Impact of the shedding level on transmission of persistent infections in Mycobacterium avium subspecies paratuberculosis (MAP)
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Abstract
Supershedders are infectious individuals that contribute a disproportionate amount of infectious pathogen load to the environment. A supershedder host may produce up to 10 000 times more pathogens than other infectious hosts. Supershedders have been reported for multiple human and animal diseases. If their contribution to infection dynamics was linear to the pathogen load, they would dominate infection dynamics. We here focus on quantifying the effect of supershedders on the spread of infection in natural environments to test if such an effect actually occurs in Mycobacterium avium subspecies paratuberculosis (MAP). We study a case where the infection dynamics and the bacterial load shed by each host at every point in time are known. Using a maximum likelihood approach, we estimate the parameters of a model with multiple transmission routes, including direct contact, indirect contact and a background infection risk. We use longitudinal data from persistent infections (MAP), where infectious individuals have a wide distribution of infectious loads, ranging upward of three orders of magnitude. We show based on these parameters that the effect of supershedders for MAP is limited and that the effect of the individual bacterial load is limited and the relationship between bacterial load and the infectiousness is highly concave. A 1000fold increase in the bacterial contribution is equivalent to up to a 2–3 fold increase in infectiousness.
Keywords
Colony Form Unit Bacterial Load Direct Transmission Mycobacterium Avium Subspecies Paratuberculosis Infectious IndividualIntroduction
The phenomenon of supershedding, where a small fraction of individuals contributes a disproportionate load of infectious pathogens to the exposure experience of susceptible individuals, has received much attention in the past several years with respect to human disease [1] and selected livestock pathogens, as well as the virtual spread of viruses [2, 3, 4, 5]. Supershedding individuals have been reported with infections, such as Escherichia coli O157: H7 [6], paratuberculosis [7, 8], HIV [9], influenza [10], and Salmonella [11]. However, the precise contribution of individuals producing a high concentration of pathogens to infection dynamics in the population remains poorly understood.
If infection is a direct hosttohost contact process, even with all contacts being effective (i.e. every contact with a susceptible results in transmission of infection), the maximum contribution of each infectious individual is capped by contact rates. However, if infection is driven by indirect contacts where susceptible individuals are exposed to infectious organisms present in a wellmixed environment, and the contribution of an individual scales linearly with the amount of pathogen shed, individuals shedding pathogens several orders of magnitude higher than the average infectious individual would drive infection dynamics by increasing the total force of infection by a similar order. Conversely, removing these highly infectious individuals would cause a dramatic reduction of pathogen concentrations and a virtual elimination of new infections. Indeed, strategies to eliminate high shedders have been proposed as a method of preventing infections [12]. However, endemic infections persist in the presence as well as in the absence of supershedders [13, 14]. Assume for example that a supershedder produces 1000 times more infective doses than a regular infected host, and that one in 100 infected hosts is a supershedder. If the force of infection would have been linear in the pathogen load, one would obtain an approximate 10fold increase in the force of infection in the presence of supershedders in comparison with the same infection in the absence of supershedders. This 10fold larger force of infection would result in very large outbreaks. However, such an increase in infection prevalence is generally not observed or expected [13, 14], leading to the socalled supershedder paradox: an observed high amount of shedding that results in relatively little impact on infection dynamics. This unexpected limited correlation between infectious burden and force of infection may be hypothesized to be due to transmission models where the force of infection is a concave function of the pathogen load rather than a linear function. Indeed, previous works have proposed multiple types of nonlinear relationships between the force of infection and the pathogen load [15, 16]. Among those, the simplest is probably a power relationship [17], as is also used here.
We here study a detailed data set of an endemic infectious disease to test our hypothesis that transmission models have a significantly sublinear (concave) relationship between the pathogen load and the force of infection.
