Modeling variants of the COVID-19 virus in Hawai‘i and the responses to forecasting

Richard Carney; Monique Chyba; Victoria Y. Fan; Prateek Kunwar; Thomas Lee; Ionica Macadangdang; Yuriy Mileyko; Richard Carney; Monique Chyba; Victoria Y. Fan; Prateek Kunwar; Thomas Lee; Ionica Macadangdang; Yuriy Mileyko

doi:10.3934/math.2023223

AIMS Mathematics

2023, Volume 8, Issue 2: 4487-4523. doi: 10.3934/math.2023223

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Research article Special Issues

Modeling variants of the COVID-19 virus in Hawai‘i and the responses to forecasting

1.
Department of Mathematics, University of Hawai‘i at Mānoa, 2565 McCarthy Mall Honolulu, Hawai‘i 96822
2.
Thompson School of Social Work & Public Health, University of Hawai‘i at Mānoa, Honolulu, HI 96822, USA
3.
Hawaii Data Collaborative, Honolulu, HI 96813, USA

Received: 09 September 2022 Revised: 11 November 2022 Accepted: 17 November 2022 Published: 05 December 2022
MSC : 39Axx, 92D25, 92D30, 93-10

In this paper we introduce a model for the spread of COVID-19 which takes into account competing SARS-CoV-2 mutations as well as the possibility of reinfection due to fading of vaccine protection. Our primary focus is to describe the impact of the B.1.617.2 (Delta) and B.1.1.529 (Omicron) variants on the state of Hawai‘i and to illustrate how the model performed during the pandemic, both in terms of accuracy, and as a resource for the government and media. Studying the effect of the pandemic on the Hawaiian archipelago is of notable interest because, as an isolated environment, its unique geography affords partially controlled travel to and from the state. We highlight the modeling efforts of the Hawai‘i Pandemic Applied Modeling Work Group (HiPAM) which used the model presented here, and we detail the model fitting and forecasting for the periods from July 2021 to October 2021 (Delta surge) and from November 2021 to April 2022 (Omicron surge). Our results illustrate that the model was both accurate when the forecasts were built on assumptions that held true, and was inaccurate when the public response to the forecasts was to enforce safety measures that invalidated the assumptions in the model.

Keywords:

Citation: Richard Carney, Monique Chyba, Victoria Y. Fan, Prateek Kunwar, Thomas Lee, Ionica Macadangdang, Yuriy Mileyko. Modeling variants of the COVID-19 virus in Hawai‘i and the responses to forecasting[J]. AIMS Mathematics, 2023, 8(2): 4487-4523. doi: 10.3934/math.2023223

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Abstract

1. Introduction

Dual numbers were first given by Clifford (1845–1879), and some properties of those were studied in the geometrical investigation, and Kotelnikov ^[1] introduced their first applications. Study applied to line geometry and kinematics dual numbers and dual vectors ^[2]. He demonstrated that the directed lines of Euclidean 3-space and the points of the dual unit sphere in $\mathbb{D}^3$ have a one-to-one relationship. Field theory also relies heavily on these numbers ^[3]. The most intriguing applications of dual numbers in field theory are found in a number of Wald publications ^[4]. Dual numbers have contemporary applications in kinematics, dynamics, computer modeling of rigid bodies, mechanism design, and kinematics ^[5,6,7].

Complex numbers have significant advantages in derivative computations. However, the second derivative computations lost these advantages ^[8]. J. A. Fike developed the hyper-dual numbers to solve this issue ^[9]. These numbers may be used to calculate both the first and second derivatives while maintaining the benefits of the first derivative using complex numbers. Furthermore, it is demonstrated that this numerical approach is appropriate for open kinematic chain robot manipulators, sophisticated software, and airspace system analysis and design ^[10].

In the literature, sequences of integers have an important place. The most famous of these sequences have been demonstrated in several areas of mathematics. These sequences have been researched extensively because of their complex characteristics and deep connections to several fields of mathematics. The Fibonacci and Lucas sequences and their related numbers are of essential importance due to their various applications in biology, physics, statistics, and computer science ^[11,12,13]. Many authors were interested in introducing and investigating several generalizations and modifications of Fibonacci and Lucas sequences. The authors investigated two classes that generalize Fibonacci and Lucas sequences, and they utilized them to compute some radicals in reduced forms. Panwar ^[14] defined the generalized $k$ -Fibonacci sequence as

$\begin{align*} F_{k, n} = pkF_{k, n-1}+qF_{k, n-2}, \end{align*}$

with initial conditions $F_{k, 0} = a$ and $F_{k, 1} = b$ . If $a = 0, k = 2, p = q = b = 1$ , the classic Pell sequence and for $a = b = 2, k = 2, p = q = 1$ , Pell-Lucas sequences appear.

The Pell numbers are the numbers of the following integer sequence:

$0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ...$

The sequence of Pell numbers, which is denoted by ${P}_n$ is defined as the linear reccurence relation

$\begin{align*} P_n = 2P_{n-1}+P_{n-2} , \quad P_0 = 0, P_1 = 1, \ \ n\ge2. \end{align*}$

The integer sequence of Pell-Lucas numbers denoted by ${Q}_n$ is given by

$2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, ...$ ,

with the same reccurence relation

$\begin{align*} Q_n = 2Q_{n-1}+Q_{n-2} , \quad Q_0 = Q_1 = 2, \ \ n\ge2. \end{align*}$

The characteristic equation of these numbers is $x^2-2x-1 = 0$ , with roots $\alpha = 1+\sqrt{2}$ and $\beta = 1-\sqrt{2}$ and the Binet's forms of these sequences are given as^{[15,16,17,18]},

$\begin{equation} P_n = \frac{\alpha^n-\beta^n}{\alpha-\beta} \end{equation}$

(1.1)

and

$\begin{equation} Q_n = \alpha^n+\beta^n. \end{equation}$

(1.2)

