An improved inverse method for estimating disease transmission rates in epidemics with low-incidence data

The accurate estimation of time-varying transmission rates is fundamental for understanding infectious disease dynamics and implementing effective public health interventions. To this end, we propose an improved inverse method for estimating time-varying transmission rates in low-incidence settings, where conventional data pre-processing approaches often fail owing to sparse-case observations. To overcome this difficulty, we introduce an exponential B-spline interpolation approach that integrates both continuous and discrete inverse methods. In addition, we incorporate a non-standard finite-difference method, which enables the reconstruction of epidemic processes under non-equidistant time intervals caused by missing case data. This method ensures that transmission rate estimates remain non-negative and smooth, even when the observed data exhibit low cases or irregularly spaced time points. We apply this approach to several infectious disease models using real-world data from the People's Republic of China and Germany, including a scarlet fever model, a multi-strain influenza model and an age-structured influenza model, as well as a COVID-19 model for Germany. The results show that our method provides accurate transmission rate estimates, particularly in low-incidence infectious diseases, multi-group epidemic models and epidemic datasets with incomplete surveillance records, demonstrating its robustness and applicability across various epidemiological contexts. The improved inverse method offers a new perspective for epidemiological modelling and provides reliable technical support for related theoretical exploration and public health decision-making.