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simulations precisely, for R0 above and below zero, and for kØ0 and kØ∞. The expected size of minor outbreaks (Figure S2D) was plotted by solving G′ (1) numerically for a range of values of R0 and k. Branching process simulations Major outbreaks (Figures 3C, S2E, S2F): A branching process epidemic was implemented by simulation, beginning with a single infectious individual. For each infectious individual, the individual reproductive number ν was drawn from a gamma distribution with chosen values of R0 and k, using the gamrnd function in Matlab (v6.1 R13, MathWorks, Cambridge MA) adapted to allow non-integer k. The number of secondary cases Z caused by that individual was then determined by drawing a Poisson random variable with mean ν, using the Matlab function poissrnd. Each individual was infectious for only one generation, and the total number of infected individuals in each generation was summed. The first generation to reach 100 cases was used as an arbitrary benchmark of epidemic growth rate. Control policies—theoretical development We consider an epidemic that has a natural (i.e. uncontrolled) offspring distribution Z~NegB(R0,k), from which we know the probability of infecting zero others is p0=(1+R0/k) −k . Under the homogeneous partial (HP) control policy, every individual’s infectiousness is reduced by a factor c, so the expected number of HP secondary cases is reduced from ν to ν =(1-c)ν. We model transmission as a Poisson c process, Z~Poisson(ν); imposing HP control, in which each transmission event from the 139
original process is eliminated with probability c (regardless of the ν value of the source HP case) changes this to a marked Poisson process so that Z ~Poisson((1-c)ν) (Taylor and Karlin 1998). If uncontrolled individual reproductive numbers are gamma- distributed, ν~gamma(R0,k), then only the scale parameter of the resulting negative HP binomial distribution is affected by HP control and Z ~NegB((1−c)R0,k). The HP variance-to-mean ratio of Z is 1+(1−c)R/k, and decreases monotonically as control effort increases (Figure S3C). c Under random absolute (RA) control, each infected individual is controlled perfectly (such that they cause zero secondary infections) with probability c. Imposition of RA control influences transmission only for the fraction 1−p0 of individuals whose natural Z value is greater than zero—of these a fraction c have RA Zc RA =0, while the remaining fraction 1−c are unaffected and have Z =Z. Under an RA RA control policy, therefore, the proportion of cases causing zero infections is p = p0+c(1−p0) and the population mean R RA c N N 1 1 = ∑ Z i Pr( case i not controlled) = ∑ N N i= 1 c ( 1− c) Z = ( 1− c) R0 i= 1 i c c . The exact RA RA RA RA distribution of Z is defined by Pr( Z =0)= p and Pr( Z =j)=(1−c)Pr(Z=j) for all c RA j>0, i.e. the distribution of Z has an expanded zero class relative to Z, while for non- zero values its density is simply reduced by a factor (1−c) from Z~NegB(R0,k). Hence, the offspring distribution under RA control has pgf: g c RA c ⎛ R ( s) = c + 1 ⎝ k 0 0 ( 1− c) ⎜1+ ( − s) 140 ⎞ ⎟ ⎠ c −k 0
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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
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
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Page 145 and 146: an integer Z (99) such that FPoisso
Page 147: fv(u). Demographic stochasticity in
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
Page 197 and 198: 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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