A semi-stochastic model of the transmission of Escherichia coli O157 in a typical UK dairy herd: dynamics, sensitivity analysis and intervention/prevention strategies

When modelling the transmission of infection within small populations, it is necessary to consider the possibility of stochastic fade-out of infection. We present a semi-stochastic model for the transmission of a microparasite, in this case Escherichia coli O157, within a multigroup system, namely a typical UK dairy herd. 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)). We present the results of our simulation study alongside data from empirical studies and also compare simulation results with those for the corresponding deterministic model. We then examine the effects of reducing shedding in the food-producing groups on outbreak size and prevalence of infection. A sensitivity analysis of herd prevalence reveals that, for both the deterministic and the semi-stochastic model, the prevalence within the herd is most sensitive to two parameters relating to the weaned group. This supports our previously reported conclusions for the deterministic model, which were based on an analysis of the next-generation matrix. The sensitivity analysis also indicates that herd prevalence is greatly affected by two other parameters relating to the lactating group. We conclude by discussing the possible efficacy of suggested intervention strategies.     

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.     

Estimating the Hidden Burden of Bovine Tuberculosis in Great Britain

The number of cattle herds placed under movement restrictions in Great Britain (GB) due to the suspected presence of bovine tuberculosis (bTB) has progressively increased over the past 25 years despite an intensive and costly test-and-slaughter control program. Around 38% of herds that clear movement restrictions experience a recurrent incident (breakdown) within 24 months, suggesting that infection may be persisting within herds. Reactivity to tuberculin, the basis of diagnostic testing, is dependent on the time from infection. Thus, testing efficiency varies between outbreaks, depending on weight of transmission and cannot be directly estimated. In this paper, we use Approximate Bayesian Computation (ABC) to parameterize two within-herd transmission models within a rigorous inferential framework. Previous within-herd models of bTB have relied on ad-hoc methods of parameterization and used a single model structure (SORI) where animals are assumed to become detectable by testing before they become infectious. We study such a conventional within-herd model of bTB and an alternative model, motivated by recent animal challenge studies, where there is no period of epidemiological latency before animals become infectious (SOR). Under both models we estimate that cattle-to-cattle transmission rates are non-linearly density dependent. The basic reproductive ratio for our conventional within-herd model, estimated for scenarios with no statutory controls, increases from 1.5 (0.26–4.9; 95% CI) in a herd of 30 cattle up to 4.9 (0.99–14.0) in a herd of 400. Under this model we estimate that 50% (33–67) of recurrent breakdowns in Britain can be attributed to infection missed by tuberculin testing. However this figure falls to 24% (11–42) of recurrent breakdowns under our alternative model. Under both models the estimated extrinsic force of infection increases with the burden of missed infection. Hence, improved herd-level testing is unlikely to reduce recurrence unless this extrinsic infectious pressure is simultaneously addressed.     

Calculating the time to extinction of a reactivating virus, in particular bovine herpes virus

The expected time to extinction of a herpes virus is calculated from a rather simple population-dynamical model that incorporates transmission, reactivation and fade-out of the infectious agent. We also derive the second and higher moments of the distribution of the time to extinction. These quantities help to assess the possibilities to eradicate a reactivating infection. The key assumption underlying our calculations is that epidemic outbreaks are fast relative to the time scale of demographic turnover. Four parameters influence the expected time to extinction: the reproduction ratio, the reactivation rate, the population size, and the demographic turn-over in the host population. We find that the expected time till extinction is very long when the reactivation rate is high (reactivation is expected more than once in a life time). Furthermore, the infectious agent will go extinct much more quickly in small populations. This method is applied to bovine herpes virus (BHV) in a cattle herd. The results indicate that without vaccination, BHV will persist in large herds. The use of a good vaccine can induce eradication of the infection from a herd within a few decades. Additional measures are needed to eradicate the virus from a whole region within a similar time-span.     

