General description of the epidemic model of the natural course of infection

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Characteristics of the bacterial shedding

C. burnetii infection in ruminants often becomes chronic, with persistent bacterium shedding: cows can shed C. burnetii for several months [59] and goats at successive parturitions [23]. This shedding is of major importance as it contaminates the environment and can lead to the infection of both susceptible animals and humans. For cows, ewes and goats, Rodolakis et al. [131] reported that, contrary to expectation, the shedding of C. burnetii could be not related to parturition. In C. burnetii infections, a great heterogeneity between shedders has been described [11, 37, 57, 131]: the shedding duration, level (i.e. the quantities of bacteria shed) and routes are variable between animals. Infected animals can indeed shed bacteria through birth products, vaginal mucus, faeces, and milk [132]. Amongst the three latter, no predominant route was identified in 242 dairy cows from 31 herds in which abortions due to C. burnetii were reported [57]. Besides, in the same study, 65% of the shedder cows shed by only one route. However, in asymptomatic herds, cows shed more frequently in milk than in vaginal mucus or faeces [59, 131]. The shedding duration and shedding levels are also variable between animals: cows can shed from a sporadic to a three-month persistent way and the concentrations of bacteria shed in vaginal mucus or milk can vary from less than 100 bacteria/g to more than 1,000,000 b/g [59]. Although the presence of heavy shedder cows (i.e. animals shedding bacteria in higher titres and in a persistent way) was reported in milk [59], the role of these animals in C. burnetii transmission between animals and from animals to humans has not been determined yet. In goats and ewes, the same shedding routes are described and in the same way, are rarely concomitant [135]. Ewes were found to shed mostly in faeces and vaginal mucus [131] while goats were reported to shed mainly in milk and vaginal mucus [11, 131, 135]. The presence of heterogeneity in a population (e.g. variability in age, contact structure, infectiousness, etc…) is known to affect infection dynamics in many diseases. As an example, a model assuming that all farms and all animals are governed by the same underlying dynamics was unable to explain the highly overdispersed distribution of prevalences of Escherichia coli O157 shedding on Scottish farms [106]. The best fit to the prevalence data was obtained when incorporating individual variability in transmission. In many cases, the heterogeneity of shedding has indeed a great impact on both the infection dynamics and the effectiveness of control measures: in dairy cattle infected by Salmonella, the presence of host heterogeneity in infectious period and contagiousness decreased the effectiveness of population-wide control strategies, making necessary the application of strategies targeting the most contagious animals [86]. For C. burnetii infections, the influence of this heterogeneity of shedding on the infection spread has not been evaluated yet. Hence, it is necessary to take into account the variability of the shedding duration, shedding levels and shedding routes in our epidemic model when representing C. burnetii spread and testing effectiveness of control strategies.

Diagnosis

Currently, the PCR is a sensitive and rapid mean to directly detect C. burnetii and therefore to identify the shedders [39]. This technique can be used on a wide range of samples (vaginal mucus, abortion material, faeces and milk). Real-time PCR is preferable to conventional PCR as it allows high sample throughput [122] and the quantification of the bacterium in the sample. As an example, a real-time PCR assay applied to bulk tank milk samples appears to be a valuable tool to assess on a larger scale the status of herds towards C. burnetii shedding [58]. Quantitative PCR kits are now commercially available. Moreover, although immunohistochemistry was until now useful when considering potential causes of abortions in ruminants, multiplex PCR that can detect and differentiate between different abortive pathogens is now under development [24].
For the serological testing, the complement fixation (CFT) was considered the reference test for historical reasons [39]. However, the indirect immunofluorescence assay (IFA) and above all, ELISA, are now widely used. According to Kittelberger et al. [79], two commercial ELISA were more sensitive than the CFT in all panels from infected ruminants: their sensitivities were 81% for the Pourquier ELISA and 95% for the IDEXX ELISA. However, none of the tests are able to distinguish between acute and chronic infection or between vaccinated and naturally infected animals. Besides, a serological test does not give clear information about the individual infection status [39]: some infected animals are indeed seronegative while they are shedding.

