doi: 10.3934/dcdsb.2021316
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A cross-infection model with diffusion and incubation period

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

* Corresponding author

Received  July 2021 Revised  December 2021 Early access January 2022

Fund Project: This research was supported by the China Scholarship Council grant (Pang: 201906280199), the National Natural Science Foundation of China (Xiao: NSFC 11631012), and the NSERC of Canada (Zhao: RGPIN-2019-05648)

In this paper, we study a cross-infection model with diffusion and incubation period. Firstly, we prove the global attractivity of the infection-free equilibrium and infected equilibrium for the spatially homogeneous system. Secondly, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number $ \mathcal{R}_0 $. It turns out that the infection-free steady state is globally attractive if $ \mathcal{R}_0<1 $; and the system is uniformly persistent if $ \mathcal{R}_0>1 $. Finally, we explore the influence of different diffusion coefficients, spatial heterogeneity of the disease transmission rate and the incubation period on $ \mathcal{R}_0 $. Our numerical results show that $ \mathcal{R}_0 $ are decreasing functions of the diffusion coefficients and the incubation period, respectively, while it is increasing with respect to the spatial heterogeneity.

Citation: Danfeng Pang, Yanni Xiao, Xiao-Qiang Zhao. A cross-infection model with diffusion and incubation period. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021316
References:
[1]

D. J. Austin and R. Anderson, Studies of antibiotic resistance within the patient, hospitals and the community using simple mathematical models, Phil. Trans. R. Soc. Lond. Ser. B, 354 (1999), 721-738.  doi: 10.1098/rstb.1999.0425.

[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. Natl. Acad. Sci. U.S.A., 96 (1999), 6908-6913.  doi: 10.1073/pnas.96.12.6908.

[3]

Z. BaiR. Peng and X.-Q. Zhao, A reaction–diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[4]

J. M. Boyce, Environmental contamination makes an important contribution to hospital infection, J. Hosp. Infect., 65 (2007), 50-54.  doi: 10.1016/S0195-6701(07)60015-2.

[5]

J. M. BoyceG. Potter-BynoeC. Chenevert and T. King, Environmental contamination due to methicillin-resistant staphylococcus aureus possible infection control implications, Infect. Control Hosp. Epidemiol., 18 (1997), 622-627. 

[6]

CDC, Center for Disease Control and Prevention, Accessed 2015, https://www.cdc.gov/hai/data/portal/index.html.

[7]

CDC, Center for Disease Control and Prevention, Accessed March, 2014, https://www.cdc.gov/hai/dpks/hospital-infections/dpk-hai.html.

[8]

S. J. Dancer, Importance of the environment in meticillin-resistant staphylococcus aureus acquisition: The case for hospital cleaning, Lancet Infect. Dis., 8 (2008), 101-113.  doi: 10.1016/S1473-3099(07)70241-4.

[9]

S. J. Dancer, The role of environmental cleaning in the control of hospital-acquired infection, J. Hosp. Infect., 73 (2009), 378-385.  doi: 10.1016/j.jhin.2009.03.030.

[10]

H. GrundmannS. HoriB. WinterA. Tami and D. J. Austin, Risk factors for the transmission of methicillin-resistant staphylococcus aureus in an adult intensive care unit: Fitting a model to the data, J. Infect. Dis., 185 (2002), 481-488. 

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, 1988. doi: 10.1090/surv/025.

[12]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[13]

Q. HuangM. A. Horn and S. Ruan, Modeling the effect of antibiotic exposure on the transmission of methicillin-resistant staphylococcus aureus in hospitals with environmental contamination, Math. Biosci. Eng., 16 (2019), 3641-3673.  doi: 10.3934/mbe.2019181.

[14]

Q. HuangX. HuoD. Miller and S. Ruan, Modeling the seasonality of methicillin-resistant Staphylococcus aureus infections in hospitals with environmental contamination, J. Biol. Dynam., 13 (2019), 99-122.  doi: 10.1080/17513758.2018.1510049.

[15]

F. Li and X.-Q. Zhao, Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease, J. Differential Equations, 272 (2021), 127-163.  doi: 10.1016/j.jde.2020.09.019.

[16]

X. LiangL. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dynam. Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.

[17]

Y. Lou and X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[18]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[19]

R. Martin and H. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[20]

K. MischaikowH. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.

[21]

D. Pang, Y. Xiao and X.-Q. Zhao, A cross-infection model with diffusive environmental bacteria, J. Math. Anal. Appl., 505 (2022), 125637, 18 pp. doi: 10.1016/j.jmaa.2021.125637.

