The SIS model with diffusion of virus in the environment

Danfeng Pang; Yanni Xiao; Danfeng Pang; Yanni Xiao

doi:10.3934/mbe.2019141

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2852-2874. doi: 10.3934/mbe.2019141

Previous Article Next Article

Research article

The SIS model with diffusion of virus in the environment

Danfeng Pang ,
Yanni Xiao ^,

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China

Received: 09 January 2019 Accepted: 20 March 2019 Published: 02 April 2019

In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number $R_0$ which can act as threshold level to determine whether the disease persists or not. In particular, if $R_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, whereas the system is uniformly persistent for $R_0>1$. For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.
- spatial heterogeneity,
- diffusion,
- basic reproduction number,
- threshold dynamics,
- traveling wave solution
Citation: Danfeng Pang, Yanni Xiao. The SIS model with diffusion of virus in the environment[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2852-2874. doi: 10.3934/mbe.2019141

Related Papers:

Abstract

In this paper, we propose an SIS-type reaction-diffusion equations, which contains both direct transmission and indirect transmission via free-living and spatially diffusive bacteria/virus in the contaminated environment, motivated by the dynamics of hospital infections. We establish the basic reproduction number $R_0$ which can act as threshold level to determine whether the disease persists or not. In particular, if $R_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, whereas the system is uniformly persistent for $R_0>1$. For the spatially homogeneous system, we investigate the traveling wave solutions and obtain that there exists a critical wave speed, below which there has no traveling waves, above which the traveling wave solutions may exist for small diffusion coefficient by the geometric singular perturbation method. The finding implies that great spatial transmission leads to an increase in new infection, while large diffusion of bacteria/virus results in the new infection decline for spatially heterogeneous environment.

