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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021316
References:
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[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.   Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[39]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.   Google Scholar

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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