The set of dual numbers is defined as

$\begin{equation*} \mathbb{D} = \{d = a+\varepsilon a^{*}\mid a, a^{*}\in \mathbb{R}, \varepsilon ^2 = 0, \varepsilon \neq 0\}. \end{equation*}$

The set of hyper-dual numbers is

$\begin{align*} \widetilde{\mathbb{D}} = \left \{\gamma = \gamma_0+\gamma_1\varepsilon +\gamma_2\varepsilon^* +\gamma_3\varepsilon\varepsilon^* \mid \gamma_0, \gamma_1, \gamma_2, \gamma_3\in \mathbb{R}\right\}, \end{align*}$

or can be rewritten as

$\begin{align*} \widetilde{\mathbb{D}} = \left \{\gamma = d+\varepsilon^*d^* \mid d, d^*\in \mathbb{D}\right\}, \end{align*}$

where $\varepsilon$ , $\varepsilon^*$ and $\varepsilon\varepsilon^*$ are hyper-dual units that satisfy

$\begin{equation*} (\varepsilon)^2 = ({\varepsilon^*})^2 = 0, \; \varepsilon\neq\varepsilon^*\neq0, \; \varepsilon\varepsilon^* = \varepsilon^*\varepsilon. \end{equation*}$

This set forms commutative and associative algebra over both the dual and real numbers ^[8,9,10].

The square root of a hyper-dual number $\gamma$ can be defined by

$\begin{equation} \sqrt{\gamma} = \sqrt{\gamma_{0}}+\frac{\gamma_{1}}{2\sqrt{\gamma_{0}}}\varepsilon+\frac{\gamma_{2}}{2\sqrt{\gamma_{0}}}\varepsilon^*+(\frac{\gamma_{3}}{2\sqrt{\gamma_{0}}}-\frac{\gamma_{1}\gamma_{2}}{4\gamma_{0}\sqrt{\gamma_{0}}})\varepsilon\varepsilon^*. \end{equation}$

(1.3)

A hyper-dual vector is any vector of the form

$\begin{align*} \vec{\gamma} = \vec{\gamma}_{0}+\vec{\gamma}_{1}\varepsilon+\vec{\gamma}_{2}\varepsilon^*+\vec{\gamma}_{3}\varepsilon\varepsilon^*, \end{align*}$

where $\vec{\gamma}_{0}, \vec{\gamma}_{1}, \vec{\gamma}_{2}, \vec{\gamma}_{3}$ are real vectors, this vector can be rewritten as $\vec{\gamma} = \vec{d}+\varepsilon^*\vec{d}^*$ , where $\vec{d}$ and $\vec{d}^*$ are dual vectors. Let $\vec{\gamma}$ and $\vec{\delta}$ be hyper-dual vectors, then their scalar product is defined as

$\begin{align} \langle \vec{\gamma}, \vec{\delta}\rangle_{HD} = &\langle\vec{\gamma}_{0}, \vec{\delta}_{0}\rangle +\left(\langle \vec{\gamma}_{0}, \vec{\delta}_{1}\rangle+\langle \vec{\gamma}_{1}, \vec{\delta}_{0}\rangle\right)\varepsilon+\left(\langle \vec{\gamma}_{0}, \vec{\delta}_{2}\rangle+\langle \vec{\gamma}_{2}, \vec{\delta}_{0}\rangle\right)\varepsilon^*\\[6pt] &+\left(\langle \vec{\gamma}_{0}, \vec{\delta}_{3}\rangle+\langle \vec{\gamma}_{1}, \vec{\delta}_{2}\rangle+\langle \vec{\gamma}_{2}, \vec{\delta}_{1}\rangle+\langle \vec{\gamma}_{3}, \vec{\delta}_{0}\rangle\right) \varepsilon\varepsilon^*, \end{align}$

(1.4)

which continents inner products of real vectors.

Let $f(x_0+ x_1\varepsilon+ x_2\varepsilon^*+ x_3\varepsilon\varepsilon^*)$ be a hyper-dual function, then

$\begin{align} f(x_0+ x_1\varepsilon+ x_2\varepsilon^*+ x_3\varepsilon\varepsilon^*) = f(x_0)+x_1f(x_0)\varepsilon+x_2f'(x_0)\varepsilon^*+(x_3f'(x_0)+x_1x_2f''(x_0))\varepsilon\varepsilon^*. \end{align}$

(1.5)

Suppose $\vec{\gamma}$ , $\vec{\delta}$ and $\Phi$ be unit hyper-dual vectors and hyper-dual angle respectively then by using (1.5) the scalar product can be written as

$\begin{align} \langle\vec{\gamma}, \vec{\delta}\rangle_{HD} & = \cos{\Phi}\\ & = \cos\phi-\varepsilon^* \phi^{\ast }\sin\phi\\ & = (\cos\psi -\varepsilon \psi^{\ast }\sin\psi)-\varepsilon^* \phi^{\ast }(\sin\psi+\varepsilon \psi^{\ast }\cos\psi), \end{align}$

(1.6)

where $\phi$ and $\psi$ are, respectively, dual and real angles.

The norm of a hyper-dual vector $\vec{\gamma}$ is given by

$\begin{align*} \left\| \vec{\gamma}\right\|_{HD} & = \left\| \vec{\gamma}_0\right\|+\frac{\langle \vec{\gamma}_0, \vec{\gamma}_1\rangle }{\left\| \vec{\gamma}_0\right\|}\varepsilon+\frac{\langle \vec{\gamma}_0, \vec{\gamma}_2\rangle }{\left\| \vec{\gamma}_0\right\|} \varepsilon^* +\left(\frac{\langle \vec{\gamma}_0, \vec{\gamma}_3\rangle }{\left\| \vec{\gamma}_0\right\|}+\frac{\langle \vec{\gamma}_1, \vec{\gamma}_2\rangle }{\left\| \vec{\gamma}_0\right\|}-\frac{\langle \vec{\gamma}_0, \vec{\gamma}_1\rangle \langle \vec{\gamma}_0, \vec{\gamma}_2\rangle}{\left\| \vec{\gamma}_0\right\|^3}\right)\varepsilon\varepsilon^*, \end{align*}$

for $\left\| \vec{\gamma}_0\right\|\neq0$ . If $\left\| \vec{\gamma}\right\|_{HD} = 1$ that is $\left\| \vec{\gamma}_0\right\| = 1$ and $\langle \vec{\gamma}_0, \vec{\gamma}_1\rangle = \langle \vec{\gamma}_0, \vec{\gamma}_2\rangle = \langle \vec{\gamma}_0, \vec{\gamma}_3\rangle = \langle \vec{\gamma}_1, \vec{\gamma}_2\rangle = 0$ , then $\vec{\gamma}$ is a unit hyper-dual vector.