A parallel term for superspreader has been proposed for individuals that contribute to a larger number of new infections in biological diseases, as well as for virus spreading in networks [19, 20, 21, 22, 23, 24, 25]. The superspreaders are also treated as hubs, which drastically increase the spread rate in contact networks. Such superspreaders could be the result of a high number of contacts (e.g. in sexually transmitted diseases), the result of a high pathogen dose in the superspreader, or the result of a highly virulent pathogen. We here argue that in the studied case, supershedding does not induce superspreading, and use a maximum likelihood framework to demonstrate this argument.
Information on MAP shedding progression is available for experimentally and naturally infected animals [26, 27, 28]. Mathematical models that capture transmission dynamics of MAP have been developed for dairy animals in the US and EU [13, 29, 30, 31, 32] using deterministic and stochastic frameworks. The majority of such models rely on an assumption of direct transmission via the fecaloral route; however, in some models of paratuberculosis transmission, animals contribute infectious material to a common environment and this environment serves as an indirect source of transmission [31, 33, 34]. The dominant transmission strategy for MAP is not well known, and the distinction between the two transmission pathways, direct or indirect, remains poorly elucidated.
The rate of being born into the MAP infected state has been estimated to be as high as 15% of all newborn calves; a more precise estimate from the farms in our data was obtained from detailed molecular typing of isolates known to have infected both the dams and the daughters (see [12] for detailed information). Transmission based on confirmation with molecular typing of the isolated MAP organisms was estimated as approximately 1% incidence of vertical transmission among all pregnant dams, and in their farms with an approximate 25% prevalence of infection in adult animals, a 4% incidence of vertical transmission among known MAP infected dams. Hence, in the data used in our models, vertical transmission was not the dominant route of transmission and was ignored here.
We here develop a parameter estimation method to partition infection transmission into contributions from direct contact, indirect contact and an unmeasured background transmission. The estimate allows us to capture the influence of individual shedding patterns on overall transmission dynamics. We then show using this model that at least in the case studied here, supershedding does not induce superspreading.
Materials and methods
Experimental observations
We analyze MAP shedding in adult animals that calved at least once on four dairy farms (A, B, C, D). Two of these farms were located in Pennsylvania, one in Vermont and one in New York. The animals are outdoors. From all cows older than 2 years of age, fecal samples were collected at least twice a year. Fecal samples were processed using the double incubation method, 2 gms were placed into a 50 mL conical tube containing 35 mL of water. The sample was rocked for 30 min and then allowed to stand at room temperature for 30 min. Then 5 mL was transferred from the top of the tube and placed into a 50 mL conical tube containing 25 mL ½ strength BHI/0.9% HPC, these were incubated overnight at 37 °C. The samples were centrifuged for 30 min at 900 g, decanted and then the pellet was resuspended with 1 mL of antibiotic brew (Amphotericin B, Naladicix Acid, Vancoymcinin ½ strength BHI). Samples were then incubated overnight at 37 °C. Following incubation, four tubes of Herrold’s egg yolk media (BD Diagnostics) were inoculated with 0.2 mL per tube and then incubated with loosened caps in a slanted position at 37 °C. At 2 weeks the lids were tightened and the tubes placed into an upright position. The tubes were read every 2 weeks with the final reading and colony counting at 16 weeks. Total sum of colony forming units (CFU) across four tubes was multiplied by 5.3 to determine CFU of MAP/g of feces. Classically a bacteria count of more than 10^{4} CFU/g has been used as the definition of supershedding [35]. However, since the current analysis is based on explicit quantitative measurements of the shedding level, such a definition is not required for the current analysis. All fecal isolates collected from the farms were stored for future analyses. In one of these farms (A), additional information on available bacterial strain types was used (14 types). The typing method applied here is a restricted subset of a multilocus shortsequencerepeat method as previously published (Loci 1, 2, 8, 9, 10) [36]. Here, we used the two most frequent strains for an individual strain analysis, as will be further discussed. Three farms (A, B, C) have also detailed MAP ELISA data and tissue culture information from a subset of slaughtered animals [37]. One farm (D) has long term recordings of bacterial load (over 20 years) from all adult cows as previously described [14]. In farm D, the sensitivity of detection increased once in the 20year period due to the introduction of an improved diagnostic test. We, thus, separated the samples for this farm into two datasets (D1, D2), with D1 using the initial diagnostic test and D2 following application of the new testing method. All farms received all MAP culture and ELISA data and were able to use this information to improve their MAP control plans. Further detailed descriptive information on these farms appears in Additional file 1 and in previous publications [14, 38]. Obviously in some experiments, we may miss a shedding cow. However, we assume that with repeated testing over time, especially on animals with postmortem testing, we accurately identify infected cows. We also assume that the actually observed CFUs of MAP are a reflection of the true infectiousness of an infected animal. This may not be the true level of CFU of MAP, but it would represent the relative infectiousness of the infected host. Note that if there is a subestimate of the CFU level, this would mainly affect high CFU levels due to right censoring in the ability to quantify bacterial load on a culture slant. Thus our estimate of gamma is an upper bound, and the effect of supershedders may be even more limited than proposed by the current analysis.
