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The percentile intervals in Figure 2A were generated directly from the Poisson distribution, with reproductive numbers drawn from specific studies of the relevant diseases where possible (Jezek and Fenner 1988; Gani and Leach 2001; Lipsitch et al. 2003; Chowell et al. 2004), or otherwise from compiled estimates (Anderson and May 1991). These latter estimates of R0 are intended to be indicative only, since they do not necessarily describe the same population setting or disease strain as the SSEs in question. The proportion of cases causing SSEs was estimated directly from outbreak datasets for Figure 2B, using the 99 th -percentile definition of an SSE proposed in the main text. To compare these observed proportions to the Poisson and negative binomial models for Z, we established 95% confidence intervals for the SSE proportion under each of those models (using mle parameters). For instance, if Z~Poisson(R) the expected proportion of cases causing SSEs is ΨPoisson=1-FPoisson(R)(Z (99) ), where FPoisson(R)(x) is the cdf of the Poisson(R) distribution, and Z (99) is the smallest integer satisfying FPoisson(R)(Z (99) )≥0.99. For an outbreak with Z~Poisson(R) and N cases, the number of SSEs will be binomially distributed, NSSE~binomial(N,ΨPoisson). To calculate the 95% confidence interval for the observed proportion of SSEs, we find the 2.5 and 97.5 percentiles of NSSE~binomial(N,ΨPoisson) and divide by N. Then 95% of outbreaks with N cases and Z~Poisson(R) will have an observed proportion of SSEs (Ψobs) in this range. 95% confidence intervals for Z~NegB(R,k) were calculated in a precisely analogous manner. Note that the proportion of 99 th -percentile SSEs, ΨPoisson, is often less than 1%, because Poisson(R) is a discrete distribution and for arbitrary R there is unlikely to be 135
an integer Z (99) such that FPoisson(R)(Z (99) ) equals 0.99 exactly. As a result, the proportion of cases causing SSEs under the negative binomial model, ΨR,k, may approach some value less than 0.01 as kØ¶. In plotting Figure 2C, we chose values of R such that ΨPoisson=ΨR,∞=0.01 and all plotted lines approached the same asymptotic value. These values were computed simply by examining Poisson cdf’s for different R. Precise values of R in Figure 2C are 0.148, 0.436, 1.279, 2.330, 3.507, 6.099, 10.345, and 20.323. Note that this effect of the discreteness of the Poisson distribution, while a nuisance in making plots, has little practical impact because most diseases have k
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Disease transmission in heterogeneo
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Abstract Disease transmission in he
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variation is quantified from outbre
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ACKNOWLEDGEMENTS I’ve been fortun
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Engineering Research Council of Can
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Epidemic modelers face an essential
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casual contact (DCC), common usage
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HCW, and patient pools) and dynamic
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Hoffmann 2004; Shen et al. 2004)—
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Chapter Two Frequency-dependent inc
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principles and reach robust conclus
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partnership, disease and demographi
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transmission to incorporate this ph
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This result can be understood intui
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Garnett 2002). We simulated epidemi
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4. Infection-induced changes in pai
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susceptible population (Diekmann an
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epidemics, since they bring R0 clos
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equired.) For slower-moving, chroni
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addressed here. Our treatment consi
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Table 1. Results for epidemics with
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(Equations 3 and A4); the lighter l
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Figure 2 A) B) Total proportion inf
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Appendix Derivation of SI pair dens
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As described in Section 2, our goal
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can be calculated as the product of
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Chapter Three Curtailing transmissi
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egions with on-going epidemics has
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transmission to the general communi
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and Ih), as well as from off-duty t
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ours—see Diekmann & Heesterbeek (
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In all cases (Figure 2C, 2D), we id
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Sensitivity to quarantining rate (s
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precautions in the hospital (η→1
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Preventing generalized community tr
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everywhere. These robust conclusion
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increasingly significant contributi
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hospitals or wards. Future work on
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Table 1. Summary of transmission an
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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 ,
Page 93 and 94: for a stochastic epidemic: the prob
Page 95 and 96: progress through the age sub-compar
Page 97 and 98: We therefore wish to characterize t
Page 99 and 100: Figure S4 Distribution of (A) incub
Page 101 and 102: Figure S2 (A) (C) 92 (B) (D)
Page 103 and 104: Figure S4 (A) (B) Proportion of cas
Page 105 and 106: 1. Introduction During the global e
Page 107 and 108: quantity p0, the proportion of case
Page 109 and 110: model selection techniques, lending
Page 111 and 112: elatively mild variation in ν, sho
Page 113 and 114: cannot make inferences based on the
Page 115 and 116: low k extinction happens within the
Page 117 and 118: measured by the variance-to-mean ra
Page 119 and 120: succeed (Figure 3B). For a given re
Page 121 and 122: to lower infectiousness. Other rele
Page 123 and 124: observations for the diseases exami
Page 125 and 126: Pneumonic plague 6 outbreaks N=74 A
Page 127 and 128: s surveillance data. 90% CI: Bootst
Page 129 and 130: Figure captions Figure 1 Evidence f
Page 131 and 132: Figure 4 Impact of control measures
Page 133 and 134: Figure 2 Other Measles Influenza Ru
Page 135 and 136: Figure 4 C Reproductive number, R 1
Page 137 and 138: epresenting transmission yields a g
Page 139 and 140: We rescaled the AICc by subtracting
Page 141 and 142: for which the median of bootstrap e
Page 143: transmission due to the most infect
Page 147 and 148: fv(u). Demographic stochasticity in
Page 149 and 150: original process is eliminated with
Page 151 and 152: For RA control, with a random propo
Page 153 and 154: ∞ ( , ) ∫ ( ) ( ). k−1 −t w
Page 155 and 156: * and for ν < ν : ∞ ∫ ν C 1
Page 157 and 158: Contrib(control policy) = Pr(contai
Page 159 and 160: ange of the first disease generatio
Page 161 and 162: Figure S1 MLE estimates of k 10 1 0
Page 163 and 164: Figure S2 (cont) D E F k=inf k=4 k=
Page 165 and 166: Figure S4 Relative frequency 0.4 0.
Page 167 and 168: SARS (March 12) and the imposition
Page 169 and 170: Transmission was predominantly by i
Page 171 and 172: Monkeypox, Zaire 1980-1984 (Jezek e
Page 173 and 174: Rubella, Hawaii 1970 (Hattis et al.
Page 175 and 176: tracing, with the stated goal of de
Page 177 and 178: Table S1. Superspreading events in
Page 179 and 180: Measles 250 Dance party ?M First ar
Page 181 and 182: SARS 187+ Apartment block 26M Amoy
Page 183 and 184: SARS 33 Hospital 62W Undiagnosed: S
Page 185 and 186: SARS 24/2* Home, emergency room, IC
Page 187 and 188: Streptococcus group A (type 1) 100+
Page 189 and 190: References Abbot, P. and L. M. Dill
Page 191 and 192: Caswell, H. (2001) Matrix Populatio
Page 193 and 194: Evatt, B. L., W. R. Dowdle, M. John
Page 195 and 196:
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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