# Modelling the Transmission of Coxiella burnetii within a UK Dairy Herd: Investigating the Interconnected Relationship between the Parturition Cycle and Environment Contamination

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Material and Methods

#### 2.1. The C. burnetii within-Herd Transmission Model

#### 2.1.1. The Infection Cycle

#### 2.1.2. Farm Demographics

#### 2.1.3. Contamination of the Environment

#### 2.2. The Mathematical Formulation of the Model

#### 2.2.1. Model Parameterization and Parameter Reduction

#### 2.3. Initial Conditions

#### 2.4. Sensitivity Analysis

## 3. Results

#### 3.1. The Standard Case

#### 3.1.1. The Effect of Farm Demographics on the Infection Cycle

#### 3.1.2. The Heterogeneity of the Environmental Contamination

#### 3.2. Sensitivity Analysis

- ${r}_{i}$, that is, the reduction rate of transmission probability by the indoors environment, which essentially provides a measure of the uniformity of the transmission rate in the different environmental compartments: the higher the value of ${r}_{i}$ the more uniform the distribution of the transmission rates. There are currently no field or experimental data available in the literature for this parameter, therefore its uncertainty in the current study is high. Previous studies have not considered compartmentalisation of the environment, thus it is a new result highlighted by our model. Consequently, future experimental or field studies should focus on estimating this parameter. It is noted that the $SI$ of ${r}_{i}$ is positive/negative for the seropositive/seronegative prevalence, suggesting that the more uniform the transmission rates become among the different environmental compartments, the more the seropositive sub-population.
- $\tau $ and $\sigma $, expressing the rates from ${A}_{mp}$ to ${I}_{mp}$ and vice versa, i.e., asymptomatic to infected. In the current study, data available from the literature were used to calibrate these parameters [7,12,23]; yet all the previous studies were performed in French cattle herds and did not distinguish between exposed and asymptomatic cattle. Therefore, there is some need for further research to enable a potentially more accurate description of these parameters.
- $\gamma $, that is the ratio of reduced transmission from nulliparous cattle. There is no data available in the literature for this parameter, hence, it is characterised by high uncertainty. As a result, further research is required to this regard. The importance of this parameter is also underlined by the fact that previous studies conclude that vaccination should be focused on nulliparous cattle [7,34,35];

#### 3.3. The Effect of the Outdoors Environment on the Infection Cycle

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameterization of the Model

#### Appendix A.1. Transmission Related Parameters

^{−1}was directly adopted from [17], who derived the parameter value from in days from [12]. The total outdoor transmission rate constant was calibrated to be proportional to the indoors one, so that ${\beta}_{c}^{o}=0.01{\beta}_{c}^{i}$; the factor $0.01$ represents the ratio of the average available space that a cattle has when housed indoors (mean of 8.3 m

^{2}per cattle [26]) over the available space when grazing outdoors (average pasture area of 1004 m

^{2}per cattle [27]). The fractional reduction terms ${r}_{i}$ and ${r}_{o}$ were tuned to $0.8$, since no data are available in the literature for this parameter. Tuning the values ${r}_{i}$ and ${r}_{o}$ depends on the number of stages n of landscape contamination. As also discussed in Section 2.2.1, we started with $n=10$ and reduced n one-by-one until reaching $n=5$, tuning ${r}_{i}$ and ${r}_{o}$ so as the profile of the overall transmission rate $f({L}_{j}^{i},{L}_{j}^{o};t)$ in Equation (1) does not vary much quantitively.

#### Appendix A.2. Infection Cycle Related Parameters

^{−1}, since approximately 3 weeks are needed for a cattle to become antibody positive (slightly longer than the period reported for goats [21]). The transition from the asymptomatic to the infectious state and vice versa (due to intermittent shedding [22]) is modelled only for multiparous cattle, with the transition rates $\tau {A}_{mp}$ and $\sigma {I}_{mp}$, respectively. For determining $\sigma $, we directly adopted the value $0.2$ per week from [7,12], which therein expresses the transition rate constant from shedding cows with antibodies to non-shedding cows with antibodies. However, since in [7,12] there are more than one shedding states, the reverse transition rate constant from non-shedding cows with antibodies to shedding ones (mean of $0.1286$ per day) is not an appropriate selection for $\tau $, herein. Instead, we calibrated $\tau $ to match the data from the field study [23], where the ratio of shedders over non-shedders is 60/19 (see Table VIII therein for the seropositive cows on a weekly basis); thus deriving the value of $\tau =60/19\sigma =0.09023$ per day. This value is smaller, but very close to the one considered in [7,12]. Finally, a removal pathway is considered for the multiparous infected cattle ${I}_{mp}$, through the removal rate $c{I}_{mp}$. We directly adopted the culling rate constant $c=9.57\xb7{10}^{-4}$ from [7] (see the average of culling rates in Table 2 therein), effectively corresponding to a 35% per annum culling rate.