Approach for Qualitative Validation Using Aggregated Data for a Stochastic Simulation Model of the Spread of the Bovine Viral-Diarrhoea Virus in a Dairy Cattle Herd

Qualitative validation consists in showing that a model is able to mimic available observed data. In population level biological models, the available data frequently represent a group status, such as pool testing, rather than the individual statuses. They are aggregated. Our objective was to explore an approach for qualitative validation of a model with aggregated data and to apply it to validate a stochastic model simulating the bovine viral-diarrhoea virus (BVDV) spread within a dairy cattle herd. Repeated measures of the level of BVDV-specific antibodies in the bulk-tank milk (total milk production of a herd) were used to summarise the BVDV herd status. First, a domain of validation was defined to ensure a comparison restricted to dynamics of pathogen spread well identified among observed aggregated data (new herd infection with a wide BVDV spread). For simulations, scenarios were defined and simulation outputs at the individual animal level were aggregated at the herd level using an aggregation function. Comparison was done only for observed data and simulated aggregated outputs that were in the domain of validation. The validity of our BVDV model was not rejected. Drawbacks and ways of improvement of the approach are discussed.     

Expressed 2-5A synthetase genes and pseudogenes in the marine sponge Geodia barretti

In cattle, bovine leukocyte antigens (BoLAs) have been extensively used as markers for bovine diseases and immunological traits. In this study, we sequenced alleles of the BoLA class II loci, BoLA-DRB3 and BoLA-DQA1, from 650 Japanese cattle from six herds [three herds (507 animals) of Japanese Black cattle and three herds (143 animals) of Holstein cattle] using polymerase chain reaction-sequence-based typing (PCR-SBT) methods. We identified 26 previously reported distinct DRB3 alleles in the two populations: 22 in Japanese Black and 17 in Holstein. The number of DRB3 alleles detected in each herd ranged from 9 to 20. Next, we identified 15 previously reported distinct DQA1 alleles: 13 in Japanese Black and 10 in Holstein. The number of alleles in each herd ranged from 6 to 10. Thus, allelic divergence is significantly greater for DRB3 than for DQA1. A population tree on the basis of the frequencies of the DRB3 and DQA1 alleles showed that, although the genetic distance differed significantly between the two cattle breeds, it was closely related within the three herds of each breed. In addition, Wu-Kabat variability analysis indicated that the DRB3 gene was more polymorphic than the DQA1 gene in both breeds and in all herds, and that the majority of the hypervariable positions within both loci corresponded to pocket-forming residues. The DRB3 and DQA1 heterozygosity for both breeds within each herd were calculated based on the Hardy-Weinberg equilibrium. Only one Japanese Black herd showed a significant difference between the expected and observed heterozygosity at both loci. This is the first report presenting a detailed study of the allelic distribution of BoLA-DRB3 and -DQA1 genes in Japanese Black and Holstein cattle from different farms in Japan. These results may help to develop improved livestock breeding strategies in the future.     

Stochastic simulations of a multi-group compartmental model for Johne's disease on US dairy herds with test-based culling intervention

Infection elimination may be an important goal of control programs. Only in stochastic infection models can true infection elimination be observed as a fadeout. The phenomena of fadeout and variable prevalence are important in understanding the transmission dynamics of infectious diseases and these phenomena are essential to evaluate the effectiveness of control measures. To investigate the stochastic dynamics of Mycobacterium avium subsp. paratuberculosis (MAP) infection on US dairy herds with test-based culling intervention, we developed a multi-group stochastic compartmental model (a continuous time Markov chain model) with both horizontal and vertical transmission. The stochastic model predicted fadeout and within-herd prevalence to have a large variance. Although test-based culling intervention generally decreased prevalence over time, it took longer than desired by producers to eliminate the endemic MAP infection from a herd. Uncertainty analysis showed that, using annual culture test and culling of only high shedders or culling of both low and high shedders with a 12-month delay in culling of low shedders, MAP infection persisted in many herds beyond 20 years. While using semi-annual culture test and culling of low and high shedders with a 6-month delay in culling of low shedders, MAP infection in many herds would be extinct within 20 years. Sensitivity analysis of the cumulative density function of fadeout suggested that combining test-based culling intervention and reduction of transmission rates through improved management between susceptible calves and shedding animals may be more effective than either alone in eliminating endemic MAP infection. We also discussed the effects of other factors such as herd size, heifer replacement, and adult cow infection on the probability of fadeout.     