Farm descriptions
Over the course of the study, Farm A had 1044 cows available for analysis, whereas the other populations (Farms B and C, D1 and D2) had between 242 and 385 cows (Additional file 1, about 98% of the cows were continuously sampled for shedding). The duration of the period examined in each population was between 6–11 years. For each farm, there were between 1222 and 6404 fecal samples and between 12 and 70 initial shedding events. The numbers of positive ELISA tests were: 43 in farm A, 5 in farm B, and 20 in farm C. The numbers of animals with positive postmortem samples were: 66 in farm A, 13 in farm B, and 30 in farm C. For farm D, no ELISA or postmortem samples were collected, and in this farm all analyses were based on fecal sample results. The number of newly infected cows and the pathogen load varied over time in the different farms (Figure 1). This variation may be used to estimate the parameters of the infection. More information on the farms can be found in [12, 39].
Extrapolation
Infection progression description
In order to estimate the parameters of the infection, we simplified previous statetransition models [13] to three states (Figure 2B; see Additional file 3 for all symbols used): susceptible (X), latent (H), shedding (Y1), and allow a fourth nonshedding infected state (Y2). We combined initial early shedding and latent periods into one compartment (H). This H compartment was treated as uniformly noninfectious for each optimization. Since published experimental data [40, 41, 42] and field data [37] indicate that animals are not consistently detectable by fecal culture following initial shedding (Y1), we allowed animals to exit this shedding phase into a nonshedding (Y2) period. Other models were tested with sequentially broader assumptions of the size of the infectious population (see Additional files 4 and 5 for more details) but no real differences were seen. Definition of “infected” cows is the same.
We assumed that the latent period H has a length distribution f(Δt), where Δt is the time interval from infection at time i to the appearance of detectable shedding in individual animals (Y1) at time l. We used a Gaussian model (Additional files 4 and 6) for estimating the duration of the latent period. We also tested an exponential model, representing a constant probability of moving from H to Y (Additional file 4), but this alternative model did not perform better than the Gaussian model (Additional file 7). Note that these models are formulated here only for the parameter estimation using the maximum likelihood method, and are not simulated nor solved by ODEs. The model is used to define the likelihood function of an observed farm shedding pattern.
Due to testing intervals of approximately 6 months and left censoring resulting from the initiation of sampling at the start of lactation, usually at the age of 2 years, observed shedders can begin shedding before the time of detection and may keep shedding for an unknown period until the next sampling time point. We assume that animals that start shedding are both infectious (according to the appropriate model) and infected. We further assume that animals shedding MAP in consecutive sampling time points are shedding in the interim period. Animals that switch shedding states from one sampling time point to the next are assumed to have initiated or ceased shedding at the midpoint between those two measurements. We assign the population exit date at 90 days (half the sampling interval) past the last sampling time point for all individuals.
Sources of infection

A probability for a susceptible cow to get infected from the constant background infectivity term (δ), which can represent unmeasured sources like inflow of bacteria from other farms, does not depend on the amount of MAP bacteria being shed or on the number of MAPinfected cows in the herd.

