# Comparing Direct and Indirect Transmission in a Simple Model of Veterinary Disease

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Extension of S-I Model

- $N$ is the population size;
- S and I represent the number of susceptible and infected animals in the system so that $N=S+I;$
- $Q$ describes the amount of infectious environmental contamination (e.g., animal waste or contaminated water);
- $\beta $ represents the rate of direct contact-based transmission;
- $\u03f5$ represents the rate at which new infections are produced via contaminated agents in the environment (Q);
- $\alpha $ is the rate at which infectious individuals create new contaminated agents (Q); and
- $k$ is the rate at which the environment is cleared of waste (by dissipation or active cleaning).

#### 2.2. Analysis in Steady State, Analytic Solution

_{0}:

#### 2.3. Limiting Cases Outside Steady State

#### 2.4. Modeling Dose Response

_{0}, is a constant that rescales $\u03f5$. Combining constants, one could define:

#### 2.5. Numerical Simulation

## 3. Results

## 4. Discussions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Sensitivity analysis of initial conditions at t = 100 days with parameters used in Table 1. The heat map shows the fraction of infectious as a function of ${R}_{direct}=\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$, and ${R}_{indirect}=\frac{\u03f5\alpha}{\mu k}$ with the range of initial infectious fraction in the population listed as: (

**a**) ${I}_{0}=0.03$, (

**b**) ${I}_{0}=0.05$, (

**c**) ${I}_{0}=0.10$.

## References

- Centers for Disease Control and Prevention. CDC 2011 Estimates: Findings. Available online: http://www.cdc.gov/foodborneburden/2011-foodborne-estimates.html (accessed on 12 August 2019).
- European Food Safety Authority. Food-Borne Zoonotic Diseases. Available online: http://www.efsa.europa.eu/en/topics/topic/foodbornezoonoticdiseases.htm (accessed on 30 September 2013).
- Lanzas, C.; Lu, Z.; Gröhn, Y.T. Mathematical Modeling of the Transmission and Control of Foodborne Pathogens and Antimicrobial Resistance at Preharvest. Foodborne Pathog. Dis.
**2011**, 8, 1–10. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lurette, A.; Belloc, C.; Touzeau, S.; Hoch, T.; Ezanno, P.; Seegers, H.; Fourichon, C. Modelling Salmonella spread within a farrow-to-finish pig herd. Vet. Res.
**2008**, 39, 5. [Google Scholar] [CrossRef] [PubMed] - Lanzas, C.; Chen, S. Mathematical modeling tools to study preharvest food safety. Microbiol. Spectr.
**2016**, 4. [Google Scholar] [CrossRef] - Conlan, A.J.K.; Coward, C.; Grant, A.J.; Maskell, D.J.; Gog, J.R. Campylobacter jejuni colonization and transmission in broiler chickens: A modelling perspective. J. R. Soc. Interface
**2007**, 4, 819–829. [Google Scholar] [CrossRef] [PubMed] - Van Gerwe, T.; Bouma, A.; Jacobs-Reitsman, W.F.; van den Broek, J.; Klingenber, D.; Stegeman, A.; Heesterbeek, J.A.P. Quantifying transmission of Campylobacter spp. among broilers. Appl. Environ. Microbiol.
**2005**, 71, 5765–5770. [Google Scholar] [CrossRef] [PubMed] - Conlan, A.J.K.; Line, J.E.; Hiett, K.; Coward, C.; Van Diemen, P.M.; Stevens, M.P.; Jone, M.A.; Gog, J.R.; Maskell, D.J. Transmission and dose–response experiments for social animals: A reappraisal of the colonization biology of Campylobacter jejuni in chickens. J. R. Soc. Interface
**2011**, 8, 1720–1735. [Google Scholar] [CrossRef] [PubMed] - Medema, G.J.; Teunis, P.F.M.; Havelaar, A.H.; Haas, C.N. Assessment of the dose-response relationship of Campylobacter jejuni. Int. J. Food Microbiol.
**1996**, 30, 101–111. [Google Scholar] [CrossRef] - United States Environmental Protection Agency. Poultry Production. Available online: http://www.epa.gov/agriculture/ag101/printpoultry.html (accessed on 30 September 2018).
- Leibler, J.H.; Carone, M.; Silbergeld, E.K. Contribution of Company Affiliation and Social Contacts to Risk Estimates of Between-Farm Transmission of Avian Influenza. PLoS ONE
