the basic reproduction number r0 of an infectious agent is defined as the expected number of secondary cases caused by one typical infected individual in a population consisting of susceptibles only .
when an outbreak has started and the approximation that the population is fully susceptible no longer holds , one generally refers to the effective reproduction number r. the value of r0 is , as a rule , different for different infectious agents and depends among other things on the characteristics of the population that the agent invades . given this ,
it is not immediate that one can adopt previously determined values or size ranges for a new outbreak , unless many of the complicated characteristics of , for example , population composition and contact structure are comparable . for various reasons one can be interested in the value of r0 or r early in an outbreak and during the outbreak .
notably , under a homogeneous mixing assumption , the values give insight into the extent of the control problem and a means of calculating how much control effort is needed . in recent years
several new methods for estimating r0 from outbreak data have been published , either as a general tool or for specific applications .
some , like wallinga and teunis   do not need much data , but require knowledge of the generation time distribution .
most of these methods , however , are data - hungry : they either need contact information , use the whole outbreak time series ( so are effectively retrospective measures ) , or increase in accuracy as the time series becomes longer .
for example , we do not observe infections we observe detections , i.e. individuals ( people , animals , farms , plants ) exhibiting symptoms .
there is then a possibly unknown incubation period distribution that convolutes the infection process into the observed process .
moreover , r0 is a generation - based concept , but generations are not observed  a daily number of new detections is observed from possibly mixed generations .
also , heterogeneity between individual infectivity and susceptibility and in contact pattern may cause the distribution of which r0 is the mean to be highly skewed ( e.g. ) . to make matters worse
, we almost never have data from an uncontrolled situation  some control measures , effective or not , often operate from the moment of detection of the index case . despite advances , but in light of the problems encountered , many publications in which r0 is estimated from outbreak data still depend on cumulative incidence and generation interval only ( see for example ) . as a rule ,
the cumulative incidence in outbreaks of an infectious disease is observed to initially grow approximately exponentially with time ( and hence the incidence grows exponentially too ) .
a frequently used approach is to fit an exponential function to the ( cumulative ) incidence and to use the approximate relationship \documentclass[12pt]{minimal } 
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 $ $ { r_0 } \approx { e^{r{t_g}}}$$ 
 \end{document } to estimate r0 , where r is the exponential growth rate and tg is the observed mean generation interval of the epidemic . for rtg small the further approximation \documentclass[12pt]{minimal } 
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 $ $ { r_0 } \approx 1 + r{t_g}$$ 
 \end{document } is sometimes used .
for example , the  real value of r is not observed , not only because of control measures in operation but also due to the stochasticity in the early phase . in addition , the definition of the generation interval is not always consistently used and the method presupposes that the population is homogeneously mixing . still , the method is easy and intuitive and one can wonder in which circumstances it would be a  good approximation , and how large discrepancies can be when these circumstances are not met . in this paper
we now derive estimates of r0 based on specific models and compare these with the previously mentioned approximations , which we denote \documentclass[12pt]{minimal } 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } and \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g}$$ 
 \end{document}. we need to emphasise that r0 is independent of timescale , whereas tg has dimension time and r has dimension time .
we also need to emphasise that we do not know r or tg , we assume that they have been estimated from data in some way , for example by estimating the doubling time of the incidence , d , and writing r = log(2)/d .
we do not write ^r or ^tg for these estimates , as this would result in far too many hats in one paper .
the incidence of an emerging infection may be calculated from \documentclass[12pt]{minimal } 
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 $ $ i(t ) = \delta ( t ) + { { s(t ) } \over n}\int\limits_0^\infty { a(\tau ) i(t - \tau ) d\tau } $ $ 
 \end{document } where (t ) , a unit spike , is the incidence of infection at time zero , the kernel a( ) is the expected infectivity of an infected as a function of  , the time since exposure to infection .