#### Appendix A.3. Farm Demography Related Parameters

^{−1}; a parameter value that is in agreement with the total calving interval (calving interval, dry period and non-gestation period for French cattle) in [7]. In addition, we assumed that the probability of the offspring being exposed to the bacterium after birth is 30% (i.e., ${p}_{b}=0.3$), which is in agreement with the vaginal shedding parturition probabilities reported in [7,32]. Further, we modelled the removal rates of each multiparous cattle sub-population using the same rate constant $\mu $, which represents removal from the herd due to age and not isolation or culling (which is incorporated in the parameter value of c). As discussed in Section 2.1.2, in order to impose in our model the restriction of maintaining the total cattle population constant, it is implied that the farm removal rate $\mu $ is set equal to the birth rate b. Although this parameter is calibrated indirectly to our model, its value $\mu =1/600\phantom{\rule{4pt}{0ex}}$day

^{−1}agrees with the natural death rate estimated in [7,12,17], summed up with a removal rate corresponding to cows over 6 years old being removed from the herd. Finally, we assumed that after the event of the first parturition, a nulliparous heifer becomes a multiparous cow and modelled this progression to occur at a continuous transition rate, depending on the parameter $\delta $, which expresses the percentage of multiparous and nulliparous cattle within the herd. For calibrating the progression rate $\delta $, we considered the average percentage of nulliparous and multiparous cattle in a UK’s cattle herd, which according to [28] is 44% for cattle younger than 2 years old and 56% for the remaining ones. In order for our model to match these values, the progression rate was directly set to $\delta =1.28b$. This value corresponds to heifers staying at the nulliparous state for about 470 days before the event of first parturition, that is in close agreement with [30] where the age before first parturition was estimated at about 410 days.

#### Appendix A.4. Environmental Contamination Related Parameters

^{−1}from [17] (who derived the parameter value from [7,12]), since they incorporated the same rate of inflow of the bacteria into the environment by shedding cattle (thus assuming a shedding rate). However, in [17] only one environmental landscape is considered with a shedding rate in the form ${\eta}_{i}{I}_{mp}$. In this work, since we consider multiple stages of landscape contamination, the same parameter value can be considered, this time with a shedding rate ${\eta}_{i}{I}_{mp}{L}_{j}^{i}$ (because ${L}_{j}^{i}$ are unitless and normalized to $[0,1]$). As for the indoor birth shedding rate, no direct parameter value was available in the literature. Thus, we calibrated ${\eta}_{b}$ so that the ratio of indoor shedding and birth shedding rate constants is ${\eta}_{i}/{\eta}_{b}=1/(2.7\times {10}^{-6})$ which is the ratio, found in [16], of the bacterium’s excretion in partus material per parturition over that in faeces and urine per day. Since in [16] this ratio is measured per parturition, we additionally multiplied ${\eta}_{b}$ with b; that is the inverse of the parturition interval. It is important here to note that in [16] the analysis was performed in Dutch dairy goat herds, thus this parameter is characterized by increased uncertainty. Nonetheless, its value compared to ${\eta}_{i}$ indicates a much higher excretion of the bacterium from partus rather than that from faeces. Similarly and for the same reasons, we adopted the framework of [17] to model the natural indoor environmental decay with the rate ${K}_{i}{L}_{j}^{i}$ and the active cleaning rate ${\u03f5}_{i}{L}_{j}^{i}$. We calibrated the parameter value of ${K}_{i}$ to match the one provided in [17] when considering only one landscape. Thus, we considered ${K}_{i}=0.0083/n\phantom{\rule{4pt}{0ex}}$day

^{−1}, in the sense that when $n=1$, one retrieves the same parameter value. As for the active cleaning rate constant, we directly adopted the value ${\u03f5}_{i}=0.1\phantom{\rule{4pt}{0ex}}$day

^{−1}from [17]. Finally, since the outdoor environment operates similarly to the indoor environment and since no available data exist, we assumed that the outdoor shedding rate constant and the outdoor natural decay rate constant to have the same values with the indoors environment; i.e., ${\eta}_{o}={\eta}_{i}$ and ${K}_{o}={K}_{i}$, respectively.

