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Original Articles

Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease

, &
Pages 128-155 | Received 10 Jan 2014, Accepted 11 Aug 2014, Published online: 08 Sep 2014

References

  • L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology Second Edition, CRC Press, Boca Raton, FL, 2010.
  • L.J.S. Allen and G.E. LahodnyJr., Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn. 6 (2012), pp. 590–611.
  • L.J.S. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Math. Biosci. 243 (2013), pp. 99–108.
  • K.B. Athreya and P.E. Ney, Branching Processes, Springer-Verlag, New York, 1972.
  • M. Bani-Yaghoub, R. Gautam, D. Döpfer, C.W. Kaspar, and R. Ivanek, Effectiveness of environmental decontamination in control of infectious diseases, Epidemiol. Infect. 140 (2012), pp. 542–553.
  • M. Bani-Yaghoub, R. Gautam, Z. Shuai, P. van den Driessche, and R. Ivanek, Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn. 6 (2012), pp. 923–940.
  • M.S. Bartlett, Stochastic Population Models in Ecology and Epidemiology, Methuen, London 1960.
  • A.D.C. Berriman, D. Clancy, H.E. Clough, and R.M. Christley, Semi-stochastic models for Salmonella infection within finishing pig units in the UK, Math. Biosci. 245 (2013), pp. 148–156.
  • D.J. Bolton, C.M. Byrne, J.J. Sheridan, D.A. McDowell, and I.S. Blair, The survival characteristics of a non-toxigenic strain of Escherichia coli O157:H7, J. Appl. Microbiol. 86 (1999), pp. 407–411.
  • S.A. Boone and C.P. Gerba, Significance of fomites in the spread of respiratory and enteric viral disease, Appl. Environ. Microbiol. 73 (2007), pp. 1687–1696.
  • L. Bourouiba, A. Teslya, and J. Wu, Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: Insights from dynamic modeling, J. Theor. Biol. 271 (2011), pp. 181–201.
  • R. Breban, J.M. Drake, and P. Rohani, A general multi-strain model with environmental transmission: Invasion conditions for the disease-free and endemic states, J. Theor. Biol. 264 (2010), pp. 729–736.
  • S. Cairncross, J. Ensink, and T. Kahawita, Evaluation of the WASH activities undertaken to prevent and control cholera outbreaks in Guinea-Conakry & Guinea-Bissau – systematic literature review (June 2009-December 2009), 2009. Available at http://www.unicef.org/evaldatabase/files/LSHTM_WASH_Evaluation_last_Guinea-Bissau.pdf (accessed 22 May 2014).
  • T. Caraco and I.-N. Wang, Free-living pathogens: Life-history constraints and strain competition, J. Theor. Biol. 250 (2008), pp. 569–579.
  • Centers for Disease Control and Prevention, Preliminary FoodNet data on the incidence of infection with pathogens transmitted commonly through food – 10 states, 2009, MMWR 59 (2010), pp. 418–422.
  • Centers for Disease Control and Prevention, Vital signs: Incidence and trends of infection with pathogens transmitted commonly through food – foodborne diseases active surveillance network, 10 U.S. sites, 1996–2010, MMWR 60 (2011), pp. 749–755.
  • Centers for Disease Control and Prevention, Cholera in Haiti, 2013. Available at http://wwwnc.cdc.gov/travel/notices/watch/haiti-cholera (accessed 22 May 2014).
  • P.P. Chapagain, J.S. van Kessel, J.S. Karns, D.R. Wolfgang, E. Hovingh, K.A. Nelen, Y.H. Schukken, and Y.T. Grohn, A mathematical model of the dynamics of Salmonella Cerro infection in a US dairy herd, Epidemiol. Infect. 136 (2008), pp. 263–272.
  • C.T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infect. Dis. (2001). doi:10.1186/1471-2334-1-1.
  • O. Diekmann, J.A.P. Heesterbeek, and J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), pp. 365–382.
  • K.S. Dorman, J.S. Sinsheimer, and K. Lange, In the garden of branching processes, SIAM Rev. 46 (2004), pp. 202–229.
  • P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48.
  • P. van den Driessche and J. Watmough, Further notes on the basic reproduction number, in Mathematical Epidemiology, F. Brauer, P. van den Driessche, and J. Wu, eds., Lecture Notes in Mathematics, Vol. 1945, Springer, Berlin, 2008, pp. 159–178.
  • R. Gautam, M. Bani-Yaghoub, W.H. Neill, D. Döpfer, C. Kaspar, and R. Ivanek, 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 (2011), pp. 10–21.
  • R. Gautam, M. Kulow, D. Park, T. Gonzales, J. Dahm, M. Shiroda, A.J. Stasic, D. Döpfer, C. Kaspar, and R. Ivanek, Transmission of Escherichia coli O157:H7 among cattle is influenced by the level of environmental contamination, Epidemiol. Infect. (2014). doi:10.1017/S0950268814000867.
