Many diseases are characterized by a long and varying sub-clinical period. Two main mechanisms can explain such periods: a slow progress toward disease or a sudden transition from a healthy state to a disease state induced by internal or external events. We here survey epidemiological features of the amount of bacteria shed during

Mycobacterium Avium subspecies Paratuberculosis (MAP) is characterized by a long sub-clinical period. Cows are typically infected early in life, but show clinical signs of disease only a number of years later. Such a sub-clinical to clinical transition is observed in many human and animal diseases, including, among many others, Human Immuno-deficiency Virus (HIV) infections [

Different infectious diseases have been characterized to show in clinical progression one of the two scenarios defined above. Most chronic diseases, such as Hepatitis C Virus (HCV) and HIV show slow progression to a clinical stage (Acquired Immune Deficiency Syndrome (AIDS) in HIV and cirrhosis in HCV) [

The classification of infectious diseases into these two general categories has important therapeutic implications. If a disease belongs to the slowly progressing group, then treatment before the clinical stage can prevent or delay further deterioration. If on the contrary the transition to the clinical stage is sudden, the optimal scheme to prevent clinical disease would be prevention of events that can induce the transition to the clinical stage.

This distinction also has implications for the predictability of future stages of clinical disease. In a slowly progressing disease, the probability of future clinical signs can be estimated from the position in the path to a full blown disease; while in a sudden transition, the probability for disease should be estimated by the frequency of events that can induce such a transition.

Infections of dairy cows with MAP are generally assumed to occur at a very young age [

In order to evaluate which of these two disease progression phenomena is most likely to occur with MAP infections, we propose a relatively simple mathematical model, the transition model, and compare it to a set of observations regarding infection and disease dynamics. We suggest here that a similar approach, assuming the presence of similar longitudinal data, may be applied to other infectious diseases.

In the following section we discuss a set of observations, and then compare multiple models to explain these observations:

A Markov model with three different possible states.

A deterministic model for the growth of the bacterial population.

Different versions of a stochastic dynamics as described by Stochastic Differential Equations (SDE).

While all models presented here are simplistic models, they can serve to differentiate between general scenarios. Given the complexity of any disease and the number of free parameters that can be introduced in mathematical models, we can almost always find a complex enough model that could explain a set of observations. We intend to show here that a very simple model can explain multiple observed phenomena depending on the choice of parameters and the value of the chosen parameters. Therefore the model proposed here aims to be a flexible yet realistic model describing real life phenomena.

The ODE were solved numerically using Matlab fourth/five order Runge Kutta, as applied in the MATLAB, ode45 function assuming non-stiff equations.

The SDE is modeled as an ODE with Ito noise unless stated otherwise. It was solved using Matlab when after each step an Ito noise is calculated. Specifically, a normal random variable with a zero mean and a variance of ^{2}

The Markov models were solved numerically using Matlab, where the probability for each cow to die is taken from a binomial distribution. The initial number of the cows was 1000, and the probability to die was set to 0.001.

In order to study the transition to the disease state, we analyzed three farms with natural infections, and examined the time course of bacterial shedding for over 1000 cows. Note that experimental and natural infections vary in many aspects [

Data for this study were gathered from three longitudinal field studies, one longitudinal follow up in an experimentally aggregated population and multiple experimental infection trials.

Field study 1 comprised three dairy farms (100, 150, 300 lactating animals per farm) in the Northeastern US [

Field study 2 followed animals on a single dairy farm with approximately 100 lactating Guernsey breed cattle in Pennsylvania (US) for a period of 20 years during an intervention program. Details of farm size, MAP prevalence and study design are available in previously published work [

Field study 3 followed animals on 17 Dutch dairy farms (32 to 104 animals per farm with a total of 1072 cows) during a national monitoring program over the course of 3.5 years. Animals were tested by ELISA and fecal culture at 6 month intervals [

In order to simplify the analysis, we defined for each cow three possible states: A) Non Shedding, B) Mild shedding, and C) High shedding. We defined the last stage to be any value above or equal to 50 Colony-Forming Units (CFU) per gram of feces, and the mild stage to be between 1 and 50 CFU per gram of feces. Generally, cows that are shedding high numbers of bacteria show or will show clinical signs of Johne’s disease.

In the current analysis, shedding time-series had typical intervals of 90–180 days, and the vast majority of cows (94.5%) never reached high shedding. In the cows that never reached high shedding (189/3397 ~ 5.5%), the vast majority of cows (>90%) never went back to mild or low shedding and had high shedding values until they were removed from the herd due to culling or death.

A large fraction of the cows never presenting high shedding levels may actually have been infected at least for some time. Among the cows never producing high shedding levels, 10% had some evidence of infection (Blood/Milk Enzyme-Linked ImmunoSorbent Assay (ELISA), tissue samples, or intermittent low or fluctuating intermediate levels of shedding).

