Journal of Mathematical Biology

, Volume 69, Issue 2, pp 501–532

A stochastic model for transmission, extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices

• Xueying Wang
• Raju Gautam
• Pablo J. Pinedo
• Linda J. S. Allen
• Renata Ivanek
Article

Abstract

Many infectious agents transmitting through a contaminated environment are able to persist in the environment depending on the temperature and sanitation determined rates of their replication and clearance, respectively. There is a need to elucidate the effect of these factors on the infection transmission dynamics in terms of infection outbreaks and extinction while accounting for the random nature of the process. Also, it is important to distinguish between the true and apparent extinction, where the former means pathogen extinction in both the host and the environment while the latter means extinction only in the host population. This study proposes a stochastic-differential equation model as an approximation to a Markov jump process model, using Escherichia coli O157:H7 in cattle as a model system. In the model, the host population infection dynamics are described using the standard susceptible-infected-susceptible framework, and the E. coli O157:H7 population in the environment is represented by an additional variable. The backward Kolmogorov equations that determine the probability distribution and the expectation of the first passage time are provided in a general setting. The outbreak and apparent extinction of infection are investigated by numerically solving the Kolmogorov equations for the probability density function of the associated process and the expectation of the associated stopping time. The results provide insight into E. coli O157:H7 transmission and apparent extinction, and suggest ways for controlling the spread of infection in a cattle herd. Specifically, this study highlights the importance of ambient temperature and sanitation, especially during summer.

Keywords

Escherichia coli O157:H7 Stochastic $${SIS}_\mathrm{E} {{ model}}$$ Kolmogorov equations Extinction outbreak

Mathematics Subject Classification (2000)

92D30 60H10 60H30

Notes

Acknowledgments

We thank three anonymous referees and the editor for their suggestions that improved this paper. This work was supported by the National Science Foundation grant NSF-EF-0913367 to RI funded under the American Recovery and Reinvestment Act of 2009. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This publication is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).

Supplementary material

285_2013_707_MOESM1_ESM.pdf (746 kb)
Supplementary material 1 (pdf 745 KB)