References

[1]	J. M. Boyce, G. Potter-Bynoe, C. Chenevert, et al., Environmental contamination due to methicillin-resistant Staphylococcus aureus: possible infection control implications, Infect. Control Hosp. Epidemiol., 18 (1997), 622–627.
[2]	J. M. Boyce, Environmental contamination makes an important contribution to hospital infection, J. Hosp. Infect., 65 (2007), 50–54.
[3]	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.
[4]	S. J. Dancer, The role of environmental cleaning in the control of hospital acquired infection, J. Hosp. Infect., 73 (2009), 378–385.
[5]	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.
[6]	A. Rampling, S. Wiseman, L. Davis, et al., Evidence that hospital hygiene is important in the control of methicillin-resistant Staphylococcus aureus, J. Hosp. Infect., 49 (2001), 109–116.
[7]	D. J. Austin, M. J. Bonten, R. A. Weinstein, et al., Vancomycin-resistant enterococci in intensivecare hospital settings:transmission dynamics, persistence, and the impact of infection control programs, Proc. Natl. Acad. Sci., 96 (1999), 6908–6913.
[8]	B. S. Cooper, G. F. Medley and G. M. Scott, Preliminary analysis of the transmission dynamics of nosocomial infections:stochastic and management effects, J. Hosp. Infect., 43 (1999), 131–147.
[9]	H. Grundmann, S. Hori, B. Winter, et al., Risk factors for the transmission of methicillin-resistant Staphylococcus aureusin an adult intensive care unit:fitting a model to the data, J. Infect. Dis. 185 (2002), 481–488.
[10]	J. Raboud, R. Saskin, A. Simor, et al., Modeling transmission of methicillin-resistant Staphylococcus aureusamong patients admitted to a hospital, Infect. Control Hosp. Epidemiol., 26 (2005), 607–615.
[11]	V. Sebille and A. J. Valleron, A computer simulation model for the spread of nosocomial infections caused by multidrug-resistant pathogens, Comput. Biomed. Res., 30 (1997), 307–322.
[12]	V. Sebille, S. 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.
[13]	X.Wang, Y. Xiao, J.Wang, et al., A mathematical model of effects of environmental contamination and presence of volunteers on hospital infections in China, J. Theor. Biol., 293 (2012), 161–173.
[14]	X. Wang, Y. Xiao, J. Wang, et al., Stochastic disease dynamics of a hospital infection model, Math. Biosci., 241 (2013), 115–124.
[15]	X. Wang, Y. Chen, W. Zhao, et al., A data-driven mathematical model of multi-drug resistant Acinetobacter baumannii transmission in an intensive care unit, Sci. Rep., 5 (2015), 9478.
[16]	L. J. S. Allen, B. M. Bolker, Y. Lou, et al., Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1–20. 17. K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929–946.
[17]	18. R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, J. Diff. Equ., 247 (2009), 1096–1119.
[18]	19. R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239–247.
[19]	20. R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reactiondi ffusion model: effects of epidemic risk and population movement, Phys. D, 259 (2013), 8–25.
[20]	21. R. Peng and X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2), 1451–1471.
[21]	22. Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Diff. Equ., 261 (2016), 4424–4447.
[22]	23. W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652–1673.
[23]	24. X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2017.
[24]	25. L. Zhang, Z. C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Diff. Equ., 258 (2015), 3011–3036.
[25]	26. L. Zhang and Z.C. Wang, A time-periodic reaction-diffusion epidemic model with infection period, Z. Angew. Math. Phys., 67 (2016), 117, 14 pp.
[26]	27. R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Diff. Equ., (2016), 3305–3343.
[27]	28. R. Cui, K. Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Diff. Equ., 263 (2017), 2343–2373.
[28]	29. K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of the endemic equilibria of a reactiondi ffusion-advection SIS epidemic model, Calc. Var., 56 (2017), 1–
[29]	30. H. Li, R. Peng and F.-B. Wang, Vary total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Diff. Equ., 262 (2017), 885–913.
[30]	31. H. Li, R. Peng and Z.-A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129–2153.
[31]	32. S. Bonhoeffer, R. M. May, G. M. Shaw, et al., Virus dynamics and drug therapy, Proc. Natl. Acad. Sci., 94 (1997), 6971–6976.
[32]	33. M. A. Nowak, S. Bonhoeffer, A. M. Hill, et al., Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci., 93 (1996), 4398–4402.
[33]	34. C. M. Brauner, D. Jolly, L. Lorenzi, et al., Heterogeneous viral environment in a HIV spatial model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 545–572.
[34]	35. H. Pourbashash, S. S. Pilyugin, P. De Leenheer, et al., Global analysis of within host virus models with cell-to-cell viral transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3–3357.
[35]	36. X. Ren, Y. Tian, L. Liu, et al., A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1–42.
[36]	37. F. Wang, Y. Huang and X. Zou, Global dynamics of a PDE in-host viral model, Appl. Anal., 93 (2014), 2312–2329.
[37]	38. K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78–95.
[38]	39. K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36–44.
[39]	40. J. Wang, J. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-tocell transmission, J. Math. Anal. Appl., 444 (2016), 1542–1564.
[40]	41. Y. Wu and X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Diff. Equ., 264 (2018), 4989–5024.
[41]	42. Y. Yang, J. Zhou, X. Ma, et al., Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions, Comput. Math. Appl., 72 (2016), 1013–1020.
[42]	43. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41 (1995). http://b-ok.cc/book/2625388/ 3a5dac
[43]	44. R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. AMS., 321 (1990), 1–44.
[44]	45. 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.
[45]	46. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer- Verlag, New York, 1984.
[46]	47. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[47]	48. S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dyn. Differ. Equ., 23 (2011), 817–842.
[48]	49. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
[49]	50. P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251–275.
[50]	51. S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Diff. Equ., 255 (2013), 265–297.
[51]	52. H. R. Thieme, Convergence results and a Poincar -Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763.
[52]	53. H. L. Smith and X-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169–6179.
[53]	54. P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Diff. Equ., 92 (1991), 252–281.
[54]	55. W.Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97–112.
[55]	56. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equ., 31 (1979), 53–98.
[56]	57. P. Hartman, Ordinary Differential Equations, Wiley and Sons, Baltimore, 1973. http://b-ok. cc/book/981126/b95a4b
[57]	58. X. Lai and X. Zou, Repulsion Effect on Superinfecting Virions by Infected Cells, Bull. Math. Biol., 76 (2014), 2806–2833.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)