the number in the population susceptible at time t is \documentclass[12pt]{minimal } 
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 $ $ s(t ) =
n - \int\limits_0^t { i(u){\rm{d}}u } $ $ 
 \end{document } for an emerging infection we assume the entire population to be susceptible at time zero .
if this is not the case , we take n to be the size of the susceptible population prior to infection . as a first step towards developing a model , we specify the general form of the kernel a( ) .
we write a( ) = r0f( ) , where r0 is the basic reproduction number that we wish to estimate , and f( ) is the infectivity kernel , which is also the probability distribution of the generation interval . for an emerging infection
we have little information about f. we may have observations of the latent period ( the time from exposure to infection to becoming infectious , te ) ; the incubation period ( the time from exposure to infection to the onset of symptoms ) which we may in some cases assume to equal te ; or the infectious period ti . given these
we may wish to impose a particular form on the kernel , and use our limited knowledge to estimate parameter values for the distribution .
one quantity of interest is the mean generation interval of the epidemic , which is taken here to be the mean time from an individual s exposure to infection to exposing others to infection ( see , for an insightful exposition ) .
we refer not to the time to the first occurrence of a secondary infection , but to the average time to all secondary infections . alternatively , and equivalently
, it can be defined as the expected duration of the primary infection at the time that a secondary infection occurs ( see ) .
the mean generation interval may be determined from the formula \documentclass[12pt]{minimal } 
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 $ $ { t_g } = \int\limits_0^\infty { tf(t){\rm{d}}t } $ $ 
 \end{document } given a probability distribution for the generation interval , f( ) , and an estimated initial rate of exponential increase for the epidemic , r , we approximate the initial stages of the epidemic by i(t ) = e with s(t )  n. equation ( 1 ) then leads to a model - consistent estimate of the basic reproduction number via the formula \documentclass[12pt]{minimal } 
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 $ $ { r_0}\int\limits_0^\infty { { e^ { - rt}}f(t){\rm{d}}t } = 1$$ 
 \end{document } ( see ) . if f(t ) were a delta function , then eqs .
( 2 , 3 ) would lead to the estimate r0 = r0 + . for the sir model , where f(t )
( 2 , 3 ) lead to the estimate r0 = r0. we now computer0 for three distribution functions which may be used as kernels : those with a fixed , exponentially or trapezoidally distributed infectious period ( see fig .
1 ) . we refer to these as r0fix , r0exp and r0trap , respectively .
we also compute r0 for the model with latent and infectious periods that each have gamma distributions , referred to as r0(m , n ) .
we have r0exp = r0(1,1 ) } and r0fix = limm , nr0(m , n ) . 
 fig . 1a
selection of kernels , f( ) that may be used to model emerging infections : kernels due to fixed and exponentially distributed latent and infectious periods are shown as dashed and dashed / dotted lines , respectively ; and the trapezium kernel is shown as a solid line .
all kernels are scaled to have integral one . a kernels suitable for modelling influenza , with latent period te = 1.6 days and infectious period ti = 4.0 days .
b kernels suitable for modelling sars , with latent period te = 5.5 days and infectious period ti = 7.0 days a selection of kernels , f( ) that may be used to model emerging infections : kernels due to fixed and exponentially distributed latent and infectious periods are shown as dashed and dashed / dotted lines , respectively ; and the trapezium kernel is shown as a solid line .
a kernels suitable for modelling influenza , with latent period te = 1.6 days and infectious period ti = 4.0 days .