[22]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[24]

J. RaboudR. SaskinA. SimorM. LoebK. GreenD. E. Low and A. McGeer, Modeling transmission of methicillin-resistant staphylococcus aureus among patients admitted to a hospital, Infect. Control Hosp. Epidemiol., 26 (2005), 607-615. 

[25]

A. RamplingS. WisemanL. DavisA. HyettA. WalbridgeG. Payne and A. Cornaby, Evidence that hospital hygiene is important in the control of methicillin-resistant staphylococcus aureus, J. Hosp. Infect., 49 (2001), 109-116.  doi: 10.1053/jhin.2001.1013.

[26]

V. SebilleS. Chevret and A.-J. Valleron, Modeling the spread of resistant nosocomial pathogens in an intensive-care unit, Infect. Control Hosp. Epidemiol., 18 (1997), 84-92. 

[27]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[28]

H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[29]

H. R. Thieme, Convergence results and a poincare-bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[30]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[31]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[32]

L. Wang and S. Ruan, Modeling nosocomial infections of methicillin-resistant staphylococcus aureus with environment contamination, Scientific Reports, 7 (2017), 1-12.  doi: 10.1038/s41598-017-00261-1.

[33]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[34]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.

[35]

X. WangY. ChenW. ZhaoY. WangQ. SongH. LiuJ. ZhaoX. HanX. Hu and H. Grundmann et al., A data-driven mathematical model of multi-drug resistant acinetobacter baumannii transmission in an intensive care unit, Scientific Reports, 5 (2015), 1-8.  doi: 10.1038/srep09478.

[36]

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. Theor. Biol., 293 (2012), 161-173.  doi: 10.1016/j.jtbi.2011.10.009.

[37]

X. WangY. XiaoJ. Wang and X. Lu, Stochastic disease dynamics of a hospital infection model, Math. Biosci., 241 (2013), 115-124.  doi: 10.1016/j.mbs.2012.10.002.

[38]

D. J. Weber and W. A. Rutala, Role of environmental contamination in the transmission of vancomycin-resistant enterococci, Infect. Control Hosp. Epidemiol., 18 (1997), 306-309. 

[39]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[40]

R. Wu and X.-Q. Zhao, A reaction–diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.

[41]

Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2615-2634.  doi: 10.3934/dcdsb.2012.17.2615.

[42]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[43]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edtion, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[44]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canadian Appl. Math. Quarterly, 4 (1996), 421-444. 

show all references

References:
[1]

D. J. Austin and R. Anderson, Studies of antibiotic resistance within the patient, hospitals and the community using simple mathematical models, Phil. Trans. R. Soc. Lond. Ser. B, 354 (1999), 721-738.  doi: 10.1098/rstb.1999.0425.

[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. Natl. Acad. Sci. U.S.A., 96 (1999), 6908-6913.  doi: 10.1073/pnas.96.12.6908.

[3]

Z. BaiR. Peng and X.-Q. Zhao, A reaction–diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[4]

J. M. Boyce, Environmental contamination makes an important contribution to hospital infection, J. Hosp. Infect., 65 (2007), 50-54.  doi: 10.1016/S0195-6701(07)60015-2.

[5]

J. M. BoyceG. Potter-BynoeC. Chenevert and T. King, Environmental contamination due to methicillin-resistant staphylococcus aureus possible infection control implications, Infect. Control Hosp. Epidemiol., 18 (1997), 622-627. 

[6]

CDC, Center for Disease Control and Prevention, Accessed 2015, https://www.cdc.gov/hai/data/portal/index.html.

[7]

CDC, Center for Disease Control and Prevention, Accessed March, 2014, https://www.cdc.gov/hai/dpks/hospital-infections/dpk-hai.html.

[8]

S. J. Dancer, Importance of the environment in meticillin-resistant staphylococcus aureus acquisition: The case for hospital cleaning, Lancet Infect. Dis., 8 (2008), 101-113.  doi: 10.1016/S1473-3099(07)70241-4.

[9]

S. J. Dancer, The role of environmental cleaning in the control of hospital-acquired infection, J. Hosp. Infect., 73 (2009), 378-385.  doi: 10.1016/j.jhin.2009.03.030.

[10]

H. GrundmannS. HoriB. WinterA. Tami and D. J. Austin, Risk factors for the transmission of methicillin-resistant staphylococcus aureus in an adult intensive care unit: Fitting a model to the data, J. Infect. Dis., 185 (2002), 481-488. 

[11]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, 1988. doi: 10.1090/surv/025.

