Convergence of two-stage iterative scheme for $K$-weak regular splittings of type II with application to Covid-19 pandemic model

TL;DR

The paper develops convergence theory for a two-stage iterative scheme applied to $K$-weak regular splittings of type II within proper cone frameworks, extending classical results to nonstandard cones and inner-iteration splittings. It proves that, under a commutativity condition $VF^{-1}G=GF^{-1}V$, the stationary and non-stationary two-stage methods converge and the induced splitting remains $K$-weak regular type II, with the key matrices $T_{s(k)}$ and $\widehat{T}_{s(k)}$ being similar. It also derives comparison results showing how certain orderings of the splittings yield faster convergence, and establishes a monotone convergence theorem for $K$-nonnegative, $K$-monotone settings. The methods are then applied to a COVID-19 age-structured next-generation-matrix model to efficiently compute $R_0=\rho(\mathcal{B}\mathcal{A}^{-1})$, with numerical experiments demonstrating performance gains of the two-stage approach for both small and large-scale systems. Overall, the work provides a rigorous framework for fast, monotone convergence in cone-structured linear systems and practical tools for epidemiological risk assessment via next-generation matrices.

Abstract

Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, $K$-monotonicity is sufficient for the convergence of $K$-regular and $K$-weak regular splittings, where $K$ is a proper cone in $\mathbb{R}^n$. However, the convergence theory of a two-stage iteration scheme in general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., $K=\mathbb{R}^n_+$), is open. To this end, we propose convergence theory of two-stage iterative scheme for $K$-weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a $K$-regular splitting and then establish some comparison theorems. We also study $K$-monotone convergence theory of the stationary two-stage iterative method in case of a $K$-weak regular splitting of type II. The most interesting and important part of this work is on $M$-matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model.

Figures (1)

• Figure 1: Monotonically increasing and decreasing convergence pattern of the iterative solution to the exact solution from two different initial approximated vector, which are nonnegative.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• Lemma 3.2
• proof