b kernels suitable for modelling sars , with latent period te = 5.5 days and infectious period ti = 7.0 days given fixed latent and infectious periods , te and ti respectively , and assuming f constant when non - zero , we have f( ) = 1/ti for te <  < te+ ti and f( ) = 0 otherwise
. for this distribution tg = te + ti/2 and \documentclass[12pt]{minimal } 
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 $ $ { r_0 } = r_0^{{\rm{fix } } } = { { r\left ( { { t_g } - { t_e } } \right ) } \over { \sinh r\left ( { { t_g } - { t_e } } \right)}}{e^{r{t_g}}}$$ 
 \end{document}r0 + is useful as an estimator for r0fix when the latent period may be regarded as fixed and the infectious period is short relative to the timescale 1/r ( rti is small ) . as sinh
x > 
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 $ $ { \lim _ { x \to 0}}{{\sinh x } \over x } = 1$$ 
 \end{document } we have r0fix  r0 + , and \documentclass[12pt]{minimal } 
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 $ $ { \lim _ { { t_e }
\to { t_g}}}r_0^{{\rm{fix } } } = r_0^ + $ $ 
 \end{document}. wallinga and lipsitch   showed that r0 + is an upper bound on estimates of r0 for any distribution f(t ) .
consider the kernel \documentclass[12pt]{minimal } 
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 $ $ f(\tau ) = \left\ { { \matrix { { { 1 \over { { t_i}}}{{\tau - { \tau _ a } } \over { { \tau _ b } - { \tau _ a } } } } & : & { \tau \in \left ( { { \tau _ a},{\tau _ b } } \right ) } \cr { { 1 \over { { t_i } } } } & : & { \tau \in \left ( { { \tau _ b},{\tau _ c } } \right ) } \cr { { 1 \over { { t_i}}}{{{\tau _
d } - \tau } \over { { \tau _ d } - { \tau _ c } } } } & : & { \tau \in \left ( { { \tau _ a},{\tau _ b } } \right ) } \cr 0 & : & { { \rm{otherwise } } } \cr } } \right.$$ 
 \end{document } this is a suitable approximation to an infectivity function where nobody is infectious before a time units or after d time units post - exposure , maximum infectivity occurs between b and c time units after exposure , and contact rates are constant . the distribution is consistent with a mean latent period of \documentclass[12pt]{minimal } 
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 $ $ { t_e } = { { { \tau _ a } + { \tau _ b } } \over 2}$$ 
 \end{document } , a mean infectious period of \documentclass[12pt]{minimal } 
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 $ $ { t_i } = \left ( { { \tau _ d } + { \tau _ c } - { \tau _ b } - { \tau _ a } } \right)/2$$ 
 \end{document } and a mean generation interval of \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { { t_i } } \over 2 } + { { { { \left ( { { \tau _ d } - { \tau _ c } } \right)}^2 } - { { \left ( { { \tau _ b } - { \tau _ a } } \right)}^2 } } \over { 12\left ( { { \tau _ d } + { \tau _ c } - { \tau _ b } - { \tau _ a } } \right)}}$$ 
 \end{document } hence , if the trapezium is symmetric \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { { t_i } } \over 2}$$ 
 \end{document } , which is the same relationship as that for the fixed infectious period .
the basic reproduction number solves r0trapf(r ) = 1 , where f(s ) is the laplace transform of f(t ) ( see appendix 1 ) in an extended seir differential equation model the population of size n is made up of s susceptibles , e that have been exposed to infection but are not yet infectious , i infectious and r that have been infected and recovered .
if the epidemic processes have a much faster timescale than the demographic processes , we obtain the equations \documentclass[12pt]{minimal } 
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 $ $ \eqalign { & { { d{e_1 } } \over { dt } } = \beta { s \over n}\sum\limits_{j = 1}^n { { i_j } - mv{e_1 } } \cr & { \rm{for } } i
{ { d{e_i } } \over { dt } } = mv{e_{i - 1 } } - mv{e_i } \cr & { { d{i_1 } } \over { dt } } = mv{e_m } - n\gamma { i_1 } \cr & { \rm{for } } j = 2, ... ,n{\rm { } } { { d{i_j } } \over { dt } } = n\gamma { i_{j - 1 } } - n\gamma { i_j } \cr & { { dr } \over { dt } } = n\gamma { i_n } \cr } $ $ 
 \end{document } the exposed and infectious classes have been subdivided e = i=1mei and i = j=1nij , respectively .
