Human/Swine H1N1 Influenza

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HistoryEdited Apr 30, 2009 5:48 PM by Andrew Rambaut…

Relationship between R0 and the epidemic growth rate

Posted by Nick Grassly

The basic reproduction number of the swine influenza epidemic can be estimated from its initial rate of spread. If we assume roughly exponential growth then the basic reproductive number is related to the growth rate by the so-called Lotka-Euler estimating equation:

eqn1


where r is the rate of exponential growth and w(t) the generation time distribution [1]. The generation time distribution can be thought of as the probability density function describing the distribution of times between successive infections in a chain of transmission.

The estimate of the basic reproductive number is therefore dependent not just on an estimate of r , but also a good estimate of the generation time distribution [2]. In the case of swine influenza the generation time distribution is unclear, but data appear consistent with seasonal influenza that has a mean generation time of approximately 3 days.

Analytical solutions for R0 can be derived for different assumed generation time distributions using the Lotka-Euler estimating equation (which is essentially a moment generating function). If we assume a generation time distribution that follows the gamma distribution, then

eqn2


where a and b are the parameters of the gamma distribution (a = m^2/s^2 and b=m/s^2where m and s are the mean and standard deviation of the distribution respectively).

Estimates of R0 based on the estimates of r reported by Andrew Rambaut are given in Table 1. Obviously r can also be estimated from epidemiological case data and this may give different results. These analyses are ongoing.

r (per day with 95% HPD) generation time distribution
(and parameters in days)		 R0
0.053 (0.0014, 0.12) gamma (m=3, s=2) 1.17 (1.00 - 1.40)
" exponential (m=3) (i.e. SIR model) 1.16 (1.00 - 1.36)



Citations

1. Wallinga J, Lipsitch M. How generation intervals shape the relationship between growth rates and reproductive numbers. Proc Roy Soc Lond B 2007;274:599-604

2. Grassly NC, Fraser C. Mathematical models of infectious disease transmission. Nat Rev Microbiol. 2008;6:477-487

Comments

Andy Brown
May 1, 2009

This estimate fits well with the estimate obtained for the 1976 Fort Dix outbreak by Lessler et al (2007)
http://rsif.royalsocietypublishing.org/content/4/15/755.long