[12]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[13]

Q. HuangM. A. Horn and S. Ruan, Modeling the effect of antibiotic exposure on the transmission of methicillin-resistant staphylococcus aureus in hospitals with environmental contamination, Math. Biosci. Eng., 16 (2019), 3641-3673.  doi: 10.3934/mbe.2019181.

[14]

Q. HuangX. HuoD. Miller and S. Ruan, Modeling the seasonality of methicillin-resistant Staphylococcus aureus infections in hospitals with environmental contamination, J. Biol. Dynam., 13 (2019), 99-122.  doi: 10.1080/17513758.2018.1510049.

[15]

F. Li and X.-Q. Zhao, Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease, J. Differential Equations, 272 (2021), 127-163.  doi: 10.1016/j.jde.2020.09.019.

[16]

X. LiangL. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dynam. Differential Equations, 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.

[17]

Y. Lou and X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[18]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[19]

R. Martin and H. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[20]

K. MischaikowH. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.

[21]

D. Pang, Y. Xiao and X.-Q. Zhao, A cross-infection model with diffusive environmental bacteria, J. Math. Anal. Appl., 505 (2022), 125637, 18 pp. doi: 10.1016/j.jmaa.2021.125637.

[22]

R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.  doi: 10.1088/0951-7715/25/5/1451.

[23]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[24]

J. RaboudR. SaskinA. SimorM. LoebK. GreenD. E. Low and A. McGeer, Modeling transmission of methicillin-resistant staphylococcus aureus among patients admitted to a hospital, Infect. Control Hosp. Epidemiol., 26 (2005), 607-615. 

[25]

A. RamplingS. WisemanL. DavisA. HyettA. WalbridgeG. Payne and A. Cornaby, Evidence that hospital hygiene is important in the control of methicillin-resistant staphylococcus aureus, J. Hosp. Infect., 49 (2001), 109-116.  doi: 10.1053/jhin.2001.1013.

[26]

V. SebilleS. Chevret and A.-J. Valleron, Modeling the spread of resistant nosocomial pathogens in an intensive-care unit, Infect. Control Hosp. Epidemiol., 18 (1997), 84-92. 

[27]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.

[28]

H. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[29]

H. R. Thieme, Convergence results and a poincare-bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.

[30]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[31]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA, 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[32]

L. Wang and S. Ruan, Modeling nosocomial infections of methicillin-resistant staphylococcus aureus with environment contamination, Scientific Reports, 7 (2017), 1-12.  doi: 10.1038/s41598-017-00261-1.

[33]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[34]

W. Wang and X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015), 1142-1170.  doi: 10.1137/140981769.

[35]

X. WangY. ChenW. ZhaoY. WangQ. SongH. LiuJ. ZhaoX. HanX. Hu and H. Grundmann et al., A data-driven mathematical model of multi-drug resistant acinetobacter baumannii transmission in an intensive care unit, Scientific Reports, 5 (2015), 1-8.  doi: 10.1038/srep09478.

[36]

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. Theor. Biol., 293 (2012), 161-173.  doi: 10.1016/j.jtbi.2011.10.009.

[37]

X. WangY. XiaoJ. Wang and X. Lu, Stochastic disease dynamics of a hospital infection model, Math. Biosci., 241 (2013), 115-124.  doi: 10.1016/j.mbs.2012.10.002.

[38]

D. J. Weber and W. A. Rutala, Role of environmental contamination in the transmission of vancomycin-resistant enterococci, Infect. Control Hosp. Epidemiol., 18 (1997), 306-309. 

[39]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[40]

R. Wu and X.-Q. Zhao, A reaction–diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.

[41]

Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2615-2634.  doi: 10.3934/dcdsb.2012.17.2615.

[42]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[43]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edtion, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

[44]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Canadian Appl. Math. Quarterly, 4 (1996), 421-444. 

Figure 1.  Schematic diagram for transmission of HCWs, patients and environmental bacteria in hospital
Figure 2.  Persistence of contaminated health-care workers, the infectious patients and the environmental bacteria when $ \mathcal{R}_0 = 1.5544>1 $
Figure 3.  Elimination of the contaminated health-care workers, the infectious patients and the environmental bacteria when $ \mathcal{R}_0 = 0.4984<1 $. Here $ \gamma = 240 $, $ c = 7 $
Figure 4.  $ \mathcal{R}_0 $ vs $ D_H, D_P $ and $ D_W $
Figure 5.  The effects of spatial heterogeneity and incubation period on $ \mathcal{R}_0 $
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