In this paper, we introduce the hyper-dual Pell and the hyper-dual Pell-Lucas numbers, which provide a natural generalization of the classical Pell and Pell-Lucas numbers by using the concept of hyper-dual numbers. We investigate some basic properties of these numbers. We also define a new vector and angle, which are called hyper-dual Pell vector and angle. We give properties of these vectors and angles to exert in the geometry of hyper-dual space.

2. Hyper-dual Pell and Hyper-dual Pell-Lucas numbers

In this section, we define the hyper-dual Pell and hyper-dual Pell-Lucas numbers and then demonstrate their fundamental identities and properties.

Definition 2.1. The $n^{th}$ hyper-dual Pell ${\mathrm{HP}}_n$ and hyper-dual Pell-Lucas ${\mathrm{HQ}}_n$ numbers are defined respectively as

$\begin{align} {\mathrm{HP}}_n = \mathrm{P}_n+\mathrm{P}_{n+1}\varepsilon+\mathrm{P}_{n+2}\varepsilon^*+\mathrm{P}_{n+3}\varepsilon\varepsilon^* \end{align}$

(2.1)

and

$\begin{align} \mathrm{HQ}_n = \mathrm{Q}_n+\varepsilon\mathrm{Q}_{n+1}+\varepsilon^*\mathrm{Q}_{n+2}+\varepsilon\varepsilon^*\mathrm{Q}_{n+3}, \end{align}$

(2.2)

where $\mathrm{P}_n$ and $\mathrm{Q}_n$ are $n^{th}$ Pell and Pell-Lucas numbers.

The few hyper-dual Pell and hyper-dual Pell-Lucas numbers are given as

${\mathrm{HP}}_1 = 1+2\varepsilon+5\varepsilon^*+12\varepsilon\varepsilon^*, {\mathrm{HP}}_2 = 2+5\varepsilon+12\varepsilon^*+29\varepsilon\varepsilon^*, ...$

and

${\mathrm{HQ}}_1 = 2+6\varepsilon+14\varepsilon^*+34\varepsilon\varepsilon^*, {\mathrm{HQ}}_2 = 6+14\varepsilon+34\varepsilon^*+82\varepsilon\varepsilon^*, ...$

Theorem 2.1. The Binet-like formulas of the hyper-dual Pell and hyper-dual Pell-Lucas numbers are given, respectively, by

$\begin{align} {\mathrm{HP}}_n = \frac{\varphi^n\;\underline{\varphi}-\psi^n\;\underline{\psi}}{\varphi-\psi} \end{align}$

(2.3)

and

$\begin{align} {\mathrm{HQ}}_n = \varphi^n\;\underline{\varphi}+\psi^n\;\underline{\psi}, \end{align}$

(2.4)

where

$\begin{align} \underline{\varphi} = 1+\varphi\varepsilon+\varphi^{2}\varepsilon^*+\varphi^{3}\varepsilon\varepsilon^*, \; \underline{\psi} = 1+\psi\varepsilon+\psi^{2}\varepsilon^*+\psi^{3}\varepsilon\varepsilon^*. \end{align}$

(2.5)

Proof. From (2.1) and the Binet formula of Pell numbers, we obtain

$\begin{align*} {\mathrm{HP}}_n& = \mathrm{P}_n+\mathrm{P}_{n+1}\varepsilon+\mathrm{P}_{n+2}\varepsilon^*+\mathrm{P}_{n+3}\varepsilon\varepsilon^*\\[6pt] & = \frac{\varphi^n-\psi^n}{\varphi-\psi}+\frac{\varphi^{n+1}-\psi^{n+1}}{\varphi-\psi}\varepsilon+\frac{\varphi^{n+2}-\psi^{n+2}}{\varphi-\psi}\varepsilon^*+\frac{\varphi^{n+3}-\psi^{n+3}}{\varphi-\psi}\varepsilon\varepsilon^*\\[6pt] & = \frac{\varphi^n(1+\varphi\varepsilon+\varphi^{2}\varepsilon^*+\varphi^{3}\varepsilon\varepsilon^*)}{\varphi-\psi}-\frac{\psi^n(1+\psi\varepsilon+\psi^{2}\varepsilon^*+\psi^{3}\varepsilon\varepsilon^*)}{\varphi-\psi} \\[6pt] & = \frac{\varphi^n\; \underline{\varphi}-\psi^n\; \underline{\psi}}{\varphi-\psi}. \end{align*}$

On the other hand, using (2.2) and the Binet formula of Pell-Lucas numbers we obtain

$\begin{align*} {\mathrm{HQ}}_n& = \mathrm{Q}_n+\mathrm{Q}_{n+1}\varepsilon+\mathrm{Q}_{n+2}\varepsilon^*+\mathrm{Q}_{n+3}\varepsilon\varepsilon^*\\[6pt] & = \left( \varphi^n+\psi^n \right)+\left( \varphi^{n+1}+\psi^{n+1} \right)\varepsilon+\left( \varphi^{n+2}+\psi^{n+2} \right)\varepsilon^*+\left( \varphi^{n+3}+\psi^{n+3} \right)\varepsilon\varepsilon^*\\[6pt] & = \varphi^n(1+\varphi\varepsilon+\varphi^{2}\varepsilon^*+\varphi^{3}\varepsilon\varepsilon^*)+\psi^n(1+\psi\varepsilon+\psi^{2}\varepsilon^*+\psi^{3}\varepsilon\varepsilon^*) \\[6pt] & = \varphi^n\; \underline{\varphi}+\psi^n\; \underline{\psi}. \end{align*}$

□ The proof is completed.

