Since the 1960s, increasingly virulent strains have caused continued poultry industry production losses worldwide. To understand the mechanisms of this virulence evolution and to evaluate the epidemiological consequences of putative control strategies, it is imperative to understand how virulence is defined and how this correlates with host mortality and infectiousness during MDV infection. We present a mathematical approach to quantify key epidemiological parameters. Host lifespan, virus latent periods and host viral shedding rates were estimated for unvaccinated and vaccinated birds, infected with one of three MDV strains. The strains had previously been pathotyped to assign virulence scores according to pathogenicity of strains in hosts. Results Our analyses show that strains of higher virulence have a higher viral shedding rate, and more rapidly kill hosts. Vaccination enhances host life expectancy but does not significantly reduce the shedding rate of the virus. While the primary latent period of the virus does not vary with challenge strain nor vaccine treatment of host, the time until the maximum viral shedding rate is increased with vaccination. Conclusions Our approach provides the tools necessary for a formal analysis of the evolution of virulence in MDV, and potentially simpler and cheaper approaches to comparing the virulence of MDV strains.

Malarial infection is associated with complex immune and erythropoietic responses in the host. A quantitative understanding of these processes is essential to help inform malaria therapy and for the design of effective vaccines. In this study, we use a statistical model-fitting approach to investigate the immune and erythropoietic responses in Plasmodium chabaudi infections of mice. Three mouse phenotypes (wildtype, T-cell-deficient nude mice, and nude mice reconstituted with T-cells taken from wildtype mice) were infected with one of two parasite clones (AS or AJ). Under a Bayesian framework, we use an adaptive population-based Markov chain Monte Carlo method and fit a set of dynamical models to observed data on parasite and red blood cell (RBC) densities. Model fits are compared using Bayes' factors and parameter estimates obtained. We consider three independent immune mechanisms: clearance of parasitised RBCs (pRBC), clearance of unparasitised RBCs (uRBC), and clearance of parasites that burst from RBCs (merozoites). Our results suggest that the immune response of wildtype mice is associated with less destruction of uRBCs, compared to the immune response of nude mice. There is a greater degree of synchronisation between pRBC and uRBC clearance than between either mechanism and merozoite clearance. In all three mouse phenotypes, control of the peak of parasite density is associated with pRBC clearance. In wildtype mice and AS-infected nude mice, control of the peak is also associated with uRBC clearance. Our results suggest that uRBC clearance, rather than RBC infection, is the major determinant of RBC dynamics from approximately day 12 post-innoculation. During the first 2–3 weeks of blood-stage infection, immune-mediated clearance of pRBCs and uRBCs appears to have a much stronger effect than immune-mediated merozoite clearance. Upregulation of erythropoiesis is dependent on mouse phenotype and is greater in wildtype and reconstitited mice. Our study highlights the informative power of statistically rigorous model-fitting techniques in elucidating biological systems., Author Summary Malaria is a disease caused by a protozoan parasite of the genus Plasmodium. Every year there are around 250 million human cases of malaria, resulting in around a million deaths. Most of the severe cases and deaths are due to Plasmodium falciparum, which is endemic in much of sub-Saharan Africa and other tropical areas. The pathology of malaria is related to the asexual stage of the parasite. Understanding the infection dynamics during this stage is therefore essential to inform malaria treatment and vaccine design. Experimental infections of rodents represent an important first step towards understanding the more complicated human infections. We developed a series of models representing different hypotheses about the main processes regulating the infection dynamics during the asexual stage. Models were fit to data on Plasmodium chabaudi infections of mice, using a Bayesian statistical framework. The accuracy of different models in explaining the RBC and parasite densities was quantified. We identify the role of different types of immune-mediated mechanism, and show that RBC production (erythropoiesis) increases during infection. Differences between mouse phenotypes are explained. Our study highlights the informative power of model-fitting techniques in explaining biological systems.

