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RA The variance-to-mean ratio of Z can be calculated from 2 ( g ′ () + g′ () 1 − ( g′ () 1 ) ) g′ ( 1) RA 1 RA RA RA c ′ (Harris 1989) and shown to equal 1+R0/k+cR0, which increases monotonically as c increases. For direct comparison with other offspring distributions, this composite distribution under RA control can be RA RA RA approximated by a new negative binomial distribution, Z ~NegB( R , k ) where RA Rc RA is given above and k is estimated using the proportion of zeros method as the solution to p RA 0 0 c ( ) ( ) RA RA RA −k c 1− p = 1+ R k = p + c . This approximation yields better 0 c than 95% overlap with the exact distribution for k§1, and better than 85% overlap for almost all of parameter space (Figure S3A). (The proportion of overlap is calculated as 1 ⎛ ⎜ ⎝ − ∑ ∞ i= 0 c ⎞ Z i, exact − Zi , approx ⎟ 2 , which scales from 0 to 1 as the two distributions go from ⎠ completely non-overlapping to identical.) The approximation approaches exactness for cØ0 and cØ1, and is least accurate for large values of k because it is unable to mimic RA the bimodal distribution of Z (Figure S3B). The approximated shape parameter c decreases monotonically as control effort c increases (Figure S3C). Relative efficacy of control policies For HP control, with all individuals’ transmission reduced by a factor c, the HP offspring distribution is Z ~NegB((1−c)R0,k) and has pgf: c g HP ⎛ ( s) = ⎜1+ ⎝ R k 0 ( 1− c) ( 1− s) . 141 c ⎞ ⎟ ⎠ −k c c RA kc
For RA control, with a random proportion c of individuals controlled absolutely, the pgf is as given above: g RA ( s) = c + ⎛ ⎜ ⎝ R k ⎞ ⎟ ⎠ −k 0 ( 1− c) 1+ ( 1− s) . Claim: For all c œ (0,1-1/R0), the probability of extinction is always greater under RA control than under HP control. ( ) k − R Proof of claim: Define G x) = 1+ x ( 1− s) ( 0 where X is a Bernoulli random variable k with a probability 1−c of success. Since G is a convex function, Jensen’s inequality implies that ( E( X ) ) < E( G( X ) ) g ( s) g HP ( s) = G = (*) RA whenever c œ (0,1) and s œ [0,1). Furthermore, for the n th iterates of the pgf we have from (*) that g ( 0) < gRA, HP, n n so the probability of disease extinction by the n th generation is always greater under RA control. Thus if c œ (0,1-1/R0), the probability of ultimate extinction under RA control is greater than that under HP control, i.e. q RA > q HP . If c > 1-1/R0, then R HP RA c = Rc so that q RA = q HP < 1 = 1. ( 0) To consider the efficacy of control policies targeting the more infectious individuals in a population, we consider a general branching process whose pgf is given by 142
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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 ,
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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
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 and 144: transmission due to the most infect
Page 145 and 146: an integer Z (99) such that FPoisso
Page 147 and 148: fv(u). Demographic stochasticity in
Page 149: original process is eliminated with
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
Page 197 and 198: Kretzschmar, M. (2000). Sexual netw
Page 199 and 200: McCallum, H., N. Barlow and J. Hone
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Taylor, H. M. and S. Karlin (1998).
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