Theorem 2.2. (Vajda-like identities) For non-negative integers m, n, and r, we have

$\begin{align*} &{\mathrm{HP}}_{m}{\mathrm{HP}}_{n}-{\mathrm{HP}}_{m-r}{\mathrm{HP}}_{n+r} = (-1)^{n+1}\mathrm{P}_{m-n-r}\mathrm{P}_{r}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right), \\[6pt] &{\mathrm{HQ}}_{m}{\mathrm{HQ}}_{n}-{\mathrm{HQ}}_{m-r}{\mathrm{HQ}}_{n+r} = (-1)^{n}\; \mathrm{Q}_{m-n}-(-1)^{n+r}\; \mathrm{Q}_{m-n-2r}\; \left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Proof. By using the Binet-like formula of hyper-dual Pell numbers, we obtain

$\begin{align*} {\mathrm{HP}}_{m}{\mathrm{HP}}_{n}-{\mathrm{HP}}_{m-r}{\mathrm{HP}}_{n+r}& = \left(\frac{\; \varphi^{m}\underline{\varphi}-\; \psi^{m}\underline{\psi}}{\varphi-\psi}\right)\left(\frac{\; \varphi^{n}\underline{\varphi}-\; \psi^{n}\underline{\psi}}{\varphi-\psi}\right) -\left(\frac{\; \varphi^{m-r}\underline{\varphi}-\; \psi^{m-r}\underline{\psi}}{\varphi-\psi}\right)\left(\frac{\; \varphi^{n+r}\underline{\varphi}-\; \psi^{n+r}\underline{\psi}}{\varphi-\psi}\right)\\[6pt] & = \frac{(\varphi^{r}-\psi^{r})(\varphi^{n}\psi^{m-r}-\psi^{n}\varphi^{m-r})}{(\varphi-\psi)^2}\underline{\varphi}\; \underline{\psi}\\[6pt] & = -\frac{(\varphi^{m-n-r}-\psi^{m-n-r})(\varphi^{r}-\psi^{r})}{(\varphi-\psi)^2}\underline{\varphi}\; \underline{\psi}, \end{align*}$

and by using (1.1), we obtain

$\begin{equation*} {\mathrm{HP}}_{m}{\mathrm{HP}}_{n}-{\mathrm{HP}}_{m-r}{\mathrm{HP}}_{n+r} = (-1)^{n+1}\mathrm{P}_{m-n-r}\mathrm{P}_{r}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{equation*}$

Similarly for hyper-dual Pell-Lucas numbers, we can obtain

$\begin{align*} {\mathrm{HQ}}_{m}{\mathrm{HQ}}_{n}-{\mathrm{HQ}}_{m-r}{\mathrm{HQ}}_{n+r}& = \left(\; \varphi^{m}\underline{\varphi}+\; \psi^{m}\underline{\psi}\right)\left(\; \varphi^{n}\underline{\varphi}+\; \psi^{n}\underline{\psi}\right)\\&\quad-\left(\; \varphi^{m-r}\underline{\varphi}+\; \psi^{m-r}\underline{\psi}\right)\left(\; \varphi^{n+r}\underline{\varphi}+\; \psi^{n+r}\underline{\psi}\right) \\& = \underline{\varphi}\; \underline{\psi}\left(\varphi^{m-n}+\psi^{m-n}-\varphi^{m-n-2r}-\psi^{m-n-2r}\right). \end{align*}$

Using (1.2) and (2.5),

$\begin{align*} {\mathrm{HQ}}_{m}{\mathrm{HQ}}_{n}-{\mathrm{HQ}}_{m-r}{\mathrm{HQ}}_{n+r} = (-1)^{n}\; \mathrm{Q}_{m-n}-(-1)^{n+r}\; \mathrm{Q}_{m-n-2r}\; \left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Thus, we obtain the desired results. □

Theorem 2.3. (Catalan-like identities) For non negative integers n and r, with $n\ge r$ , we have

$\begin{align*} &{\mathrm{HP}}_{n-r}{\mathrm{HP}}_{n+r}-{{\mathrm{HP}}}_{n}^2 = (-1)^{n-r}\mathrm{P}_{r}^{2}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right), \\[6pt] &{\mathrm{HQ}}_{n-r}{\mathrm{HQ}}_{n+r}-{\mathrm{HQ}}_{n}^2 = 8(-1)^{n-r}\; \mathrm{P}_{r}^{2}\; \left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Proof. From (2.3), we obtain

$\begin{align*} {\mathrm{HP}}_{n-r}{\mathrm{HP}}_{n+r}-{\mathrm{HP}}_{n}^2& = \left(\frac{\; \varphi^{n-r}\underline{\varphi}-\; \psi^{n-r}\underline{\psi}}{\varphi-\psi}\right)\left(\frac{\; \varphi^{n+r}\underline{\varphi}-\; \psi^{n+r}\underline{\psi}}{\varphi-\psi}\right)-\left(\frac{\; \varphi^n\underline{\varphi}-\; \psi^n\underline{\psi}}{\varphi-\psi}\right)^2\\[6pt] & = \frac{\varphi^{n}\psi^{n}}{8}\; \underline{\varphi}\; \underline{\psi}\left(2-\psi^{r}\varphi^{-r}-\psi^{-r}\varphi^{r}\right)\\[6pt] & = (-1)^{n-r}\; \underline{\varphi}\; \underline{\psi}\left(\frac{\varphi^{r}-\psi^{r}}{\varphi-\psi}\right)^{2}, \end{align*}$

and by using (1.1) and (2.5), we will have

$\begin{equation*} {\mathrm{HP}}_{n-r}{\mathrm{HP}}_{n+r}-{\mathrm{HP}}_{n}^2 = (-1)^{n-r}\mathrm{P}_{r}^{2}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{equation*}$

