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process models centers on the probability generating function (pgf) of the offspring distribution, g(s): g s ∑ p ∞ ( ) , |s|≤1 = k = 0 k k s Two important properties of the epidemic process follow directly from g(s). The basic reproductive number, R0, is by definition the mean value of Z, and is equal to g′(1). The probability that an infectious individual will cause no secondary infections, p0=Pr(Z=0), is g(0). Thus a great deal can be learned about an outbreak from the y- intercept of the pgf and its slope at s=1. The n th iterate of the pgf, gn(s), is the pgf of Zn, the number of cases in the n th generation, and is defined as follows: g0(s)=s, g1(s)=g(s), and gn+1(s)=g(gn(s)) for n=1,2,3,… (Harris 1989). The probability that the epidemic has gone extinct by the n th generation is thus gn(0). We denote the probability of extinction as nØ∞ by q, then q is a solution to the equation q=g(q), which from monotonicity and convexity of g(s) has at most one solution on the interval (0,1) (Harris 1989). When R0≤1, the only solution to q=g(q) is q=1 and disease extinction is certain; when R0>1, there is a unique positive solution less than one (Harris 1989). Finally, the pgf for the total number of individuals infected in all generations of a minor outbreak (i.e. one that goes extinct) is defined implicitly as G(s)=sg(G(s)) (Harris 1989). The expected size of a minor outbreak is then G′ (1), and can be calculated numerically for a given g(s). For our treatment of the transmission process, we assume that each infectious case has an individual reproductive number ν, drawn from some distribution with pdf 137
fv(u). Demographic stochasticity in transmission is then represented by a Poisson process, as is standard in branching process treatments of epidemics (Diekmann and Heesterbeek 2000). This yields the following pgf for a Poisson distribution with mean ν distributed as fv(u): If ν is a constant, R0, then the pgf is: g ( ) ∫ ∞ − u 1−s ( s) = e fν ( u) du 0 g( s) = e −R0 ( 1−s ) If ν is exponentially distributed with mean R0, the resulting offspring distribution is geometric with mean R0 (Taylor and Karlin 1998) and pgf: ( ( ) ) 1 1 − + R − ) ( g s = s If ν is gamma distributed, with mean R0 and shape parameter k, the resulting offspring distribution is negative binomial, also with mean R0 and shape parameter k (Taylor and Karlin 1998), with pgf: 0 1 ⎛ R0 g( s) = ⎜1+ 1 ⎝ k ( − s) This expression was applied in all of the general branching process results shown above to derive our results. The expression q=g(q) was solved numerically to generate Figures 3B and S2B, showing the dependence of the extinction probability on R0 and k. The negative binomial pgf itself is plotted in Figure S2A, showing how the probability of infecting zero others (p0) increases sharply with k for a given R0. The probability of extinction in the n th generation (Figure S2C) was calculated using gn(0)−gn−1(0). These numerical solutions match the averaged output of many 138 ⎞ ⎟ ⎠ −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 ,
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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 and 144: transmission due to the most infect
Page 145: an integer Z (99) such that FPoisso
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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