A model of bovine tuberculosis control in domesticated cattle herds.

A typical strategy for disease control in domesticated animals involves regular field tests and quarantine of infected herds. This prevents disease spread beyond the herd, while slaughter of diseased animals removes the infection from within the herd. A model of bovine tuberculosis (Tb) control in cattle is examined, which includes ''test and slaughter'' combined with herd isolation and vaccination. Herd status is represented by an integral equation expressing the duration of herd isolation. The current Tb situation in New Zealand is used as an example, and vaccination strategy discussed. Extrapolation of existing management strategies indicate that a vaccine of efficacy greater than 96% would be required, reaching 95% of target Tb levels within six years. These results suggest that a complementary strategy of vaccination and vector control may be more promising than vaccination alone.     

Distribution of transposable elements in prokaryotes

We consider models for the distribution of the number of elements per host genome for families of transposable elements (TEs). The hosts are assumed to be prokaryotes. These models assume a constant rate of infection of uninfected hosts by TEs, replicative transposition within each host, and a reduction of the fitness of a host dependent on the number of TEs it contains. No provision was made for the deletion of individual TEs within a host or for recombination, since both are relatively rare events in prokaryotes. These models mostly assume that the TE performs no function for the host, and that the reduction in fitness with increased copy number is due to effects such as the impairment of beneficial genes by transposition or homologous recombination. We also consider a model in which the TEs can convey a selective advantage to the host. The equilibrium distributions of copy number are determined for these models, and are of a variety of classical types. Relevant parameters of the models are estimated using data on the distribution of insertion sequences in natural isolates of Escherichia coli.     

Variation in the load of the horn fly, Haematobia irritans, in cattle herds is determined by the presence or absence of individual heifers

The distribution of horn flies, Haematobia irritans (L.) (Diptera: Muscidae), in herds of Danish Holstein-Friesian cattle was investigated in two studies conducted during two field seasons. In the first study, highly significant differences in fly distribution between the most and the least fly-susceptible heifers were observed. In one herd, the mean difference between the most fly-susceptible and the most fly-resistant heifers was 268 Ha. irritans specimens. The highest ratio between upper and lower mean fly number was 64.1:1, whereas the lowest was 3.1:1. In the second year, it was demonstrated that the heifers kept their rank in fly attraction over time. The trial clearly demonstrated that some heifers were attracting flies, whereas others, even in the same herd, only carried a few. In the second study, heifers were moved in and out of herds in an attempt to manipulate fly loads in the herds. In year 1, one herd (herd A) received four fly-resistant heifers from another herd (herd B), resulting in a drop in the mean number of flies, whereas herd B received four fly-susceptible heifers from herd A, resulting in an elevation of the mean number of flies. In year 2, a similar pattern emerged using herds C and D, and when the cattle were later returned to their original herds, the fly loads returned to their original distribution. The data presented here show unequivocally that, for horn flies, there can be considerable differences in fly loads for individual heifers within the Holstein-Friesian breed. Furthermore, the overall fly load within herds can be manipulated, and can be reversed. Thus, the distribution in the number of flies within a herd appears to depend on the number of fly-resistant or fly-susceptible heifers. The possible role of chemical factors emitted by heifers, i.e. volatile semiochemicals, in determining differences in fly loads is discussed, whereby attractants are emitted by fly-susceptible heifers and enable flies to locate their host, and repellents are emitted by fly-resistant heifers such that the flies are actively repelled from the herd.     