References

1. Alejandro RC (2004) Flushing and scraping freestalls and drylot pens. In: News D (ed) University of California cooperative extension, CAGoogle Scholar
2. Allen L (2007) An introduction to mathematical biology. Prentice-Hall, Upper Saddle RiverGoogle Scholar
3. Allen LJS, Allen EJ, Jonsson CB (2006) The impact of environmental variation on hantavirus infection in rodents. In: Gumel AB, Castillo-Chavez C, Mickens RE, Clemence DP (eds) Contemporary mathematics series. Proceedings of the joint summer research conference on modeling the dynamics of human diseases: emerging paradigms and challenges, vol 410. AMS, Providence, pp 1–15Google Scholar
4. Allen LJS, Lahodny GE Jr (2012) Extinction thresholds in deterministic and stochastic epidemic models. J Biol Dyn 6:590–611
5. Ayscue P, Lanzas C, Ivanek R, Grohn YT (2009) Modeling on-farm Escherichia coli O157:H7 population dynamics. Foodborne Pathog Dis 6:461–470
6. Bani-Yaghoub M, Gautam R, Döpfer D, Kaspar CW, Ivanek R (2012a) Effectiveness of environmental decontamination in control of infectious diseases. Epidemiol Infect 140(3):542–553
7. Bani-Yaghoub M, Gautam R, Shuai Z, van den Driessche P, Ivanek R (2012b) Reproduction numbers for infections with free-living pathogens growing in the environment. J Biol Dyn 6(2):923–940Google Scholar
8. Barkocy-Gallagher GA, Arthur TM, Rivera-Betancourt M, Nou X, Shackelford SD, Wheeler TL, Koohmaraie M (2003) Seasonal prevalence of Shiga toxin-producing Escherichia coli, including O157:H7 and non-O157 serotypes, and Salmonella in commercial beef processing plants. J Food Prot 66:1978–1986Google Scholar
9. Berry ED, Miller DN (2005) Cattle feedlot soil moisture and manure content: II. Impact on Escherichia coli O157. J Environ Qual 34:656–663
10. Chase-Topping M, Gally D, Low C, Matthews L, Woolhouse M (2008) Super-shedding and the link between human infection and livestock carriage of Escherichia coli O157. Nat Rev Microbiol 6:904–912
11. Clancy D (2005) A stochastic SIS infection model incorporating indirect transmission. J Appl Probab 42:725–737
12. Cray WC et al (1998) Effect of dietary stress on fecal shedding of Escherichia coli O157:H7 in calves. Appl Environ Microbiol 64:1975–1979Google Scholar
13. Cushing JM, Costantino RF, Dennis B, Desharnais RA, Henson SM (2003) Chaos in ecology. Academic Press, San DiegoGoogle Scholar
14. Davis MA, Cloud-Hansen KA, Carpenter J, Hovde CJ (2005) Escherichia coli O157:H7 in environments of culture-positive cattle. Appl Environ Microbiol 71:6816–6822
15. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio $$R_0$$ in models for infectious diseases in heterogeneous population. J Math Biol 28:365–382
16. Döpfer D, Geue L, Schares S, Mintel B, Hoffmann B, Fischer EAJ (2012) Dynamics of shiga-toxin producing Escherichia coli (STEC) and their virulence factors in cattle. Prev Vet Med 103(1):22–30
17. Durrett R (1996) Probability: theory and examples. Duxbury Press, BelmontGoogle Scholar
18. Ewald PW (2004) Evolution of virulence. Infect Dis Clin North Am 18:1–15
19. Ferrari MJ, Grais RF, Bharti N, Conlan AJK, Bjornstad ON, Wolfson LJ, Guerin PJ, Djibo A, Grenfell BT (2008) The dynamics of measles in sub-Saharan Africa. Nature 451:679–684
20. Friedman A (1964) Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs
21. Gardiner CW (2009) Handbook of Stochastic Methods for physics, chemistry, and the natural sciences, 4th edn. Springer, BerlinGoogle Scholar
22. Gautam R, Bani-Yaghoub M, Neill WH, Döpfer D, Kaspar CW, Ivanek R (2011) Modeling the effect of seasonal variation in ambient temperature on the transmission dynamics of a pathogen with a free-living stage: example of Escherichia coli O157:H7 in a dairy herd. Prev Vet Med 102(1):10–21
23. Gillespie DT (2002) The chemical Langevin equation and Fokker–Planck equation for the reverisble isomerization reaction. J Phys Chem A 106:5063–5071
24. Heesterbeek JAP, Roberts MG (2007) The type-reproduction number T in models for infectious disease control. Math Biosci 206:3–10
25. Hussein HS, Sakuma T (2005) Prevalence of shiga toxin-producing Escherichia coli in dairy cattle and their products. J Dairy Sci 88:450–465
26. It$$\hat{\text{ o }}$$ K (1944) Stochastic Integral. Proc Imp Acda Tokyo 20:519–524Google Scholar
27. Jiang X, Morgan J, Doyle MP (2002) Fate of Escherichia coli O157:H7 in manure-amended soil. Appl Environ Microbiol 68:2605–2609
28. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Springer, New York
29. Karmali MA, Petric M, Lim C, Fleming PC, Arbus GS, Lior H (1985) The association between idiopathic hemolytic uremic syndrome and infection by verotoxin-producing Escherichia coli. Inf Dis 151:775–782
30. Kudva IT, Blanch K, Hovde CJ (1998) Analysis of Escherichia coli O157:H7 survival in ovine or bovine manure and manure slurry. Appl Environ Microbiol 64(9):3166–3174Google Scholar
31. Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 8:344–356
32. Kurtz TG (1981) Approximation of population processes. In: CBMS-NSF regional conference series in applied mathematics, vol 36. SIAM, PhiladelphiaGoogle Scholar
