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considerably longer (or shorter) than D time units. For both classes of STD the full- system simulation approaches the timescale-approximated solution as pairing and unpairing rates (1/D) increase to infinity. Exact correspondence with frequency- dependent transmission thus is obtained only for instantaneous contact duration. For finite partnership (and between-partnership) durations, the epidemic always proceeds more slowly, since opportunities to transmit infection are more limited (Kretzschmar 2000). The results are radically different for the two disease classes, owing to their distinct intrinsic timescales. For the faster bacterial STDs, the epidemics predicted by the full-system simulation and by frequency-dependent transmission diverge rapidly as D increases (Figure 2A). For mean durations of 1 day or less (extremely fast partner change) the epidemics are roughly equivalent—differing by less than 10% in final prevalence and 25% in time to half-maximum prevalence. When D increases to just 1 week the full-system curve is hardly recognizable. The viral STD (Figure 2B), with its slower disease processes and higher reproductive number, is more forgiving of slower pair dynamics. For mean partnership durations of up to 1 month the full-system epidemic is roughly equivalent (as defined above) to the frequency-dependent approximation, and even with D=3 months the epidemics are qualitatively similar (though their rates of growth differ considerably). As a general rule, we have found that the agreement between the full-system and frequency-dependent models depends on both the relative timescales of disease and pairing dynamics and the reproductive number of the disease: slower disease dynamics and higher values of R0 permit longer- term partnerships to be modeled accurately by frequency-dependence. 21
4. Infection-induced changes in pairing behavior For scenarios where the timescale approximation is reasonable (discussed in Section 5), we now extend our derivation in keeping with strong evidence that the infection status of individuals influences their contact behavior. We introduce different rates kS and kI at which susceptible and infected individuals enter partnerships, and different rates lSS, lSI and lII at which the three types of partnerships dissolve (see Equation (A2) for implementation details). In the appendix, we derive analytic * solutions for the steady-state SI pair density from the resulting equations. As above we can use these in conjunction with Equation (2) to express the total incidence rate in closed-form expressions. By defining the dimensionless proportions s=S/N and i=I/N, and representing our contact parameters by the vector all solutions share a generalized frequency-dependent form: P SI * PSI = φκ (s,i) SI N κ = (k S ,k I ,l SS ,l SI ,l II ) we find that where φκ(s,i) is a function of time-varying values of s and i (though note s+i=1 for all time) and the pair formation and dissolution rates κ. The φκ(s,i) for four different behavioral cases are shown in Table 1, encompassing situations where individual infection status influences either pair-formation rates or partnership durations or both, as well as the baseline case of no behavioral shifts. The φκ(s,i) are independent of total density N, as expected since the pairing process can be described entirely in terms of frequencies (by dividing both sides of Equation (A2) by N). Hence, as for the simple case in Section 2, we find that εrapid pairing∝SI/N∝N, i.e. that the total incidence increases linearly with N. Therefore the per 22 (4)
Page 1 and 2: Disease transmission in heterogeneo
Page 3 and 4: Abstract Disease transmission in he
Page 5 and 6: variation is quantified from outbre
Page 7 and 8: ACKNOWLEDGEMENTS I’ve been fortun
Page 9 and 10: Engineering Research Council of Can
Page 11 and 12: Epidemic modelers face an essential
Page 13 and 14: casual contact (DCC), common usage
Page 15 and 16: HCW, and patient pools) and dynamic
Page 17 and 18: Hoffmann 2004; Shen et al. 2004)—
Page 19 and 20: Chapter Two Frequency-dependent inc
Page 21 and 22: principles and reach robust conclus
Page 23 and 24: partnership, disease and demographi
Page 25 and 26: transmission to incorporate this ph
Page 27 and 28: This result can be understood intui
Page 29: Garnett 2002). We simulated epidemi
Page 33 and 34: susceptible population (Diekmann an
Page 35 and 36: epidemics, since they bring R0 clos
Page 37 and 38: equired.) For slower-moving, chroni
Page 39 and 40: addressed here. Our treatment consi
Page 41 and 42: Table 1. Results for epidemics with
Page 43 and 44: (Equations 3 and A4); the lighter l
Page 45 and 46: Figure 2 A) B) Total proportion inf
Page 47 and 48: Appendix Derivation of SI pair dens
Page 49 and 50: As described in Section 2, our goal
Page 51 and 52: can be calculated as the product of
Page 53 and 54: Chapter Three Curtailing transmissi
Page 55 and 56: egions with on-going epidemics has
Page 57 and 58: transmission to the general communi
Page 59 and 60: and Ih), as well as from off-duty t
Page 61 and 62: ours—see Diekmann & Heesterbeek (
Page 63 and 64: In all cases (Figure 2C, 2D), we id
Page 65 and 66: Sensitivity to quarantining rate (s
Page 67 and 68: precautions in the hospital (η→1
Page 69 and 70: Preventing generalized community tr
Page 71 and 72: everywhere. These robust conclusion
Page 73 and 74: increasingly significant contributi
Page 75 and 76: hospitals or wards. Future work on
