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Global dynamics of a delay virus model with recruitment and saturation effects of immune responses
Structural calculations and propagation modeling of growing networks based on continuous degree
a.  Department of Mathematics, Shanghai University, Shanghai 200444, China 
b.  Complex Systems Research Center, Shanxi University, Taiyuan 030051, Shanxi, China 
When a network reaches a certain size, its node degree can be considered as a continuous variable, which we will call continuous degree. Using continuous degree method (CDM), we analytically calculate certain structure of the network and study the spread of epidemics on a growing network. Firstly, using CDM we calculate the degree distributions of three different growing models, which are the BA growing model, the preferential attachment accelerating growing model and the random attachment growing model. We obtain the evolution equation for the cumulative distribution function $F(k,t)$, and then obtain analytical results about $F(k,t)$ and the degree distribution $p(k,t)$. Secondly, we calculate the joint degree distribution $p(k_1, k_2, t)$ of the BA model by using the same method, thereby obtain the conditional degree distribution $p (k_1k_2) $. We find that the BA model has no degree correlations. Finally, we consider the different states, susceptible and infected, according to the node health status. We establish the continuous degree SIS model on a static network and a growing network, respectively. We find that, in the case of growth, the new added health nodes can slightly reduce the ratio of infected nodes, but the final infected ratio will gradually tend to the final infected ratio of SIS model on static networks.
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S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Structure of growing networks with preferential linking, Physical Review Letters, 85 (2000), 4633. doi: 10.1103/PhysRevLett.85.4633. 
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P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Physical Review Letters, 85 (2000), 4629. 
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J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011), 143164. doi: 10.1007/s0028501003312. 
[16] 
C. Liu, J. Xie, H. Chen, H. Zhang and M. Tang, Interplay between the local information based behavioral responses and the epidemic spreading in complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 103111, 7 pp. doi: 10.1063/1.4931032. 
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S. Milgram, The small world problem, Psychology Today, 2 (1967), 6067. 
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J. C. Miller, A. C. Slim and E. M. Volz, Edgebased compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012), 890906. 
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J. C. Miller and I. Z. Kiss, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena, 9 (2014), 442. doi: 10.1051/mmnp/20149202. 
[20] 
Y. Moreno, R. PastorSatorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal BCondensed Matter and Complex Systems, 26 (2002), 521529. doi: 10.1140/epjb/e20020122. 
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M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167256. doi: 10.1137/S003614450342480. 
[22] 
R. PastorSatorras and A. Vespignani, Epidemic spreading in scalefree networks, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. 
[23] 
D. Shi, Q. Chen and L. Liu, Markov chainbased numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140. 
[24] 
E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293310. doi: 10.1007/s0028500701164. 
[25] 
D. J. Watts and S. H. Strogatz, Collective dynamics of "smallworld" networks, Nature, 393 (1998), 440442. 
[26] 
H. Zhang, J. Xie, M. Tang and Y. Lai, Suppression of epidemic spreading in complex networks by local information based behavioral responses, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014), 043106, 7 pp. doi: 10.1063/1.4896333. 
show all references
References:
[1] 
R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 4797. doi: 10.1103/RevModPhys.74.47. 
[2] 
A. L. Barabási, R. Albert and H. Jeong, Meanfield theory for scalefree random networks, Physica A: Statistical Mechanics and its Applications, 272 (1999), 173187. 
[3] 
A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509512. doi: 10.1126/science.286.5439.509. 
[4] 
S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Structure of growing networks with preferential linking, Physical Review Letters, 85 (2000), 4633. doi: 10.1103/PhysRevLett.85.4633. 
[5] 
S. N. Dorogovtsev and J. F. F. Mendes, Scaling properties of scalefree evolving networks: Continuous approach, Physical Review E, 63 (2001), 056125. doi: 10.1103/PhysRevE.63.056125. 
[6] 
S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW, Oxford University Press, New York, 2013. 
[7] 
P. Erdős and A. Rényi, On the strength of connectedness of a random graph, Acta Mathematica Hungarica, 12 (1961), 261267. doi: 10.1007/BF02066689. 
[8] 
M. Faloutsos, P. Faloutsos and C. Faloutsos, On powerlaw relationships of the internet topology, ACM SIGCOMM Computer Communication Review, 29 (1999), 251262. doi: 10.1145/316188.316229. 
