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May  2017, 22(3): 1145-1166. doi: 10.3934/dcdsb.2017056

Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

This paper is an invited contribution to the special issue in honor of Steve Cantrell

Received  September 2015 Revised  December 2016 Published  January 2017

Mathematical models of antibiotic resistant infection epidemics in hospital intensive care units are developed with two modeling methods, individual based models and differential equations based models. Both models dynamically track uninfected patients, patients infected with a nonresistant bacterial strain not on antibiotics, patients infected with a nonresistant bacterial strain on antibiotics, and patients infected with a resistant bacterial strain. The outputs of the two modeling methods are shown to be complementary with respect to a common parameterization, which justifies the differential equations modeling approach for very small patient populations present in an intensive care unit. The model outputs are classified with respect to parameters to distinguish the extinction or endemicity of the bacterial strains. The role of stewardship of antibiotic use is analyzed for mitigation of these nosocomial epidemics.

Citation: Glenn F. Webb. Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1145-1166. doi: 10.3934/dcdsb.2017056
References:
[1]

E. Armeanu and M. Bonten, Control of vancomycin-resistant Enterococci : One size fits all?, Clin. Inf. Dis, 41 (2005), 210-216. doi: 10.1086/431206. Google Scholar

[2]

D. J. AustinM. J. BontenR. A. WeinsteinS. Slaughter and R. M. Anderson, Vancomycin-resistant Enterococci in intensive-care hospital settings: Transmission dynamics, persistence, and the impact of infection control programs, Proc. Nat. Acad. Sci. USA, 96 (1999), 6908-6913. doi: 10.1073/pnas.96.12.6908. Google Scholar

[3]

S. BarnesB. Golden and E. Wasil, MRSA transmission reduction using agent-based modeling and simulation, INFORMS J. Comput., 22 (2010), 635-636. doi: 10.1287/ijoc.1100.0386. Google Scholar

[4]

S. BarnesB. GoldenE. Wasil and J. Furuno, Contribution of interfacility patient movement to overall methicillin-resistant Staphylococcus aureus prevalence levels, Infect. Cont. Hosp. Epid., 32 (2011), 1073-1078. doi: 10.1086/662375. Google Scholar

[5]

B. BoldinM. J. Bonten and O. Diekmann, Relative effects of barrier precautions and topical antibiotics on nosocomial bacterial transmission: results of multi-compartment models, Bull. Math. Biol., 69 (2007), 2227-2248. doi: 10.1007/s11538-007-9205-1. Google Scholar

[6]

S. BonhoefferM. Lipsitch and B. R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Nat. Acad. Sci. USA, 94 (1997), 1206-1211. doi: 10.1073/pnas.94.22.12106. Google Scholar

[7]

M. J. BontenD. C. BergmannsH. Speijer and E. Stobberingh, Characteristics of polyclonal endemicity of Pseudomonas aeruginosa colonization in intensive care units, Amer. J. Resp. Crit. Care. Med., 160 (1999), 1212-1219. Google Scholar

[8]

M. C. BootsmaO. Diekmann and M. J. Bonten, Controlling methicillin-resistant Staphylococcus aureus: Quantifying the effects of interventions and rapid diagnostic testing, Proc. Nat. Acad. Sci. USA, 103 (2006), 5620-5625. doi: 10.1073/pnas.0510077103. Google Scholar

[9]

A. Boyer, A. Doussau, R. Thièbault, A. G. Venier, V. Tran and H. Boulestreau, Pseudomonas aeruginosa acquisition in an intensive care unit: Relationship between antibiotic selective pressure and patients' environment, Crit. Care Med. , 15 (2011), R55.Google Scholar

[10]

F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology Lecture Notes in Mathematics, 45 Springer, New York, 2008.Google Scholar

[11]

C. Browne and G. F. Webb, A nosocomial epidemic model with infection of patients due to contaminated rooms, Math. Biosci. Eng., 12 (2015), 761-787. doi: 10.3934/mbe.2015.12.761. Google Scholar

[12]

L. Caudill and B. Lawson, A hybrid agent-based and differential equations model for simulating antibiotic resistance in a hospital ward, Proc. 2013 Winter Simul. Conf. IEEE, Piscataway, NJ, (2013), 1419-1430.Google Scholar

