An Improved Inverse Method for Estimating Disease Transmission Rates in Low-Prevalence Epidemics

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-prevalence settings, where conventional data preprocessing approaches often fail due to sparse case observations. To overcome this difficulty, we introduce an exponential B-spline interpolation approach that integrates both continuous and discrete inverse methods. This method ensures that transmission rate estimates remain non-negative and smooth, even when the observed data exhibit low cases. We apply this approach to several infectious disease models using real-world data from China, including a scarlet fever model, a multi-strain influenza model, and an age-structured influenza model. The results show that our method provides accurate transmission rate estimates, particularly in low-prevalence infectious diseases and multi-group epidemic models, demonstrating its robustness and applicability across various epidemiological contexts. The improved inverse method offers a new perspective for epidemiological modeling and provides reliable technical support for related theoretical exploration and public health decision-making.

💡 Research Summary

The paper addresses a critical challenge in epidemiological modeling: estimating time‑varying transmission rates β(t) for diseases that exhibit low prevalence, where case counts are sparse and traditional preprocessing methods often produce negative or unstable estimates. To overcome this, the authors propose an “exponential B‑spline inverse method” that integrates continuous‑time and discrete‑time inverse frameworks while guaranteeing non‑negative, smooth estimates of new infections.

The methodology proceeds in two main steps. First, observed new‑case counts yₖ (≥0) are log‑transformed to Yₖ = ln yₖ. A B‑spline basis of chosen order and knot configuration is then fitted to the logarithmic data, yielding a smooth interpolant f(t) = ∑₁ᵐ Pᵢ Bᵢ,ₚ(t). Because the spline operates on log‑values, it preserves the sign of the data and avoids the oscillations that plague direct polynomial or spline interpolation of very small numbers. Second, the interpolant is exponentiated: ỹ(t) = exp f(t). This step enforces ỹ(t) ≥ 0 for all t, providing a biologically plausible estimate of the continuous stream of new infections.

In the continuous inverse setting, the authors embed ỹ(t) into a non‑autonomous SEIR model. The model equations are standard:

dS/dt = Λ − β(t) S I/N − dS,
dE/dt = β(t) S I/N − (σ + d)E,
dI/dt = σE − (γ + d)I,
dR/dt = γI − dR.

Using the identity σE(t) = ỹ(t) and differentiating, the authors define H(t) = dE/dt = (1/σ) dỹ/dt. Closed‑form solutions for N(t), I(t), and R(t) are derived analytically, while S(t) follows from the conservation relation S = N − E − I − R. Substituting these expressions into the second SEIR equation yields an explicit formula for β(t):

β(t) =