The aim of this analysis is to evaluate how generic properties of networks of livestock farms connected by movements of cattle impact on the potential for spread of infectious diseases. We focus on endemic diseases with long infectious periods in affected cattle, such as bovine tuberculosis. Livestock farm networks provide a rare example of large but fully specified directed contact networks, allowing investigations into how properties of such networks impact the potential for spread of infections within them. Here we quantify the latter in terms of the basic reproduction number, R0, and partition the contributions to R0 from first order moments (mean contact rates) and second order moments (variances and covariances of contact rates) of the farm contact matrices. We find that the second order properties make a substantial contribution to the magnitude of R0, similarly to that reported for other populations. Importantly, however, we find that the magnitude of these effects depends on exactly how the contacts between farms are defined or weighted. We note that the second order properties of a directed contact network may vary through time even with little change in the mean contact rates or in overall connectedness of the network. Keywords: Basic reproduction number, Infectious disease, Heterogeneity, 20â80 rule, Contact network, Bovine tuberculosis

This paper investigates the early viral dynamics of foot-and-mouth disease (FMD) within infected pigs. Using an existing within-host model, we investigate whether individual variation can be explained by the effect of the initial dose of FMD virus. To do this, we consider the experimental data on the concentration of FMD virus genomes in the blood (viral load). In this experiment, 12 pigs were inoculated with one of three different doses of FMD virus: low; medium; or high. Measurements of the viral load were recorded over a time course of approximately 11 days for every 8 hours. The model is a set of deterministic differential equations with the following variables: viral load; virus in the interstitial space; and the proportion of epithelial cells available for infection, infected and uninfected. The model was fitted to the data for each animal individually and also simultaneously over all animals varying only the initial dose. We show that the general trend in the data can be explained by varying only the initial dose. The higher the initial dose the earlier the development of a detectable viral load.

Rapid detection of infectious disease outbreaks is often crucial for their effective control. One example is highly pathogenic avian influenza (HPAI) such as H5N1 in commercial poultry flocks. There are no quantitative data, however, on how quickly the effects of HPAI infection in poultry flocks can be detected. Here, we study, using an individual-based mathematical model, time to detection in chicken flocks. Detection is triggered when mortality, food or water intake or egg production in layers pass recommended thresholds suggested from the experience of past HPAI outbreaks. We suggest a new threshold for caged flocks—the cage mortality detection threshold—as a more sensitive threshold than current ones. Time to detection is shown to depend nonlinearly on R 0 and is particularly sensitive for R 0 R 0 HPAI strain such as H7N7 over a wide range of flock sizes. Time to detection of the effects of a low R 0 HPAI strain such as H5N1 can be significantly improved, particularly for large flocks, by lowering detection thresholds, and this can be accomplished without causing excessive false alarms in uninfected flocks. The results are discussed in terms of optimizing the design of disease surveillance programmes in general.

Since 2001 models of the spread of foot-and-mouth disease, supported by the data from the UK epidemic, have been expounded as some of the best examples of problem-driven epidemic models. These claims are generally based on a comparison between model results and epidemic data at fairly coarse spatio-temporal resolution. Here, we focus on a comparison between model and data at the individual farm level, assessing the potential of the model to predict the infectious status of farms in both the short and long terms. Although the accuracy with which the model predicts farms reporting infection is between 5 and 15%, these low levels are attributable to the expected level of variation between epidemics, and are comparable to the agreement between two independent model simulations. By contrast, while the accuracy of predicting culls is higher (20–30%), this is lower than expected from the comparison between model epidemics. These results generally support the contention that the type of the model used in 2001 was a reliable representation of the epidemic process, but highlight the difficulties of predicting the complex human response, in terms of control strategies to the perceived epidemic risk.

Livestock movement is thought to be a risk factor for the transmission of infectious diseases of farm animals. Simple mathematical models were constructed for the transmission of Escherichia coli serogroup O157 between Scottish cattle farms, and the models were used in a preliminary exploration of factors contributing to the levels of infection reported in the field. The results suggest that cattle movement can make a significant contribution to the observed prevalence of E. coli O157-positive farms, but is not by itself sufficient for the persistence of E. coli O157. The results also suggest that cattle movements involving infected farms with cattle shedding an exceptional amount of E. coli O157, ‘super-shedders’, also make a substantial contribution to the prevalence of infected farms. Simulations indicate that E. coli O157 could have reached the currently observed prevalence levels in less than a decade. Implications and findings from our models are discussed in relation to possible control of E. coli O157 in Scottish cattle.