[13]

Centers for Disease Control and Prevention, Antibiotic/Antimicrobial Resistance, http://www.cdc.gov/drugresistance/, (2015), accessed May 12,2015.Google Scholar

[14]

E. M. AgataM. A. Horn and G. F. Webb, The impact of persistent gastrointestinal colonization on the transmission dynamics of vancomycin-resistant Enterococci, J. Infect. Dis., 185 (2002), 766-773. Google Scholar

[15]

E. M. AgataM. A. Horn and G. F. Webb, A mathematical model quantifying the impact of antibiotic exposure, surveillance cultures and other interventions on the prevalence of vancomycin-resistant Enterococci, J. Infect. Dis., 192 (2005), 2004-2011. Google Scholar

[16]

E. M. AgataP. MagalS. Ruan and G. F. Webb, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Nat. Acad. Sci. USA, 102 (2005), 13343-13348. Google Scholar

[17]

E. M. AgataP. MagalS. Ruan and G. F. Webb, Asymptotic behavior in nonsocomial epidemic models with antibiotic resistance, Dif. Int. Eqs., 19 (2006), 573-600. Google Scholar

[18]

E. M. AgataM. A. Horn and G. F. Webb, Quantifying the impact of bacterial fitness and repeated antimicrobial exposure on the emergence of multidrug-resistant gram-negative bacilli, Math. Mod. Nat. Phen., 2 (2007), 129-142. doi: 10.1051/mmnp:2008014. Google Scholar

[19]

E. M. AgataP. MagalD. OlivierS. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theoret. Biol., 249 (2007), 487-499. doi: 10.1016/j.jtbi.2007.08.011. Google Scholar

[20]

E. M. AgataM. A. HornR. MoelleringS. Ruan and G. F. Webb, Modeling the invasion of community-acquired methicillin-resistant Staphylococcus aureus into the hospital setting, Clin. Infect. Dis., 48 (2009), 274-284. Google Scholar

[21]

E. M. AgataM. A. HornR. MoelleringS. Ruan and G. F. Webb, Competition of hospital-acquired and community-acquired methicillin-resistant Staphylococcus aureus strains in hospitals, J. Biol. Dyn., 4 (2010), 115-129. doi: 10.1080/17513750903026411. Google Scholar

[22]

E. M. AgataJ. Pressley and G. F. Webb, Rapid emergence of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains in the hospital setting, Math. Mod. Nat. Phen., 5 (2010), 76-93. doi: 10.1051/mmnp/20105306. Google Scholar

[23]

E. M. AgataJ. Pressley and G. F. Webb, The effect of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains on competitive exclusion, J. Theoret. Biol., 264 (2010), 645-656. Google Scholar

[24]

E. M. AgataM. A. HornS. RuanJ. R. Wares and G. F. Webb, Efficacy of infection control interventions in reducing the spread of multidrug-resistant organisms in the hospital setting, PLoS One, 7 (2012). Google Scholar

[25]

T. H. DellitR. C. OwensJ. E. McGowanD. N. GerdingR. A. WeinsteinJ. P. BurkeW. C. Huskins and D. L. Paterson, Infectious Diseases Society of America and the Society for Healthcare Epidemiology of America guidelines for developing an institutional program to enhance antimicrobial stewardships, Clin. Infect. Dis., 44 (2007), 159-177. Google Scholar

[26]

J. FerrerS. Maëlle and L. Termine, NosoLink: An agent-based approach to link patient flows and staff organization with the circulation of nosocomial pathogens in an intensive care unit, Procedia Comp. Sci., 18 (2013), 1485-1494. doi: 10.1016/j.procs.2013.05.316. Google Scholar

[27]

D. T. Grima, E. M. Agata and G. F. Webb, Mathematical model of the impact of a non-antibiotic treatment for Clostridium difficile on the endemic prevalence of vancomycin-resistant Enterococci in a hospital setting, Comp. Math. Meth. Med. (2012).Google Scholar

[28]

M. HaberB. Levin and P. Kramarz, Antibiotic control of antibiotic resistance in hospitals: A simulation study, BMC Infect. Dis., 10 (2010), 254-263. doi: 10.1186/1471-2334-10-254. Google Scholar