The cowtocow direct infection term (β) is based on the assumption that each herd is well mixed and that transmission is direct, and that the contribution of each infected cow is not affected by the total number of cows. This term is proportional to the number of infected cows (a density dependent transmission) which is denoted by y. This value is obviously time dependent.

The indirect infection term (α) is based on the assumption that transmission is proportional to the available MAP bacterial burden (w). The force of infection is then proportional to the amount of bacteria raised to the power γ [denoted by αw(γ)]. This term is also time dependent. Specifically, the probability of being infected is proportional to a power of the single cow’s total MAP amount \(w\left( \gamma \right) = \sum\nolimits_{i \in cows} {w_{i}^{\gamma } }\). We have also tested an alternative model where the infection is proportional to the total amount of free bacteria to the power of γ: \(w\left( \gamma \right) = (\sum\nolimits_{i \in cows} {w_{i} )^{\gamma } }\), as discussed in the Additional file 4. This alternative model did not produce better results. The persistence/survival of the bacteria in the environment is of limited practical importance, since we use intervals of 180 days, and the bacteria survives typically less than 6 months in the environment.
In all following equations, we will assume a dependence of the number of infected cows, infection probability and amount of shed bacteria, also when not explicitly stating the time dependence.
Maximum likelihood fit
Note that the term \(\left( {\begin{array}{*{20}c} {S_{l} } \\ {y_{l} } \\ \end{array} } \right)\) is not affected by the parameters of the model, and is ignored in all following computations.
The first sampling time point for each farm was not considered, since there was no reliable information on the time when the cows that started shedding in this first sample were first infected (Figure 1D).
When analyzing a single farm, the score obtained from the optimization is simply the log likelihood as in Equation 3. When analyzing all the farms together, the total score is the sum of the costs for each individual farm. The cost is minus the maximum likelihood. Thus, a lower cost is a better likelihood. A better fit for a model with a set of parameters (higher likelihood) represents a better solution. A numerical optimization (NelderMead [43] with 1000 random initial conditions) is performed in Matlab to find the parameter set producing the highest log likelihood. We first fit the data of each farm separately with its own set of parameters α, β, γ, δ, μ, σ. For each farm we obtained a different cost.
We then tested for the optimal score, when we set all parameters (α, β, γ, δ, μ, σ) to be equal among all farms. Such an assumption provides an average contribution of each infection term in each farm: the cowtocow direct term, the indirect term via bacteria shed and the constant term.
In order to compare between the different models, we used the likelihood ratio test [44]. We calculated the test statistic which is twice the log of the likelihoods ratio (or twice the difference in loglikelihoods). We then calculated the Chi squared value of that difference with the number of degrees of freedom between the two models, and checked significance.
Contribution of each transmission route to the total infectivity
Once the best parameters were obtained, we analyzed the contribution of each term of transmission to the total force of infection for each farm. This was done by calculating the average value per day of the indirect transmission term [αw(γ)] and of the direct cowtocow transmission term (β). The constant infection pressure contribution is given simply by the value of δ.
Strainlevel analysis
Simulation
On the first day, we chose 20 cows to start shedding. The initial number of shedding cows had no effect on the steady state results. Every infected cow starts shedding µ intervals after the initial “infection”. It continues shedding until it reaches the cutoff value and then dies. In addition, every cow dies at the “age” of 7 intervals (3 ½ years), which is the mean lifetime of cows in the real farms. The shedding level distribution is taken from the observed shedding level distribution (Additional file 2). When a cow dies, another cow is born to maintain a constant herd size. The simulation was performed for 350 intervals (each representing a period of 180 days). We calculated the average number of MAP infected cows and the average number of culled cows in equilibrium (after the first 50 intervals). The number of infected cows was averaged among 200 realizations of the simulation. We computed the number of infected and culled cows as a function of the culling level (the shedding level above which cows are culled).