**2010**, 5, e9888. [Google Scholar] [CrossRef] - World Health Organization. Salmonella and Campylobacter in Chicken Meat. 2009. Available online: http://www.who.int/foodsafety/publications/micro/MRA19.pdf (accessed on 30 September 2018).
- Newell, D.G.; Fearnley, C. Sources of Campylobacter Colonization in Broiler Chickens. Appl. Environ. Microbiol.
**2003**, 69, 4343–4351. [Google Scholar] [CrossRef] - Humphrey, T.J.; Henley, A.; Lanning, D.G. The colonization of broiler chickens with Campylobacter jejuni: Some epidemiological investigations. Epidemiol. Infect.
**1993**, 110, 601–607. [Google Scholar] [CrossRef] [PubMed] - Food Standards Agency. Research Project B15004: Measures and Best Practice to Minimise Infection of Remaining Birds with Campylobacter when Broiler Flocks Are Thinned. Available online: http://www.foodbase.org.uk//admintools/reportdocuments/191-1-325_B15004_Final_report_track_changes_accepted_1.3.07.pdf (accessed on 30 September 2018).
- Prabakaran, R. Good Practices in Planning and Management of Integrated Commercial Poultry Production in South Asia; Food and Agricultural Organization of the United Nations: Rome, Italy, 2003; 97p. [Google Scholar]
- van Wagenberg, C.P.A.; van Horne, P.L.M.; Sommer, H.M.; Nauta, M.J. Cost-effectiveness of Campylobacter interventions on broiler farms in six European countries. Microb. Risk Anal.
**2016**, 2, 53–62. [Google Scholar] [CrossRef] - Doerr, D.; Hu, K.; Renly, S.; Edlund, S.; Davis, D.; Lessler, J.; Filter, M.; Kasbohrer, A.; Appel, B.; Kaufman, J. Accelerating investigation of food-borne disease outbreaks using pro-active geospatial of food supply chains. In Proceedings of the First ACM SIGSPATIAL International Workshop on the Use of GIS in public health, Redondo Beach, CA, USA, 6 November 2012. [Google Scholar]
- Van Gerwe, T.; Miflin, J.K.; Templeton, J.M.; Bouma, A.; Wagenaar, J.A.; Jacobs-Reitsman, W.F.; Stegeman, A.; Klingenber, D. Quantifying transmission of Campylobacter jejuni in commercial broiler flocks. Appl. Environ. Microbiol.
**2009**, 75, 625–628. [Google Scholar] [CrossRef] [PubMed] - Ross, T. A simple, spread-sheet based, food safety risk assessment tool. Int. J. Food Microbiol.
**2002**, 77, 39–53. [Google Scholar] [CrossRef] - Cousins, M.; Sargeant, J.M.; Fishman, D.; Greer, A.L. Modelling the transmission dynamics of Campylobacter in Ontario, Canada, assuming house flies, Musca domestica, are a mechanical vector of disease transmission. R. Soc. Open Sci.
**2019**, 6, 181394. [Google Scholar] [CrossRef] [PubMed] - Liao, S.J.; Marshall, J.; Hazelton, M.L.; French, N.P. Extending statistical models for source attribution of zoonotic diseases: A study of campylobacteriosis. J. R. Soc. Interface
**2019**, 6, 20180534. [Google Scholar] [CrossRef] [PubMed] - Rushton, S.; Humphrey, T.; Shirley, M.; Bull, S.; Jørgensen, F. Campylobacter in housed broiler chickens: A longitudinal study of risk factors. Epidemiol. Infect.
**2009**, 137, 1099–1110. [Google Scholar] [CrossRef] [PubMed] - Sibanda, N.; McKenna, A.; Richmond, A.; Ricke, S.C.; Callaway, T.; Stratakos, A.C.; Gundogdu, O.; Corcionivoschi, N. A review of the effect of management practices on campylobacter prevalence in poultry farms. Front. Microbiol.
**2008**, 9, 2002. [Google Scholar] [CrossRef] [PubMed] - Teunis, P.; Van den Brandhof, W.; Nauta, M.; Wagenaar, J.; Van den Kerkhof, H.; Van Pelt, W. A reconsideration of the Campylobacter dose-response relation. Epidemiol. Infect.
**2005**, 133, 583–592. [Google Scholar] [CrossRef] [PubMed] - Liu, W.-M.; Hethcote, H.W.; Levin, S.A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol.
**1987**, 25, 359–380. [Google Scholar] [CrossRef] [PubMed] - Eclipse Foundation. Spatio Temporal Epidemiological Modeler project. Available online: www.eclipse.org/STEM (accessed on 28 August 2019).

