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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. [2] D. J. Austin, M. J. Bonten, R. A. Weinstein, S. 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. [3] S. Barnes, B. 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. [4] S. Barnes, B. Golden, E. 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. [5] B. Boldin, M. 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. [6] S. Bonhoeffer, M. 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. [7] M. J. Bonten, D. C. Bergmanns, H. 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. [8] M. C. Bootsma, O. 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. [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. [10] F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology Lecture Notes in Mathematics, 45 Springer, New York, 2008. [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. [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. [13] Centers for Disease Control and Prevention, Antibiotic/Antimicrobial Resistance, http://www.cdc.gov/drugresistance/, (2015), accessed May 12,2015. [14] E. M. Agata, M. 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. [15] E. M. Agata, M. 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. [16] E. M. Agata, P. Magal, S. Ruan and G. F. Webb, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Nat. Acad. Sci. USA, 102 (2005), 13343-13348. [17] E. M. Agata, P. Magal, S. Ruan and G. F. Webb, Asymptotic behavior in nonsocomial epidemic models with antibiotic resistance, Dif. Int. Eqs., 19 (2006), 573-600. [18] E. M. Agata, M. 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. [19] E. M. Agata, P. Magal, D. Olivier, S. 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. [20] E. M. Agata, M. A. Horn, R. Moellering, S. 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. [21] E. M. Agata, M. A. Horn, R. Moellering, S. 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. [22] E. M. Agata, J. 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. [23] E. M. Agata, J. 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. [24] E. M. Agata, M. A. Horn, S. Ruan, J. 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). [25] T. H. Dellit, R. C. Owens, J. E. McGowan, D. N. Gerding, R. A. Weinstein, J. P. Burke, W. 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. [26] J. Ferrer, S. 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. [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). [28] M. Haber, B. 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. [29] J. R. Hotchkiss, D. G. Strike, D. A. Simonson, A. 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. [30] A. Hurford, A. M. Morris, D. 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. [31] M. Lipsitch, C. 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. [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. [33] Y. Meng, R. Davies, K. Hardy and P. Hawkey, An application of agent-based simulation to the management of hospital-acquired infection, J. Simulation, 4 (2010), 60-67. [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. [35] L. Pelupessy, M. 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. [36] N. Plipat, I. H. Spicknall, J. 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. [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. [38] D. P. Raymond, S. J. Pelletier, T. D. Crabtree, T. G. Gleason, L. L. Hamm, T. 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. [39] D. L. Smith, J. Dushoff, E. N. Perencevich, A. 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. [40] L. Termine, L. Kardas-Sloma, L. Kardas, L. Opatowski, C. Brun-Buisson, P.-Y. Böelle and D. Guillemot, NosoSim: An agent-based model of nosocomial pathogens circulation in hospitals, Procedia Comp. Sci., 1 (2010), 2245-2252. [41] M. Thuong, K. Arvaniti, R. Ruimy, P. de la Salmoniére, A. Scanvic-Hameg, J. 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. [42] E. van Kleef, J. V. Robotham, M. Jit, S. R. Deeny and W. J. Edmunds, Modelling the transmission of healthcare associated infections: A systematic review, BMC Infect. Dis., 13 (2013), 294-307. [43] J. L. Vincent, Nosocomial infections in adult intensive-care units, Lancet, 361 (2003), 2068-2077. doi: 10.1016/S0140-6736(03)13644-6. [44] X. Wang, Y. Xiao, J. 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. [45] D. J. Weber, R. Raasch and W. A. Rutala, Nosocomial infections in the ICU -the growing importance of antibiotic-resistant pathogens, Chest, 115 (1999), 34S-41S. [46] World Health Organization, Antimicrobial resistance: Global report on surveillance ISBN 978 92 4 156474 8, accessed April 30,2014.

