A history-dependent approach for accurate initial condition estimation in epidemic models

Mathematical modeling is a powerful tool for understanding and predicting complex dynamical systems, ranging from gene regulatory networks to population-level dynamics. However, model predictions are highly sensitive to initial conditions, which are often unknown. In infectious disease models, for instance, the initial number of exposed individuals (E) at the time the model simulation starts is frequently unknown. This initial condition has often been estimated using an unrealistic, history-independent assumption for simplicity: the chance that an exposed individual becomes infectious is the same regardless of the timing of their exposure (i.e., exposure history). Here, we show that this history-independent method can yield serious bias in the estimation of the initial condition. To address this, we developed a history-dependent initial condition estimation method derived from a master equation expressing the time-varying likelihood of becoming infectious during a latent period. Our method consistently outperformed the history-independent method across various scenarios, including those with measurement errors and abrupt shifts in epidemics, for example, due to vaccination. In particular, our method reduced estimation error by 55% compared to the previous method in real-world COVID-19 data from Seoul, Republic of Korea, which includes likely infection dates, allowing us to obtain the true initial condition. This advancement of initial condition estimation enhances the precision of epidemic modeling, ultimately supporting more effective public health policies. We also provide a user-friendly package, Hist-D, to facilitate the use of this history-dependent initial condition estimation method.Author summary: Accurately predicting infectious disease spread requires knowing the initial number of individuals in the exposed compartment at the start of the simulation (E(t0)), but this number is usually unknown. A common method to estimate E(t0) assumes that the chance of an exposed individual becoming infectious is the same, regardless of when they were exposed. However, this unrealistic assumption can lead to serious errors in the estimation of E(t0). To solve this problem, we developed a method that considers exposure timing. Our method successfully estimated E(t0) even with measurement errors or sudden changes in outbreak conditions. In particular, our approach accurately estimated E(t0) for COVID-19 data from Seoul that includes likely infection dates, which allowed us to obtain the true initial condition. This advancement of initial condition will help improve epidemic predictions and public health strategies. Our method can also be applied to estimate initial conditions in systems where timing or history matters, such as protein maturation or cell degradation pathways. To facilitate the broad adoption of our method, we have also developed and released Hist-D, a user-friendly software package.