On the other hand, from (2.4) and (2.5) we obtain

$\begin{align*} {\mathrm{HQ}}_{n-r}{\mathrm{HQ}}_{n+r}-{\mathrm{HQ}}_{n}^2& = \left(\; \varphi^{n-r}\underline{\varphi}+\; \psi^{n-r}\underline{\psi}\right)\left(\; \varphi^{n+r}\underline{\varphi}+\; \psi^{n+r}\underline{\psi}\right)-\left(\; \varphi^n\underline{\varphi}+\; \psi^n\underline{\psi}\right)^2 \\& = \underline{\varphi}\; \underline{\psi}\left(\varphi^{n-r}\psi^{n+r}+\varphi^{n+r}\psi^{n-r}-2\psi^{n}\varphi^{n}\right)\\ & = 8(-1)^{n-r}\; \underline{\varphi}\; \underline{\psi}\left(\frac{\varphi^{r}-\psi^{r}}{\varphi-\psi}\right)^{2}\\ & = 8(-1)^{n-r}\; \mathrm{P}_{r}^{2}\; \left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

□

Corollary 2.1. (Cassini-like identities) For non-negative integer n, we have

$\begin{align*} &\mathrm{HP}_{n-1}\; \mathrm{HP}_{n+1}-\mathrm{HP}_{n}^2 = (-1)^{n-1}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right), \\[6pt] &\mathrm{HQ}_{n-1}\; \mathrm{HQ}_{n+1}- \mathrm{HQ}_{n}^2 = 8(-1)^{n-1}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Proof. We can get the result by taking $r = 1$ in Theorem 2.3. □

Theorem 2.4. (d'Ocagne-like identities) For non-negative integers n and m,

$\begin{align*} &\mathrm{HP}_{m+1}\; \mathrm{HP}_{n}-\mathrm{HP}_{m}\; \mathrm{HP}_{n+1} = \left( -1 \right)^{m}\mathrm{P}_{n-m}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right), \\[6pt] &\mathrm{HQ}_{m+1}\; \mathrm{HQ}_{n}-\mathrm{HQ}_{m}\; \mathrm{HQ}_{n+1} = 8\left( -1 \right)^{n}\mathrm{P}_{m-n}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Proof. Using (1.1), (2.3), and (2.5), we have

$\begin{align*} \mathrm{HP}_{m+1}\; \mathrm{HP}_{n}-\mathrm{HP}_{m}\; \mathrm{HP}_{n+1}& = \left(\frac{\; \varphi^{m+1}\underline{\varphi}-\; \psi^{m+1}\underline{\psi}}{\varphi-\psi}\right)\left(\frac{\; \varphi^{n}\underline{\varphi}-\; \psi^{n}\underline{\psi}}{\varphi-\psi}\right)\\ &\quad-\left(\frac{\; \varphi^{m}\underline{\varphi}-\; \psi^{m}\underline{\psi}}{\varphi-\psi}\right)\left(\frac{\; \varphi^{n+1}\underline{\varphi}-\; \psi^{n+1}\underline{\psi}}{\varphi-\psi}\right)\\ & = (\varphi-\psi)(\varphi^{n}\psi^{m}-\varphi^{m}\psi^{n}) \underline{\varphi}\; \underline{\psi} \\ & = \left( -1 \right)^{m}\mathrm{P}_{n-m}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

Using (1.2), (2.4) and (2.5), we have

$\begin{align*} \mathrm{HQ}_{m+1}\; \mathrm{HQ}_{n}-\mathrm{HQ}_{m}\; \mathrm{HQ}_{n+1} = 8\left( -1 \right)^{n}\mathrm{P}_{m-n}\left( 1+2\varepsilon+6\varepsilon^*+12\varepsilon\varepsilon^* \right). \end{align*}$

□

3. Hyper dual Pell vectors and angle

In this section, we introduce hyper-dual Pell vectors and hyper-dual Pell angle. We will give geometric properties of them.

Definition 3.1. The $n^{th}$ hyper-dual Pell vector is defined as

$\begin{align*} \overrightarrow{{\mathrm{HP}}}_n = \vec{\mathrm{P}}_n+\vec{\mathrm{P}}_{n+1}\varepsilon+\vec{\mathrm{P}}_{n+2}\varepsilon^*+\vec{\mathrm{P}}_{n+3}\varepsilon\varepsilon^*, \end{align*}$

where $\vec{\mathrm{P}}_{n} = (\mathrm{P}_n, \mathrm{P}_{n+1}, \mathrm{P}_{n+2})$ is a real Pell vector. The hyper-dual Pell vector $\overrightarrow{\mathrm{HP}}_n$ can be rewritten in terms of dual Pell vectors $\vec{\mathbb{P}}_n$ and $\vec{\mathbb{P}} ^*_{n}$ as

$\begin{align*} \nonumber \overrightarrow{\mathrm{HP}}_n & = (\vec{\mathrm{P}}_n+\vec{\mathrm{P}}_{n+1}\varepsilon)+(\vec{\mathrm{P}}_{n+2}+\vec{\mathrm{P}}_{n+3}\varepsilon)\varepsilon^* \\ & = \vec{\mathbb{P}}_n+\varepsilon^*\vec{\mathbb{P}} ^*_{n}. \end{align*}$

Theorem 3.1. The scalar product of hyper-dual Pell vectors $\overrightarrow{{\mathrm{HP}}}_n$ and $\overrightarrow{{\mathrm{HP}}}_m$ is

