Mathematical model of the acute inflammatory response to Escherichia coli in intramammary challenge

A mathematical model to study resistance and tolerance to infection at the animal and population levels: application to E. coli mastitis

A mathematical model is proposed that describes the colonization of host tissues by a contagious pathogen and the early nonspecific immune response, the impact of the infection on the performances of the host, and the spread of the infection in the population. The model obeys specific biological characteristics: Susceptible hosts are infected after contact with an infected one. The number of pathogenic units that invade a susceptible host is dependent on the infectious dose provided by the infected host and on the ability of the susceptible host to resist the invasion. After entry in host, pathogenic changes over time are expressed as the difference between the intrinsic logistic growth rate and the Holling type II kill rate provided by the immune response cells. Hosts have different ability to restrict reproduction of the pathogen units. The number of response cells actively recruited to the site of infection depends on the number of the pathogenic units. Response cells are removed after having killed a fixed number of pathogenic units. The effects of the number of pathogenic units on the performances of the host depend upon its levels of tolerance to the deleterious effects of both pathogenic and response cells. Pre-infection costs are associated to tolerance and resistance levels. Estimates of most biological parameters of the model are based on published experimental studies while resistance/tolerance parameters are varied across their allowable ranges. The model reproduces qualitatively realistic outcomes in response to infection: healthy response, recurrent infection, persistent infectious and non-infectious inflammation, and severe immunodeficiency. Evolution across time at the animal and population levels is presented. Effects on animal performances are discussed with respect to changes in resistance/tolerance parameters and selection strategies are suggested.

Effectiveness analysis of resistance and tolerance to infection

Background

Tolerance and resistance provide animals with two distinct strategies to fight infectious pathogens and may exhibit different evolutionary dynamics. However, few studies have investigated these mechanisms in the case of animal diseases under commercial constraints.

Methods

The paper proposes a method to simultaneously describe (1) the dynamics of transmission of a contagious pathogen between animals, (2) the growth and death of the pathogen within infected hosts and (3) the effects on their performances. The effectiveness of increasing individual levels of tolerance and resistance is evaluated by the number of infected animals and the performance at the population level.

Results

The model is applied to a particular set of parameters and different combinations of values. Given these imputed values, it is shown that higher levels of individual tolerance should be more effective than increased levels of resistance in commercial populations. As a practical example, a method is proposed to measure levels of animal tolerance to bovine mastitis.

Conclusions

The model provides a general framework and some tools to maximize health and performances of a population under infection. Limits and assumptions of the model are clearly identified so it can be improved for different epidemiological settings.

Bistability and bacterial infections

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.

Modelling the dynamics of intramammary E. coli infections in dairy cows: understanding mechanisms that distinguish transient from persistent infections

The majority of intramammary infections with Escherichia coli in dairy cows result in transient infections with duration of about 10 days or less, although more persistent infections (2 months or longer) have been identified. We apply a mathematical model to explore the role of an intracellular mammary epithelial cell reservoir in the dynamics of infection. We included biological knowledge of the bovine immune response and known characteristics of the bacterial population in both transient and persistent infections. The results indicate that varying the survival duration of the intracellular reservoir reproduces the data for both transient and persistent infections. Survival in an intracellular reservoir is the most likely mechanism that ensures persistence of E. coli infections in mammary glands. Knowledge of the pathogenesis of persistent infections is essential to develop preventive and treatment programmes for these important infections in dairy cows.

Translational potential of systems-based models of inflammation

A critical goal of translational research is to convert basic science to clinically relevant actions related to disease prevention, diagnosis, and eventually enable physicians to identify and evaluate treatment strategies. Integrated initiatives are identified as valuable in uncovering the mechanism underpinning the progression of human diseases. Tremendous opportunities have emerged in the context of systems biology that aims at the deconvolution of complex phenomena to their constituent elements and the quantification of the dynamic interactions between these components through the development of appropriate computational and mathematical models. In this review, we discuss the potential role systems-based translation research can have in the quest to better understand and modulate the inflammatory response.