Results
Comparison between sources of infection in each farm separately
Parameters obtained for the best fit.
Cost  α  µ  σ  δ  γ  β  

Farm A  170.1717  3.49E−05  1  0.100166  0  0.551659  0 
Farm B  28.042  0  2.8563  0.16409  0.000227  0.19292  0.00029 
Farm C  113.82  6.59E−06  1.0064  0.13209  0  0  0.000138 
Farm D1  178.2996  0.002539  1.04284  0.21597  0  6.66E−08  0 
Farm D2  109.6386  0.000117  4.710256  0.105494  0.097014  0.00797  0 
All farms together  653.77  0.000607  1.173  0.15691  0.000851  0.11596  0 
All farms together analysis
In order to obtain a more robust estimate of the possible transmission models, we computed the optimal parameters, assuming all farms have equal parameters. The bestfitting ML model included indirect transmission and a constant infectivity (Figure 4B), but not the direct transmission term. The resulting model was significantly better than the null model with constant infectivity (p = 3.5 × 10^{−7}). This can be also seen from the important contribution of the indirect transmission term to the infectivity in all farms (Figure 4B). We performed an additional optimization of the same model using a minimal least square regression with similar results (Additional files 10 and 11). We have also tested multiple alternative models, but none of them was as significant as the model with both indirect transmission and a constant infectivity term. While the direct transmission term could be removed with no effect on the resulting likelihood, removing the constant term significantly decreased the model likelihood.
The model in which transition to high shedding has a constant probability over time (fit with exponential rather than Gaussian, see Additional file 4) produced a lower likelihood and a large contribution of the constant infection pressure (δ) in all the models (Additional file 7). The model with the power over the total shed bacteria (and not over the bacteria shed by each cow, see Additional file 4) also produced a lower score and a large contribution of the constant term.
Power of indirect transmission term
Individual strain analysis
Best parameters obtained for multistrain farm for the different models (using two types of strains).
Strain 1  Strain 2  

Cost  113.289  89.383 
α  4.27E−05  0.0003 
µ  1.19  1.000007 
σ  0.106  0.176 
δ  0  0 
γ  0.473  6.77E−08 
β  0.001  0 
Contribution α  0.0007  0.001 
Contribution β  0.003  0 
Contribution δ  0  0 
Effect of supershedding on superspreading and culling strategies
We have shown above that the power of shed bacteria in the infection model was low. For example, increasing the amount of bacteria shed by a factor of 10, increases the force of infection by a factor of less than 1.2–2.2. Thus, supershedding does not lead to superspreading. In order to test the implication of this power on culling strategies, we simulated a herd with the maximum likelihood parameters, and tested the effect of culling cows shedding above a given level. Specifically, we tested a culling strategy of removing all cows shedding above a given level, and tested the effect of this strategy on the steady state level of infected cows. We assume all cows have a limited lifespan of 3–4 years, and only removed cows from the herd when they were observed to shed above a certain level. The shedding level distributions as well as all other parameters were based on the observed farm parameters.