the times spent in the exposed and infectious classes are gamma distributed with means te = 1/ and ti = 1/ , respectively , andr0 = /. the mean generation interval is \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { n + 1 } \over { 2n}}{t_i}$$ 
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 $ $ r_0^{(m , n ) } = { { { { 2nr } \over { n + 1}}\left ( { { t_g } - { t_e } } \right){{\left ( { 1 + { r \over m}{t_e } } \right)}^m } } \over { 1 - { { \left ( { 1 + { { 2r } \over { n + 1}}\left ( { { t_g } - { t_e } } \right ) } \right)}^ { - n}}}}$$ 
 \end{document } this result is derived in appendix 2 , where it is also shown that given values of r , te and tg , r0(m , n ) is an increasing function of both m and n. the well - known seir differential equation model is the special case of eqs .
the times spent in the exposed and infectious classes are exponentially distributed with means te = 1/ and ti = 1/ respectively , and the appropriate kernel function in eqs .
( 2 , and 3 ) is \documentclass[12pt]{minimal } 
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 $ $ f(\tau ) = { { \gamma v } \over { \gamma - v}}\left ( { { e^ { - v\tau } } - { e^ { - \gamma \tau } } } \right)$$ 
 \end{document } ( see ) .
the mean generation interval is \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { t_i}$$ 
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 $ $ r_0^{\exp } = r_0^{(1,1 ) } = 1 + r\left ( { { 1 \over v } + { 1 \over \gamma } } \right ) + { { { r^2 } } \over { v\gamma } } = 1 + r{t_g } + { r^2}{t_e}\left ( { { t_g } - { t_e } } \right)$$ 
 \end{document } the approximation \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g } \le r_0^{\exp } $ $ 
 \end{document } is appropriate for the sir model , for which    , te  0 and tg  ti = 1/. hence r0 performs best as an estimate when either the latent period te or the infectious period ti is small compared to tg , and performs worst when they are equal .
given fixed latent and infectious periods , te and ti respectively , and assuming f constant when non - zero , we have f( ) = 1/ti for te <  < te+ ti and f( ) = 0 otherwise . for this distribution tg
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 $ $ { r_0 } = r_0^{{\rm{fix } } } = { { r\left ( { { t_g } - { t_e } } \right ) } \over { \sinh r\left ( { { t_g } - { t_e } } \right)}}{e^{r{t_g}}}$$ 
 \end{document}r0 + is useful as an estimator for r0fix when the latent period may be regarded as fixed and the infectious period is short relative to the timescale 1/r ( rti is small ) .
as sinh x > 
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 $ $ { \lim _ { x \to 0}}{{\sinh x } \over x } = 1$$ 
 \end{document } we have r0fix  r0 + , and \documentclass[12pt]{minimal } 
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 $ $ { \lim _ { { t_e }
\to { t_g}}}r_0^{{\rm{fix } } } = r_0^ + $ $ 
 \end{document}. wallinga and lipsitch   showed that r0 + is an upper bound on estimates of r0 for any distribution f(t ) .