Page 77 and 78: Table 1. Summary of transmission an
Page 79 and 80: formulation of our model allows for
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Figure 3 (A) Increase in cumulative
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Figure 1 74
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Figure 3 (A) (B) (C) Cumulative num
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Appendix Sensitivity to population
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symptomatic HCWs is low. Indeed, th
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E I I I I m, i m, 1 m, 2 m , 3 m ,
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for a stochastic epidemic: the prob
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progress through the age sub-compar
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We therefore wish to characterize t
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Figure S4 Distribution of (A) incub
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Figure S2 (A) (C) 92 (B) (D)
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Figure S4 (A) (B) Proportion of cas
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1. Introduction During the global e
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quantity p0, the proportion of case
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model selection techniques, lending
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elatively mild variation in ν, sho
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cannot make inferences based on the
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low k extinction happens within the
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measured by the variance-to-mean ra
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succeed (Figure 3B). For a given re
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to lower infectiousness. Other rele
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observations for the diseases exami
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Pneumonic plague 6 outbreaks N=74 A
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s surveillance data. 90% CI: Bootst
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Figure captions Figure 1 Evidence f
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Figure 4 Impact of control measures
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Figure 2 Other Measles Influenza Ru
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Figure 4 C Reproductive number, R 1
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epresenting transmission yields a g
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We rescaled the AICc by subtracting
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for which the median of bootstrap e
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transmission due to the most infect
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an integer Z (99) such that FPoisso
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fv(u). Demographic stochasticity in
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original process is eliminated with
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For RA control, with a random propo
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∞ ( , ) ∫ ( ) ( ). k−1 −t w
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* and for ν < ν : ∞ ∫ ν C 1
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Contrib(control policy) = Pr(contai
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ange of the first disease generatio
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Figure S1 MLE estimates of k 10 1 0
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Figure S2 (cont) D E F k=inf k=4 k=
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Figure S4 Relative frequency 0.4 0.
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SARS (March 12) and the imposition
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Transmission was predominantly by i
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Monkeypox, Zaire 1980-1984 (Jezek e
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Rubella, Hawaii 1970 (Hattis et al.
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tracing, with the stated goal of de
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Table S1. Superspreading events in
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Measles 250 Dance party ?M First ar
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SARS 187+ Apartment block 26M Amoy
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SARS 33 Hospital 62W Undiagnosed: S
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SARS 24/2* Home, emergency room, IC
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Streptococcus group A (type 1) 100+
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References Abbot, P. and L. M. Dill
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Caswell, H. (2001) Matrix Populatio
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Evatt, B. L., W. R. Dowdle, M. John
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Health Canada (2003) Summary of Sev
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Kretzschmar, M. (2000). Sexual netw
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McCallum, H., N. Barlow and J. Hone
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Taylor, H. M. and S. Karlin (1998).
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