33. Lande R, Engen S, Saether B-E (2003) Stochastic population dynamics in ecology and conservation. Oxford University Press, Oxford
34. LeJeune JT, Wetzel AN (2007) Preharvest control of Escherichia coli O157 in cattle. J Anim Sci 85:E73–E80
35. Liu WC, Jenkins C, Shaw DJ et al (2005) Modelling the epidemiology of verocytotoxin-producing Escherichia coli serogroups in young calves. Epidemiol Infect 133:449–458
36. Liu WC, Shaw DJ, Matthews L et al (2007) Modelling the epidemiology and transmission of verocytotoxin-producing Escherichia coli serogroups O26 and O103 in two different calf cohorts. Epidemiol Infect 135:1316–1323
37. MacDiarmid BN, Watkin BR (1972) The cattle dung patch. 3. Distribution and rate of decay of dung patches and their influence on grazing behaviour. J Br Grossl Soc 27:48–54
38. Maule A (2000) Survival of verocytotoxigenic Escherichia coli O157 in soil, water and on surfaces. Symp Ser (Soc Appl Microbiol) 29:71–78
39. Matthews L, Low JC, Gally DL et al (2006a) Heterogeneous shedding of Escherichia coli O157 in cattle and its implications for control. Proc Natl Acad Sci 103:547–552
40. Matthews L, McKendrick IJ, Ternent HE, Gunn GJ, Synge BA, Woolhouse MEJ (2006b) Super-shedding cattle and the transmission dynamics of Escherichia coli O157. Epidemiol Infect 134:131–142
41. McGee P, Bolton DJ, Sheridan JJ, Earley B, Leonard L (2001) The survival of Escherichia coli O157:H7 in slurry from cattle fed different diets. Lett Appl Microbiol 32:152–155
42. Mead PS, Griffin PM (1998) Escherichia coli O157-H7. Lancet 352:1207–1212
43. Mechie SC, Chapman PA, Siddons CA (1997) A fifteen month study of Escherichia coli O157:H7 in a dairy herd. Epidemiol Infect 118:17–25
44. Mitchell AR, Griffiths DF (1980) The finite difference method in partial differential equations. Wiley, New York
45. Murray JD (2002) Mathematical biology, vol I. Springer, Berlin
46. Nataro JP, Kaper JB (1998) Diarrheagenic Escherichia coli. Clin Microbiol Rev 11:142–201Google Scholar
47. Ripa J (2012) Stochasticity, environmental. In: Hastings A, Gross LJ (eds) Encyclopedia of theoretical ecology. University of California Press, Berkeley, pp 712–718Google Scholar
48. Roberts MG, Heesterbeek JAP (2003) A new method for estimating the effort required to control an infectious disease. Proc R Soc Lond B 270:1359–1364
49. Shane EM, Endres MI, Janni KA (2010) Alternative bedding materials for compost bedded pack barns in Minnesota: a descriptive study. Appl Eng Agric 26:465–473
50. Schuss Z (2009) Theory and applications of stochastic processes. In: Springer series on applied mathematical sciences, vol 170Google Scholar
51. Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, Cambridge
52. Turner J, Begon M, Bowers RG, French NP (2003) A model appropriate to the transmission of a human food-borne pathogen in a multi-group managed herd. Prev Vet Med 57:175–198
53. Turner J, Bowers RG, Begon M, Robinson SE, French NP (2006) A semi-stochastic model of the transmission of Escherichia coli O157 in a typical UK dairy herd: dynamics, sensitivity analysis and intervention/prevention strategies. J Theor Biol 241:806–822
54. Turner J, Bowers RG, Clancy D, Behnke MC, Christley RM (2008) A network model of E.coli O157 transmission within a typical UK dairy herd: the effect of heterogeneity and clustering on the prevalence of infection. J Theor Biol 254:45–54
55. van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48
56. Verhulst PF (1838) Notice sur la loi que la population poursuit dans son accroissement. Corresp Math Phys 10:113–121Google Scholar
57. Vital M, Hammes F, Egli T (2008) Escherichia coli O157 can grow in natural freshwater at low carbon concentrations. Environ Microbiol 10:2387–2396
58. Vosough Ahmadi B, Frankena K, Turner J, Velthuis AGJ, Hogeveen H, Huirne RBM (2007) Effectiveness of simulated interventions in reducing the estimated prevalence of Escherichia coli O157:H7 in lactating cows in dairy herds. Vet Res 38:755–771
59. Wang G, Zhao T, Doyle MP (1996) Fate of enterohemorrhagic Escherichia coli O157:H7 in bovine feces. Appl Environ Microbiol 62:2567–2570Google Scholar
60. Wood JC, McKendrick IJ, Gettinby G (2007) A simulation model to assess herd-level intervention strategies against Escherichia coli O157. Epidemiol Infect 135:749–764
61. Zhang XS, Chase-Topping ME, McKendrick IJ, Savill NJ, Woolhouse MEJ (2010) Spread of Escherichia coli O157:H7 infection among Scottish cattle farms: stochastic models and model selection. Epidemics 2:11–20

Authors and Affiliations

• Xueying Wang
• 1
Email author
• Raju Gautam
• 2
• Pablo J. Pinedo
• 3
• Linda J. S. Allen
• 4
• Renata Ivanek
• 2
1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
2. 2.Department of Veterinary Integrative Biosciences, College of Veterinary Medicine and Biomedical SciencesTexas A&M UniversityCollege StationUSA
3. 3.Ruminant Animal Health, Amarillo Texas AgriLife Research Center, Veterinary Medicine & Biomedical SciencesTexas A&M UniversityCollege StationUSA
4. 4.Department of Mathematics and StatisticsTexas Tech UniversityLubbockUSA