consider the kernel \documentclass[12pt]{minimal } 
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 $ $ f(\tau ) = \left\ { { \matrix { { { 1 \over { { t_i}}}{{\tau - { \tau _ a } } \over { { \tau _ b } - { \tau _ a } } } } & : & { \tau \in \left ( { { \tau _ a},{\tau _ b } } \right ) } \cr { { 1 \over { { t_i } } } } & : & { \tau \in \left ( { { \tau _ b},{\tau _ c } } \right ) } \cr { { 1 \over { { t_i}}}{{{\tau _ d } - \tau } \over { { \tau _
d } - { \tau _ c } } } } & : & { \tau \in \left ( { { \tau _ a},{\tau _ b } } \right ) } \cr 0 & : & { { \rm{otherwise } } } \cr } } \right.$$ 
 \end{document } this is a suitable approximation to an infectivity function where nobody is infectious before a time units or after d time units post - exposure , maximum infectivity occurs between b and c time units after exposure , and contact rates are constant . the distribution is consistent with a mean latent period of \documentclass[12pt]{minimal } 
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 $ $ { t_e } = { { { \tau _ a } + { \tau _ b } } \over 2}$$ 
 \end{document } , a mean infectious period of \documentclass[12pt]{minimal } 
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 $ $ { t_i } = \left ( { { \tau _ d } + { \tau _ c } - { \tau _ b } - { \tau _ a } } \right)/2$$ 
 \end{document } and a mean generation interval of \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { { t_i } } \over 2 } + { { { { \left ( { { \tau _ d } - { \tau _ c } } \right)}^2 } - { { \left ( { { \tau _ b } - { \tau _ a } } \right)}^2 } } \over { 12\left ( { { \tau _ d } + { \tau _ c } - { \tau _ b } - { \tau _ a } } \right)}}$$ 
 \end{document } hence , if the trapezium is symmetric \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { { t_i } } \over 2}$$ 
 \end{document } , which is the same relationship as that for the fixed infectious period . the basic reproduction number solves r0trapf(r ) = 1 , where f(s ) is the laplace transform of f(t ) ( see appendix 1 )
in an extended seir differential equation model the population of size n is made up of s susceptibles , e that have been exposed to infection but are not yet infectious , i infectious and r that have been infected and recovered .
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 $ $ \eqalign { & { { d{e_1 } } \over { dt } } = \beta { s \over n}\sum\limits_{j = 1}^n { { i_j } - mv{e_1 } } \cr & { \rm{for } } i = 2, ... ,m{\rm { } } { { d{e_i } } \over { dt } } = mv{e_{i - 1 } } - mv{e_i } \cr & { { d{i_1 } } \over { dt } } = mv{e_m } - n\gamma { i_1 } \cr & { \rm{for } } j = 2, ... ,n{\rm { } } { { d{i_j } } \over { dt } } = n\gamma { i_{j - 1 } } - n\gamma { i_j } \cr & { { dr } \over { dt } } = n\gamma { i_n } \cr } $ $ 
 \end{document } the exposed and infectious classes have been subdivided e = i=1mei and i = j=1nij , respectively .
the times spent in the exposed and infectious classes are gamma distributed with means te = 1/ and ti = 1/ , respectively , andr0 = /. the mean generation interval is \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { { n + 1 } \over { 2n}}{t_i}$$ 
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 $ $ r_0^{(m , n ) } = { { { { 2nr } \over { n + 1}}\left ( { { t_g } - { t_e } } \right){{\left ( { 1 + { r \over m}{t_e } } \right)}^m } } \over { 1 - { { \left ( { 1 + { { 2r } \over { n + 1}}\left ( { { t_g } - { t_e } } \right ) } \right)}^ { - n}}}}$$ 
 \end{document } this result is derived in appendix 2 , where it is also shown that given values of r , te and tg , r0(m , n ) is an increasing function of both m and n.
the well - known seir differential equation model is the special case of eqs . ( 6 ) with m = n = 1 . for this model the times spent in the exposed and infectious classes
are exponentially distributed with means te = 1/ and ti = 1/ respectively , and the appropriate kernel function in eqs .
( 2 , and 3 ) is \documentclass[12pt]{minimal } 
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 $ $ f(\tau ) = { { \gamma v } \over { \gamma - v}}\left ( { { e^ { - v\tau } } - { e^ { - \gamma \tau } } } \right)$$ 
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the mean generation interval is \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { t_i}$$ 
 \end{document } , and given r we have \documentclass[12pt]{minimal } 
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 $ $ r_0^{\exp } = r_0^{(1,1 ) } = 1 + r\left ( { { 1 \over v } + { 1 \over \gamma } } \right ) + { { { r^2 } } \over { v\gamma } } = 1 + r{t_g } + { r^2}{t_e}\left ( { { t_g } - { t_e } } \right)$$ 
 \end{document } the approximation \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g } \le r_0^{\exp } $ $ 
 \end{document } is appropriate for the sir model , for which    , te  0 and tg  ti = 1/. hence r0 performs best as an estimate when either the latent period te or the infectious period ti is small compared to tg , and performs worst when they are equal .
we assumed that we had estimated values of the initial rate of exponential increase of infection incidence , r , the mean latent period , te , and the mean generation interval , tg .
