A nivel individual

The force of infection in a compartmental model is the incidence rate at which susceptible individuals in the population become infected. It depends on the amount of infectious units to which the susceptible individuals are exposed. In the case of ovine paratuberculosis in a pastoral environment, the infectious compartment is represented by the amount of MAP organisms to which the animals are orally exposed.

We fitted the probability of active infection over time using time-dependant data of

histological lesions at post-mortem to find parameters to use in the calculation of the force of

infection. We derived three parameters for the force of infection: rate parameters from a

Weibull distribution (α, k) and a dose-dependent plateau of active infectionƒš൫ሺ ൏ –ሻ൯.

The first appearance of histological lesions or the onset of faecal shedding can both be used as a proxy for entering the active infection stage. A preliminary review (Chapter 4) suggested that the presence of histological lesions in the intestinal wall and MAP faecal excretion are concomitant, especially at an early phase of transient shedding. Descriptive results in this meta-analysis confirm the concomitance of shedding and histological lesions in the guts (Figure 6-6).

We analysed both the onset of shedding (using survival analysis) and the presence of

histological lesions at post-mortem (using GAMM). We concluded that data of histological

lesions at post-mortem were more robust and only the analysis of these data are reported in

the results of this manuscript and were used to estimate parameters for the mathematical model. The quality of the histological lesions dataset was considered better than longitudinal shedding data due to the following reasons:

- More power to detect and adjust for confounders: the meta-analysis had 627 sheep with

histological lesions versus 159 with longitudinal shedding data. The former presented a greater range of covariate pattern allowing for more meaningful model building.

- The presence of histological lesions is considered a gold standard whereas detection of

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6.4.1.5 show that when using shedding data, the outcome depended heavily on the diagnostic test used.

- 50% of sheep with longitudinal shedding data only had 3 successive observations, and

intervals between observations were sometimes very long, hence the difficulty to determine the onset of shedding using survival analysis.

Although the analysis using longitudinal faecal shedding were dismissed in this manuscript, they confirmed that the risk factors for onset of shedding were similar to those for onset of histological lesions, reinforcing the inferences obtained in 6.4.1.2. In particular, sheep infected with the bovine strain shed significantly earlier than sheep infected with the ovine strain, in line with results about the occurrence of histological lesions. Moreover, the time from challenge to shedding was significantly shorter in lambs than in animals older than three months old. This finding of transient shedding in lambs agrees with Mitchell et al. (2015) that transient shedding occurs in calves, and that this may affect the dynamics of PTB infection. We used GAMM (logistic regression) to model the presence of histological data over time since inoculation, and secondarily derived the incidence of histological lesions based on the shape predicted by the model. The advantage of using logistic regression was that we did not need to assume that the time of observation was the time of onset, since we modelled the prevalence. The classical approach to estimate rates is survival analysis (modelling incidence), but this would require assuming that the time of observation was the time of onset of histological lesions. This, however, would be totally biased for observations occurring several months or years after inoculation. Our data consisted of point observations, which are not well suited for survival analysis. GAMM uses optimal smoothing procedures with penalised likelihood to evaluate the shape of the probability over time (Wood, 2006). Since the smooth term is non- parametric, it does not require an assumption about the shape, which is therefore purely data driven while accounting for the effect of other covariates in the parametric part of the model. Once this shape was known, we made an approximation based on the look of the data (early steep increase followed by a plateau of infection) to determine the part of the curve corresponding to cumulative incidence and used this part only to calculate parameters for the rate of infection and the plateau of infection. An advantage of using GAMM was that despite using one part of the obtained curve for parameters estimation, all other (later time points) observations were part of the model, and contributed to the estimation of the effect of other covariates. The part of the curve to use was data-driven rather than relying on an assumption.

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The sigmoid shape of the predictive probability of histology lesions versus time (part before the plateau, Figure 6-8) was evocative of a Weibull distribution and not-compatible with an exponential distribution of time to active infection. There may be various other distributions that would provide a good fit, but the Weibull fit was excellent. With a parameter (k) greater than one, the Weibull fit indicated that the rate of onset of lesions increased over time. The pattern was independent of the age of infection. Because ODE cannot handle a rate varying with time since exposure, one would have to use partial differential equations or individual based stochastic modelling to model this process. To approximate the Weibull distribution, we therefore fitted a constant exponential rate instead. Detailed results from this alternative approach are described in Appendix B.

The meta-analysis revealed that, independently of the rate of infection, there existed a dose- dependent overall probability of “success” of infection with MAP, represented by the dose-

dependent plateauƒš൫ሺ ൏ –ሻ൯. Thus, the cumulative probability of active infection

remains below one for low to medium doses and higher inoculum doses are required to infect all sheep. Our results (Figure 6-8 c) suggest a progressive effect of the dose. This is biologically plausible; it should be incorporated in compartmental, SIR-type model. The results indicate that there is considerable animal variation in the probability of becoming infected at a specific dose.