Effect of adaptive rewiring delay in an SIS network epidemic model

Jing Li; Zhen Jin; Yuan Yuan; Jing Li; Zhen Jin; Yuan Yuan

doi:10.3934/mbe.2019407

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 8092-8108. doi: 10.3934/mbe.2019407

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Research article

Effect of adaptive rewiring delay in an SIS network epidemic model

Jing Li ^1,2,3,
Zhen Jin ^{2,3
,
,},
Yuan Yuan ^{2,4
,
,}

1.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan Shanxi 030006, China;
2.
Complex Systems Research Center, Shanxi University, Taiyuan Shanxi 030006, China;
3.
Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, Shanxi 030006, China;
4.
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, A1C 5S7, Canada

Received: 01 February 2019 Accepted: 12 July 2019 Published: 06 September 2019

In the real world, in order to avoid the infection risk, people tend to cut off the links with their infected neighbors, then look for other susceptible individuals to rewire. However, the rewiring process does not occur immediately, but takes some time. We therefore establish a delayed SIS network model with adaptive rewiring mechanism and analyze the long-term steady states for the system with and without the rewiring delay. We find that with the rewiring time, there are infinite equilibria lie on a line in a high-dimensional state space, which is quite different from normal delayed model. The numerical simulation results show that the system approaches to different steady state on the line under the same initial values and different rewiring delays, and the stable limit cycle can appear with the increase of rewiring delay. These surprising results may provide new insights into the study of delayed network epidemic model.
- SIS network epidemic model,
- adaptive rewiring,
- time delay,
- line equilibria
Citation: Jing Li, Zhen Jin, Yuan Yuan. Effect of adaptive rewiring delay in an SIS network epidemic model[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8092-8108. doi: 10.3934/mbe.2019407

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Abstract

In the real world, in order to avoid the infection risk, people tend to cut off the links with their infected neighbors, then look for other susceptible individuals to rewire. However, the rewiring process does not occur immediately, but takes some time. We therefore establish a delayed SIS network model with adaptive rewiring mechanism and analyze the long-term steady states for the system with and without the rewiring delay. We find that with the rewiring time, there are infinite equilibria lie on a line in a high-dimensional state space, which is quite different from normal delayed model. The numerical simulation results show that the system approaches to different steady state on the line under the same initial values and different rewiring delays, and the stable limit cycle can appear with the increase of rewiring delay. These surprising results may provide new insights into the study of delayed network epidemic model.

