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Influence of non-homogeneous mixing on final epidemic size in a meta-population model. (English) Zbl 1447.92410

Summary: In meta-population models for infectious diseases, the basic reproduction number \(\mathcal{R}_0\) can be as much as 70% larger in the case of preferential mixing than that in homogeneous mixing [J. W. Glasser et al., “The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study”, Lancet ID 16, No. 5, 599–605 (2016; doi:10.1016/S1473-3099(16)00004-9)]. This suggests that realistic mixing can be an important factor to consider in order for the models to provide a reliable assessment of intervention strategies. The influence of mixing is more significant when the population is highly heterogeneous. In this paper, another quantity, the final epidemic size (\(\mathcal{F}\)) of an outbreak, is considered to examine the influence of mixing and population heterogeneity. Final size relation is derived for a meta-population model accounting for a general mixing. The results show that \(\mathcal{F}\) can be influenced by the pattern of mixing in a significant way. Another interesting finding is that, heterogeneity in various sub-population characteristics may have the opposite effect on \(\mathcal{R}_0\) and \(\mathcal{F}\).

MSC:

92D30 Epidemiology
92D25 Population dynamics (general)
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[1] R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Cambridge University Press, New York, 1991. [Google Scholar]
[2] V. Andreasen, The final size of an epidemic and its relation to the basic reproduction number, Bull. Math. Biol. 73 (2011), pp. 2305-2321. doi: 10.1007/s11538-010-9623-3[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1334.92388
[3] J. Arino, F. Brauer, P. van den Driessche, J. Watmough, and J. Wu, A final size relation for epidemic models, Math. Biosci. Eng. 4 (2007), pp. 159-175. doi: 10.3934/mbe.2007.4.159[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1123.92030
[4] F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol. 70 (2008), pp. 1869-1885. doi: 10.1007/s11538-008-9326-1[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1147.92033
[5] F. Brauer, C. Castillo-Chavez, and Z. Feng, Mathematical Models in Epidemiology, Springer, 2018, to appear [Google Scholar] · Zbl 1433.92001
[6] S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk-and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol. 8 (1991), pp. 1-29. doi: 10.1093/imammb/8.1.1[Crossref], [PubMed], [Google Scholar] · Zbl 0764.92017
[7] T. Chen, R. Liu, and Q. Wang, Application of susceptible-infected-recovered model in dealing with an outbreak of acute hemorrhagic conjunctivitis on one school, Chin. J. Epidemiol. 32 (2011), pp. 723-726. [Google Scholar]
[8] T. Chen and R. Liu, Study of the efficacy of quarantine during outbreaks of acute hemorrhagic conjunctivitis at schools through the susceptive-infective-quarantine-removal-model, Chin. J. Epidemiol. 34 (2013), pp. 75-79. [Google Scholar]
[9] G. Chowell, E. Shim, F. Brauer, P. Diaz-Dueñas, J.M. Hyman, and C. Castillo-Chavez, Modelling the transmission dynamics of acute haemorrhagic conjunctivitis: Application to the 2003 outbreak in Mexico, Stat. Med. 25 (2006), pp. 1840-1857. doi: 10.1002/sim.2352[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[10] Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosci. Eng. 4 (2007), pp. 675-686. doi: 10.3934/mbe.2007.4.675[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1142.92036
[11] Z. Feng, A.N. Hill, P.J. Smith, and J.W. Glasser, An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing, J. Theor. Biol. 386 (2015), pp. 177-187. doi: 10.1016/j.jtbi.2015.09.006[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1343.92472
[12] Z. Feng, A.N. Hill, A.T. Curns, and J.W. Glasser, Evaluating targeted interventions via meta-population models with multi-level mixing, Math. Biosci. 287 (2017), pp. 93-104. doi: 10.1016/j.mbs.2016.09.013[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1377.92068
[13] J.W. Glasser, Z. Feng, S.B. Omer, P.J. Smith, and L.E. Rodewald, The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: A modelling study, Lancet ID 16 (2016), pp. 599-605. doi: 10.1016/S1473-3099(16)00004-9. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[14] J.A. Jacquez, C.P. Simon, J. Koopman, L. Sattenspiel, and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Math. Biosci. 92 (1988), pp. 119-199. doi: 10.1016/0025-5564(88)90031-4[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0686.92016
[15] J. Ma and D.J. Earn, Generality of the final size formula for an epidemic of a newly invading infectious disease, Bull. Math. Biol. 68 (2006), pp. 679-702. doi: 10.1007/s11538-005-9047-7[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1334.92419
[16] P. Magal, O. Seydi, and G. Webb, Final size of an epidemic for a two-group SIR model, SIAM J. Appl. Math. 76 (2016), pp. 2042-2059. doi: 10.1137/16M1065392[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1350.92053
[17] A. Nold, Heterogeneity in disease-transmission modeling, Math. Biosci. 52 (1980), pp. 227-240. doi: 10.1016/0025-5564(80)90069-3[Crossref], [Web of Science ®], [Google Scholar] · Zbl 0454.92020
[18] G. Poghotanyan, Z. Feng, J.W. Glasser, and A.N. Hill, Constrained minimization problems for the reproduction number in meta-population models, J. Math. Biol. (2018), pp. 1-37. doi: 10.1007/s00285-018-1216-z. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1406.37065
[19] M.J. Tildesley and M.J. Keeling, Is ##img####img####img##R0 a good predictor of final epidemic size: Foot-and-mouth disease in the UK, J. Theor. Biol. 258 (2009), pp. 623-629. doi: 10.1016/j.jtbi.2009.02.019[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1402.92414
[20] D. Tilman and P. Kareiva, Spatial Ecology, Princeton University Press, Princeton, NJ, 1998. [Crossref], [Google Scholar]
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