GLOBAL ANALYSIS FOR A DELAYED SIV MODEL WITH DIRECT AND ENVIRONMENTAL TRANSMISSIONS

Tongqian Zhang; Xinzhu Meng; Tonghua Zhang

doi:10.11948/2016035

2016 Volume 6 Issue 2

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Tongqian Zhang, Xinzhu Meng, Tonghua Zhang. GLOBAL ANALYSIS FOR A DELAYED SIV MODEL WITH DIRECT AND ENVIRONMENTAL TRANSMISSIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 479-491. doi: 10.11948/2016035

Citation:

Tongqian Zhang, Xinzhu Meng, Tonghua Zhang. GLOBAL ANALYSIS FOR A DELAYED SIV MODEL WITH DIRECT AND ENVIRONMENTAL TRANSMISSIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 479-491. doi: 10.11948/2016035

GLOBAL ANALYSIS FOR A DELAYED SIV MODEL WITH DIRECT AND ENVIRONMENTAL TRANSMISSIONS

1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, P. R. China;
2 State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, P. R. China;
3 Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC 3122, Australia

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Abstract

Abstract

In this paper, we propose a new SIV epidemic model with time delay, which also involves both direct and environmental transmissions. For such model, we first introduce the basic reproduction number R by using the next generation matrix. And then global stability of the equilibria is discussed by means of Lyapunov functionals and LaSalle's invariance principle for delay differential equations, which shows that the infection-free equilibrium of the system is globally asymptotically stable if R<1 and the epidemic equilibrium of the system is globally asymptotically stable for R>1:
- Global stability /
- environmental transmission /
- time delay /
- Lyapunov functional
MSC: 34D23;34K20;92D30