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. [2] D. J. Austin, M. J. Bonten, R. A. Weinstein, S. 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. [3] S. Barnes, B. 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. [4] S. Barnes, B. Golden, E. 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. [5] B. Boldin, M. 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. [6] S. Bonhoeffer, M. 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. [7] M. J. Bonten, D. C. Bergmanns, H. 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. [8] M. C. Bootsma, O. 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. [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. [10] F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology Lecture Notes in Mathematics, 45 Springer, New York, 2008. [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. [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. [13] Centers for Disease Control and Prevention, Antibiotic/Antimicrobial Resistance, http://www.cdc.gov/drugresistance/, (2015), accessed May 12,2015. [14] E. M. Agata, M. 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. [15] E. M. Agata, M. 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. [16] E. M. Agata, P. Magal, S. Ruan and G. F. Webb, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Nat. Acad. Sci. USA, 102 (2005), 13343-13348. [17] E. M. Agata, P. Magal, S. Ruan and G. F. Webb, Asymptotic behavior in nonsocomial epidemic models with antibiotic resistance, Dif. Int. Eqs., 19 (2006), 573-600. [18] E. M. Agata, M. 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. [19] E. M. Agata, P. Magal, D. Olivier, S. 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. [20] E. M. Agata, M. A. Horn, R. Moellering, S. 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. [21] E. M. Agata, M. A. Horn, R. Moellering, S. 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. [22] E. M. Agata, J. 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. [23] E. M. Agata, J. 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. [24] E. M. Agata, M. A. Horn, S. Ruan, J. 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). [25] T. H. Dellit, R. C. Owens, J. E. McGowan, D. N. Gerding, R. A. Weinstein, J. P. Burke, W. 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. [26] J. Ferrer, S. 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. [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). [28] M. Haber, B. 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. [29] J. R. Hotchkiss, D. G. Strike, D. A. Simonson, A. 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. [30] A. Hurford, A. M. Morris, D. 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. [31] M. Lipsitch, C. 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. [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. [33] Y. Meng, R. Davies, K. Hardy and P. Hawkey, An application of agent-based simulation to the management of hospital-acquired infection, J. Simulation, 4 (2010), 60-67. [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. [35] L. Pelupessy, M. 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. [36] N. Plipat, I. H. Spicknall, J. 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. [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. [38] D. P. Raymond, S. J. Pelletier, T. D. Crabtree, T. G. Gleason, L. L. Hamm, T. 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. [39] D. L. Smith, J. Dushoff, E. N. Perencevich, A. 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. [40] L. Termine, L. Kardas-Sloma, L. Kardas, L. Opatowski, C. Brun-Buisson, P.-Y. Böelle and D. Guillemot, NosoSim: An agent-based model of nosocomial pathogens circulation in hospitals, Procedia Comp. Sci., 1 (2010), 2245-2252. [41] M. Thuong, K. Arvaniti, R. Ruimy, P. de la Salmoniére, A. Scanvic-Hameg, J. 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. [42] E. van Kleef, J. V. Robotham, M. Jit, S. R. Deeny and W. J. Edmunds, Modelling the transmission of healthcare associated infections: A systematic review, BMC Infect. Dis., 13 (2013), 294-307. [43] J. L. Vincent, Nosocomial infections in adult intensive-care units, Lancet, 361 (2003), 2068-2077. doi: 10.1016/S0140-6736(03)13644-6. [44] X. Wang, Y. Xiao, J. 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. [45] D. J. Weber, R. Raasch and W. A. Rutala, Nosocomial infections in the ICU -the growing importance of antibiotic-resistant pathogens, Chest, 115 (1999), 34S-41S. [46] World Health Organization, Antimicrobial resistance: Global report on surveillance ISBN 978 92 4 156474 8, accessed April 30,2014.
Schematic diagram of the IBM patient compartments and parameters. All exiting patients are replaced immediately by an uninfected patient
An example of the IBM with parameters $N_H = 4$ $N_P$ = 10, $T_V = 4 hr$ $N_V = 2$, $\omega_{Noff}=0.9$, $\omega_{Non}=0.9$, $\omega_R=0.9$, $\pi_N=0.9$, $\pi_R=0.9$, $\beta_{on}=0.6$, $\beta_{off}=0.6$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$
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
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
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
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
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
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
The DEM in the case that both strains extinguish. The parameters match the parameters for the IBM in Example 2: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$
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
The DEM in the case that only the nonresistant strain extinguishes. The parameters match the parameters for the IBM in Example 3: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$
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 $(5.56098, 11.122, 0)$
The DEM in the case that both strains become endemic. The parameters match the parameters for the IBM in Example 4: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1=1/T_R$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$
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
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 $R_{01}=1.0$
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 $R_{01}=1.0$
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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