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Projected Spread Models

Jung-Chao Ban, Jyy-I Hong, Cheng-Yu Tsai, Yu-Liang Wu

TL;DR

The paper addresses how testing limitations and incubation dynamics obscure heterogeneity in disease spread by introducing projected spread models that map hidden types to explicit types via a map $\Phi$. It systematically develops both topological and random formulations, deriving complete formulas for projected spread rates in terms of the left eigenvector of the appropriate $\xi$-matrix, across $1$-spread, $m$-spread, and $k$-block-code configurations. The main contributions include explicit, unified expressions for $s_p(a,\cdot)$ and its random analogs, proofs that the projected rates depend only on eigenvector weights and not on initial hidden types (on non-extinction), and comprehensive numerical examples that validate the theory for all three relative cases ($m-1=k$, $m-1<k$, $m-1>k$). The results have practical significance for pandemic modeling under imperfect testing and incubation scenarios, enabling accurate prediction of observable spread rates for explicitly defined groups and informing targeted interventions.

Abstract

We present a disease transmission model that considers both explicit and non-explicit factors. This approach is crucial for accurate prediction and control of infectious disease spread. In this paper, we extend the spread model from our previous works \cite{ban2021mathematical,ban2023randomspread, ban2023mathematical, ban2023spread} to a projected spread model that considers both hidden and explicit types. Additionally, we provide the spread rate for the projected spread model corresponding to the topological and random models. Furthermore, examples and numerical results are provided to illustrate the theory.

Projected Spread Models

TL;DR

The paper addresses how testing limitations and incubation dynamics obscure heterogeneity in disease spread by introducing projected spread models that map hidden types to explicit types via a map . It systematically develops both topological and random formulations, deriving complete formulas for projected spread rates in terms of the left eigenvector of the appropriate -matrix, across -spread, -spread, and -block-code configurations. The main contributions include explicit, unified expressions for and its random analogs, proofs that the projected rates depend only on eigenvector weights and not on initial hidden types (on non-extinction), and comprehensive numerical examples that validate the theory for all three relative cases (, , ). The results have practical significance for pandemic modeling under imperfect testing and incubation scenarios, enabling accurate prediction of observable spread rates for explicitly defined groups and informing targeted interventions.

Abstract

We present a disease transmission model that considers both explicit and non-explicit factors. This approach is crucial for accurate prediction and control of infectious disease spread. In this paper, we extend the spread model from our previous works \cite{ban2021mathematical,ban2023randomspread, ban2023mathematical, ban2023spread} to a projected spread model that considers both hidden and explicit types. Additionally, we provide the spread rate for the projected spread model corresponding to the topological and random models. Furthermore, examples and numerical results are provided to illustrate the theory.
Paper Structure (22 sections, 13 theorems, 102 equations, 16 figures, 2 tables)

This paper contains 22 sections, 13 theorems, 102 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Projected spread models
  3. $1$-spread model and $0$-block code
  4. $1$-spread model and $k$-block code for $k\geq 1$
  5. $m$-spread model and $0$-block code for $m\geq 1$
  6. General cases: $m$-spread model and $k$-block code
  7. The case where $m-1=k$
  8. The case where $m-1<k$
  9. The case where $m-1>k$
  10. Random projected spread models
  11. Random $1$-spread model and $0$-black code
  12. Random $1$-spread model and $k$-block code for $k\geq 1$
  13. Examples with numerical results
  14. Examples for Topological models
  15. The case where $m-1=k$
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $\mathcal{S=}\{p_{i}\}_{i=1}^{L}$ be a spread model over $\mathcal{B}$, and write $\mathcal{B}=\{b_{j}\}_{j=1}^{K}$. Suppose $M$ is the associated $\xi$-matrix, and $\rho =\rho _{M}$ is the maximal eigenvalue of $M$ with positive left eigenvector $w=(w(b_{j}))_{j=1}^{K}$ and $\sum_{j=1}^K \omega

Figures (16)

  • Figure 1: The graph of $0$-block code $\Phi:\{a_1,a_2,a_3,a_4,b_1,b_2\}\to \{a,b\}$ defined by $\Phi(a_i)=a$ and $\Phi(b_i)=b$ in $\mathbb{N}.$
  • Figure 2: The graph of $0$-block code $\Phi:\{a_1,a_2,a_3,a_4,b_1,b_2\}\to \{a,b\}$ defined by $\Phi(a_i)=a$ and $\Phi(b_i)=b$ in $\mathbb{N}^2.$
  • Figure 3: The potential $1$-patterns
  • Figure 4: A potential $2$-pattern
  • Figure 5: Not a potential $2$-pattern
  • ...and 11 more figures

Theorems & Definitions (22)

  • Theorem 2.1: Proposition 2 ban2023spread
  • Lemma 2.2: Lemma 2 ban2023mathematical
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6: Spread rate for $m$-spread model
  • Theorem 2.7
  • ...and 12 more