[29]

J. R. HotchkissD. G. StrikeD. A. SimonsonA. F. Broccard and P. S. Crooke, An agent based and spatially explicit model of pathogen dissemination in the intensive care unit, Crit. Care Med., 33 (2005), 168-176. doi: 10.1097/01.CCM.0000150658.05831.D2. Google Scholar

[30]

A. HurfordA. M. MorrisD. N. Fisman and J. Wu, Linking antimicrobial prescribing to antimicrobial resistance in the ICU: Before and after an antimicrobial stewardship program, Epidemics, 4 (2012), 203-210. doi: 10.1016/j.epidem.2012.12.001. Google Scholar

[31]

M. LipsitchC. T. Bergstrom and B. R. Levin, The epidemiology of antibiotic resistance in hospitals: Paradoxes and prescriptions, Proc. Nat. Acad. Sci. USA, 97 (2000), 1938-1943. doi: 10.1073/pnas.97.4.1938. Google Scholar

[32]

E. S. McBryde and D. L. McElwain, A mathematical model investigating the impact of an environmental reservoir on the prevalence and control of vancomycin-resistant Enterococci, J. Infect. Dis., 193 (2006), 1473-1474. doi: 10.1086/503439. Google Scholar

[33]

Y. MengR. DaviesK. Hardy and P. Hawkey, An application of agent-based simulation to the management of hospital-acquired infection, J. Simulation, 4 (2010), 60-67. Google Scholar

[34]

Office of Disease Prevention and Health Promotion, National Action Plan to Prevent Health Care-Accociated Infections: Road Map to Elimination, http://www.health.gov/hcq/prevent_hai.asp, (2015), accessed May 12,2015.Google Scholar

[35]

L. PelupessyM. Bonten and O. Diekmann, How to access the relative importance of different colonization routes of pathogens within hospital settings, Proc. Nat. Acad. Sci. USA, 99 (2002), 5601-5605. Google Scholar

[36]

N. PlipatI. H. SpicknallJ. S. Koopman and J. N Eisenberg, The dynamics of methicillin-resistant Staphylococcus aureus exposure in a hospital model and the potential for environmental intervention, BMC Inf. Dis., 13 (2013), 595. Google Scholar

[37]

K. Prabaker and R. A. Weinstein, Trends in antimicrobial resistance in intensive care units in the United States, Curr. Opin. Crit. Care, 17 (2011), 474-479. doi: 10.1097/MCC.0b013e32834a4b03. Google Scholar

[38]

D. P. RaymondS. J. PelletierT. D. CrabtreeT. G. GleasonL. L. HammT. L. Pruett and R. G. Sawyer, Epidemiology of Pseudomonas aeruginosa and risk factors for carriage acquisition in an intensive care unit, J. Hosp. Infect., 53 (2003), 274-282. Google Scholar

[39]

D. L. SmithJ. DushoffE. N. PerencevichA. D. Harris and S. A. Levin, Persistent colonization and the spread of antibiotic resistance in nosocomial pathogens: resistance is a regional problem, Proc. Nat. Acad. Sci. USA, 101 (2004), 3709-3714. doi: 10.1073/pnas.0400456101. Google Scholar

[40]

L. TermineL. Kardas-SlomaL. KardasL. OpatowskiC. Brun-BuissonP.-Y. Böelle and D. Guillemot, NosoSim: An agent-based model of nosocomial pathogens circulation in hospitals, Procedia Comp. Sci., 1 (2010), 2245-2252. Google Scholar

[41]

M. ThuongK. ArvanitiR. RuimyP. de la SalmoniéreA. Scanvic-HamegJ. C. Lucet and B. Régnier, Impact of a rotating empiric antibiotic schedule on infectious mortality in an intensive care unit, Crit. Care Med., 29 (2013), 1101-1108. Google Scholar

[42]

E. van KleefJ. V. RobothamM. JitS. R. Deeny and W. J. Edmunds, Modelling the transmission of healthcare associated infections: A systematic review, BMC Infect. Dis., 13 (2013), 294-307. Google Scholar

[43]