Magnitude and distribution in the European Union

There are currently no EU rules about the notification and surveillance of C. burnetii infection and/or Q fever in domestic ruminants. Based on available data [39], C. burnetii is present in most, if not all, member states. It does not appear to be an increase in Q fever prevalence/incidence but comparison over time and between countries is problematic as there are considerable differences in testing protocol and data availability. In Gran Canaria island (Spain), 34.7% of 1,249 randomly selected ruminants (60.4% of goats, 31.7% of sheep and 12.2% of cattle) were reported seropositive using an indirect ELISA kit [134]. In northern Spain, a serosurvey was carried out in 1,379 sheep (42 flocks), 626 beef cattle (46 herds) and 115 goats (11 herds) [138]: ELISA anti-C. burnetii antibody prevalence was slightly higher in sheep (11.8 ± 2.0%) than in goats (8.7 ± 5.9%) and beef cattle (6.7 ± 2.0%); herd prevalence was 74% for ovine flocks, 45% for goat flocks and 43% for cattle. In Denmark, a study based on bulk tank milk samples from 100 randomly selected dairy herds demonstrated a prevalence of 59% antibody positive herds [3].

Roles of epidemiological modelling

A model is a simplified representation of a complex phenomenon. By definition, all the models are “wrong” because they make simplifying assumptions [77]. However, modelling is an essential tool, particularly useful every time a major public health issue is raised. As an example, the infection by the Human Immunodeficiency Virus started being perceived in a different way when biomathematicians showed that observed data were compatible with the assumption that 100% of infected people develop the disease. Before, as only a small part of seropositive people showed clinical signs, the asymptomatic seropositive people were considered as healthy carriers [158]. In epidemiology, models have different roles. Prediction is the most obvious one and often aims at guiding policy decisions [77]. For quantitative prediction purposes, the model has to be accurate and validated (i.e. with the smallest uncertainty which could impact the conclusions). This objective is most of the time difficult to reach. When a new infection is introduced in a former susceptible area, no historical data are available. In addition, an epidemic reference situation in the absence of control measure or with a perfectly known control programme is rarely recorded, especially for animal infectious diseases [45]. Thus, most of the time, although models used to evaluate control strategies are sophisticated and parameter-rich, model conclusions are not quantitative. However, qualitative outputs are enough for a large range of purposes and especially for the comparison of different scenarios (e.g. spread of an infection in different regions, for different pathogen strains, for different control measures, etc…). A model also helps understanding how an infectious disease spreads in the real world [77]. It provides the modeller a virtual world in which everything can be recorded and every factor can be examined. For example, it is possible to explore the effects of variable numbers of partners on the spread of sexually transmitted diseases or the effects of neighbourhood contacts or animal purchases on the spread of livestock diseases. Besides, a model can allow estimating non observable parameters. For instance, some events can be very rare in the real world but have major consequences in public health. Their frequency has then to be assessed but this is hardly feasible through field observations. As an example, the residual risk of HIV infection through blood transfusion is now very low. It is unfeasible to assess by a comparative experiment the potential benefit of an additional prevention strategy. Instead, a model can be used to estimate this infection risk and simulate control scenarios [158]. Although estimation of parameters can be considered as a role in itself, it also helps understanding the infection process and is a prerequisite for prediction purposes. At last, modelling allows highlighting gaps of knowledge: to develop models, modellers need quantitative data whereas most of the time, only qualitative data is available in the literature or known by experts. Models are then a mean to critically evaluate the range of information available on the modelled system.
It has to be highlighted that modelling and field or experimental work are complementary approaches (Figure 1.2). On one hand, as we will see later, data is required to conceptualize the model, estimate the value of parameters and validate the model. On the other hand, models help testing biological assumptions, optimizing experiments protocols or identifying gaps of knowledge.