( 4 , and 7 ) to calculate model - based estimates of the basic reproduction number using the assumptions of a fixed or exponentially distributed infectious period , leading to r0fix and r0exp , respectively .
we did this for values of the ratio of the latent period to the generation interval , te / tg , in the range zero to 0.99 .
the values of r0fix and r0exp are plotted as functions of te / tg for rtg = 0.5 , 1.0 , 1.5 , 2.0 in fig . 2 , and compared with the values of the estimators \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g}$$ 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } in those cases . when rtg = 0.5 , 1.0 , 1.5 or 2.0 we have r0 = 1.5 , 2.0 , 2.5 or 3.0 and r0 + = 1.65 , 2.72 , 4.48 or 7.39 , respectively . 

2estimates of the basic reproduction number r0 as a function of the ratio of the latent period to the mean generation interval , te / tg .
the solid plots are a , b fixed infectious period : r0fix with rtg = 0.5 ( lower ) , rtg = 1.5 ( upper ) ; and rtg = 1.0 ( lower ) , rtg = 2.0 ( upper ) respectively , and c , d exponentially distributed infectious period : r0exp with rtg = 0.5 ( lower ) , rtg = 1.5 ( upper ) ; and rtg = 1.0 ( lower ) , rtg = 2.0 ( upper ) , respectively .
the horizontal dotted and dashed lines indicate the corresponding values of \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g}$$ 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } , respectively estimates of the basic reproduction number r0 as a function of the ratio of the latent period to the mean generation interval , te / tg .
the solid plots are a , b fixed infectious period : r0fix with rtg = 0.5 ( lower ) , rtg = 1.5 ( upper ) ; and rtg = 1.0 ( lower ) , rtg = 2.0 ( upper ) respectively , and c , d exponentially distributed infectious period : r0exp with rtg = 0.5 ( lower ) , rtg = 1.5 ( upper ) ; and rtg = 1.0 ( lower ) , rtg = 2.0 ( upper ) , respectively .
the horizontal dotted and dashed lines indicate the corresponding values of \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g}$$ 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } , respectively the results shown in fig . 2 illustrate that for fixed values of r , te and tg , the values of r0 and r0 + are lower and upper bounds , respectively for both r0exp and r0fix . in sects .
( 2.1 and 2.4 ) it was shown that r0fix  r0 + and r0  r0exp , respectively .
it is proved in appendix 2 that r0(m , n ) 0 is an increasing function of both m and n. putting these results together we obtain the inequality \documentclass[12pt]{minimal } 
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 $ $ r_0^ - \le r_0^{\exp } \le r_0^{(m , n ) } < r_0^{{\rm{fix } } } \le r_0^ + $ $ 
 \end{document } where m and n are any finite positive integers .
table 1 was constructed to illustrate the results that may be obtained for some specific infections .
parameters were chosen from the literature to be representative of influenza , severe acute respiratory syndrome ( sars ) , smallpox   and foot and mouth disease ( fmd ) . for each infection a trapezium distribution was constructed for f( ) , and used together with an estimate of r0 to calculate an estimate of the initial exponential increase , r. the function f( ) was also used to calculate values for the mean generation interval tg , the mean latent period te and the mean infectious period ti .