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References

[1]	R. M. Anderson, C. A. Donnelly, N. M. Ferguson et al., Transmission dynamics and epidemiology of bse in british cattle, Nature, 382(1996)(6594), 779-788. Google Scholar
[2]	M. Andraud, M. Dumarest, R. Cariolet et al., Direct contact and environmental contaminations are responsible for hev transmission in pigs, Veterinary Research, 44(2013)(1), 102. Google Scholar
[3]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an sir epidemic model with distributed time delay, Nonlinear Analysis:Theory, Methods and Applications, 47(2001)(6), 4107-4115. Google Scholar
[4]	J. A. Blanchong, M. D. Samuel, D. R. Goldberg et al., Persistence of pasteurella multocida in wetlands following avian cholera outbreaks, Journal of Wildlife Diseases, 42(2006)(1), 33-39. Google Scholar
[5]	K. L. Cooke and P. Van Den Driessche, Analysis of an seirs epidemic model with two delays, Journal of Mathematical Biology, 35(1996)(2), 240-260. Google Scholar
[6]	M. J. Corbel, Brucellosis in humans and animals, Tech. rep., World Health Organization, 2006. Google Scholar
[7]	D. H. D'Souza, A. Sair, K. Williams et al., Persistence of caliciviruses on environmental surfaces and their transfer to food, International Journal of Food Microbiology, 108(2006)(1), 84-91. Google Scholar
[8]	H. Field, P. Young, J. M. Yob et al., The natural history of hendra and nipah viruses, Microbes and Infection, 3(2001)(4), 307-314. Google Scholar
[9]	J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar
[10]	J. Henning, J. Meers, P. R. Davies and R. S. Morris, Survival of rabbit haemorrhagic disease virus (rhdv) in the environment, Epidemiology and Infection, 133(2005)(4), 719-730. Google Scholar
[11]	Q. Hou, X. Sun, Y. Wang et al., Global properties of a general dynamic model for animal diseases:A case study of brucellosis and tuberculosis transmission, Journal of Mathematical Analysis and Applications, 414(2014)(1), 424-433. Google Scholar
[12]	G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured hiv infection model, SIAM Journal on Applied Mathematics, 72(2012)(1), 25-38. Google Scholar
[13]	G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay sir and seir epidemic models with nonlinear incidence rate, Bulletin of Mathematical Biology, 72(2010)(5), 1192-1207. Google Scholar
[14]	B. A. Jones, D. Grace, R. Kock and etc., Zoonosis emergence linked to agricultural intensification and environmental change, Proceedings of the National Academy of Sciences, 110(2013)(21), 8399-8404. Google Scholar
[15]	A. A. King, E. L. Ionides, M. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics, Nature, 454(2008)(7206), 877-880. Google Scholar
[16]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993. Google Scholar
[17]	M. M. Pascual, X. Rodó, S. P. Ellner et al., Cholera dynamics and el ninosouthern oscillation, Science, 289(2000)(5485), 1766-1769. Google Scholar
[18]	W. Ma, M. Song and Y. Takeuchi, Global stability of an sir epidemicmodel with time delay, Applied Mathematics Letters, 17(2004)(10), 1141-1145. Google Scholar
[19]	W. Ma, Y. Y. Takeuchi, T. Hara and E. E. Beretta, Permanence of an sir epidemic model with distributed time delays, Tohoku Mathematical Journal, 54(2002), 581-581. Google Scholar
[20]	C. C. McCluskey, Complete global stability for an sir epidemic model with delay!'distributed or discrete, Nonlinear Analysis:Real World Applications, 11(2010), 55-59. Google Scholar
[21]	X. Meng, L. Chen and B. Wu, A delay sir epidemic model with pulse vaccination and incubation times, Nonlinear Analysis:Real World Applications, 11(2010)(1), 88-98. Google Scholar
[22]	M. W. Miller, N. T. Hobbs and S. J. Tavener, Dynamics of prion disease transmission in mule deer, Ecological Applications, 16(2006)(6), 2208-2214. Google Scholar
[23]	V. J. Munster, C. Baas, P. Lexmond and etc., Spatial, temporal, and species variation in prevalence of influenza a viruses in wild migratory birds, PLoS pathogens, 3(2007)(5), e61. Google Scholar
[24]	B. Roche, J. M. Drake and P. Rohani, A general multi-strain model with environmental transmission:invasion conditions for the disease-free and endemic states, Journal of Theoretical Biology, 264(2010)(3), 729-736. Google Scholar
[25]	B. Roche, J. M. Drake and P. Rohani, The curse of the pharaoh revisited:evolutionary bi-stability in environmentally transmitted pathogens, Ecology Letters, 14(2011)(6), 569-575. Google Scholar
[26]	P. Rohani, R. Breban, D. E. Stallknecht and J. M. Drake, Environmental transmission of low pathogenicity avian influenza viruses and its implications for pathogen invasion, Proceedings of the National Academy of Sciences, 106(2009)(25), 10365-10369. Google Scholar
[27]	M. H. Roper, J. H. Vandelaer and F. L. Gasse, Maternal and neonatal tetanus, The Lancet, 370(2007)(9603), 1947-1959. Google Scholar
[28]	E. J. Routh, W. K. Clifford, W. K. Sturm and M. Boche, Stability of Motion, Taylor and Francis, London, 1975. Google Scholar
[29]	P. Van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180(2002)(1), 29-48. Google Scholar
[30]	C. T. Webb, C. P. Brooks, K. L. Gage and M. F. Antolin, Classic flea-borne transmission does not drive plague epizootics in prairie dogs, Proceedings of the National Academy of Sciences, 103(2006)(16), 6236-6241. Google Scholar
[31]	M. Webster, R Gand Peiris, H. Chen and Y. Guan, H5n1 outbreaks and enzootic influenza, Biodiversity, 7(2006)(1), 51-55. Google Scholar
[32]	Y. Xiao, R. G. Bowers, D. Clancy and N. P. French, Dynamics of infection with multiple transmission mechanisms in unmanaged/managed animal populations, Theoretical Population Biology, 71(2007)(4), 408-423. Google Scholar
[33]	J. Zhang, G. Q. Sun, X. D. Sun et al., Prediction and control of brucellosis transmission of dairy cattle in zhejiang province, china, PloS One, 9(2014)(11), e108592. Google Scholar
[34]	T. Zhang and H. Zang, Delay-induced turing instability in reaction-diffusion equations, Physical Review E, 90(2014), 052908. Google Scholar