## Appendix B. The Equilibria of the Q Fever Model

## Appendix C. Confidence Level of the Model’s Response

**Figure A1.**Prevalence (

**left panel**) of seronegative (SN) and seropositive (SP) cattle for the continuous housing case, as in the left panel of Figure 3. Nulliparous (

**middle panel**) and multiparous (

**right panel**) sub-population profiles for the continuous housing case, as in the left panel of Figure 4. The solution profiles are derived on the basis of the 1000 parameter sets considered, with the mean values shown in solid and the 95% confidence intervals in dashed.

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**Figure 1.**Flow diagram representation of the C. burnetii transmission model. The sub-populations of the model within the nulliparous and multiparous compartments are denoted by rounded squares, except from the magenta squares that denote the indoors and outdoors environment compartments. The transition rates between the sub-populations are indicated by the black arrows (solid/dashed when related to the infection cycle/demographics) with their associated parameters. The dashed red arrows denote transition rates incorporating contributions from/to the environment.

**Figure 2.**Flow diagram representation of the indoor environment. The multiple stages of contamination ${L}_{j}^{i}$ for $j=0,\cdots ,n$ are denoted by round squares, which are connected by the black solid arrows, indicating the incremental decay of contamination, environmental hygiene and total indoor shedding output of the herd. The double arrows denote the accumulated contamination input to the ${L}_{j}^{i}$ stage by all less contaminated stages ${L}_{0}^{i},\dots ,{L}_{j-1}^{i}$.

**Figure 3.**Prevalence of seronegative (SN, ${S}_{np}+{E}_{np}+{S}_{mp}+{E}_{mp}$) and seropositive (SP, ${A}_{np}+{A}_{mp}+{I}_{mp}$) cattle, with the seropositive ones being either non-shedders (${A}_{np}+{A}_{mp}$) or shedders (${I}_{mp}$). Both continuous (

**left**) and seasonal (

**right**) housing are considered; in the latter case the herd remains indoors/outdoors during the first/second half of each year.

**Figure 4.**Evolution of the sub-populations profiles for the continuous (

**left**) and seasonal (

**right**) housing cases considered; in the latter case the herd remains indoors/outdoors during the first/second half of each year. The nulliparous sub-populations are depicted with sold curves, while the multiparous ones with dashed curves.

**Figure 5.**Environmental contamination to the infection spread provided by the landscapes ${L}_{j}^{i}$ and ${L}_{j}^{o}$ of the indoor (

**left**) and outdoor (

**right**) compartments, respectively. The contamination landscape in the continuous housing case is depicted with solid curves, while that of the seasonal housing is depicted with dotted curves; in the latter case the herd remains indoors/outdoors during the first/second half of each year. The outdoor environment is only activated in the seasonal housing case.

**Figure 6.**Visualised sensitivities $SI$ of the model outputs for each parameter p calculated by Equation (8) after one and five years (blue and red bars, respectively), for the continuous housing case. The model outputs are the prevalences of seronegative, seropositive, seropositive non-shedding and seropositive shedding cattle and the related sensitivities $S{I}_{SN}$, $S{I}_{SP}$, $S{I}_{SP\phantom{\rule{3.33333pt}{0ex}}NS}$ and $S{I}_{SP\phantom{\rule{3.33333pt}{0ex}}S}$ are shown from the left panel to right one. The indices $SI$ are sorted in descending order on the basis of the 5-year output.

**Figure 7.**Comparison of the extended seasonal (ES) and limited seasonal (LS) housing cases with the regular seasonal (RS) case. The

**left**panel depicts the prevalence of seronegative (SN) and seropositive (SP) cattle, accounting for non-shedders and shedders. The

**right**panel shows the outdoor environmental load of the contaminated landscapes ${L}_{1,2,3,4}^{o}$ and the contamination free landscape ${L}_{0}^{o}$.