Background A key challenge for modelling infectious disease dynamics is to understand the spatial spread of infection in real landscapes. This ideally requires a parallel record of spatial epidemic spread and a detailed map of susceptible host density along with relevant transport links and geographical features. Results Here we analyse the most detailed such data to date arising from the UK 2001 foot and mouth epidemic. We show that Euclidean distance between infectious and susceptible premises is a better predictor of transmission risk than shortest and quickest routes via road, except where major geographical features intervene. Conclusion Thus, a simple spatial transmission kernel based on Euclidean distance suffices in most regions, probably reflecting the multiplicity of transmission routes during the epidemic.

Stem cells in the basal layer of human interfollicular epidermis form clusters that can be reconstituted in vitro. In order to supply the interfollicular epidermis with differentiated cells, the size of these clusters must be controlled. Evidence suggests that control is regulated via differentiation of stem cells on the periphery of the clusters. Moreover, there is growing evidence that this regulation is mediated by the Notch signalling pathway. In this paper, we develop theoretical arguments, in conjunction with computer simulations of a model of the basal layer, to show that regulation of differentiation is the most likely mechanism for cluster control. In addition, we show that stem cells must adhere more strongly to each other than they do to differentiated cells. Developing our model further we show that lateral-induction, mediated by the Notch signalling pathway, is a natural mechanism for cluster control. It can not only indicate to cells the size of the cluster they are in and their position within it, but it can also control the cluster size. This can only be achieved by postulating a secondary, cluster wide, differentiation signal, and cells with high Delta expression being deaf to this signal.

Dispersing predators and prey can exhibit complex spatio-temporal wave-like patterns if the interactions between them cause oscillatory dynamics. We study the effect of these predator- prey density waves on the competition between prey populations and between predator popu- lations with different dispersal strategies. We first describe 1- and 2-dimensional simulations of both discrete and continuous predator-prey models. The results suggest that any population that diffuses faster, disperses farther, or is more likely to disperse will exclude slower diffus- ing, shorter dispersing, or less likely dispersing populations, everything else being equal. It also appears that it does not matter whether time, space, or state are discrete or continuous, nor what the exact interactions between the predators and prey are. So long as waves exist the competition between populations occurs in a similar fashion. We derive a theory that qualitatively explains the observed behaviour and calculate approximate analytical solutions that describe, to a reasonable extent, these behaviours. Predictions about the cost of dispersal are tested. If strong enough, cost can reverse the populations' relative competitive strengths or lead to coexistence because of the effect of spiral wave cores. The theory is also able to explain previous results of simulations of coexistence in host-parasitoid models (Comins, H. N., and Massell, M. P., 1996, J. Theor. Biol. 183, 19-28).

In a discrete-generation, individual-oriented model of predator-prey interactions that exhibits oscillations, we show that the self-structuring of the populations into spiral waves induces a selection pressure for ever-increasing dispersal distances in both populations. As the dispersal distances increase, the sizes of the spatial patterns increase, until they are too large to fit into the limited space. The patterns are then lost and the predators go extinct. This scenario, is, however, not the only outcome. A second selection pressure induced by the spatial boundary can cause reduction of the dispersal distances. Depending on the relative strengths of the two selection pressures, the predators and prey may speciate to give coexistence between short-dispersing boundary quasi-species and far-dispersing spiral quasi-species. Now, when pattern loss occurs, the predators switch to predating on the boundary prey quasi-species and do not go extinct. Also, if the populations reproduce sexually, local gene flow can inhibit the evolution of increasing dispersal distances, and hence the spatial patterns are not lost. Speciation and coexistence can also occur in the sexually reproducing species.

The process by which one may take a discrete model of a biophysical process and construct a continuous model based upon it is of mathematical interest as well as being of practical use. In this work, we take the extended Potts model applied to biological cell movement to its continuous limit. Beginning with a single cell moving in one dimension on a lattice and obeying Potts model rules of movement we develop an expression for the diffusion coefficient of a collection of noninteracting cells which depends explicitly on the Potts model parameters. We show how this coefficient varies when the Potts parameters for cell membrane elasticity and cell-medium adhesion are varied, and perform computer simulations which support our theoretical result. We explain the relationship between the probability of occupancy of lattice points and the density profile in the continuous limit, and extend our analysis by including interactions between the cells. In so doing we are able to develop a set of coupled ordinary differential equations showing the evolution of a density profile in the presence of significant cell-cell adhesion, and show how increases in the strength of this adhesion modulates diffusion. In so doing we develop some insights into how continuous models of physical systems can be based upon discrete models which describe the same system.