Estimating degree of mastitis from time-series measurements in milk: a test of a model based on lactate dehydrogenase measurements

The aim of this study was to test a model for mastitis detection using a logic that allows examination of time-related changes and a progressive scale of mastitis state (i.e., not using specificity/sensitivity). The model produces a mastitis risk (MR) for individual cows on a scale from 0 (completely healthy) to 1 (full-blown mastitis). The main model input was lactate dehydrogenase (LDH; mumol/min per L) x milk yield. Test data containing 253 mastitis cases were used. Proportional samples were collected from each cow at each milking and analyzed for LDH and somatic cell count (SCC). The basis for the health definitions was veterinary treatment records. A refinement of the basic health definitions was made using systematic positive deviations in log(SCC) to indicate untreated infections. Two subsets of cows were identified: mastitic cows and cows completely free of mastitis (healthy controls). The time-profiles of these 2 groups in a 60-d window relative to day of veterinary treatment were examined. Model reliability throughout all stages of lactation and degrees of infection was examined using SCC as a continuous measure of degree of mastitis. The time-profile for the health controls was flat throughout the 60-d window with a median MR of 0.02. In contrast, the profile of the mastitic cows increased above the control cows' baseline from about -6 d, rising to a MR value of 0.20 at d 0, and declining to the control level after treatment. There were significant differences between mastitic and healthy cows from -4 to +2 d relative to veterinary treatment. When cases were time-aligned to peak of infection, rather than veterinary treatment, there was a much sharper peak to the time-profile of mastitic cows. The median MR at peak was 0.62 and the mean was 0.80. Using these data, the MR value of 0.62 had a <1% likelihood of actually coming from a healthy control. Testing against SCC, on the whole data set, showed that only 2.1% of all MR values had an error >0.7. These estimates of model reliability are comparable with the greatest values reported in the literature and, additionally, the model was able to detect significant differences between mastitic and healthy cows 4 d before treatment. It was also found that specificity/sensitivity calculations are inappropriate for evaluating time-related changes and a progressive scale of predicted mastitis state.

Cumulative physiological events influence the inflammatory response of the bovine udder to Escherichia coli infections during the transition period

A high proportion of intramammary coliform infections present at parturition develop disease characterized by severe inflammatory signs and sepsis during the first 60 to 70 d of lactation. In the lactating bovine mammary gland, the innate immune system plays a critical role in determining the outcome of these infections. Since the beginning of the 1990s, research has increased significantly on bovine mammary innate defense mechanisms in connection with the pathogenesis of coliform mastitis. Neutrophils are key effector cells of the innate immune response to intramammary infection, and their function is influenced by many physiological events that occur during the transition period. Opportunistic infections occur when the integrity of the host immune system is compromised by physical and physiological conditions that make the host more susceptible. The innate immune system of many periparturient cows is immunocompromised. It is unlikely that periparturient immunosuppression is the result of a single physiological factor; more likely, several entities act in concert, with profound effects on the function of many organ systems of the periparturient dairy cow. Their defense system is unable to modulate the complex network of innate immune responses, leading to incomplete resolution of the pathogen and the inflammatory reaction. During the last 30 yr, most efforts have been focused on neutrophil diapedesis, phagocytosis, and bacterial killing. How these functions modulate the clinical outcome of coliform mastitis, and how they can be influenced by hormones and metabolism has been the subject of intensive research and is the focus of this review. The afferent (sensing) arm of innate immunity, which enables host recognition of a diverse array of pathogens, is the subject of intense research interest and may contribute to the variable inflammatory response to intramammary infections during different stages of lactation. The development of novel interventions that modulate the inflammatory response or contribute to the elimination of the pathogen or both may offer therapeutic promise in the treatment of mastitis in periparturient cows.