Improving epidemic size prediction through stable reconstruction of disease parameters by reduced iteratively regularized Gauss–Newton algorithm

Abstract

Classical compartmental epidemic models of infectious diseases track the dynamic transition of individuals between different epidemiological states or risk groups. Reliable quantification of various transmission pathways in these models is paramount for optimal resource allocation and successful design of public health intervention programs. However, with limited epidemiological data available in the case of an emerging disease, simple phenomenological models based on a smaller number of parameters can play an important role in our quest to make forward projections of possible outbreak scenarios. In this paper, we employ the generalized Richards model for stable numerical estimation of the epidemic size (defined as the total number of infections throughout the epidemic) and its turning point using case incidence data of the early epidemic growth phase. The minimization is carried out by what we call the Reduced Iteratively Regularized Gauss–Newton (RIRGN) algorithm, a problem-oriented numerical scheme that takes full advantage of the specific structure of the non-linear operator at hand. The convergence analysis of the RIRGN method is suggested and numerical simulations are conducted with real case incidence data for the 2014–15 Ebola epidemic in West Africa. We show that the proposed RIRGN provides a stable algorithm for early estimation of turning points using simple phenomenological models with limited data.

On iteratively regularized predictor–corrector algorithm for parameter identification *

“…The experiments have shown that our new method is accurate and stable for a broad range of initial values and regularization sequences. As compared to the 'traditional' approach [25], which consists in minimizing B(u) − d δ 2 with respect to θ, while solving the equation G(θ, u) = g numerically at every step of the iterative process, for the two inverse problems considered, method (1.8) and (1.9) is more time efficient and more reliable in its estimation of the unknown parameters.…”

“…It has been observed in [25], that estimation of the unknown parameters, p, r, a, and K, from problem (3.4) and (3.5) in a 'traditional way' given early incidence data, d δ , that is, minimizing dC dt − d δ 2 with respect to p, r, a, and K while solving equation (3.5) numerically at every step of the iterative process, is extremely unstable in K, leading to very inaccurate predictions of disease capacity. Therefore, in this subsection, we use our newly proposed PCA (1.8) and (1.9) in order to approximate a solution to (3.4) and (3.5) in a stable manner for t ∈ [t 1 , t n ] with t n < B.…”

A primer on stable parameter estimation and forecasting in epidemiology by a problem-oriented regularized least squares algorithm

“…This implies the Hessian approximation which we reduce further by dropping and setting This permits to lower the condition number and to enforce the symmetric non-negative definite structure. Dividing both sides of the modified linear system by , which enables us to move all noise from the matrix to the right-hand side, one arrives at what we call Reduced IRGN (RIRGN) ( Smirnova et al., 2017 ) where , and …”

“…For the early transmission period, phenomenological models of a logistic type, describing the progression of the epidemic in terms of the cumulative number of reported cases, C , provide a simple alternative. In what follows, we employ the generalized Richards model ( Chowell et al., 2016 , Smirnova et al., 2017 , Turner et al., 1976 ) to estimate crucial disease parameters, such as the intrinsic growth rate ( r ), the deceleration of growth ( p ), the final size of the epidemic ( K ), the disease turning point (τ), and the extent of deviation from the S-shaped dynamics of the classical logistic-growth curve ( a ), see Fig. 1 .…”