J. L. Vincent, Nosocomial infections in adult intensive-care units, Lancet, 361 (2003), 2068-2077. doi: 10.1016/S0140-6736(03)13644-6. Google Scholar

[44]

X. WangY. XiaoJ. Wang and X. Lu, A mathematical model of effects of environmental contamination and presence of volunteers on hospital infections in China, J. Theoret. Biol., 293 (2012), 161-173. doi: 10.1016/j.jtbi.2011.10.009. Google Scholar

[45]

D. J. WeberR. Raasch and W. A. Rutala, Nosocomial infections in the ICU -the growing importance of antibiotic-resistant pathogens, Chest, 115 (1999), 34S-41S. Google Scholar

[46]

World Health Organization, Antimicrobial resistance: Global report on surveillance ISBN 978 92 4 156474 8, accessed April 30,2014.Google Scholar

show all references

References:
[1]

E. Armeanu and M. Bonten, Control of vancomycin-resistant Enterococci : One size fits all?, Clin. Inf. Dis, 41 (2005), 210-216. doi: 10.1086/431206. Google Scholar

[2]

D. J. AustinM. J. BontenR. A. WeinsteinS. Slaughter and R. M. Anderson, Vancomycin-resistant Enterococci in intensive-care hospital settings: Transmission dynamics, persistence, and the impact of infection control programs, Proc. Nat. Acad. Sci. USA, 96 (1999), 6908-6913. doi: 10.1073/pnas.96.12.6908. Google Scholar

[3]

S. BarnesB. Golden and E. Wasil, MRSA transmission reduction using agent-based modeling and simulation, INFORMS J. Comput., 22 (2010), 635-636. doi: 10.1287/ijoc.1100.0386. Google Scholar

[4]

S. BarnesB. GoldenE. Wasil and J. Furuno, Contribution of interfacility patient movement to overall methicillin-resistant Staphylococcus aureus prevalence levels, Infect. Cont. Hosp. Epid., 32 (2011), 1073-1078. doi: 10.1086/662375. Google Scholar

[5]

B. BoldinM. J. Bonten and O. Diekmann, Relative effects of barrier precautions and topical antibiotics on nosocomial bacterial transmission: results of multi-compartment models, Bull. Math. Biol., 69 (2007), 2227-2248. doi: 10.1007/s11538-007-9205-1. Google Scholar

[6]

S. BonhoefferM. Lipsitch and B. R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Nat. Acad. Sci. USA, 94 (1997), 1206-1211. doi: 10.1073/pnas.94.22.12106. Google Scholar

[7]

M. J. BontenD. C. BergmannsH. Speijer and E. Stobberingh, Characteristics of polyclonal endemicity of Pseudomonas aeruginosa colonization in intensive care units, Amer. J. Resp. Crit. Care. Med., 160 (1999), 1212-1219. Google Scholar

[8]

M. C. BootsmaO. Diekmann and M. J. Bonten, Controlling methicillin-resistant Staphylococcus aureus: Quantifying the effects of interventions and rapid diagnostic testing, Proc. Nat. Acad. Sci. USA, 103 (2006), 5620-5625. doi: 10.1073/pnas.0510077103. Google Scholar

[9]

A. Boyer, A. Doussau, R. Thièbault, A. G. Venier, V. Tran and H. Boulestreau, Pseudomonas aeruginosa acquisition in an intensive care unit: Relationship between antibiotic selective pressure and patients' environment, Crit. Care Med. , 15 (2011), R55.Google Scholar

[10]

F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology Lecture Notes in Mathematics, 45 Springer, New York, 2008.Google Scholar

[11]

C. Browne and G. F. Webb, A nosocomial epidemic model with infection of patients due to contaminated rooms, Math. Biosci. Eng., 12 (2015), 761-787. doi: 10.3934/mbe.2015.12.761. Google Scholar

[12]

L. Caudill and B. Lawson, A hybrid agent-based and differential equations model for simulating antibiotic resistance in a hospital ward, Proc. 2013 Winter Simul. Conf. IEEE, Piscataway, NJ, (2013), 1419-1430.Google Scholar

[13]

Centers for Disease Control and Prevention, Antibiotic/Antimicrobial Resistance, http://www.cdc.gov/drugresistance/, (2015), accessed May 12,2015.Google Scholar

