Abstract
The effective degree SIR model describes the dynamics of diseases with lifetime acquired immunity on a static random contact network. It is typically modeled as a system of ordinary differential equations describing the probability distribution of the infection status of neighbors of a susceptible node. Such a construct may not be used to study networks with an infinite degree distribution, such as an infinite scale-free network. We propose a new generating function approach to rewrite the effective degree SIR model as a nonlinear transport type partial differential equation. We show the existence and uniqueness of the solutions the are biologically relevant. In addition we show how this model may be reduced to the Volz model with the assumption that the infection statuses of the neighbors of an susceptible node are initially independent to each other. This paper paves the way to study the stability of the disease-free steady state and the disease threshold of the infinite dimensional effective degree SIR models.
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References
Eames KTD, Keeling MJ (2002) Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc Natl Acad Sci 99:13330–13335
Hartman P (1982) Ordinary Differential Equations, 2nd edn. Birkhäuser, Boston, Basel, Stuttgart
Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653
Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions, vol 444. John Wiley & Sons, New Jersey
Kermack W, McKendrick A (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 115:700–721
Kiss IZ, Miller JC, Simon PL (2017) Mathematics of Epidemics on Networks. Springer International Publishing AG, Switzerland
Lindquist J, Ma J, van den Driessche P, Willeboordse F (2011) Effective degree network disease models. J Math Biol 62:143–164
Miller JC (2011) A note on a paper by Erik Volz: SIR dynamics in random networks. J Math Biol 62:349–358
Miller JC, Kiss IZ (2014) Epidemic spread in networks: Existing methods and current challenges. Math Model Nat Phenom 9:4–42
Pastor-Satorras R, Vespignani A (2002) Epidemic dynamics in finite size scale-free networks. Phys Rev E 65:035108(R)
van den Driessche P, Watmough J (2002) Reproduction numbers and subthreshold endemic equilibria of compartmental models for disease transmission. Math Biosci 180:29–48
Volz E (2008) SIR dynamics in random networks with heterogeneous connectivity. J Math Biol 56:293–310
Acknowledgements
J.M. was supported by National Natural Science Foundation of China 412 No. 11771075 and a NSERC Discovery grant. K.M. was supported by a NSERC CGS-M grant. S.I. was supported by NSERC research 371637-2019. The authors would like to thank the anonymous referees for their constructive comments. The alternative variable change discussed in Remark 4 is suggested by a referee.
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Ibrahim, S., Ma, J. & Manke, K. Generating function approach to the effective degree SIR model. J. Math. Biol. 84, 59 (2022). https://doi.org/10.1007/s00285-022-01764-w
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DOI: https://doi.org/10.1007/s00285-022-01764-w