Some of the cows show an initial low shedding stage before moving on to high shedding values. However, the average time from the first non-zero level to high shedding is one sample (less than 180 days) with a narrow distribution (Figure

Before high shedding started, the fraction of cows expressing a first high shedding event from the cows that were still in the herd at a given age was computed, by dividing the fraction of cows expressing first high shedding at a given time point (Figure

One can thus summarize the epidemiology of MAP by the following stylized facts:

Most MAP infected cows never reach high shedding.

Within the MAP infected cows reaching high shedding, the vast majority of cows never go back to low/no shedding.

In the group of cows that are high shedders, these animals reached the stage of high shedding fast after initial shedding, compared to the length of the sub-clinical period.

The ratio to reach high shedding is constant after approximately an age of 1000 days.

Most cows are infected and some occasionally shed low levels of bacteria.

We have here equated clinical signs to high shedding levels of MAP. While such shedding levels are often seen in cows with clinical signs defining Johne’s disease, the presence of high shedding is not completely equivalent to a transition to a clinical stage. Still, a clear relation between high-shedding and clinical signs has been reported. A much more detailed description of the epidemiology and clinical signs can be found in the accompanying paper by Mitchell et al. [

The epidemiology of MAP as described above can be represented as a three state model: The first state is healthy, uninfected (H). The second state is sub-clinical with potential low or intermediate shedding (S) and the third state is high shedding with potentially signs of clinical disease (C). The transitions in this model would be from H to S and possibly back to H and from S to C, with no possible transition back from C to S (Figure

In order to compare the two models, we propose a generic ODE and SDE framework to study the parameters required by each type of model and to determine which model is more plausible.

A Markov model can reproduce many of the observed features. The fraction of cows that reach high shedding is determined by

One could amend these two weaknesses by explicitly incorporating the difference between experimental and natural infections into the model, and assuming that the transition probabilities are determined by environmental and internal elements. In such a model, the transition probabilities would be much higher for experimental than for natural infections. The lack of an initial refractory period can be amended by introducing a larger number of intermediate stages between the S and C stages.

Given enough such intermediate stages, the Markov chain behavior approaches a random variable. Thus, while technically correct, we will show that a random variable description provides a simpler description of such a stochastic process.

The second model can be studied using a standard ODE approach, since it does not contain stochastic elements. The simplest model would be constant reproduction and destruction rates for the bacteria in a single cow. For the sake of simplicity, let us model the bacteria level within a given cow, and denote it by x. Let us assume that the bacteria are destroyed by the immune system or cleared by any other mechanism at a rate of

In this model, only two solutions would be possible: either the bacteria are cleared from the host, or the bacteria are growing exponentially and the high shedding occurs probably with the onset of the clinical signs. We do not explicitly state the properties of the bacterial dynamics, once high shedding is reached, but the dynamics at this stage have no significant effect on the conclusions, since we assume that once this high shedding is reached, the cow cannot go back to the transient or healthy state. A simple description of the dynamics beyond this stage can be through logistic growth:

For negative values of

Such a model is inconsistent with multiple observations:

In this model we do not expect cows not eventually becoming ill to have bacteria in them after some stage, since the bacteria frequency is expected to decrease over time in these cows.

The ratio is not expected to be approximately uniform in time, since there is no

In this model, the disease probability is not affected by the initial bacterial dose during infection, in contrast with the clear difference in the frequency of sick cows in experimental and natural infection studies, as mentioned above.

The two approaches can be combined through a slightly more complex model that includes two realistic features. The first feature to include is an explicit non linear growth rate in addition to the elements above. The power of the non-linear growth rate can be any power above one. We here use a power of two for the sake of simplicity. This would represent a positive feedback of the bacteria on itself. Such a feedback can occur if for example, bacteria survive better within granuloma, which in turn are produced by the bacteria. The model would then become:

Note that many different positive feedback loops can produce a similar behavior, beyond the possible effect of granuloma.

In contrast with the model of Equation (

However, this model still suffers from two problems discussed for the model in Equation (

In this model we do not expect cows not eventually becoming ill to have bacteria in them, since the population that will never be sick has low

The ratio is not expected to be uniform, since there is no

These two limitations can be solved using two slight modifications to the model: the introduction of a constant source of bacteria (

For the appropriate parameter values (as will be further discussed), this model has two attractors: a low shedding attractor determined by the value of

If

Within the cows reaching high shedding, the vast majority of cows never go back to low/no shedding. This asymmetry in the transition is a result of the different properties of the two attractors. When the system resides in the low bacterial level attractor, it has a constant probability of moving to the high attractor, and a large enough fluctuation is enough to move it to the high attractor. The opposite is not true. When the system resides at the high attractor, it will diverge in a finite short time, and will thus never be able to return to its initial state.