$\begin{align} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_m \rangle = &\frac{7\mathrm{Q}_{n+m+2}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m}}{8} +(\frac{7\mathrm{Q}_{n+m+3}}{4}-\frac{(-1)^{m}\mathrm{Q}_{n-m}}{4})\varepsilon \\ &+(\frac{7\mathrm{Q}_{n+m+4}}{4}-\frac{3(-1)^{m}\mathrm{Q}_{n-m}}{4})\varepsilon^* +(\frac{7\mathrm{Q}_{n+m+5}}{2}-\frac{3(-1)^{m}\mathrm{Q}_{n-m}}{2})\varepsilon\varepsilon^*. \end{align}$

(3.1)

Proof. By using (1.4), we can write

$\begin{align} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_{m}\rangle = &\langle\vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m}\rangle +\left(\langle \vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m+1}\rangle+\langle \vec{\mathrm{P}}_{n+1}, \vec{\mathrm{P}}_{m}\rangle\right)\varepsilon+\left(\langle\vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m+2}\rangle+\langle \vec{\mathrm{P}}_{n+2}, \vec{\mathrm{P}}_{m}\rangle\right)\varepsilon^*\\[6pt] &+\left(\langle \vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m+3}\rangle+\langle \vec{\mathrm{P}}_{n+1}, \vec{\mathrm{P}}_{m+2}\rangle+\langle \vec{\mathrm{P}}_{n+2}, \vec{\mathrm{P}}_{m+1}\rangle+\langle \vec{\mathrm{P}}_{n+3}, \vec{\mathrm{P}}_{m}\rangle\right) \varepsilon\varepsilon^*. \end{align}$

(3.2)

Now we calculate the above inner products for real Pell vectors $\vec{\mathrm{P}}_{n}$ and $\vec{\mathrm{P}}_{m}$ by using Binet's formula of Pell numbers as

$\begin{align*} \langle \vec{\mathrm{P}}_n, \vec{\mathrm{P}}_{m}\rangle& = \mathrm{P}_{n}\mathrm{P}_{m}+\mathrm{P}_{n+1}\mathrm{P}_{m+1}+ \mathrm{P}_{n+2}\mathrm{P}_{m+2} \\ & = \left(\frac{\varphi^{n}-\psi^{n}}{\varphi-\psi}\right)\left(\frac{\varphi^{m}-\psi^{m}}{\varphi-\psi}\right)+\left(\frac{\varphi^{n+1}-\psi^{n+1}}{\varphi-\psi}\right)\left(\frac{\varphi^{m+1}-\psi^{m+1}}{\varphi-\psi}\right) +\left(\frac{\varphi^{n+2}-\psi^{n+2}}{\varphi-\psi}\right)\left(\frac{\varphi^{m+2}-\psi^{m+2}}{\varphi-\psi}\right)\\ & = \frac{\varphi^{n+m}+\psi^{n+m}}{(\varphi-\psi)^2}+\frac{\varphi^{n+m+2}+\psi^{n+m+2}}{(\varphi-\psi)^2} +\frac{\varphi^{n+m+4}+\psi^{n+m+4}}{(\varphi-\psi)^2}-\frac{\left( \varphi^{n}\psi^{m}+\varphi^{m}\psi^{n} \right)\varphi^{-m}\psi^{-m}}{(\varphi-\psi)^2\; \varphi^{-m}\psi^{-m}} \\ & = \frac{1}{8}\left( \mathrm{Q}_{n+m}+\mathrm{Q}_{n+m+2}+\mathrm{Q}_{n+m+4}+(-1)^{m}\mathrm{Q}_{n-m} \right)\\ & = \frac{7\mathrm{Q}_{n+m+2}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m}}{8}. \end{align*}$

□ Similarly,

$\begin{align*} \langle \vec{\mathrm{P}}_n, \vec{\mathrm{P}}_{m+1} \rangle& = \frac{7\mathrm{Q}_{n+m+3}}{8}+\frac{(-1)^{m}\mathrm{Q}_{n-m-1}}{8}, \\ \langle \vec{\mathrm{P}}_{n+1}, \vec{\mathrm{P}}_{m} \rangle& = \frac{7\mathrm{Q}_{n+m+3}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m+1}}{8}, \\ \langle \vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m+2} \rangle& = \frac{7\mathrm{Q}_{n+m+4}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m-2}}{8}, \\ \langle \vec{\mathrm{P}}_{n+2}, \vec{\mathrm{P}}_{m} \rangle& = \frac{7\mathrm{Q}_{n+m+4}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m+2}}{8}, \\ \langle \vec{\mathrm{P}}_{n}, \vec{\mathrm{P}}_{m+3} \rangle& = \frac{7\mathrm{Q}_{n+m+5}}{8}+\frac{(-1)^{m}\mathrm{Q}_{n-m-3}}{8}, \\ \langle \vec{\mathrm{P}}_{n+1}, \vec{\mathrm{P}}_{m+2} \rangle& = \frac{7\mathrm{Q}_{n+m+5}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m-1}}{8}, \\ \langle \vec{\mathrm{P}}_{n+2}, \vec{\mathrm{P}}_{m+1} \rangle& = \frac{7\mathrm{Q}_{n+m+5}}{8}+\frac{(-1)^{m}\mathrm{Q}_{n-m+1}}{8}, \\ \langle \vec{\mathrm{P}}_{n+3}, \vec{\mathrm{P}}_{m} \rangle& = \frac{7\mathrm{Q}_{n+m+5}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m+3}}{8}. \end{align*}$

By substituting these equalities in (3.2), we obtain the result.

