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. 2009 Jun 29:6:11.
doi: 10.1186/1742-4682-6-11.

Models of epidemics: when contact repetition and clustering should be included

Timo Smieszek  1 , Lena Fiebig, Roland W Scholz
Affiliations

Models of epidemics: when contact repetition and clustering should be included

Timo Smieszek et al. Theor Biol Med Model. .

Abstract

Background: The spread of infectious disease is determined by biological factors, e.g. the duration of the infectious period, and social factors, e.g. the arrangement of potentially contagious contacts. Repetitiveness and clustering of contacts are known to be relevant factors influencing the transmission of droplet or contact transmitted diseases. However, we do not yet completely know under what conditions repetitiveness and clustering should be included for realistically modelling disease spread.

Methods: We compare two different types of individual-based models: One assumes random mixing without repetition of contacts, whereas the other assumes that the same contacts repeat day-by-day. The latter exists in two variants, with and without clustering. We systematically test and compare how the total size of an outbreak differs between these model types depending on the key parameters transmission probability, number of contacts per day, duration of the infectious period, different levels of clustering and varying proportions of repetitive contacts.

Results: The simulation runs under different parameter constellations provide the following results: The difference between both model types is highest for low numbers of contacts per day and low transmission probabilities. The number of contacts and the transmission probability have a higher influence on this difference than the duration of the infectious period. Even when only minor parts of the daily contacts are repetitive and clustered can there be relevant differences compared to a purely random mixing model.

Conclusion: We show that random mixing models provide acceptable estimates of the total outbreak size if the number of contacts per day is high or if the per-contact transmission probability is high, as seen in typical childhood diseases such as measles. In the case of very short infectious periods, for instance, as in Norovirus, models assuming repeating contacts will also behave similarly as random mixing models. If the number of daily contacts or the transmission probability is low, as assumed for MRSA or Ebola, particular consideration should be given to the actual structure of potentially contagious contacts when designing the model.

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Figures

Figure 1
Figure 1
State transitions and contact structures. Subfigure a: Two transitions are allowed between three different states an individual can take: (S)usceptible to (I)nfectious and (I)nfectious to (R)ecovered. β denotes the transmission probability of one susceptible-infectious pair per time step. i stands for the number of infectious contacts that a specific susceptible individual has at the current time step. t gives the current simulation time, whereas tinf gives the time step at which the individual was infected. τ is the infectiousperiod. Subfigure b: We compare two model types: the contacts in the first type change daily while those in the second type are constant over time. The second model type assuming repetitive contacts exists in the two variants 2a and 2b.
Figure 2
Figure 2
Model differences depending on τ, n and β. Subfigures a-c show the difference in the total outbreak size between a pure random mixing model and a model assuming complete repetitiveness (without clustering) relative to the population size N. Contour plots are interpolated from a grid of measurement points using Microsoft® Office Excel 2003. (a) infectious period: 2 ≤ τ ≤ 14, step width (sw): sw = 1; daily number of contacts: 4 ≤ n ≤ 20, sw = 2; per-contact transmission probability: β·n·τ = 1.6. (b) 1.2 ≤ β·n·τ ≤ 4.0, sw = .2; 4 ≤ n ≤ 20, sw = 2; τ = 14. (c) 1.2 ≤ β·n·τ ≤ 4.0, sw = .2; 2 ≤ τ ≤ 14, sw = 1; n = 4.
Figure 3
Figure 3
Ratio of the basic reproduction numbers. Subfigure a shows the ratio R0,rep/R0,ran (as defined in equation 1) for 1 ≤ n ≤ 20 (number of daily contacts) and τ = 14 (infectious period). Triangles stand for β·n·τ = R0,ran = 2.4, squares for R0,ran = 1.8 and circles for R0,ran = 1.2. Subfigure b gives R0,rep/R0,ran depending on the infectious period τ. Red lines and symbols are for n = 4, and blue lines stand for n = 10, whereas green lines represent n = 16. The meaning of the symbols is identical as in subfigure a.
Figure 4
Figure 4
Dampening effect of clustering. Subfigures a-d show the difference in the total outbreak size between a pure random mixing model and a model assuming complete repetitiveness (with different levels of clustering) relative to the population size N for 4 ≤ n ≤ 20, 1.2 ≤ β·n·τ ≤ 4.0 and τ = 14. Subfigure 4a is identical with subfigure 2b. The clustering coefficient CC is increased picture-wise in steps of .2.
Figure 5
Figure 5
Mixed models. Subfigures a-p show the decrease of the total outbreak size relative to the size of the total population when the fraction of repetitive and clustered contacts is increased. 25% rep means that one fourth of all contacts on a given day repeat every day but that three fourths of the contacts on a given day are unique. Clustering coefficients CC are only defined and calculated for the repetitive fraction of the contacts. All simulations were calculated for an infectious period of 14 days. Orange circles stand for β·n·τ = 1.2, red squares for β·n·τ = 1.8, blue triangles for β·n·τ = 2.4 and green rhombi for β·n·τ = 3.0. The number of daily contacts n increases in steps of 4 per line of the subfigures, beginning with n = 8 in the first line. The first column of the subfigures shows CC = .0, the second column CC = .2, the third column CC = .4 and the fourth column CC = .6.
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References

    1. Koopman JS. Infection transmission science and models. Jpn J Infect Dis. 2005;58:S3–8. - PubMed
    1. Hethcote HW, Yorke JA. Gonorrhea transmission dynamics and control. Berlin: Springer; 1984. - PubMed
    1. Anderson RM, May RM. Infectious diseases of humans: dynamics and control. Oxford, UK: Oxford University Press; 1991.
    1. Duerr HP, Schwehm M, Leary CC, De Vlas SJ, Eichner M. The impact of contact structure on infectious disease control: influenza and antiviral agents. Epidemiol Infect. 2007;135:1124–1132. doi: 10.1017/S0950268807007959. - DOI - PMC - PubMed
    1. Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys Rev Lett. 2001;86:3200–3203. doi: 10.1103/PhysRevLett.86.3200. - DOI - PubMed

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