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D. Smith1, Dhanasekaran Vijaykris...Falserambaut2009-07-08T14:53:06+00:00Shanghai/71T/09 does not have PB2-627Klycett2009-06-28 16:11:04+00:002009-06-28 16:11:04updated7lycett2009-06-28 16:06:53+00:002009-06-28 16:06:53updated6lycett2009-06-19 07:03:23+00:002009-06-19 07:03:23updated5Added tag - hotlycett2009-06-19 07:03:22+00:002009-06-19 07:03:22addTag4A/Shanghai/71T/2009 & PB2-627Klycett2009-06-18 22:40:51+00:002009-06-18 22:40:51updated3PB2-627K on A/Shanghai/71T/2009lycett2009-06-18 22:34:10+00:002009-06-18 22:34:10updated2First additionlycett2009-06-18 22:12:59+00:002009-06-18 22:12:59created1wiki2009-06-28T16:11:04+00:00groups/influenza/wiki/8bc58FalsePB2-627K & PB2-701N in A/Shanghai/71T/2009/groups/influenza/wiki/8bc58/PB2627K__PB2701N_in_AShanghai71T2009.htmlSam Lycett7 updatesPB2-627K & PB2-701N in A/Shanghai/71T/2009 Notes on A/Shanghai/71T/2009 by Sam Lycett Update 28 June 09 The sequences A/Shanghai/71T/2009 were updated on 24...Falselycett2009-06-28T16:11:04+00:00comas2009-05-10 21:28:29+00:002009-05-10 21:28:29updated6comas2009-05-10 21:27:07+00:002009-05-10 21:27:07updated5comas2009-05-10 21:26:45+00:002009-05-10 21:26:45updated4Added tag - hotcomas2009-05-10 21:26:43+00:002009-05-10 21:26:43addTag3comas2009-05-10 21:26:25+00:002009-05-10 21:26:25updated2First additioncomas2009-05-10 21:13:16+00:002009-05-10 21:13:16created1wiki2009-05-10T21:28:29+00:00groups/influenza/wiki/aba06FalseUpdated Median Joining network (HA genes) - 8 May 2009 - Iñaki Comas/groups/influenza/wiki/aba06/Updated_Median_Joining_network_HA_genes__8_May_2009__Iaki_Comas.htmlIñaki Comas6 updatesUpdated Median Joining network (HA genes) - 8 May 2009 - Iñaki Comas Here is an update of the median joining network with all swine samples available at GISAID as of 8 of May. The alignment was provided by Oliver Pyb...Falsecomas2009-05-10T21:28:29+00:00pybus2009-05-08 15:45:54+00:002009-05-08 15:45:54updated6pybus2009-05-06 19:32:17+00:002009-05-06 19:32:17updated5Added tag - selectionpybus2009-05-06 18:30:11+00:002009-05-06 18:30:11addTag4Added tag - hotpybus2009-05-06 18:29:59+00:002009-05-06 18:29:59addTag3pybus2009-05-06 18:29:25+00:002009-05-06 18:29:25updated2First additionpybus2009-05-06 18:18:26+00:002009-05-06 18:18:26created1wiki2009-05-08T15:45:54+00:00groups/influenza/wiki/87337FalseAmino acid changes in HA, NA & MP leading to outbreak 6 May 2009 - Oliver Pybus and Samir Bhatt/groups/influenza/wiki/87337/Amino_acid_changes_in_HA_NA__MP_leading_to_outbreak_6_May_2009__Oliver_Pybus_and_Samir_Bhatt.htmlOliver Pybus6 updatesAmino acid changes in HA, NA & MP leading to outbreak 6 May 2009 - Oliver Pybus and Samir Bhatt So, I hear you ask, what amino acid changes have occurred along those long branches leading to the outbreak strain? Right, we've gone and ...Falsepybus2009-05-08T15:45:54+00:00comas2009-05-08 12:59:46+00:002009-05-08 12:59:46updated7rambaut2009-05-08 11:40:12+00:002009-05-08 11:40:12updated6Added tag - genetic structurecomas2009-05-08 10:49:11+00:002009-05-08 10:49:11addTag5Added tag - transmissioncomas2009-05-08 10:48:52+00:002009-05-08 10:48:52addTag4comas2009-05-08 10:45:56+00:002009-05-08 10:45:56updated3Added tag - hotcomas2009-05-08 10:44:35+00:002009-05-08 10:44:35addTag2First additioncomas2009-05-08 10:37:39+00:002009-05-08 10:37:39created1wiki2009-05-08T12:59:46+00:00groups/influenza/wiki/8e6d0FalseMedian joining network (HA gene) - 7May2009 - Iñaki Comas/groups/influenza/wiki/8e6d0/Median_joining_network_HA_gene__7May2009__Iaki_Comas.htmlIñaki Comas7 updatesMedian joining network (HA gene) - 7May2009 - Iñaki Comas I have created a median joining network with the HA sequences present in the NCBI as of 7May2009. This includes 49 isolates from different locat...Falsecomas2009-05-08T12:59:46+00:00hot/groups/influenza/search/index.rss?sort=modifiedDate&kind=all&sortDirection=reverse&excludePages=wiki/welcomelist/groups/influenza/search/?sort=modifiedDate&kind=all&sortDirection=reverse&excludePages=wiki/welcomeRecent ChangesRecentChangesListUpdates?sort=modifiedDate&kind=all&sortDirection=reverse&excludePages=wiki/welcome0/groups/influenza/sidebar/RecentChangesListmodifiedDateallRecent ChangesRecentChangesListUpdateswiki/welcomeNo recent changes.reverse5search