[14]

E. M. AgataM. A. Horn and G. F. Webb, The impact of persistent gastrointestinal colonization on the transmission dynamics of vancomycin-resistant Enterococci, J. Infect. Dis., 185 (2002), 766-773. Google Scholar

[15]

E. M. AgataM. A. Horn and G. F. Webb, A mathematical model quantifying the impact of antibiotic exposure, surveillance cultures and other interventions on the prevalence of vancomycin-resistant Enterococci, J. Infect. Dis., 192 (2005), 2004-2011. Google Scholar

[16]

E. M. AgataP. MagalS. Ruan and G. F. Webb, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Nat. Acad. Sci. USA, 102 (2005), 13343-13348. Google Scholar

[17]

E. M. AgataP. MagalS. Ruan and G. F. Webb, Asymptotic behavior in nonsocomial epidemic models with antibiotic resistance, Dif. Int. Eqs., 19 (2006), 573-600. Google Scholar

[18]

E. M. AgataM. A. Horn and G. F. Webb, Quantifying the impact of bacterial fitness and repeated antimicrobial exposure on the emergence of multidrug-resistant gram-negative bacilli, Math. Mod. Nat. Phen., 2 (2007), 129-142. doi: 10.1051/mmnp:2008014. Google Scholar

[19]

E. M. AgataP. MagalD. OlivierS. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theoret. Biol., 249 (2007), 487-499. doi: 10.1016/j.jtbi.2007.08.011. Google Scholar

[20]

E. M. AgataM. A. HornR. MoelleringS. Ruan and G. F. Webb, Modeling the invasion of community-acquired methicillin-resistant Staphylococcus aureus into the hospital setting, Clin. Infect. Dis., 48 (2009), 274-284. Google Scholar

[21]

E. M. AgataM. A. HornR. MoelleringS. Ruan and G. F. Webb, Competition of hospital-acquired and community-acquired methicillin-resistant Staphylococcus aureus strains in hospitals, J. Biol. Dyn., 4 (2010), 115-129. doi: 10.1080/17513750903026411. Google Scholar

[22]

E. M. AgataJ. Pressley and G. F. Webb, Rapid emergence of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains in the hospital setting, Math. Mod. Nat. Phen., 5 (2010), 76-93. doi: 10.1051/mmnp/20105306. Google Scholar

[23]

E. M. AgataJ. Pressley and G. F. Webb, The effect of co-colonization with community-acquired and hospital-acquired methicillin-resistant Staphylococcus aureus strains on competitive exclusion, J. Theoret. Biol., 264 (2010), 645-656. Google Scholar

[24]

E. M. AgataM. A. HornS. RuanJ. R. Wares and G. F. Webb, Efficacy of infection control interventions in reducing the spread of multidrug-resistant organisms in the hospital setting, PLoS One, 7 (2012). Google Scholar

[25]

T. H. DellitR. C. OwensJ. E. McGowanD. N. GerdingR. A. WeinsteinJ. P. BurkeW. C. Huskins and D. L. Paterson, Infectious Diseases Society of America and the Society for Healthcare Epidemiology of America guidelines for developing an institutional program to enhance antimicrobial stewardships, Clin. Infect. Dis., 44 (2007), 159-177. Google Scholar

[26]

J. FerrerS. Maëlle and L. Termine, NosoLink: An agent-based approach to link patient flows and staff organization with the circulation of nosocomial pathogens in an intensive care unit, Procedia Comp. Sci., 18 (2013), 1485-1494. doi: 10.1016/j.procs.2013.05.316. Google Scholar

[27]

D. T. Grima, E. M. Agata and G. F. Webb, Mathematical model of the impact of a non-antibiotic treatment for Clostridium difficile on the endemic prevalence of vancomycin-resistant Enterococci in a hospital setting, Comp. Math. Meth. Med. (2012).Google Scholar

[28]

M. HaberB. Levin and P. Kramarz, Antibiotic control of antibiotic resistance in hospitals: A simulation study, BMC Infect. Dis., 10 (2010), 254-263. doi: 10.1186/1471-2334-10-254. Google Scholar

[29]