Discussion
We have here developed a theoretical basis for the limited effect of the supershedder paradox in MAP. The potential drastic effect of supershedders in MAP and the resulting paradox has been studied in previous models [12]. This paradox stems from the presence of highly infectious individuals that are transiently present but do not overwhelm the system with new infections. We specifically address the issue of highly infectious individuals through the mechanism of increased probability of successful transmission. Although this question has been approached in E.coli O157:H7 infection in feedlot environments, the majority of studies on infectious individuals have focused on those individuals with unusually high rates of contacts [19, 45, 46, 47]. One possible conclusion from the presence of supershedders was that they are indeed superspreaders and that the disease dynamics are dictated by their presence. Such a conclusion has led to models of network vaccination, where the removal of hubs in the network would prevent the spread of epidemics [48, 49]. Another possible solution for the supershedder paradox is that indirect transmission is not the main source of new infections and that direct contact between infectious and susceptible hosts is required for transmission. In such direct transmission models, if the direct transmission probability is not proportional to the level of shed bacteria, supershedders would have no increased impact on population transmission rates. A recent study in E.coli O157:H7 [50] found that high shedding individuals only modestly increased the risk of transmission. They also found no evidence that environmental contamination by faeces of shedding cattle contributed to transmission over timescales longer than 3 days [50]. We here propose a similar approach where the rate of transmission is not linear with the absolute number of pathogens present in the environment, and thus supershedders are not superspreaders. We showed that this is indeed the case through a detailed analysis of longterm observational data on a natural spread of MAP in multiple herds, since MAP is an infectious disease with slow progression [39]. For this pathogen, it was shown here that indirect transmission determines the force of infection, but a drastic increase in the number of bacteria present due to supershedders resulted in only a minor increase in the force of infection. Neither direct transmission nor a force of infection which is linear with the individual animal bacterial load was a better predictor of infection dynamics than models in which indirect transmission that used total bacterial burden as an indirect transmission term. Note that models which contain a direct transmission component may also explain the observed dynamics, although they are less plausible. However, the conclusion of such models would be similar to the conclusions above. We suggest here that this principle may be true for other infectious diseases where the range of free pathogen numbers produced by each infected host varies over multiple orders of magnitude [1, 6, 19, 20]. However, precise longitudinal data, similar to the data used in our analyses, will be necessary to distinguish between transmission models in other diseases.
The nonlinearity of the force of infection with total bacterial load indicates that, while the number of shed bacteria may be important, its contribution to the force of infection is much less than linear. A cow which has a 1000fold increase in MAP bacteria has at most a 10fold increase in contribution to force of infection, and probably only a 2fold increase (based on the most probable model). When properly generalized to include the effect of vaccination on the infection probability via each pathway, these results open the way for detailed optimization models for control strategies. For example, when environmental saturation occurs at a relatively low concentration of bacteria, removing only high shedding animals from the population may not be successful in eliminating infection. Under these conditions, the additional importance of supershedders relative to “average” shedders is relatively minor. As mentioned before, this results appears to fit observational data in many populations where selective elimination of supershedders does not result in elimination of infection [1, 2, 10, 14, 27]. Our findings could have major implication for control programs that previously often focused solely on identifying and eliminating high and supershedding hosts. Note that since the force of infection is not highly affected by the amount of shed bacteria, it is hard to distinguish between direct transmission and indirect transmission which is highly sublinear. The exact transmission routes are not well defined, but if there is indirect transmission, it is highly sublinear.
Beyond the common effect of direct transmission, different farms had very different average forces of infection in the optimal results (Figure 4) as a result of different infection prevalence. This different force of infection can be due to the hygiene conditions in the farm or to the farm size, but also due to varying measures of infected and infectious cows, as can be observed for example in farm D over different time periods (D1 and D2).The reduction in the force of infection (the probability that a cow becomes infected in a given day from any source) can in such a case represent a decrease in the definition of newly infected cows.
MAP is an infectious disease with a slow progression. The fraction of sick cows in our data, as well as in our simulation, is 1–5% of the total number of cows. This is in good agreement with the literature [39]. Intervention programs that were used were insufficient to address longterm persistence of MAP [12].
Notes
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Conceived and designed the experiments: YL and YHS. Analyzed the data: NS. Contributed reagents/materials/analysis tools: RHW, TF, AKP, RMM and EK. Wrote the paper: YL, RMM, YHS and NS. All authors read and approved the final manuscript.
Acknowledgements
The authors acknowledge the support of the Withinhost 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 DBI1300426, with additional support from The University of Tennessee, Knoxville. We acknowledge the long term funding of the Regional Dairy Quality Management Alliance through a collaborative contract with the USDA agricultural research services. The support by the Johne’s Disease Integrated Program is acknowledged. Financial Support: USDA JDIP, USDA NIFA Grant #201005149 (RMM), NIMBioS. We thank Miriam Beller for the text editing of the current manuscript.
Supplementary material
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