table 1estimates of r0 that could be made for emerging infectionsinfectionrtgtetir0r0+r0fixr0expr0trapinfluenza0.1983.651.604.101.722.062.001.852.00sars0.1349.005.507.002.213.343.222.553.20smallpox0.057620.515.011.02.183.263.202.453.20fmd0.1656.002.008.001.992.702.512.212.50the initial rate of exponential increase ( r day ) , mean generation interval ( tg days ) , mean latent period ( te days ) and mean infectious period ( ti days ) that could be observed for epidemics of influenza , sars , smallpox ; and for foot andmouth disease ( fmd ) spreading between farms ; together with the corresponding estimates of the basic reproduction numbermade using the approximations r0 = 1+rtg and \documentclass[12pt]{minimal } 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } , or assuming a rectangular , exponential or trapeziodal distribution for the infectious period , leading to r0fixr0exp and r0trap , respectively estimates of r0 that could be made for emerging infections the initial rate of exponential increase ( r day ) , mean generation interval ( tg days ) , mean latent period ( te days ) and mean infectious period ( ti days ) that could be observed for epidemics of influenza , sars , smallpox ; and for foot andmouth disease ( fmd ) spreading between farms ; together with the corresponding estimates of the basic reproduction numbermade using the approximations r0 = 1+rtg and \documentclass[12pt]{minimal } 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document } , or assuming a rectangular , exponential or trapeziodal distribution for the infectious period , leading to r0fixr0exp and r0trap , respectively if these values of r , tg and te had been estimated from data , and then r0 had been estimated by r0 , r0 + , r0fix , r0exp or r0trap , the estimates presented in table 1 would have resulted . by construction , the estimated value of r0trap then corresponds to the assumed value of r0 .
hence table 1 must be regarded as a comparison of estimates that may be made ; the relative values are important rather than the absolute values .
we have derived and discussed model - consistent methods for estimating the basic reproduction number ( r0 ) for an infectious disease from the initial rate of exponential growth of incidence of infection ( r ) at the beginning of an epidemic .
these methods can only be applied to incidence data from the period where it is reasonable to assume that the whole population may be regarded as susceptible ( s(t ) } ~ n ) . among the first pieces of information obtained for an emerging infection are observations of the latent and infectious periods .
these may be used as estimates to construct a rectangular kernel for an integral equation model , or to derive rate parameters for a differential equation model .
examples of infectivity kernels with the same latent and infectious periods are shown in fig .
1 . these kernels appear to be quite different , with the exponential kernel allowing some transmission of infection from time zero , and exhibiting a long infection tail .
the transmission at early times mitigates against the success of control methods based on contact tracing , or any other method with inherent delays .
the tail can lead to transmission appearing to continue in the model long after control measures should have eliminated the infection .
1 allow for some variability in the fixed and latent periods to be incorporated in the model .
1a ) is consistent with a latent period uniformly distributed between 1.2 and 2.0 days , and an infectious period of 4.0 days .
1b ) is consistent with nobody being infectious before 4 days , everybody infectious by 7 days , everybody still infectious at 11 days and nobody infectious after 14 days ; with the proportion infectious at intermediate times determined by linear interpolation . as well as allowing for some variability , the trapezoidal kernel has the advantage over the rectangular one that it is a continuous function , and this avoids problems with numerical schemes that do not allow discontinuities . of course , if further information is available , then other distributions may be more appropriate .
figure 2 compares estimates of the basic reproduction number based on fixed ( fig . 2a , 2b ) and exponentially distributed ( fig .
2c , 2d ) infectious periods , r0fix andr0exp , respectively , with the estimates based only on mean generation interval , \documentclass[12pt]{minimal } 
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 $ $ r_0^ - = 1 + r{t_g}$$ 
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 $ $ r_0^ + = { e^{r{t_g}}}$$ 
 \end{document}. the estimate r0 is inaccurate whenever rtg is not small . using the fixed infectious period model , r0 + approximates r0fix when te / tg is near to one , that is when the infection has a long latent period and a short infectious period .
the estimate r0 is an approximation to r0exp if either the latent period or infectious period are very short , but r0 + is never a good estimator for r0exp and it s use is therefore inconsistent with an seir model .
the estimates of r0 derived in this paper have the ordering r0  r0exp  r0(m , n ) < r0fix  r0 + .
this inequality establishes that , given the same values of r , te and ti , r0 provides a closer estimate for r0exp than for r0fix , and r0 + provides a closer estimate for r0fix than for r0exp .
even though results derived from the gamma distributed kernel , r0(m , n ) , are not displayed in fig .
