Abstract
Behavioral responses triggered by the perceived risk of experiencing the disease represent a key ingredient in the spread of epidemics across human population. In this paper, two forms of individual awareness (i.e., the risk perception of an emerging epidemic) are addressed: Contact awareness that increases with individual contact number, and local awareness that increases with the fraction of infected contacts. By extending the probability generating functionology, the author shows that it is possible to track the evolution of the degree distributions among susceptible and infected individuals when the underlying network of contacts is represented by a semi-random configuration model. It is hopefully to shed some light on the dynamic aspects of networked epidemiological models.
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References
Kermack W O and McKendrick A G, A contribution to the mathematical theory of epidemics, R. Soc. Lond. Proc. Ser. A, 1927, 115: 700–721.
Brauer F and Castillo-Chávez C, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001.
Shang Y, A Lie algebra approach to susceptible-infected-susceptible epidemics, Electron. J. Differ. Equ., 2012, 2012(233): 1–7.
Allard A, Noël P A, Dubé L J, and Pourbohloul B, Heterogeneous bond percolation on multitype networks with an application to epidemic dynamics, Phys. Rev. E, 2009, 79: 036113.
Durrett R, Random Graph Dynamics, Cambridge University, Cambridge, 2006.
Keeling M, The implications of network structure for epidemic dynamics, Theor. Popul. Biol., 2005, 67: 1–8.
Keeling M and Eames K T D, Networks and epidemic models, J. R. Soc. Interface, 2005, 2: 295–307.
Meyers L A, Pourbohloul B, Newman M E J, Skowronski D M, and Brunham R C, Network theroy and SARS: Predicting outbreak diversity, J. Theor. Biol., 2005, 232: 71–81.
Pautasso M, Moslonka-Lefebvre M, and Jeger M J, The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks, Ecol. Complexity, 2010, 7: 424–432.
Shang Y, Asymptotic behavior of estimates of link probability in random networks, Rep. Math. Phys., 2011, 67: 255–257.
Altmann M, Susceptible-infected-removed epidemic models with dynamic partnerships, J. Math. Biol., 1995, 33: 661–675.
Bauch C T, A versatile ODE approximation to a network model for the spread of sexually transmitted diseases, J. Math. Biol., 2002, 45: 375–395.
Eames K T D and Keeling M J, Modeling dynamic and network heterogeneities in the spread of sexually transmitted disease, Proc. Natl. Acad. Sci. USA, 2002, 99: 13330–13335.
Simon P L, Taylor M, and Kiss I Z, Exact epidemic models on graphs using graph-automorphism driven lumping, J. Math. Biol., 2011, 62: 479–508.
Barthélémy M, Barrat A, Pastor-Satorras R, and Vespignani A, Dynamical patterns of epidemic outbreaks in complex heterogeneous networks, J. Theor. Biol., 2005, 235: 275–288.
Pastor-Satorras R and Vespignani A, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 2001, 86: 3200–3203.
Volz E, SIR dynamics in random networks with heterogeneous connectivity, J. Math. Biol., 2008, 56: 293–310.
Molloy M and Reed B, A critical point for random graphs with a given degree sequence, Random Struct. Algor., 1995, 6: 161–179.
Shang Y, Mixed SI(R) epidemic dynamics in random graphs with general degree distributions, Appl. Math. Comput., 2013, 219: 5042–5048.
Shang Y, Distribution dynamics for SIS model on random networks, J. Biol. Syst., 2012, 20: 213–220.
Ferguson N, Capturing human behaviour, Nature, 2007, 446(7137): 733.
Reluga T C, Game theory of social distancing in response to an epidemic, PLoS Comput. Biol., 2010, 6: e1000793.
Tracht S M, Del Valle S Y, and Hyman J M, Mathematical modelling of the effectiveness of facemasks in reducing the spread of novel influenza A(H1N1), PLoS One, 2010, 5: e9018.
Funk S, Gilad E, Watkins C, and Jansen V A A, The spread of awareness and its impact on epidemic outbreaks, Proc. Natl. Acad. Sci. U.S.A, 2009, 106: 6872–6877.
Bagnoli F, Liò P, and Sguanci L, Risk perception in epidemic modeling, Phys. Rev. E, 2007, 76: 061904.
Kitchovitch S and Liò P, Risk perception and disease spread on social networks, Procedia Comput. Sci., 2010, 1: 2339–2348.
Wu Q, Fu X, Small M, and Xu X J, The impact of awareness on epidemic spreading in networks, Chaos, 2012, 22: 013101.
Shang Y, Modeling epidemic spread with awareness and heterogeneous transmission rates in networks, J. Biol. Phys., 2013, 39(3): 489–500.
Funk S, Salathé M, and Jansen V A A, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J. R. Soc. Interface, 2010, 7: 1247–1256.
Funk S, Gilad E, and Jansen V A A, Endemic disease, awareness, and local behavioural response, J. Theor. Biol., 2010, 264: 501–509.
Janson S, The probability that a random multigraph is simple, Comb. Probab. Comput., 2009, 18: 205–225.
Shang Y, Efficient strategies for attack via partial information in scale-free networks, Inf. Sci. Lett., 2012, 1: 1–5.
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This research was supported by Air Force Office of Scientific Research under Grant No. FA9550-09-1-0165.
This paper was recommended for publication by Editor SUN Liuquan.
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Shang, Y. Degree distribution dynamics for disease spreading with individual awareness. J Syst Sci Complex 28, 96–104 (2015). https://doi.org/10.1007/s11424-014-2186-x
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DOI: https://doi.org/10.1007/s11424-014-2186-x