An example of epidemiological model in animal health

In livestock, epidemic models have been developed for various infectious diseases such as tuberculosis [71], brucellosis [40], BSE [6], Bovine Viral Diarrhoea [162], Escherichia coli infections [90, 157, 172], Salmonella infections [85], paratuberculosis [101], Contagious Bovine Pleuropneumonia [102], bluetongue [148] or foot-and-mouth disease [76].
A well-known example of mathematical model in domestic ruminants is the approach developed by Keeling et al. [74] representing the 2001 foot-and-mouth disease (FMD) outbreak. The aim of this study was to understand how spatial and individual heterogeneities influenced the course of the epidemic and to compare different vaccination and culling scenarios. The FMD outbreak was characterised by both a high probability of local spread and less frequent longer-range transmission due to fomites and personnel movements. There was additional heterogeneity in farm size and species composition. In the model, the susceptibility and infectiousness of a farm were assumed varying according to its size and species composition. The model was fit to data, and a good agreement was obtained. It is worthy that the inclusion of the host species and herd-size heterogeneities in transmission were required to reach this good agreement and to properly simulate the spatial aggregation of cases. Large farms, and especially cattle ones, were indeed found to have a key role in the infection spread. Modeling demonstrated that culling infected farms, their direct contacts and contiguous farms was much more effective in reducing both the number of cases and the total number of culled farms than culling infected farms only. The delay from infection report to culling was also an important factor influencing the effectiveness of the control measures. Moreover, Keeling et al. considered the potential impact of both reactive and prophylactic vaccination on future FMD epidemics in the United Kingdom [75]. Mass prophylactic vaccination campaign could reduce the size and duration of the epidemic and vaccinating above 80,000 farms (over the 100,000 cattle farms in the UK) would even prevent almost all major epidemics. In addition, at the start of an outbreak, mass reactive vaccination, in combination with culling and animal movements restrictions might also control ongoing epidemics. On the contrary, ring vaccination would have a limited effectiveness.
This example shows (i) how important is to take into account the presence of host heterogeneity in a population, and (ii) how models can help understanding the course of the infection and guide decisions makers. From a practical point of view, this model played an important role in the formulation of the DEFRA’s (Department of Environment, Food and Rural Affairs of the United Kingdom) contingency plan published in 2004 [77]. However, before model conclusions can inform policy, economical, sociological and logistical constraints have to be taken into account [75]. For example, farmers will have to be sure that vaccination will not devalue or limit the sale of their stock, or that they will get compensation. Besides, as vaccination will undoubtedly suppress clinical disease but not always infection, careful surveillance will therefore be required. Thus, model results have to be set back in the real world before being used for decision purposes. At last, this model was based on extensive data of a single epidemic. For an outbreak with a different strain (i.e. with different transmission properties and host specificities) or for an outbreak in another location (i.e. with different farming practices or weather conditions), the model will have to be adapted to the new situation before conclusions could be drawn from its outputs.

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Choice of the mathematical formalism