  • R. Gautam, G.E. Lahodny Jr., M. Bani-Yaghoub, P.S. Morley, and R. Ivanek, Understanding the role of cleaning in the control of Salmonella Typhimurium in grower-finisher pigs: A modelling approach, Epidemiol. Infect. (2013). doi:10.1017/s0950268813001805.
  • N.S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974.
  • P. Haccou, P. Jagers, and V.A. Vatutin, Branching Processes Variation, Growth, and Extinction of Populations, Cambridge Studies in Adaptive Dynamics, Cambridge Univ. Press, Cambridge, MA, 2005.
  • T.E. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin, 1963.
  • J.B. Harris, R.C. LaRocque, F. Qadri, E.T. Ryan, and S.B. Calderwood, Cholera, Lancet 379 (2012), pp. 2466–2476.
  • D.M. Hartley, J.G. MorrisJr., and D.L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Med. 3 (2006), pp. 63–69.
  • L.C. Heider, R.W. Meiring, A.M. Hoet, W.A. Gebreys, J.A. Funk, and T.E. Wittum, Evaluation of vaccination with a commercial subunit vaccine on shedding of Salmonella enterica in subclinically infected dairy cows, J. Am. Vet. Med. Assoc. 233 (2008), pp. 466–469.
  • O.A. van Herwaarden and J. Grasman, Stochastic epidemics: Major outbreaks and the duration of the endemic period, J. Math. Biol. 33 (1995), pp. 581–601.
  • A.A. Hill, E.L. Snary, M.E. Arnold, L. Alban, and A.J.C. Cook, Dynamics of Salmonella transmission on a British pig grower-finisher farm: A stochastic model, Epidemiol. Infect. 136 (2008), pp. 320–333.
  • J.K. House and B.P. Smith, Profitable strategies to control salmonellosis in dairy cattle, Proceedings of the WBC Congress, Quebec, Canada.
  • J.K. House, M.M. Ontiveros, N.M. Blackmer, E.L. Dueger, J.B. Fitchhorn, G.R. McArthur, and B.P. Smith, Evaluation of an autogenous Salmonella bacterin and a modified live Salmonella serotype Choleraesuis vaccine on a commercial dairy farm, Am. J. Vet. Res. 62 (2001), pp. 1897–1902.
  • P. Jagers, Branching Processes with Biological Applications, Wiley Series in Probability and Mathematical Statistics – Applied Probability and Statistics, Wiley-Intersci, London, 1975.
  • M.A. Jensen, S.M. Faruque, J.J. Mekalanos, and B.R. Levin, Modeling the role of bacteriophage in the control of cholera outbreaks, Proc. Natl. Acad. Sci. 103 (2006), pp. 4652–4657.
  • S. Kabir, Cholera vaccines: The current status and problems, Rev. Med. Microbiol. 16 (2005), pp. 101–116.
  • S. Karlin and H. Taylor, A First Course in Stochastic Processes, Academic Press, New York, 1975.
  • D.G. Kendall, Deterministic and stochastic epidemics in closed populations, Proc. Symp. Math. Stat. Prob. 3rd, Berkeley 4, 1956.
  • M. Kimmel and D. Axelrod, Branching Processes in Biology, Springer-Verlag, New York, 2002.
  • C. Lanzas, S. Briend, R. Ivanek, Y. Lo, P.P. Chapagain, K.A. Ray, P. Ayscue, L.D. Warnick, and Y.T. Gröhn, The effect of heterogeneous infectious period and contagiousness on the dynamics of Salmonella transmission in dairy cattle, Epidemiol. Infect. 136 (2008), pp. 1496–1510.
  • C. Lanzas, L.D. Warnick, R. Ivanek, P. Ayscue, D.V. Nydam, and Y.T. Gröhn, The risk and control of Salmonella outbreaks in calf-raising operations: A mathematical modeling approach, Vet. Res. (2008). doi:10.1051/vetres:2008038.
  • J.T. LeJeune, T.E. Besser, and D.D. Handcock, Cattle water troughs as reservoirs of Escherichia coli O157:H7, Appl. Environ. Microbiol. 67 (2001), pp. 3053–3057.
  • M. Lélu, M. Langlais, M.-L. Poulle, and E. Gilot-Fromont, Transmission dynamics of Toxoplasma gondii along an urban-rural gradient, Theor. Popul. Biol. 78 (2010), pp. 139–147.
  • Z. Lu, Y.T. Grohn, R.L. Smith, D.R. Wolfgang, J.A.S. van Kessel, and Y.H. Schukken, Assessing the potential impact of Salmonella vaccines in an endemically infected dairy herd, J. Theor. Biol. 259 (2009), pp. 770–784.
  • D. Ludwig, Persistence of dynamical systems under random perturbations, SIAM Rev. 17 (1975), pp. 605–640.
  • J. Mee, Management of infectious diseases in split-calving dairy herds, 2009. Available at http://www.teagasc.ie/publications/2009/20091105/management_of_infectious_diseases.pdf (accessed 22 May 2014).
