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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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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. Google Scholar 
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S. Milgram, The small world problem, Psychology Today, 2 (1967), 6067. Google Scholar 
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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. Google Scholar 
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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. Google Scholar 
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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. Google Scholar 
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M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167256. doi: 10.1137/S003614450342480. Google Scholar 
[22] 
R. PastorSatorras and A. Vespignani, Epidemic spreading in scalefree networks, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. Google Scholar 
[23] 
D. Shi, Q. Chen and L. Liu, Markov chainbased numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140.Google Scholar 
[24] 
E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293310. doi: 10.1007/s0028500701164. Google Scholar 
[25] 
D. J. Watts and S. H. Strogatz, Collective dynamics of "smallworld" networks, Nature, 393 (1998), 440442. Google Scholar 
[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. Google Scholar 
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. Google Scholar 
[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. Google Scholar 
[3] 
A. L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509512. doi: 10.1126/science.286.5439.509. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[13] 
P. L. Krapivsky, S. Redner and F. Leyvraz, Connectivity of growing random networks, Physical Review Letters, 85 (2000), 4629.Google Scholar 
[14] 
P. L. Krapivsky and S. Redner, Organization of growing random networks, Physical Review E, 63 (2001), 066123. doi: 10.1103/PhysRevE.63.066123. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[17] 
S. Milgram, The small world problem, Psychology Today, 2 (1967), 6067. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[21] 
M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167256. doi: 10.1137/S003614450342480. Google Scholar 
[22] 
R. PastorSatorras and A. Vespignani, Epidemic spreading in scalefree networks, Physical Review Letters, 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200. Google Scholar 
[23] 
D. Shi, Q. Chen and L. Liu, Markov chainbased numerical method for degree distributions of growing networks, Physical Review E, 71 (2005), 036140.Google Scholar 
[24] 
E. Volz, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 56 (2008), 293310. doi: 10.1007/s0028500701164. Google Scholar 
[25] 
D. J. Watts and S. H. Strogatz, Collective dynamics of "smallworld" networks, Nature, 393 (1998), 440442. Google Scholar 
[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. Google Scholar 
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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