Seroepidemiology of bovine viral diarrhoea virus (BVDV) in the Adamawa Region of Cameroon and use of the SPOT test to identify herds with PI calves

Bovine viral diarrhoea, caused by the bovine viral diarrhoea virus (BVDV) in the Pestivirus genus of the Flaviviridae, is one of the most important diseases of cattle world wide causing poor reproductive performance in adult cattle and mucosal disease in calves. In addition it causes immunosuppression and increased susceptibility to other infections, the impact of which is uncertain, particularly in sub-Saharan Africa where animals are exposed to a much wider range and higher intensity of infections compared to Europe. There are no previous estimates of the seroprevalence of BVDV in cattle in Cameroon. This paper describes the serological screening for antibodies to BVDV and antigen of BVDV in a cattle population in the Adamawa Region of Cameroon in 2000. The estimates of herd-level and within herd seroprevalences adjusted for test imperfections were 92% and 30% respectively and 16.5% of herds were classed as having a persistently infected calf (PI) in the herd within the last year based on the "spot" test approach. There was evidence of clustering of herds with PI calves across the north and west of the Region which corresponds with the higher cattle density areas and of self-clearance of infection from herds. A multivariable model was developed for the risk of having a PI calf in the herd; proximity to antelope, owning a goat, mixing with > 10 other herds at grazing and the catchment area of the veterinary centre the herd was registered at were all significant risk factors. Very little is known about BVDV in sub-Saharan Africa and these high seroprevalences suggest that there is a large problem which may be having both direct impacts on fertility and neonate mortality and morbidity and also indirect effects through immunosuppression and susceptibility to other infections. Understanding and accounting for BVDV should be an important component of epidemiological studies of other diseases in sub-Saharan Africa.     

Higher non-return rate associated with Mycobacterium avium subspecies paratuberculosis infection at early stage in Holstein dairy cows

The effects of infection by Mycobacterium avium paratuberculosis (Map) on dairy cows are poorly documented and quite controversial. This retrospective study aimed at quantifying the variation in non-return to service of Holstein dairy cows according to their Map-infection status. Three different statuses were defined based on both individual and herd tests results: ELISA positive cow, all tests negative cow in a negative herd and all tests negative cow in a positive herd. Whatever the age at Map testing, the status was attributed to a cow from its first lactation onwards. Non-return to service was determined at 200 days after first and second services. The study was performed from 1999 to 2007 on 185,950 AI from 48,914 cows in early stage of the infection in 1069 herds by logistic regression controlling for known factors influencing non-return rate. Non-return rate was higher for infected cows compared to negative cows from negative herds (RR of 1.10 or +3.9 points of % of non-return rate). The effect was significant for parities 1 and 2 (RR of 1.11 and 1.12, respectively) but not for higher ones. This effect was lower when comparing positive cows to negative cows in the same herds but relative risks were still above 1. The hypothesis that the effect of Map on non-return depends upon the stage of infection is formulated.     

Estimates for local and movement-based transmission of bovine tuberculosis in British cattle

Both badgers and livestock movements have been implicated in contributing to the ongoing epidemic of bovine tuberculosis (BTB) in British cattle. However, the relative contributions of these and other causes are not well quantified. We used cattle movement data to construct an individual (premises)-based model of BTB spread within Great Britain, accounting for spread due to recorded cattle movements and other causes. Outbreak data for 2004 were best explained by a model attributing 16% of herd infections directly to cattle movements, and a further 9% unexplained, potentially including spread from unrecorded movements. The best-fit model assumed low levels of cattle-to-cattle transmission. The remaining 75% of infection was attributed to local effects within specific high-risk areas. Annual and biennial testing is mandatory for herds deemed at high risk of infection, as is pre-movement testing from such herds. The herds identified as high risk in 2004 by our model are in broad agreement with those officially designated as such at that time. However, border areas at the edges of high-risk regions are different, suggesting possible areas that should be targeted to prevent further geographical spread of disease. With these areas expanding rapidly over the last decade, their close surveillance is important to both identify infected herds qucikly, and limit their further growth.     