References

[1]	M. Faloutsos, P. Faloutsos and C. Faloutsos, On power-law relationships of the internet topology, Comput. Commun. Rev., 29 (1999), 251-262.
[2]	V. Colizza, A. Barrat, M. Barthélemy, et al., The role of the airline transportation network in the prediction and predictability of global epidemics, Proc. Nati. Acad. Sci. USA, 103 (2006), 2015-2020.
[3]	M. E. J. Newman, The structure of scientific collaboration networks, Proc. Nati. Acad. Sci. USA, 98 (2001), 404-409.
[4]	M. E. J. Newman, Scientific collaboration networks. I. Network construction and fundamental results, Phys. Rev. E, 64 (2001), 016131.
[5]	P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Phys. Rev. Lett., 85 (2000), 4629-4632.
[6]	S. N. Dorogovtsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach, Phys. Rev. E, 63 (2001), 056125.
[7]	G. Kossinets and D. J. Watts, Empirical analysis of an evolving social network, Science, 311(2006), 88-90.
[8]	A. Clauset, C. R. Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Rev., 51 (2009), 661-703.
[9]	M. Kivela, A. Arenäs, M. Barthélemy, et al., Multilayer network, J. Comp. Net., 2 (2014), 203-271.
[10]	S. Boccaletti, G. Bianconi, R. Criado, et al., The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), 1-122.
[11]	V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks, Phys. Rev. Lett., 99 (2007), 148701.
[12]	N. Masuda, Effects of diffusion rates on epidemic spreads in metapopulation networks, New J. Phys., 12 (2010), 093009.
[13]	W. K. V. Chan and C. Hsu, Service scaling on hyper-networks, Serv. Science, 1 (2009), 17-31.
[14]	A. Trilla, G. Trilla and C. Daer, The 1918 "Spanish flu" in Spain, Clin. Infect. Dis., 47 (2008), 668-673.
[15]	M. A. Marra, S. J. Jones, C. R. Astell, et al., The genome sequence of the SARS-associated coronavirus, Science, 300 (2003), 1399-1404.
[16]	J. S. Peiris, S. T. Lai, L. L. Poon, et al., Coronavirus as a possible cause of severe acute respiratory syndrome, Lancet, 361 (2003), 1319-1325. doi: 10.1016/S0140-6736(03)13077-2
[17]	T. R. Frieden, I. Damon, B. P. Bell, et al., Ebola 2014-new challenges, new global response and responsibility, New Engl. J. Med., 371 (2014), 1177-1180.
[18]	W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721.
[19]	N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, Hafner Press, New York, 1975.
[20]	R. Anderson and R. May, Infectious disease of human: dynamic and control, Press Oxford, Oxford University, 1991.
[21]	M. Martcheva, Introduction to Mathematical Epidemiology, Springer-Verlag, New York, 2010.
[22]	I. Al-Darabsah and Y. Yuan, A time-delayed epidemic model for Ebola disease transmission, Appl. Math. Comput., 290 (2016), 307-325.
[23]	A. S. Klovdahl, Social networks and the spread of infectious diseases£the AIDS example, Soc. Sci. Med., 21 (1985), 1203-1216.
[24]	R. M. May and R. M. Andemon, Transmission dynamics of HIV infection, Nature, 326 (1987), 137-142.
[25]	O. Diekmann, M. C. M. De Jong and J. A. J. Metz, A deterministic epidemic model taking account of repeated contacts between the same Individuals, J. Appl. Prob., 35 (1998), 448-462.
[26]	Y. Moreno, R. Pastor-Satorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, Euro. Phys. J. B, 26 (2002), 521-529.
[27]	E. Volz, SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 56(2008) 293-310.
[28]	C. T. Bauch, The spread of infectious diseases in spatially structured populations: an invasory pair approximation, Math. Biosci., 198 (2005), 217-237.
[29]	C. Moore and M. E. J. Newman, Epidemics and percolation in small-world networks, Phys. Rev. E, 61 (2000), 5678-5682.
[30]	R. Pastor-Satorras and A. Vespignani, Epidemic dynamics and endemic states in complex networks, Phys. Rev. E, 63 (2001) 066117.
[31]	N. M. Ferguson, D. A. T. Cummings, C. Fraser, et al., Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452.
[32]	T. Gross, C. J. D. DLimaa and B. Blasius, Epidemic Dynamics on an Adaptive Network, Phys. Rev. Lett. 96 (2006), 208701.
[33]	D. H. Zanette and S. Risau-Gusmán, Infection spreading in a population with evolving contacts, J. Biol. Phys., 34 (2008), 135-148.
[34]	L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Phys. Rev. E 77(2008), 066101.
[35]	A. Szabó-Solticzky, L. Berthouze, I. Z. Kiss, et al., Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis, J. Math. Biol. 72 (2016), 1153-1176.
[36]	J. Li, Z. Jin, Y. Yuan, et al., A non-Markovian SIR network model with fixed infectious period and preventive rewiring, Comput. Math. Appl., 75 (2018), 3884-3902.
[37]	I. Z. Kiss, L. Berthouze, T. J. Taylor, et al., Modelling approaches for simple dynamic networks and applications to disease transmission models, Proc. R. Soc. A, 468 (2012), 1332-1355.
[38]	T. Rogers, W. Clifford-Brown, C. Mills, et al., Stochastic oscillations of adaptive networks: application to epidemic modelling, J. Stat. Mech., 2012 (2012), P08018.
[39]	S. Risau-Gusman and D. H. Zanette, Contact switching as a control strategy for epidemic outbreaks, J. Theor. Biol., 257 (2009), 52-60.
[40]	M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proc. R. Soc. Lond. B, 266 (1999), 859-867.
[41]	X. Zhang, C. Shan, Z. Jin, et al., Complex dynamics of epidemic models on adaptive networks,Journal of differential equations, J. Differ. Equations, 266 (2019), 803-832.
[42]	J. Graef, M. Li and L. Wang, A study on the effects of disease caused death in a simple epidemic model, in Dyn. Syst. Differ. Equations (eds. W. Chen and S Hu), Southwest Missouri State University Press, (1998), 288-300.
[43]	M. Y. Li, W. Liu, C. Shan, et al., Turning Points And Relaxation Oscillation Cycles in Simple Epidemic Models, SIAM J. Appl. Math., 76 (2016), 663-687.

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