_{0}from Early Epidemic Growth Data

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The basic reproduction number, _{0}, a summary measure of the transmission potential of an infectious disease, is estimated from early epidemic growth rate, but a likelihood-based method for the estimation has yet to be developed. The present study corrects the concept of the actual reproduction number, offering a simple framework for estimating _{0} without assuming exponential growth of cases. The proposed method is applied to the HIV epidemic in European countries, yielding _{0} values ranging from 3.60 to 3.74, consistent with those based on the Euler-Lotka equation. The method also permits calculating the expected value of _{0} using a spreadsheet.

The basic reproduction number, _{0}, of an infectious disease is the average number of secondary cases generated by a single primary case in a fully susceptible population [_{0} is the most widely used epidemiological measurement of the transmission potential in a given population. Statistical estimation of _{0} has been performed for various infectious diseases [_{0} has been used for determining the minimum coverage of immunization, because the threshold condition to prevent a major epidemic in a randomly-mixing population is given by 1-1/_{0} [_{0} gives an estimate of the so-called final size,

Methodological discussions concerning the statistical inference of _{0} are still in progress, and it is recognized that the estimate is very sensitive to dispersal (or underlying epidemiological assumptions) of the progression of a disease [_{0} using epidemic data of an emerging (or exotic) disease, the exponential growth rate, _{0} using that knowledge (see below). That is, the conventional estimation technique has required two statistical steps, namely, first estimate _{0}.

The estimation method can be illustrated by employing a simple renewal process which adheres to the original definition of _{0} [

Since _{0} represents the total number of secondary cases that a primary case generates during the entire course of infection, the estimate is given by ([

When

Since the density function of the generation time,

Replacing _{0} is obtained [

The estimation of _{0} is achieved by measuring the exponential growth rate _{0} (_{0}, has been missing). Moreover, _{0} using her/his own data.

The purpose of the present study is to offer an improved framework for estimating _{0} from early epidemic growth data, which may be slightly more tractable among non-experts as compared with the above mentioned estimator (5). A likelihood-based approach is proposed to permit derivation of the uncertainty bounds of _{0}. For an exposition of the proposed method, the incidence data of the HIV epidemic in Europe is explored.

In addition to _{0}, a different measurement of the transmission potential using widely available epidemiological data, the actual reproduction number, _{a}, has been proposed for HIV/AIDS [_{a} is much simpler than _{0} in that _{a} is defined as a product of the mean duration of infectiousness and the ratio of incidence to prevalence [

The actual reproduction number _{a} is written as:
_{a} = _{0} is given by:

_{a} coincides with _{0} as long as _{a} has been emphasized to have an application in HIV/AIDS [_{a} tends to yield a biased estimate (if _{a} is regarded as a proxy for _{0}), and the estimate of _{a} is not as robust as that is obtained with

Here, the above mentioned negative aspect of _{a} is reconsidered by correcting the definition of _{a}. The disease of interest in the present study is HIV. The frequency of secondary transmissions relative to infection-age _{1} = 0.24 years), followed by a long asymptomatic period with a low frequency of secondary transmissions (for _{2} = 8.38 years) [_{3} = 0.75 years until death or until the infected individual ceases risky sexual contact due to AIDS [_{1}, _{2} and _{3} have been estimated at 1.30, 0.05 and 0.36 per year [

Here the concept of _{a} is corrected. The

The numerator represents the number of new infections at calendar time _{0}, the concept of the denominator is better replaced by “the total number of effective contacts (which can potentially lead to secondary transmissions with an equal probability)”. Therefore, the corrected _{a} is better written as:

Replacing

Thus, the estimator of corrected _{a} in (11) is identical to that of _{0}. In other words, _{0} can be estimated from the incidence data and the generation time without assuming exponential growth of cases during the early phase of an epidemic. It should be noted that the ratio of prevalence to mean generation time

Here the epidemic data of HIV/AIDS in three European countries: France, the Western part of Germany (

_{0} is estimated using two different methods, one employing the estimator (5) and another using the corrected _{a}. For the former approach, the exponential growth rate is estimated via a pure birth process [_{t} cases in year _{0} mirrors the uncertainty in the estimate of _{0} via (5) is made by using

The latter method, proposed in the present study, employs the estimator of corrected _{a} in (11). Since the data are yearly, the _{s} is assumed to be given by _{s} = _{0} is calculated as a normalized yearly average, because _{1} is as short as 0.24 years.

