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The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the randomized SI (susceptible/infected) model. We obtain the probability density function for the early behavior of the epidemic. This provides an appropriate distribution for the initial conditions with which to describe the subsequent deterministic evolution of the system. We apply the method of matching asymptotic expansions to link the two stages. This allows us to estimate the standard deviation of the number of infectives in the developed outbreak, and the statistical characteristics of the outbreak time. The potential trajectories caused by the stochastic nature of the contamination stage show greatest divergence at the initial and fade-out stages and coincide most tightly just after the peak of the epidemic. The time to the peak of the outbreak is not strongly dependent on the initial trajectory., (Copyright © 2011 Elsevier Inc. All rights reserved.)

A 1D lattice of coupled susceptible/infected/removed (SIR) epidemic centres is considered numerically and analytically. We describe a mechanism for the interaction between nodes in an SIR network, i.e. for the migration process of individuals between epidemic centres with a finite-characteristic time. More specifically, we study a model for a weakly coupled population distributed between the interacting centres with a diffusion-type migration process. A 1D lattice of SIR nodes is studied numerically. Travelling wave-like solutions preserving their shape and speed are found over a wide parameter range. For weak coupling, the main part of the travelling wave is well approximated by the limiting SIR solution. Explicit formulae are found for the speed of the travelling waves and compared with the results of numerical simulation.

Concurrent infection of cattle with bovine viral diarrhoea virus (BVDV) and Mycobacterium bovis is considered to be a possible risk factor for onward transmission of bovine tuberculosis (BTB) in infected cattle and is known to compromise diagnostic tests. A comparison is made here of M. bovis shedding (i.e. release) characteristics from 12 calves, six experimentally co-infected with BVDV and six infected with M. bovis alone, using simple models of bacterial replication. These statistical and mathematical models account for the intermittent or episodic nature of shedding, the dynamics of within-host bacterial proliferation and the sampling distribution from a given shedding episode. We show that while there are distinct differences among the shedding patterns of calves given the same infecting dose, there is no statistically significant difference between the two groups of calves. Such differences as there are, can be explained solely in terms of the shedding frequency, but with all calves potentially excreting the same amount of bacteria in a given shedding episode post-infection. The model can be thought of as a process of the bacteria becoming established in a number of discrete foci of colonization, rather than as a more generalized infection of the respiratory tract. In this case, the variability in the shedding patterns of the infected calves can be explained solely by differences in the number of foci established and shedding being from individual foci over time. Should maximum exposure on a particular occasion be a critical consideration for cattle-to-cattle transmission of BTB, cattle that shed only intermittently may still make an important contribution to the spread and persistence of the disease.

In accordance with a policy to eliminate all transmissible spongiform encephalopathies from the food chain, a national untargeted ram breeding programme to eliminate scrapie in the UK is in the final stages of planning. Here we formulate a model of flock-to-flock scrapie transmission, in order to consider the effect of a targeted breeding programme which is in the early stages of consideration. We estimate the size of the susceptible flock population, and discuss implications for potential control programmes. Targeting all rams and ewes in highly susceptible flocks rather than rams in all flocks will eradicate scrapie more quickly, and so is likely to be beneficial as long as suitable penalties or incentives are available to facilitate their identification. A more restricted programme aimed only at highly affected flocks would be much easier to implement and crucially will eradicate scrapie just as quickly. This will leave behind a residue population of susceptible sheep, which could then be gradually removed by a more general breeding programme.

Within-host mathematical models of Eimeria maxima and Eimeria praecox infections of the chicken are presented and used to investigate the role of host cell availability as a possible determinant of the so-called 'crowding effect'; whereby the fecundity of the parasites decreases as infectious dose increases. Assumptions about the number of available host cells, the average lifespan of these cells and the age structure within the host-cell population were made and mathematical models were constructed and combined with experimental data to test whether these conditions could reproduce the crowding effect in the two species. Experimental data demonstrated that crowding during in vivo infections was apparent following very low infectious doses, but none of the models could adequately reproduce crowding at the same doses while maintaining realistic estimates of the dynamics of the enterocyte pool. However, both the size and lifespan of the enterocyte pool were demonstrated to have substantial effects on the fecundity of the infections, particularly at higher doses. These data indicate that host cell availability cannot be solely responsible for the crowding effect. Alternative factors such as the influence of the primary immune response to the parasite may also be explored using within-host models and other applications of these models are discussed.

Responses to an anonymous postal survey concerning scrapie are analysed. Risk factors associated with farms that have had scrapie are identified as size, geographical region, lambing practices and holding of certain breeds. Further analysis of farms that have scrapie only in bought-in animals reveals that such farms tend to breed a smaller proportion of their replacement animals than farms without scrapie. Farms that have had scrapie in home-bred animals have attributes associated with breeding many animals: large numbers of rams bought, few ewes bought, and many animals that are home-bred. The demography of British sheep farms as described by size, breeds, purchasing behaviour, age structure and proportion of animals that are home-bred is summarized. British farms with scrapie reveal certain special features: they have more sheep that are found dead, more elderly ewes and more cases of scab.

Observations that growth of Plasmodium falciparum in vitro is inhibited by high temperatures have led to hypotheses that malaria fever may influence the parasite population dynamics, regulating parasite density and synchronizing parasite growth. In order to investigate the fever hypotheses, we have developed an age-structured coupled Markov chain model that describes the parasite erythrocyte cycle and its interaction with the host fever response. We estimated the model parameters using data collected from laboratory parasite cultures that were exposed to febrile or normal temperature. Using the experimental parameter values, quantitative predictions were made of the effect of fever in determining the parasite population dynamics. It was concluded from the model behaviour that, during the primary infection of a non-immune host, a typical episode of fever can effect density-dependent regulation of the parasite population, maintaining cycles of parasitaemia and promoting synchronous parasite growth.

Classical studies of non-immune individuals infected with Plasmodium falciparum reveal that the infection may be regulated for long periods at a relatively stable parasite density, despite the enormous growth potential of a parasite that continually replicates within host erythrocytes. This suggests that the parasite population may be controlled by density-dependent mechanisms, and in theory the most obvious of these is competition between parasites for host erythrocytes. Here we evaluate the role of this mechanism in the regulation of parasitaemia, by modelling the basic population interaction between parasites and erythrocytes in a form that allows all the essential parameters to be estimated from clinical data. Our results show that competition cannot account for the total regulation of P. falciparum, but when combined with immune mechanisms it may play a more important role than is generally supposed. Further analysis of the model indicates that in the long term, parasite replication at low parasite densities can contribute significantly to the high degree of anaemia observed in natural infection, a conclusion which is not obvious from simple clinical observation.