Example 3.1. Let $\overrightarrow{{\mathrm{HP}}}_1 = (1, 2, 5)+(2, 5, 12)\varepsilon+(5, 12, 29)\varepsilon^*+(12, 29, 70)\varepsilon\varepsilon^*$ and $\overrightarrow{{\mathrm{HP}}}_0 = (0, 1, 2)+(1, 2, 5)\varepsilon+(2, 5, 12)\varepsilon^*+(5, 12, 29)\varepsilon\varepsilon^*$ be the hyper-dual Pell vectors. The scalar product of $\overrightarrow{{\mathrm{HP}}}_1$ and $\overrightarrow{{\mathrm{HP}}}_0$ are

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_1, \overrightarrow{\mathrm{HP}}_0 \rangle & = \frac{7\mathrm{Q}_{3}-Q_1}{8} +\frac{7\mathrm{Q}_{4}-Q_1}{4}\varepsilon +\frac{7\mathrm{Q}_{5}-3Q_1}{4}\varepsilon^* +\frac{7\mathrm{Q}_{6}-3Q_1}{2}\varepsilon\varepsilon^*\\ & = 12+59\varepsilon+142\varepsilon^*+690\varepsilon\varepsilon^*. \end{align*}$

By the other hand

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_1, \overrightarrow{\mathrm{HP}}_{0}\rangle \nonumber & = \langle\vec{\mathrm{P}}_{1}, \vec{\mathrm{P}}_{0}\rangle +\left(\langle \vec{\mathrm{P}}_{1}, \vec{\mathrm{P}}_{1}\rangle+\langle \vec{\mathrm{P}}_{2}, \vec{\mathrm{P}}_{0}\rangle\right)\varepsilon+\left(\langle\vec{\mathrm{P}}_{1}, \vec{\mathrm{P}}_{2}\rangle+\langle \vec{\mathrm{P}}_{3}, \vec{\mathrm{P}}_{0}\rangle\right)\varepsilon^*\\[6pt] &\quad+\left(\langle \vec{\mathrm{P}}_{1}, \vec{\mathrm{P}}_{3}\rangle+\langle \vec{\mathrm{P}}_{2}, \vec{\mathrm{P}}_{2}\rangle+\langle \vec{\mathrm{P}}_{3}, \vec{\mathrm{P}}_{1}\rangle+\langle \vec{\mathrm{P}}_{4}, \vec{\mathrm{P}}_{0}\rangle\right) \varepsilon\varepsilon^*\\ & = 12+(30+29)\varepsilon+(72+70)\varepsilon^*+(174+173+174+169)\varepsilon\varepsilon^*\\ & = 12+59\varepsilon+142\varepsilon^*+690\varepsilon\varepsilon^*. \end{align*}$

The results are the same as we expected.

Corollary 3.1. The norm of $\overrightarrow{{\mathrm{HP}}}_n$ is

$\begin{align} \| \overrightarrow{\mathrm{HP}}_n \|^2 = \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_n \rangle = &\frac{7\mathrm{Q}_{2n+2}}{8}-\frac{(-1)^{n}}{4} +(\frac{7\mathrm{Q}_{2n+3}}{4}-\frac{(-1)^{n}}{2})\varepsilon \\&+(\frac{7\mathrm{Q}_{2n+4}}{4}-\frac{3(-1)^{n}}{2})\varepsilon^* +(\frac{7\mathrm{Q}_{2n+5}}{2}-3(-1)^{n}\mathrm)\varepsilon\varepsilon^*. \end{align}$

(3.3)

Proof. The proof is clear from taking $m = n$ in (3.1). □

Example 3.2. Find the norm of $\overrightarrow{{\mathrm{HP}}}_1 = (1, 2, 5)+(2, 5, 12)\varepsilon+(5, 12, 29)\varepsilon^*+(12, 29, 70)\varepsilon\varepsilon^*$ .

If we take $n = 1$ in (3.3) and use (1.3), then we will get

$\begin{align*} \| \overrightarrow{\mathrm{HP}}_1 \| & = \sqrt{\frac{7\mathrm{Q}_{4}}{8}+\frac{1}{4} +(\frac{7\mathrm{Q}_{5}}{4}+\frac{1}{2})\varepsilon +(\frac{7\mathrm{Q}_{6}}{4}+\frac{3}{2})\varepsilon^* +(\frac{7\mathrm{Q}_{7}}{2}+3\mathrm)\varepsilon\varepsilon^*}\\ & = \sqrt{30+144\varepsilon+348\varepsilon^*+1676\varepsilon\varepsilon^*}\\ & = \sqrt{30}+\frac{72}{\sqrt{30}}\varepsilon+\frac{174}{\sqrt{30}}\varepsilon^*+\frac{734}{5\sqrt{30}}\varepsilon\varepsilon^*. \end{align*}$

From (1.6) and (3.1), the following cases can be given for the scalar product of hyper-dual Pell vectors $\overrightarrow{\mathrm{HP}}_n$ and $\overrightarrow{\mathrm{HP}}_m$ .

Case 3.1. Assume that $\cos\phi = 0$ and $\phi^{\ast }\neq 0$ , then $\psi = \frac{\pi}{2}$ , $\psi^* = 0$ , therefore

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_m \rangle = -\varepsilon^* \phi^*& = (\frac{7\mathrm{Q}_{m+n+4}}{4}-\frac{3(-1)^{m}\mathrm{Q}_{n-m}}{4})\varepsilon^*+(\frac{7\mathrm{Q}_{m+n+5}}{2}-\frac{3(-1)^{m}\mathrm{Q}_{n-m}}{2}\mathrm)\varepsilon\varepsilon^*, \end{align*}$

then, we get

$\begin{equation*} \phi^* = \left( -1 \right)^{m} \left( \frac{3}{2}+ \varepsilon\right)-\frac{7}{4}\left( \mathrm{Q}_{m+n+4}+2\; \varepsilon\; \mathrm{Q}_{m+n+5}\right) \end{equation*}$

and corresponding dual lines $\mathrm{}{d}_1$ and ${d}_2$ are perpendicular such that they do not intersect each other; see Figure 1.

Figure 1. Geometric representation of hyper-dual angle between the directed dual lines

${d}_1$ and

${d}_2$ .