The likelihood of estimating _{0} with (15) is proposed as follows. First, the inverse of both sides of (15) is taken:

The numerator of the right-hand side indicates the total number of effective contacts made by potential primary cases in year

The right-hand side of _{0}. This is a simple binomial sampling process. In other words, the likelihood function for estimating _{0} is:
_{0} is obtained by minimizing the negative logarithm of (17), and the 95% CI are derived from the profile likelihood.

_{0} for HIV in France, Western Germany and the UK. The maximum likelihood estimates of _{0} ranged from 3.65–4.08. Again, Western Germany yielded the highest estimate without an overlap of the uncertainty bound with the other two countries. The maximum likelihood estimate of _{0} based on the proposed new method ranged from 3.59 to 3.74. Not only were the qualitative patterns for the expected values of _{0} consistent with those based on an exponential growth assumption, but the 95% CI also broadly overlapped with those based on the other method. In particular, although _{0} in Western Germany using the proposed method is smaller than that based on an exponential growth assumption, the 95% CIs of the two methods overlapped. The 95% CI based on the proposed method appeared to be wider than those based on exponential growth assumption. Since HIV is mainly transmitted via sexual contact, the above mentioned estimate may vary with the mixing pattern and contact frequency (thus, there is no general disease-specific _{0}, especially for HIV/AIDS). At least, compared with a previous estimate of _{0} as ranging from 13.9 to 54.5 in the USA, based on an exponential growth assumption that adopted a mean infectious period of 10 years [_{0} in the present study appeared to be much smaller using a precise estimate of the generation time distribution.

The present study proposed the use of the corrected actual reproduction number, _{a}, for statistical inference of _{0} based on incidence and known relative frequency of secondary transmissions (_{0}) was required [_{0} from incidence data. The simple likelihood function employing a binomial distribution was also proposed to yield an appropriate uncertainty bound of _{0}. It should be noted that given the knowledge of _{s} and readily available incidence data, _{0} without likelihood. Such a calculation can be attained using any spreadsheet.

The usefulness of the actual reproduction number, calculated as a product of the mean generation time and the incidence-to-prevalence ratio, has been previously emphasized in assessing the epidemiological time course of an epidemic [_{a} does not precisely capture the secondary transmission if the transmission rate _{a} by the total number of potential contacts, it was shown that the _{0} derived from the renewal equation coincides with the corrected actual reproduction number, _{a}, and also that the likelihood can be quite easily derived. The corrected _{a} does not require prevalence data, and uses only the incidence data and the generation time distribution.

Many future tasks remain, however. Most importantly, the estimation of _{0} from early epidemic growth data for a heterogeneously-mixing population is called for. _{0} in the present study can even be interpreted as _{0} for a heterogeneously-mixing population (_{0} and the next-generation matrix from structured data (_{0} from early epidemic growth data, while being easily tractable and calculable among general epidemiologists.

The work of H. Nishiura was supported by the JST PRESTO program.

_{0}in models for infectious diseases in heterogeneous populations

_{0}from the initial phase of an outbreak of a vector-borne infection

The relative frequency of secondary transmissions of HIV as a function of the time since infection. A step function is employed to approximately model the frequency of secondary transmissions relative to infection-age. For _{1} years shortly after infection, the frequency _{1} is very high. Subsequently, for _{2} years (_{2} is persistently low, followed by a time period with high infectiousness _{3} for _{3} years until death or no secondary transmission due to AIDS. Following a statistical study [_{1}, _{2} and _{3} are assumed to be 0.24, 8.38 and 0.75 years. Assuming that the contact frequency does not vary as a function of time since infection, _{1}, _{2} and _{3} are estimated at 1.30, 0.05 and 0.36 per year.

Epidemic curves of HIV/AIDS in France, Western Germany and the United Kingdom (UK) from 1976–2000.

The transmission tree with _{0} = 2. (A) Black circles represent primary cases that are infectious to others at time _{0} = 2, _{0}.

Comparison of the estimates of the basic reproduction number for HIV/AIDS obtained using two different estimation methods.

^{1} |
_{0} (exponential growth) ^{2} |
_{0} (proposed likelihood) ^{3} | |
---|---|---|---|

France | 1.15 (1.12, 1.17) | 3.65 (3.64, 3.66) | 3.59 (3.38, 3.81) |

Western Germany | 2.15 (2.02, 2.29) | 4.08 (4.02, 4.14) | 3.74 (3.43, 4.08) |

UK | 1.21 (1.18, 1.25) | 3.67 (3.66, 3.69) | 3.65 (3.38, 3.96) |

The intrinsic growth rate during the exponential growth phase;

the basic reproduction number estimated using

the basic reproduction number estimated using