The third observed feature is the fast emergence of clinical signs after initial shedding. Such a fast growth is indeed expected from the non-linear growth term, which as mentioned above will lead a finite time divergence of

The non-zero value of

The ratio to reach high shedding is constant after a period and then slightly deteriorates (Figures

While this model explains most observed features, it has one weakness, which is sensitivity to the value of the

The following two sections are quite mathematical and the biological conclusions of the paper can be understood without them. We here performed a sensitivity analysis to the results of Equation (

In order to understand the relation between the probability to get infected and the parameters

In order to check that this is the case, we simulated the dynamics in Equation (

As mentioned above, in the rescaled units

However, one can assume that

The model presented here contains four elements:

linear bacterial growth (i.e. a constant term in the ODE).

Destruction of the bacteria by the immune system.

Supra-exponential bacterial growth (i.e. a supra linear term in the ODE).

Random fluctuations.

The first term is expected in any model where bacteria grow with no saturation. Similarly the second term is expected in any model where bacteria are affected by the immune response of the host, including killing of bacteria by B or T cells. The two last terms are slightly more complex.

The non-linear bacterial growth can occur whenever existing bacteria facilitate the growth of more bacteria. In other words, there is a positive feedback of the current bacteria concentration on the future bacterial growth. The opposite may also happen, where bacterial growth prevents or reduces killing of existing bacteria. Such mechanisms are actually observed in MAP where bacteria organize in large granuloma and within these granuloma, they are protected against killing [

The random fluctuations used here were multiplicative. In other words, random elements increase or decrease the net growth rate of the bacteria, either through a weakening of the immune response following other diseases or stressful events such as giving birth or transport events [

From an evolutionary point of view, latency seems to be the optimal solution for pathogens, since it ensures the long term survival of their growth environment. Virulence can thus be treated as an accident of the pathogen life cycle. Indeed, many models were developed, to explain the emergence of virulence from an evolutionary point of view (e.g. [

The role of the immune system is crucial, with a quite general agreement that sub-clinical stages are basically induced by immune control, and that the transition to clinical disease is often associated with an escape from immune control [

However, a mathematical model of the basic mechanisms driving the exit from latency in general and methods to validate these conclusions at the epidemiological level are missing. We present here a comparison of multiple mathematical models, where the objective was to model the MAP shedding patterns. Ultimately, the model is gauged against observed within host immune dynamics on the observed MAP shedding patterns in cows from real-life populations. All models that were used contained a single variable (the bacteria). All other elements, such as the immune response, were assumed to be constant. Within these models, we show a model with two attractors: one representing the low bacteria concentration state and one representing the high shedding state. Stochastic transitions between these two attractors provided the simplest of the observed features, mainly a relatively flat probability of high shedding and clinical signs after an initial low or intermittent shedding period and a rapid transition to high shedding.

Other models could also be adapted to give similar results, but this would require more complex models and assumptions on the distribution of parameters. The two-state model presented here is far from being the only possible model in this category. In recent years many mathematical models have been studied that describe a wide variety of different systems: biological systems [

While all models presented here are obviously overly simplistic, the objective of these models was to describe the essence of MAP within-host infection dynamics. More complex models may better reproduce many details of the dynamics, but will require many more, often unsupported, assumptions.

An important conclusion from these models is that the best method to prevent the transition to high shedding, in infected cows is to limit external events or other diseases. This can be checked by comparing the fraction of infected cows that develop clinical signs in different conditions.

A caveat of the proposed models is that they do not explicitly integrate the dynamics of the immune response. Thus, they cannot be directly compared to experimental observation on the relationship between immune response properties and MAP shedding patterns. A second caveat is the absence of early shedding in all models studied here. Infected calves are known to have an early shedding phase shortly after initial infection. However in all models studied here the development is unidirectional from non-infected to infected to high shedding. These two caveats can be solved using a model that includes the acquirement of a specific adaptive immune response following infection. However, there does not seem to be enough immuno-epidemiological observations at this stage to justify a more complex model.

While we here focused on MAP, the conclusions from this analysis are relevant to a large group of diseases with a similar epidemiology. We used a few criteria, such as the fraction of cows becoming high shedders, the dose response and within cow infection dynamics. It would be of interest to check if diseases can be broadly divided into groups fitting each type of model presented here.

Acquired immune deficiency syndrome

Bovine leukemia virus

Colony-forming unit

Enzyme-linked immunoSorbent assay

Hepatitis C virus

Human immuno-deficiency virus

Herpes simplex virus

Mycobacterium Avium subspecies Paratuberculosis

Ordinary differential equation

Stochastic differential equation

Tubercle vacillus

The authors declare that they have no competing interests.

HB and YL performed the theoretical computations and have written the manuscript. YS has performed the experiments. RM has performed the experimental data analysis. All authors read and approved the final manuscript.

The authors acknowledge the support of the Within-host modeling of MAP infections Working Group at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award DBI-1300426, with additional support from The University of Tennessee, Knoxville.