J. R. HotchkissD. G. StrikeD. A. SimonsonA. F. Broccard and P. S. Crooke, An agent based and spatially explicit model of pathogen dissemination in the intensive care unit, Crit. Care Med., 33 (2005), 168-176. doi: 10.1097/01.CCM.0000150658.05831.D2. Google Scholar

[30]

A. HurfordA. M. MorrisD. N. Fisman and J. Wu, Linking antimicrobial prescribing to antimicrobial resistance in the ICU: Before and after an antimicrobial stewardship program, Epidemics, 4 (2012), 203-210. doi: 10.1016/j.epidem.2012.12.001. Google Scholar

[31]

M. LipsitchC. T. Bergstrom and B. R. Levin, The epidemiology of antibiotic resistance in hospitals: Paradoxes and prescriptions, Proc. Nat. Acad. Sci. USA, 97 (2000), 1938-1943. doi: 10.1073/pnas.97.4.1938. Google Scholar

[32]

E. S. McBryde and D. L. McElwain, A mathematical model investigating the impact of an environmental reservoir on the prevalence and control of vancomycin-resistant Enterococci, J. Infect. Dis., 193 (2006), 1473-1474. doi: 10.1086/503439. Google Scholar

[33]

Y. MengR. DaviesK. Hardy and P. Hawkey, An application of agent-based simulation to the management of hospital-acquired infection, J. Simulation, 4 (2010), 60-67. Google Scholar

[34]

Office of Disease Prevention and Health Promotion, National Action Plan to Prevent Health Care-Accociated Infections: Road Map to Elimination, http://www.health.gov/hcq/prevent_hai.asp, (2015), accessed May 12,2015.Google Scholar

[35]

L. PelupessyM. Bonten and O. Diekmann, How to access the relative importance of different colonization routes of pathogens within hospital settings, Proc. Nat. Acad. Sci. USA, 99 (2002), 5601-5605. Google Scholar

[36]

N. PlipatI. H. SpicknallJ. S. Koopman and J. N Eisenberg, The dynamics of methicillin-resistant Staphylococcus aureus exposure in a hospital model and the potential for environmental intervention, BMC Inf. Dis., 13 (2013), 595. Google Scholar

[37]

K. Prabaker and R. A. Weinstein, Trends in antimicrobial resistance in intensive care units in the United States, Curr. Opin. Crit. Care, 17 (2011), 474-479. doi: 10.1097/MCC.0b013e32834a4b03. Google Scholar

[38]

D. P. RaymondS. J. PelletierT. D. CrabtreeT. G. GleasonL. L. HammT. L. Pruett and R. G. Sawyer, Epidemiology of Pseudomonas aeruginosa and risk factors for carriage acquisition in an intensive care unit, J. Hosp. Infect., 53 (2003), 274-282. Google Scholar

[39]

D. L. SmithJ. DushoffE. N. PerencevichA. D. Harris and S. A. Levin, Persistent colonization and the spread of antibiotic resistance in nosocomial pathogens: resistance is a regional problem, Proc. Nat. Acad. Sci. USA, 101 (2004), 3709-3714. doi: 10.1073/pnas.0400456101. Google Scholar

[40]

L. TermineL. Kardas-SlomaL. KardasL. OpatowskiC. Brun-BuissonP.-Y. Böelle and D. Guillemot, NosoSim: An agent-based model of nosocomial pathogens circulation in hospitals, Procedia Comp. Sci., 1 (2010), 2245-2252. Google Scholar

[41]

M. ThuongK. ArvanitiR. RuimyP. de la SalmoniéreA. Scanvic-HamegJ. C. Lucet and B. Régnier, Impact of a rotating empiric antibiotic schedule on infectious mortality in an intensive care unit, Crit. Care Med., 29 (2013), 1101-1108. Google Scholar

[42]

E. van KleefJ. V. RobothamM. JitS. R. Deeny and W. J. Edmunds, Modelling the transmission of healthcare associated infections: A systematic review, BMC Infect. Dis., 13 (2013), 294-307. Google Scholar

[43]

J. L. Vincent, Nosocomial infections in adult intensive-care units, Lancet, 361 (2003), 2068-2077. doi: 10.1016/S0140-6736(03)13644-6. Google Scholar

[44]