2 , we have established that r0fix and r0exp are upper and lower bounds respectively for r0(m , n ) .
note that for the model with a fixed infectious period , \documentclass[12pt]{minimal } 
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 $ $ { t_g } = { t_e } + { t_i}/2$$ 
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 $ $ { t_g } = { t_e } + { t_i}$$ 
 \end{document}. the inequality , and the results presented in fig .
2 are derived on the assumption that te and tg are the same in both models ; ti is defined consistently with the model , and hence differs between models .
1 , which have the same latent and infectious periods , te and ti , and hence different mean generation intervals tg . table 1 shows results that could be obtained when estimating r0 for emerging infections , with parameters suitable for these infections .
the parameter values indicated for each infection should be regarded as sensible values rather than exact estimates , and the results are presented to indicate the relevance of fig .
this is not surprising for the influenza example where the trapezium kernel has steep sides ( fig .
the results in table 1 confirm that if rtg is small , hence r0 is close to one , then r0 may be used as an estimator for r0exp when an exponential model is appropriate .
if a model with a fixed infectious period is more appropriate then r0 + is a better estimate for r0fix , especially when te /tg is closer to one : compare for example the relative values of r0 + and r0fix for smallpox where te /tg = 0.73 and fmd where te /tg
wearing et al .   compared estimates based on r0exp with results obtained using gamma - distributed infection kernels .
heffernan and wahl   also examined the problem , and provided correction factors for estimates of r0 based on both the mean and variance of observed transition times .
wallinga and lipsitch   used a similar approach to ours , and derived estimates of r0 for a selection of infectivity kernels f(t ) , including those derived from a gamma - distributed infectious period but only with te = 0 .
they also considered the case where f(t ) is a normal distribution ; if employed though this distribution should be truncated to avoid the possibility of negative generation intervals . even though we selected a number of particular kernels for our study , and these cover a reasonable range of first choices , our method is applicable to all biologically sensible kernels .
estimates made early in an epidemic are likely to be based on household studies , and may be truncated due to local saturation of contacts .
in addition , it is unclear how valid such estimates are when extrapolated to the wider community with multiple levels of mixing
as a new infection emerges the appropriate model is speculative , and in any situation there is no such thing as the correct model .
had tg = 2.6 and te = 1.48 days , and estimated r0  r0 , obtaining values in the range 1  2 . their model was more complex than those discussed here , but we have seen that for low values of rtg , r0 provides a reasonable estimate .
.   used an seir model , and estimated r from r0exp which is consistent with their model .
our approach is to advocate using an estimate of r0 that is consistent with the model used to evaluate control strategies .
 we investigate the merit of deriving an estimate of the basic reproduction number \documentclass[12pt]{minimal } 
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 $ $ 
 \mathcal{r}_0 
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 \end{document } early in an outbreak of an ( emerging ) infection from estimates of the incidence and generation interval only . 
 we compare such estimates of \documentclass[12pt]{minimal } 
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 $ $ 
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 \end{document } with estimates incorporating additional model assumptions , and determine the circumstances under which the different estimates are consistent . 
 we show that one has to be careful when using observed exponential growth rates to derive an estimate of \documentclass[12pt]{minimal } 
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 $ $ 
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 , and we quantify the discrepancies that arise . 
1 Introduction
2 Model-consistent estimation of 
2.1 Fixed infectious period
2.2 Trapezoidal infection kernel
2.3 SEIR differential equation models
2.4 Exponentially distributed infectious period
3 Method and results
4 Conclusions and discussion