Models can be deterministic, describing average dynamics, or stochastic, considering that chance can have a great impact on the infection dynamics [163]. As an example, in a SIR deterministic model, the rate at which individuals recover is fixed: for a given state of variable I, there is always the same number of individuals going from I to R during a time step. For a SIR stochastic model, I individuals have a given probability of transition from I to R and, due to random draws, the number of individuals going from I to R during a time step is variable from one model repetition to the other. Deterministic formulations are suited for large populations where randomness has relatively little overall impact, whereas stochastic models are more appropriate for small populations and rare events where the fluctuations have larger effects [45].
The scale and unit of modelling have also to be defined. The spread of a livestock infectious disease can indeed be modelled within a herd (the unit of modelling is then the animal) or between herds (the unit of modelling is then the farm). The scale is closely related to the research question. In most circumstances, disease transmission is a localized process. If, for instance, the study is aimed at exploring the infection spread in a school after an infected child is introduced, the unit and scale should be the individual and the population respectively. However, movements of individuals between human or animal populations facilitate the geographical spread of infectious diseases [77]. If the study is focused, for example, on determining the influence of neighbouring relationships and animal movements on the infection dynamics, a between herd scale should be considered in the model. For such models, the within farm infection dynamics is explicitly represented or not. In the previously described FMD model (see introduction IV.3.), the unit of modelling is the farm: due to the rapid transmission of the virus between animals situated at the same location, the within herd infection dynamics can be assumed negligible. Therefore, the whole farm is considered infected as soon as an animal is infected. On the contrary, for similar time scales but moderately spreading pathogens like the Bovine Viral Diarrhoea Virus, the within herd dynamics should be represented in details since it is unrealistic to assume that the entire farm is infected as soon as a single animal is infected [34]. For models describing the spread of a pathogen at a larger scale (region, country, or world), the spatial positions of hosts are often taken into account. The previously cited FMD model aimed at simulating the spread of the FMD over the whole United Kingdom, it was a spatial model [74]: the locations of all farms were explicitly represented and the rate of transmission between two farms was expressed as a function of the distance.
Models can be compartmental or individual-based. In individual-based models, the health state of each individual is monitored over time, whereas compartmental models track the infection process for the individuals of a same health state collectively [163]. Individual-based models are often more computationally intensive but can also provide finer information. When host characteristics (such as age, sex, gestation status, intensity of contacts with the other individuals, etc.) are supposed to have an impact on the infection dynamics and are variable between individuals, it is more appropriate to develop individual-based approaches than compartmental models with many distinct compartments.
Concerning their time dependence, models can be in continuous time (the system could then be described by differential equations) or in discrete time (difference equations could then capture the dynamics). Differential equations provide a means for avoiding the issues regarding the size of the time step by describing events occurring continuously, rather than at discrete time intervals [163]. Indeed, in discrete time models, the choice of the time step is crucial: if this latter is too large (i.e. two successive transitions between health states can occur during a single time step), the model provides inaccurate and even non sense results. The appropriate size of the time step depends on the modelled phenomena: it should generally be less than the shortest average duration that individuals spend in a given health state [163].
At last, to describe the infection dynamics on the long run, key aspects of demography of the population considered (births, deaths, and migrations) may need to be incorporated in the model. In the case of animal populations being managed by humans (e.g. pigs or cattle herds), the representation of demography can be quite complex and sometimes require the development of an elaborated population dynamics model [44, 94].

Confrontation of the model to data

A second step after the model elaboration consists in determining appropriate and plausible values for model parameters. This can be done qualitatively based on information from the literature or expert’s opinions. If data are available, model parameters can be quantitatively assessed by fitting model predictions to data. However, sometimes this assessment is not feasible from existing knowledge or data and new data sets need to be collected and analysed using statistical methods. Although generic models can provide an intuitive explanation of the transmission of infectious diseases, it is only through detailed parameterization and rigorous assignment of numerical values to parameters that useful public health guidance can be generated [77].
When performing parameter inference from data different statistical techniques could be used. A widely known approach is that of “least squares”. The sum of squares of the difference between the model predictions and the observed data is calculated in order to determine the parameter values which lead to the smallest value for this sum [163]. Another well-defined and widely-applied approach when fitting a model to data is that of “maximum likelihood”: for a given set of parameters, the dynamics predicted by the model is determined. Then, the likelihood (i.e. the probability) that the observed data come from such dynamics is calculated. The best-fitting parameters are those which maximize this likelihood: the model is in closest agreement with the available data [77].