X. WangY. XiaoJ. Wang and X. Lu, A mathematical model of effects of environmental contamination and presence of volunteers on hospital infections in China, J. Theoret. Biol., 293 (2012), 161-173. doi: 10.1016/j.jtbi.2011.10.009. Google Scholar

[45]

D. J. WeberR. Raasch and W. A. Rutala, Nosocomial infections in the ICU -the growing importance of antibiotic-resistant pathogens, Chest, 115 (1999), 34S-41S. Google Scholar

[46]

World Health Organization, Antimicrobial resistance: Global report on surveillance ISBN 978 92 4 156474 8, accessed April 30,2014.Google Scholar

Figure 1.  Schematic diagram of the IBM patient compartments and parameters. All exiting patients are replaced immediately by an uninfected patient
Figure 2.  An example of the IBM with parameters $N_H = 4$ $N_P$ = 10, $T_V = 4 hr$ $N_V = 2$, $\omega_{Noff}=0.9$, $\omega_{Non}=0.9$, $\omega_R=0.9$, $\pi_N=0.9$, $\pi_R=0.9$, $\beta_{on}=0.6$, $\beta_{off}=0.6$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$
Figure 3.  Example 2 of the IBM with parameters $N_H = 10, $ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$. All three infected patient compartments extinguish
Figure 4.  Graphs of 50 runs of the IBM with parameters and initial conditions in Example 2. The thicker curves represent the averages of the 50 runs in each compartment. The averages of the simulations approach extinction in 30 days, but some individual runs do not
Figure 5.  Example 3 of the IBM with parameters $N_H = 10, $ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$. Only the resistant strain is prevalent after 30 days
Figure 6.  Graphs of 50 runs of the IBM with parameters and initial conditions in Example 3. The thicker curves represent the averages of the 50 runs in each compartment. The average of the resistant patient populations approaches extinction over 30 days, but the averages of the nonresistant patient populations do not
Figure 7.  Example 4 of the IBM with parameters $N_H = 10, $ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1$. Both the nonresistant and resistant strains are prevalent after 30 days
Figure 8.  Graphs of 50 runs of the IBM with parameters and initial conditions in Example 4. The thicker curves represent the averages of the 50 runs in each compartment. The averages of both nonresistant and resistant strains are prevalent after 30 days
Figure 9.  The DEM in the case that both strains extinguish. The parameters match the parameters for the IBM in Example 2: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$
Figure 10.  Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 5. The thick black curve corresponds to the graphs in Fig. 9. Trajectories oscillate as they converge to $(0, 0, 0)$. An initial increase or decrease in infected patient populations could be misinterpreted as a long-term trend
Figure 11.  The DEM in the case that only the nonresistant strain extinguishes. The parameters match the parameters for the IBM in Example 3: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$
Figure 12.  Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 6. The thick black curve corresponds to the graphs in Fig. 11. All trajectories converge to $(5.56098, 11.122, 0)$
Figure 13.  The DEM in the case that both strains become endemic. The parameters match the parameters for the IBM in Example 4: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1=1/T_R$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$
Figure 14.  Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 7. The thick black curve corresponds to the graphs in Fig. 13. All trajectories converge with oscillations to the same limiting value
Figure 15.  Graph of $R_{01}$ as a function of $\beta_{off}$ and $\pi_R$. All other parameters are as in Example 7. $R_{01}$ increases linearly with increasing $\pi_R$, but nonlinearly with increasing $\beta_{off}$. Consequently, there is advantage in stopping AB use in patients infected with the resistant strain as soon as possible. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
Figure 16.  Graph of $R_{01}$ as a function of $T_{Non}$ and $T_R$. All other parameters are as in Example 7. $R_{01}$ increases sub-linearly with increasing $T_{Non}$ and $T_R$. Consequently, there is greater effect in reducing the LOS of patients infected with the nonresistant strain on AB and patients infected with the resistant strain (also on AB) as soon as possible. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
Figure 17.  Graph of $R_{01}$ as a function of $T_V / 48$ days = visit time intervals in hours and $N_P$. All other parameters are as in Example 7. $R_{01}$ increases super-linearly with decreasing $T_V$ and $N_P$. Increasing either $T_V$ or $N_P$ results in fewer patient-HCW visits. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
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