A clarification of transmission terms in host-microparasite models: numbers, densities and areas

Viral infection in internally structured hosts. I. Conditions for persistent infection

For a virus population within its host, two important levels of structure can be considered: multiple cell types which can be infected, and tissue types or body compartments which may be coupled via movement. We develop a model with both types of structure. Migration between compartments can create "sources" and "sinks" within the virus population, where realized viral growth rate and abundance is lowered in some compartments compared to what would be observed in isolation. Using both analytical and numerical methods, we investigate how this within-host spatial structure affects the conditions for persistent viral infection. We find that migration between compartments makes the establishment of infection more difficult than it would be in the absence of migration, implying that within-host spatial structure combined with viral movement decreases the likelihood of viral establishment. If migration is symmetrical and compartments are heterogeneous, an increase in migration rates between compartments generally makes establishment less likely. This may help to explain the tissue specificity observed for many viruses. There are, however, important exceptions to this result. These include circumstances where the virus initially invades a compartment that is unfavorable to population growth and migration is necessary to infect other parts of the host body. Stochastic aspects of viral establishment may also favor increased migration as it tends to dampen the amplitude of fluctuations in population size during the initial transient phase of establishment.

A model of the spread of the bovine viral-diarrhoea virus within a dairy herd

Wet BVDSim (a stochastic simulation model) was developed to study the dynamics of the spread of the bovine viral-diarrhoea virus (BVDV) within a dairy herd. This model took into account herd-management factors (common in several countries), which influence BVDV spread. BVDSim was designed as a discrete-entity and discrete-event simulation model. It relied on two processes defined at the individual-animal level, with interactions. The first process was a semi-Markov process and modelled the herd structure and dynamics (demography, herd management). The second process was a Markov process and modelled horizontal and vertical virus transmission. Because the horizontal transmission occurs by contacts (nose-to-nose) and indirectly, transmission varied with the separation of animals into subgroups. Vertical transmission resulted in birth of persistently infected (PI) calves. Other possible consequences of a BVDV infection during the pregnancy period were considered (pregnancy loss, immunity of calves). The outcomes of infection were modelled according to the stage of pregnancy at time of infection. BVDV pregnancy loss was followed either by culling or by a new artificial insemination depending on the modelled farmer's decision. Consistency of the herd dynamics in the absence of any BVDV infection was verified. To explore the model behaviour, the virus spread was simulated over 10 years after the introduction of a near-calving PI heifer into a susceptible 38 cow herd. Different dynamics of the virus spread were simulated, from early clearance to persistence of the virus 10 years after its introduction. Sensitivity of the model to the uncertainty on transmission coefficient was analysed. Qualitative validation consisted in comparing the bulk-milk ELISA results over time in a sample of herds detected with a new infection with the ones derived from simulations.

The evolutionary economics of immunity

How much of its resources should an individual invest in a costly immune system? In this article, we apply an evolutionarily stable strategy analysis to an epidemic model to answer this question. On the one hand, an investment in immune function confers protection to infectious agents by reducing host susceptibility, pathogen virulence, or the length of the infectious period. On the other hand, an immune system is costly since it absorbs resources that otherwise might be invested in increasing the host's fertility or longevity. In addition, an active immune system may be able to clear pathogens efficiently but at the same time may result in immunopathology. By means of a reproductive value approach, we show how to compare the costs and benefits of an immune system systematically and how to derive the evolutionarily stable level of immune function. We then apply these methods to various plausible scenarios. The analysis reveals that the relationship between the life span of an organism and the optimal level of investment in immune function is less straightforward than one might expect. First, the prevalence of infection is reduced to the lowest possible level only under special circumstances. Second, members of a long-lived species do not necessarily have to invest more in immune function than those of a short-lived species. In fact, the opposite may be true. Third, the outcome of evolution can be contingent on the initial conditions. Depending on its initial investment strategy, a population may evolve to a state where very much or almost nothing is invested in a costly immune system.

A model appropriate to the transmission of a human food-borne pathogen in a multigroup managed herd

We describe a model of microparasite transmission within a multigroup managed farming system. The model was formulated to represent transmission of Escherichia coli O157 within a typical UK dairy herd and was used to suggest possible on-farm control strategies. The model includes birth, death, maturation, the dry/lactating cycle and various types of transmission (i.e. direct, pseudovertical (representing direct faecal-oral transmission between dam and calf within the first 48 h) and indirect (via free-living infectious units in the environment)). A combination of numerical and analytical techniques was used to analyse the model. We found that pseudovertical transmission and indirect transmission via infectious units in the 'general' environment can lead to more groups being affected, but otherwise have relatively little effect on the invasion criteria. To reduce infection within the herd, we suggest that efforts be directed at reducing the opportunity for group-specific indirect transmission-particularly within the weaned group.

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.