Improved estimation of the effective reproduction number with heterogeneous transmission rates and reporting delays

In the face of an infectious disease, a key epidemiological measure is the basic reproduction number, which quantifies the average secondary infections caused by a single case in a susceptible population. In practice, the effective reproduction number, denoted as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:math> , is widely used to assess the transmissibility of the disease at a given time t. Real-time estimating this metric is vital for understanding and managing disease outbreaks. Traditional statistical inference often relies on two assumptions. One is that samples are assumed to be drawn from a homogeneous population distribution, neglecting significant variations in individual transmission rates. The other is the ideal case reporting assumption, disregarding time delays between infection and reporting. In this paper, we thoroughly investigate these critical factors and assess their impact on estimating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:math> . We first introduce negative binomial and Weibull distributions to characterize transmission rates and reporting delays, respectively, based on which observation and state equations are formulated. Then, we employ a Bayesian filtering for estimating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mi>t</mml:mi></mml:msub> </mml:math> . Finally, validation using synthetic and empirical data demonstrates a significant improvement in estimation accuracy compared to conventional methods that ignore these factors.