Parameter | Unit | Description | Source | |
---|---|---|---|---|

${p}^{i}$, ${p}^{o}$ | - | Proportion of time per year for which cattle stay indoors and outdoors | Equation (6) | |

${\beta}_{j}^{i}$, ${\beta}_{j}^{o}$ | day^{−1} | Transmission rates of ${S}_{np}/{S}_{mp}$ becoming ${E}_{np}/{E}_{mp}$ from each indoor and outdoor contaminated landscape | Equation (7) | |

${r}_{i}$, ${r}_{o}$ | $0.8$ | - | Reduction rate of transmission probability by indoor and outdoor environment | Assumed |

${\beta}_{c}^{i}$ | $0.0943$ | day^{−1} | Total indoor transmission rate | [12,17] |

${\beta}_{c}^{o}$ | $0.01{\beta}_{c}^{i}$ | day^{−1} | Total outdoor transmission rate | CtM [12,26,27] |

$\gamma $ | $0.1$ | - | Ratio of reduced transmission rate for nulliparous cattle | Assumed |

$\rho $ | $0.1$ | day^{−1} | Transition rate of ${E}_{np}/{E}_{mp}$ eliminating the disease | [7,12] |

$\alpha $ | $0.04762$ | day^{−1} | Transition rate of ${E}_{np}/{E}_{mp}$ becoming ${A}_{np}/{A}_{mp}$ | [21] |

$\tau $ | $0.09023$ | day^{−1} | Transition rate of ${A}_{mp}$ becoming ${I}_{mp}$ | CtM [7,12,23] |

$\sigma $ | $0.02857$ | day^{−1} | Transition rate of ${I}_{mp}$ becoming ${A}_{mp}$ | [7,12] |

c | $9.57\times {10}^{-4}$ | day^{−1} | Removal rate of ${I}_{mp}$ due to culling and isolation | [7] |

$\delta $ | $0.0021$ | day^{−1} | Progression rate from nulliparous to multiparous cattle | CtM [28] |

b | $1/600$ | day^{−1} | Birth rate of multiparous cattle | [7,29,30,31] |

${p}_{b}$ | $0.3$ | - | Probability of the offspring being exposed after birth from ${A}_{mp}/{I}_{mp}$ | [7,32] |

$\mu $ | b | day^{−1} | Natural death and removal rate | CtM [7,12,17] |

${\eta}_{i}$ | $0.04$ | day^{−1} | Indoor shedding rate from ${I}_{mp}$ | [7,12,17] |

${\eta}_{b}$ | ${\eta}_{i}/2.7\times {10}^{-6}$ | - | Indoor birth shedding rate from ${I}_{mp}$ | CtM [16] |

${\eta}_{o}$ | $0.04$ | day^{−1} | Outdoor shedding rate from ${I}_{mp}$ | Assumed |

${K}_{i}$ | $0.0083/n$ | day^{−1} | Natural indoor environment decay rate | CtM [17] |

${K}_{o}$ | $0.0083/n$ | day^{−1} | Natural outdoor environment decay rate | Assumed |

${\u03f5}_{i}$ | $0.1$ | day^{−1} | Active clearing rate of contaminated indoor environment | [17] |

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**MDPI and ACS Style**

Patsatzis, D.G.; Wheelhouse, N.; Tingas, E.-A.
Modelling the Transmission of *Coxiella burnetii* within a UK Dairy Herd: Investigating the Interconnected Relationship between the Parturition Cycle and Environment Contamination. *Vet. Sci.* **2022**, *9*, 522.
https://doi.org/10.3390/vetsci9100522

**AMA Style**

Patsatzis DG, Wheelhouse N, Tingas E-A.
Modelling the Transmission of *Coxiella burnetii* within a UK Dairy Herd: Investigating the Interconnected Relationship between the Parturition Cycle and Environment Contamination. *Veterinary Sciences*. 2022; 9(10):522.
https://doi.org/10.3390/vetsci9100522

**Chicago/Turabian Style**

Patsatzis, Dimitrios G., Nick Wheelhouse, and Efstathios-Al. Tingas.
2022. "Modelling the Transmission of *Coxiella burnetii* within a UK Dairy Herd: Investigating the Interconnected Relationship between the Parturition Cycle and Environment Contamination" *Veterinary Sciences* 9, no. 10: 522.
https://doi.org/10.3390/vetsci9100522