  • C.J. Mode, Multitype Branching Processes Theory and Applications, Elsevier, New York, 1971.
  • Z. Mukandavire, S. Liao, J. Wang, H. Gaff, D.L. Smith, and J.G. MorrisJr., Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proc. Natl. Acad. Sci. USA 108 (2011), pp. 8767–8772.
  • S. Pénisson, Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis, Ph.D. diss., Mathematische Statistik und Wahrscheinlichkeitstheorie, Institut für Mathematik der Unversität, Postdam, Germany, 2010.
  • J.A. Robertsson, A.A. Lindberg, S. Hoiseth, and B.A. Stocker, Salmonella typhimurium infection in calves: Protection and survival of virulent challenge bacteria after immunization with live or inactivated vaccines, Infect. Immun. 41 (1983), pp. 742–750.
  • E. Scallan, R.M. Hoekstra, F.J. Angulo, R.V. Tauxe, M.-A. Widdowson, S.L. Roy, J.L. Jones, and P.M. Griffin, Foodborne illness acquired in the United States – major pathogens, Emerg. Infect. Dis. 17 (2011), pp. 7–15.
  • E. Scott and S.F. Bloomfield, The survival and transfer of microbial contamination via cloths, hands and utensils, J. Appl. Bacteriol. 68 (1990), pp. 271–278.
  • Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci. 234 (2011), pp. 118–126.
  • Z. Shuai, J.H. Tien, and P. van den Driessche, Cholera models with hyperinfectivity and temporary immunity, Bull. Math. Biol. 74 (2012), pp. 2423–2445.
  • L.W. Sinton, R.R. Braithwaite, C.H. Hall, and M.L. Mackenzie, Survival of indicator and pathogenic bacteria in bovine feces on pasture, Appl. Environ. Microbiol. 73 (2007), pp. 7917–7925.
  • B.P. Smith, F.G. Habasha, M. Reina-Guerra, and A.J. Hardy, Immunization of calves against salmonellosis, Am. J. Vet. Res. 41 (1980), pp. 1947–1951.
  • I. Soumpasis and F. Butler, Development and application of a stochastic epidemic model for the transmission of Salmonella Typhimurium at the farm level of the pork production chain, Risk Anal. 29 (2009), pp. 1521–1533.
  • G. Steinbach and H. Meyer, Efficacy of subcutaneous inoculation of calves with ‘Murivac’ inactivated salmonellosis vaccine, Tierarztliche Praxis. 22 (1994), pp. 529–531.
  • J.H. Tien and D.J.D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72 (2010), pp. 1506–1533.
  • J.H. Tien, H.N. Poinar, D.N. Fisman, and D.J.D. Earn, Herald waves of cholera in nineteenth century London, J. R. Soc. Int. 8 (2011), pp. 756–760.
  • M. Vital, H.P. Füchslin, F. Hammes, and T. Egli, Growth of Vibrio cholerae O1 Ogawa Eltor in freshwater, Microbiol. 153 (2007), pp. 1993–2001.
  • X. Wang, R. Gautam, P.J. Pinedo, L.J.S. Allen, and R. Ivanek, A stochastic model for transmission, extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices, J. Math. Biol. (2013). doi:10.1007/s00285-013-0707-1.
  • A. Weber, C. Bernt, K. Bauer, A. Mayr, The control of bovine salmonellosis under field conditions using herd-specific vaccines, Tierarztliche Praxis. 21 (1993), pp. 511–516.
  • P. Whittle, The outcome of a stochastic epidemic: A note on Bailey's paper, Biometrika 42 (1955), pp. 116–122.
  • World Health Organization, Cholera vaccines: WHO position paper, Wkly. Epidemiol. Rec. 85 (2010), pp. 117–128.
  • World Health Organization, Cholera in Sierra Leone: The case study of an outbreak, 2012. Available at http://www.who.int/features/2012/cholera_sierra_leone/en.
  • C. Wray, W.J. Sojka, J.A. Morris, and W.J.B. Morgan, The immunization of mice and calves with gal E mutants of Salmonella typhimurium, J. Hyg. 79 (1977), pp. 17–24.
  • Y. Xiao, R.G. Bowers, D. Clancy, and N.P. French, Understanding the dynamics of Salmonella infections in dairy herds: A modelling approach, J. Theor. Biol. 233 (2005), pp. 159–175.
  • Y. Xiao, D. Clancy, N.P. French, and R.G. Bowers, A semi-stochastic model for Salmonella infection in a multi-group herd, Math. Biosci. 200 (2006), pp. 214–233.
  • H. Xu, N. Roberts, F. Singleton, R. Attwell, D. Grimes, and R. Colwell, Survival and viability of nonculturable Escherichia coli and Vibrio cholerae in the estuarine and the marine environment, Microb. Ecol. 8 (1982), pp. 313–323.