Effects of inert volume-excluding macromolecules on protein fiber formation. I. Equilibrium models

The equilibrium Oosawa-Asakura model for nucleated assembly of rod-like protein fibers is recast in terms of dimensionless (scaled) quantities. The model is then generalized to treat arbitrarily large deviations from thermodynamic ideality arising from high fractional volume occupancy by an inert protein or polymer. Each state of association of the self-associating protein is modeled as an equivalent rigid convex particle (sphere or spherocylinder) and the crowding species is modeled either as an equivalent sphere or cylindrical rod. The resulting conservation of mass relation is readily solved to yield the fractional abundance of monomer, from which the entire equilibrium distribution of oligomeric species can be calculated, either directly or through the use of an additional scaling relationship. Results indicating the potential effect of volume occupancy on the equilibrium solubility of the self-associating protein and upon the equilibrium distribution of polymer size are presented. It is found that the fractional (logarithmic) change in both solubility and in the breadth of the polymer size distribution scale almost linearly with the fractional (logarithmic) change in the thermodynamic activity of monomer.     

Tick-borne diseases in ruminants of Central and Southern Italy: epidemiology and case reports

Sera and blood from cattle and sheep were examined for the presence of Babesia and Theileria spp by microscopy and serology at the Parasitology Department of the Istituto Zooprofilattico Sperimentale of Abruzzo and Molise (IZSAM). Of the 47 bovine herds (323 animals) tested, 15 were found positive for Babesia bigemina and 1 for Babesia bovis. Two outbreaks occurred, one caused by B. bigemina and one by B. bovis. The B. bigemina outbreak occurred in Abruzzo and has been followed for two years. The isolate of B. bigemina was very pathogenic leading to the death of two cows out of 57. The vector responsible of the transmission appeared to be Rhipicephalus bursa. Parasites were observed in the erythrocytes for 30 days whereas sera were positive to indirect immunofluorescence (IFA) for at least one year. The B. bovis outbreak occurred in the province of Mantova (Northern Italy) in a group of 70 beef cattle imported from France. The infection resulted in the death of 5 animals and severe illness in another 6. In contrast with what occurred for Babesia infection, no clinical cases were recorded in cattle when species of Theileria were detected by microscopy. Of the 24 bovine herds (252 animals) tested for Theileria, 21 were found positive for the T. "sergenti"/buffeli/orientalis group. Single and mixed infection of T. "sergenti" and T. buffeli/orientalis were detected in herds of cross-bred cattle from Abruzzo and Marche. The parasites were identified by using a polymerase chain reaction which amplified DNA encoding p32/34. Most of the collected ticks (90%) were adults of R. bursa whereas the others were adults of Hyalomma detritum. During the period the animals have been observed (18 months), no clinical cases have been recorded and no associations have been found between blood abnormalities and animals found infected with Theileria. Prevalences of subclinically infected carriers increased from February till December (95.4%) even if the animals were indoors and no ticks were present. The prevalence then dropped dramatically six months later (76.7%). In calves less than 1 year old, the prevalence of infection significantly (p<0.05) increased with age, however intraerythrocytic stages of Theileria were found in the blood of three newborn calves (<7 days of age). Of the 18 ovine flocks tested for Babesia spp. (150 animals examined), 1 was positive for B. ovis and 2 for B. motasi. B. motasi infection was not associated with symptoms, while an outbreak of babesiosis caused by B. ovis occurred in Abruzzo. The infection resulted in the death of 3 animals (0.75% of the flock), two rams (20% of the total number) and a ewe, and severe illness in another 5 ewes (2% of the flock). Specimens of R. bursa and R. turanicus were collected from the infected animals. Of the 18 flocks (150 animals) examined, 12 were microscopically positive for Theileria spp. No clinical cases were recorded and identification at species level was not possible on the basis of morphological criteria. The prevalence distribution of infected herds and infected animals within herds and flocks have been calculated by a Monte Carlo simulation model, running 10,000 iterations. The most likely levels of prevalence of infected herds and infected animals within herds found for the species observed were as follows: 20% for B. bigemina with a prevalence within herd of 27%, 11% for B. bovis (18% within herd), 10% for Babesia ovis (19% within herd), 10% for B. motasi (17.5% within herd), 63% for Theileria in cattle (66% within herd) and 51% for Theileria in sheep (55% within herd).     