[9] 
M. J. Gagen and J. S. Mattick, Accelerating, hyperaccelerating, and decelerating networks, Physical Review E, 72 (2005), 016123. doi: 10.1103/PhysRevE.72.016123. 
[10] 
T. House and M. J. Keeling, Insights from unifying modern approximations to infections on networks, Journal of The Royal Society Interface, 8 (2011), 6773. doi: 10.1098/rsif.2010.0179. 
[11] 
M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 859867. doi: 10.1098/rspb.1999.0716. 
[12] 
K. T. D. Ken and M. J. Keeling, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases, Proceedings of the National Academy of Sciences, 99 (2002), 1333013335. 
[13] 
P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Physical Review Letters, 85 (2000), 4629. 
[14] 
P. L. Krapivsky and S. Redner, Organization of growing random networks, Physical Review E, 63 (2001), 066123. doi: 10.1103/PhysRevE.63.066123. 
[15] 
J. Lindquist, J. Ma, P. van den Driessche and F. H. Willeboordse, Effective degree network disease models, Journal of Mathematical Biology, 62 (2011), 143164. doi: 10.1007/s0028501003312. 
[16] 
C. Liu, J. Xie, H. Chen, H. Zhang and M. Tang, Interplay between the local information based behavioral responses and the epidemic spreading in complex networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 25 (2015), 103111, 7 pp. doi: 10.1063/1.4931032. 
[17] 
S. Milgram, The small world problem, Psychology Today, 2 (1967), 6067. 
[18] 
J. C. Miller, A. C. Slim and E. M. Volz, Edgebased compartmental modelling for infectious disease spread, Journal of the Royal Society Interface, 9 (2012), 890906. 
[19] 
J. C. Miller and I. Z. Kiss, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena, 9 (2014), 442. doi: 10.1051/mmnp/20149202. 
[20] 
Y. Moreno, R. PastorSatorras and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical Journal BCondensed Matter and Complex Systems, 26 (2002), 521529. doi: 10.1140/epjb/e20020122. 
[21] 
M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167256. doi: 10.1137/S003614450342480. 
[22] 
R. PastorSatorras and A. Vespignani, Epidemic spreading in scalefree networks, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. 
[23] 
D. Shi, Q. Chen and L. Liu, Markov chainbased numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140. 
[24] 
E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293310. doi: 10.1007/s0028500701164. 
[25] 
D. J. Watts and S. H. Strogatz, Collective dynamics of "smallworld" networks, Nature, 393 (1998), 440442. 
[26] 
H. Zhang, J. Xie, M. Tang and Y. Lai, Suppression of epidemic spreading in complex networks by local information based behavioral responses, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014), 043106, 7 pp. doi: 10.1063/1.4896333. 
Notation  Definition 
 The total number of nodes in the initial network. 
 The total number of edges in the initial network. 
 The probability for the node 
 The total number of nodes at time 
 The total number of nodes which degree not more than 
 The cumulative distribution function of node degree, or the proportion of nodes which degree not more than 
 The degree distribution, or the probability density of node which degree equal to 
 The total number of directed edges at time 
 The total number of directed edges which degree sequentially not larger than 
 The joint cumulative distribution function at time 
 The joint degree distribution at time 
 The conditional degree distribution at time 
 The marginal distribution at time 
 The cumulative distribution function of susceptible nodes at time 
 The cumulative distribution function of infected nodes at time 
 The probability density of susceptible nodes which degree equal to 
 The probability density of infected nodes which degree equal to 
 The probability that a edge emitted by degree 
 The probability that a edge points to an infected node in degree unrelated network. 
Notation  Definition 
 The total number of nodes in the initial network. 
 The total number of edges in the initial network. 
 The probability for the node 
 The total number of nodes at time 
 The total number of nodes which degree not more than 
 The cumulative distribution function of node degree, or the proportion of nodes which degree not more than 
 The degree distribution, or the probability density of node which degree equal to 
 The total number of directed edges at time 
 The total number of directed edges which degree sequentially not larger than 
 The joint cumulative distribution function at time 
 The joint degree distribution at time 
 The conditional degree distribution at time 
 The marginal distribution at time 
 The cumulative distribution function of susceptible nodes at time 
 The cumulative distribution function of infected nodes at time 
 The probability density of susceptible nodes which degree equal to 
 The probability density of infected nodes which degree equal to 
 The probability that a edge emitted by degree 
 The probability that a edge points to an infected node in degree unrelated network. 
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