**Figure 2.**Heat map of the fraction infectious as a function of ${R}_{direct}=\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$, and ${R}_{indirect}=\frac{\u03f5\alpha}{\mu k}$, with the epidemiological values listed in Table 1. The figure shows numerical simulation near steady state (t = 1000 days). The transition from zero infection to endemic infection occurs along a line with R

_{0}= 1.

**Figure 3.**(

**a**) Heat map of the infectious fraction as a function of ${R}_{direct}=\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$, and ${R}_{indirect}=\frac{\u03f5\alpha}{\mu k}$ with the epidemiological values listed in Table 1 at t = 100 days. The numerical solution at t = 100 does not match the analytical solution any more (no longer steady state). (

**b**) 3D surface plot of the infectious fraction provides another view of the transition at t = 100 days.

**Figure 4.**The epidemic wave in I(t), Q(t), and dI/dt, in the time period 0 <= t <= 100 days at the points (

**A**), (

**B**), (

**C**), (

**D**) indicated in Figure 3. The values of the pair ($\frac{\beta}{\mu},\frac{\u03f5\alpha}{k\mu})$ are (

**A**) = (0, 1.7); (

**B**) = (2.5, 0); (

**C**) = (1.25, 85); (

**D**) = (2.5, 1.7). The figure shows the early onset of epidemic wave with respect to direct transmission, indirect transmission, and the mixed transmission modes.

**Figure 5.**Sensitivity analysis of epidemiological parameters for indirect transmission at t = 100 days. The heat map shows the fraction of infectious as a function of ${R}_{direct}=\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$, and ${R}_{indirect}=\frac{\u03f5\alpha}{\mu k}$ with the range of epidemiological values listed as: (

**a**) $\beta =\left[0,12\right]$, $\u03f5=\left[0,12\right]$, $k=1,\alpha =0.1$; (

**b**) $\beta =\left[0,12\right]$, $\u03f5=0.1$, $k=0.1$, $\alpha =\left[0,12\right]$; (

**c**) $\beta =\left[0,12\right]$, $\u03f5=\left[0,24\right]$, $k=2$, $\alpha =0.1$.

**Figure 6.**Heat map of the infectious fraction at t = 100 days including the effects of dose response with a power $\nu =1.6,\mathrm{and}{Q}_{0}=0.2$. In general, for $\nu >1$, the epidemic is delayed but the transition from I = 0 to a fully infectious population is sharpened relative to direct transmission (or indirect transmission with $\nu =1.0$). Parameter values are $\beta =\left[0,12\right]$, $\u03f5=\left[0,12\right]$, $k=1$, $\alpha =0.1$, and $\mu =1/30.$

**Table 1.**Epidemiological parameters were chosen to explore a range of reproductive number with equivalent contribution possible from direct and/or indirect transmission. The epidemiological parameters are known to vary with strain [9], but the range of values listed in the table are based on typical values from the literature [6,7,8,13,17].

Parameter | Value of Range [a,b] | Contribution to Reproductive Number | Range [a,b] |
---|---|---|---|

$\alpha $ [day^{−1}] | 0.1 | ${R}_{indirect}=\frac{\u03f5\alpha}{\mu k}$ | [0,3.6] |

$\u03f5$ [day^{−1}] | [0,0.12] | ||

$k$ [day^{−1}] | 0.1 [17] | ${R}_{direct}=\raisebox{1ex}{$\beta $}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$ | [0,3.6] |

$\beta $ [day^{−1}] | [0,0.12] [7] | ||

$\mu $ [day^{−1}] | 1/30 [13] | - | |

N | 1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yagci Sokat, K.; Edlund, S.; Clarkson, K.; Kaufman, J.
Comparing Direct and Indirect Transmission in a Simple Model of Veterinary Disease. *Mathematics* **2019**, *7*, 1039.
https://doi.org/10.3390/math7111039

**AMA Style**

Yagci Sokat K, Edlund S, Clarkson K, Kaufman J.
Comparing Direct and Indirect Transmission in a Simple Model of Veterinary Disease. *Mathematics*. 2019; 7(11):1039.
https://doi.org/10.3390/math7111039

**Chicago/Turabian Style**

Yagci Sokat, Kezban, Stefan Edlund, Kenneth Clarkson, and James Kaufman.
2019. "Comparing Direct and Indirect Transmission in a Simple Model of Veterinary Disease" *Mathematics* 7, no. 11: 1039.
https://doi.org/10.3390/math7111039