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

A stochastic model for transmission,extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices

Many infectious agents transmitting through a contaminated environment are able to persist in the environment depending on the temperature and sanitation determined rates of their replication and clearance, respectively. There is a need to elucidate the effect of these factors on the infection transmission dynamics in terms of infection outbreaks and extinction while accounting for the random nature of the process. Also, it is important to distinguish between the true and apparent extinction, where the former means pathogen extinction in both the host and the environment while the latter means extinction only in the host population. This study proposes a stochastic-differential equation model as an approximation to a Markov jump process model, using Escherichia coli O157:H7 in cattle as a model system. In the model, the host population infection dynamics are described using the standard susceptible-infected-susceptible framework, and the E. coli O157:H7 population in the environment is represented by an additional variable. The backward Kolmogorov equations that determine the probability distribution and the expectation of the first passage time are provided in a general setting. The outbreak and apparent extinction of infection are investigated by numerically solving the Kolmogorov equations for the probability density function of the associated process and the expectation of the associated stopping time. The results provide insight into E. coli O157:H7 transmission and apparent extinction, and suggest ways for controlling the spread of infection in a cattle herd. Specifically, this study highlights the importance of ambient temperature and sanitation, especially during summer.     

A contribution to the interpretation of immunological survey records

In certain infection types the use of official immunological survey figures expressed mathematically by means of exponential distribution or its distributional function may help assess the epidemiological trends in herd immunity levels. This approach to the evaluation of immunological survey records is primarily well applicable to infections that are uniform in exposure pattern, affect the entire population, and result in life-long immunity. The parameters characterizing the rise in the level of herd immunity can be determined by means of negative exponential regression. Applied to selected immunological survey data this essentially simple catalytic model of investigation may provide reliable information on the actual state of immunity in the general population, and may also help estimate its dynamics in the retrospect.     

Potential Benefits of Cattle Vaccination as a Supplementary Control for Bovine Tuberculosis

Vaccination for the control of bovine tuberculosis (bTB) in cattle is not currently used within any international control program, and is illegal within the EU. Candidate vaccines, based upon Mycobacterium bovis bacillus Calmette-Guérin (BCG) all interfere with the action of the tuberculin skin test, which is used to determine if animals, herds and countries are officially bTB-free. New diagnostic tests that Differentiate Infected from Vaccinated Animals (DIVA) offer the potential to introduce vaccination within existing eradication programs. We use within-herd transmission models estimated from historical data from Great Britain (GB) to explore the feasibility of such supplemental use of vaccination. The economic impact of bovine Tuberculosis for farmers is dominated by the costs associated with testing, and associated restrictions on animal movements. Farmers’ willingness to adopt vaccination will require vaccination to not only reduce the burden of infection, but also the risk of restrictions being imposed. We find that, under the intensive sequence of testing in GB, it is the specificity of the DIVA test, rather than the sensitivity, that is the greatest barrier to see a herd level benefit of vaccination. The potential negative effects of vaccination could be mitigated through relaxation of testing. However, this could potentially increase the hidden burden of infection within Officially TB Free herds. Using our models, we explore the range of the DIVA test characteristics necessary to see a protective herd level benefit of vaccination. We estimate that a DIVA specificity of at least 99.85% and sensitivity of >40% is required to see a protective benefit of vaccination with no increase in the risk of missed infection. Data from experimentally infected animals suggest that this target specificity could be achieved in vaccinates using a cocktail of three DIVA antigens while maintaining a sensitivity of 73.3% (95%CI: 61.9, 82.9%) relative to post-mortem detection.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.