Table of contents :

Chapter 1: General introduction
I- Pathogen characteristics and host response to infection
II- Q fever in Humans
1. Routes of transmission to humans
2. Clinical manifestations
3. Magnitude and distribution of human Q fever in the European Union
III- Q fever in domestic ruminants
1. Modes of contamination
2. Clinical manifestations
3. Characteristics of the bacterial shedding
4. Diagnosis
5. Magnitude and distribution in the European Union
6. Control
IV- Why a modelling approach to understand C. burnetii infection and assess control strategies?
1. Why not field observations?
2. Roles of epidemiological modelling
3. An example of epidemiological model in animal health
V- Objective and outline of the thesis
Chapter 2: Elaboration of a model representing the spread of C. burnetii within a dairy herd and estimation of its main parameters from data
I- Some generalities on how to build a model and to confront it to data
1. Choice of the model structure
2. Choice of the mathematical formalism
3. Confrontation of the model to data
II- The modelling of C. burnetii spread within a dairy cattle herd.
1. Description of the data set used for parameter estimation (data set A).
2. Description of the data used to define some of the prior distributions (data set B)
3. Modelling assumptions
III- Manuscript: Spread of Q fever within dairy cattle herds: key parameters inferred using a Bayesian approach
1. Abstract
2. Introduction
3. Data
4. Model and methods
a. Epidemic model
b. Bayesian framework
c. Bayesian inference: calculation of the posterior distribution of the model parameters from likelihood and prior distribution
d. Model adequacy
5. Results
a. Parameters of transition between health states
b. Environment-related parameters
c. Checking of model adequacy for the data
6. Discussion
7. Acknowledgements
8. Supplementary material
a. Data
b. Likelihood
c. Convergence of the MCMC algorithm
d. Posterior and prior distributions of transition and shedding parameters
e. Posteriors and priors of the environment
f. Summary statistics for the initial real health states
Chapter 3: Representation of the heterogeneity of shedding in the model of within herd spread of C. burnetii and identification of the most influential parameters of the infection dynamics
I- Why and how to represent heterogeneity in host population?
1. Two classic examples of heterogeneity in human diseases: sexually transmitted infections and childhood diseases
2. Superspreading events occur in many infectious diseases
II- The heterogeneity of shedding in C. burnetii infections
1. Shedding routes
2. Shedding levels
III- Why and how to perform a sensitivity analysis?
1. Aims of sensitivity analyses
2. How to perform sensitivity analysis?
3. Types of methods
IV- Manuscript: Modelling the effect of heterogeneity of shedding on the within herd Coxiella burnetii spread and identification of key parameters by sensitivity analysis
1. Abstract
2. Introduction
3. Model
a. General description
b. Initial conditions and parameter values of the standard scenario
4. Sensitivity analysis
a. Outputs and factors
b. Design of experiments
c. Analysis of the temporal outputs (of the first, second and third experiments)
d. Analysis of the extinction rate (of the second experiment)
e. Analysis of the outputs at a the time point 260 (of the first, second and third experiments)
5. Results
a. Infection dynamics of the standard scenario
b. Influence of the epidemiological factors on the model outputs
6. Discussion
7. Conclusion
8. Acknowledgements
Chapter 4: Assessment of the comparative effectiveness of three vaccination strategies in C. burnetii infection
I- Why and how to include vaccination in models?
1. Different types of vaccination programmes and their representation in epidemic modelling
2. An example of model aimed at assessing the effectiveness of vaccination
II- Manuscript: Modelling effectiveness of herd level vaccination against Q fever in dairy cattle
1. Abstract
2. Introduction
3. Materials and methods
a. General description of the epidemic model of the natural course of infection
b. Representation of the vaccination
c. Vaccination scenarios
d. Parameters and initial conditions
e. Outputs of the model
4. Results
a. Description of the herds at the start the vaccination strategy
b. Influence of the vaccination scenarios on the temporal model outputs
c. Influence of the pv values on the model dynamics
d. Influence of the vaccination scenarios and of the pv values on the extinction rate
5. Discussion
6. Acknowledgements
7. Supplementary material
Chapter 5: General discussion
I- Major findings
II- Comments on the modelling approach, inference and model analysis.
1. Choice of the mathematical formalism
2. Choice of the model structure
3. Estimation of main epidemiological parameters
4. Sensitivity analysis
5. Simulation of control strategies
III- Available and required data for model conceptualization, inference and validation
IV- Implications and perspectives
References

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