Correcting the actual reproduction number: a simple method to estimate R(0) from early epidemic growth data

Abstract

The basic reproduction number, R(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 R(0) without assuming exponential growth of cases. The proposed method is applied to the HIV epidemic in European countries, yielding R(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 R(0) using a spreadsheet.

Keywords: AIDS; HIV; basic reproduction number; epidemiology; estimation techniques; infectious diseases; statistical model; transmission.

Figure 1.

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 d₁ years shortly after infection, the frequency g₁ is very high. Subsequently, for d₂ years (i.e., during the asymptomatic period), g₂ is persistently low, followed by a time period with high infectiousness g₃ for d₃ years until death or no secondary transmission due to AIDS. Following a statistical study [20], d₁, d₂ and d₃ 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, g₁, g₂ and g₃ are estimated at 1.30, 0.05 and 0.36 per year.

Figure 2.

Epidemic curves of HIV/AIDS in France, Western Germany and the United Kingdom (UK) from 1976–2000. A. The yearly number of new HIV infections (i.e., incidence) and new AIDS cases from 1976–2000. AIDS cases are the observed data, while HIV incidence is estimated by means of a back-calculation method used by Artzrouni [23]. B. The early growth phase of the HIV epidemic. The peak incidence was observed in 1982 in Western Germany and 1983 in France and the UK.

Figure 3.

The transmission tree with R₀ = 2. (A) Black circles represent primary cases that are infectious to others at time t and white circles are secondary cases generated by the primary cases. Secondary transmissions from primary to secondary cases are given with the basic reproduction number, R₀ = 2, i.e., each primary case generates two secondary cases. (B) Reconstruction of the transmission tree. Given that all the potential contacts made by primary cases (black circles) are known using the incidence data and the generation time distribution, the probability that each potential contact resulted in a secondary transmission is given by 1/R₀.