DownLoad: Full-Size Img PowerPoint

Case 3.2. Assume that $\phi^{\ast } = 0$ and $\phi\neq0$ , then we obtain

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_m \rangle = \cos\phi& = \left( \frac{7\mathrm{Q}_{m+n+2}}{8}-\frac{(-1)^{m}\mathrm{Q}_{n-m}}{8} \right)+\left( \frac{7\mathrm{Q}_{m+n+3}}{4}-\frac{(-1)^{m}\mathrm{Q}_{n-m}}{4} \right)\varepsilon, \end{align*}$

therefore

$\begin{equation*} \phi = \arccos\left(\left( \frac{7\mathrm{Q}_{m+n+2}}{8}-\frac{(-1)^{m}{Q}_{n-m}}{8} \right)+\left( \frac{7\mathrm{Q}_{m+n+3}}{4}-\frac{(-1)^{m}{Q}_{n-m}}{4} \right)\varepsilon\right), \end{equation*}$

and corresponding dual lines ${d}_1$ and ${d}_2$ intersect each other; see Figure 2.

Figure 2. Intersection of dual lines

${d}_1$ and

${d}_2$ .

DownLoad: Full-Size Img PowerPoint

Case 3.3. Assume that $\cos\phi = 0$ and $\phi^* = 0$ , then $\psi = \frac{\pi}{2}$ and $\psi^{\ast } = 0$ , therefore

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_m \rangle & = 0, \end{align*}$

and dual lines ${d}_1$ and ${d}_2$ intersect each other at a right angle; see Figure 3.

Figure 3. Perpendicular intersection of dual lines

${d}_1$ and

${d}_2$ .

DownLoad: Full-Size Img PowerPoint

Case 3.4. Assume that $\phi = 0$ and $\phi^* = 0$ , then

$\begin{align*} \langle \overrightarrow{\mathrm{HP}}_n, \overrightarrow{\mathrm{HP}}_m \rangle & = 1, \end{align*}$

in this case corresponding dual lines ${d}_1$ and ${d}_2$ are parallel; see Figure 4.

Figure 4. Parallel of dual lines

${d}_1$ and

${d}_2$ .

DownLoad: Full-Size Img PowerPoint

4. Conclusions

In the present study, we introduce two families of hyper-dual numbers with components containing Pell and the Pell-Lucas numbers. First, we define hyper-dual Pell and Pell-Lucas numbers. Afterwards, by means of the Binet's formulas of Pell and Pell-Lucas numbers, we investigate identities such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. After that, we define hyper-dual Pell vector and angle with some properties and geometric applications related to them. In the future it would be valuable to replicate a similar exploration and development of our findings on hyper-dual numbers with Pell and Pell-Lucas numbers. These results can trigger further research on the subjects of the hyper-dual numbers, vector, and angle to carry out in the geometry of dual and hyper-dual space.

Author contributions

Faik Babadağ and Ali Atasoy: Conceptualization, writing-original draft, writing-review, editing. All authors have read and approved the final version of the manuscript for publication.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	American Hospital Association: CDC estimates 58% of Americans have been infected with SARS-CoV-2, Apr. 26, 2022. Available from: https://www.aha.org/news/headline/2022-04-26-cdc-estimates-58-americans-have-been-infected-sars-cov-2
[2]	S. Allred, M. Chyba, J. M. Hyman, Y. Mileyko, B. Piccoli, The COVID-19 Pandemic Evolution in Hawai'i and New Jersey: A Lesson on Infection Transmissibility and the Role of Human Behavior, Predicting Pandemics in a Globally Connected World, 1 (2022), 109–140. https://doi.org/10.1007/978-3-030-96562-4_4 doi: 10.1007/978-3-030-96562-4_4
[3]	N. Bellomo, M. A. J. Chaplain, Predicting Pandemics in a Globally Connected World, Volume 1 Toward a Multiscale, Multidisciplinary Framework through Modeling and Simulation, Springer International Publishing, 2022. https://doi.org/10.1007/978-3-030-96562-4
[4]	Bloomberg: Omicron Four Times More Transmissible Than Delta in New Study, By Kanoko Matsuyama, Dec. 8, 2021. Available from: https://www.bloomberg.com/news/articles/2021-12-09/omicron-four-times-more-transmissible-than-delta-in-japan-study
[5]	C. Branas, A. Rundle, S. Pei, W. Yang, B. G. Carr, S. Sims, et al., Flattening the curve before it flattens us: hospital critical care capacity limits and mortality from novel coronavirus (SARS-CoV2) cases in US counties, medRxiv, 2020.
[6]	D. Buitrago-Garcia, D. Egli-Gany, M. J. Counotte, S. Hossmann, H. Imeri, A. M. Ipekci, et al., Occurrence and transmission potential of asymptomatic and presymptomatic SARS-CoV-2 infections: A living systematic review and meta-analysis, PLOS Med., 17 (2020), e1003346. https://doi.org/10.1371/journal.pmed.1003346 doi: 10.1371/journal.pmed.1003346
[7]	M. Chyba, Y. Mileyko, O. Markovichenko, R. Carney, A. Koniges, Epidemiological Model of the Spread of COVID-19 in Hawaii's Challenging Fight Against the Disease, The Ninth International Conference on Global Health Challenges, (2020), 32–38.
[8]	M. Chyba, T. Klotz, Y. Mileyko, C. Shanbrom, A look at endemic equilibria of compartmental epidemiological models and model control via vaccination and mitigation, arXiv: 2109.01738, 2021.
[9]	M. Chyba, P. Kunwar, Y. Mileyko, A. Tong, W. Lau, A. Koniges, COVID-19 heterogeneity in islands chain environment, PLoS One, 17 (2022), e0263866. https://doi.org/10.1371/journal.pone.0263866 doi: 10.1371/journal.pone.0263866
[10]	N. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, M. Baguelin, et al., Report 9 - Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College London, 10 (2020), 491–197.
[11]	M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. Casagrandi, et al., Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484–10491. https://doi.org/10.1073/pnas.2004978117 doi: 10.1073/pnas.2004978117
[12]	G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855–860. https://doi.org/10.1038/s41591-020-0883-7 doi: 10.1038/s41591-020-0883-7
[13]	W. T. Harvey, A. M. Carabelli, B. Jackson, R. K. Gupta, E. C. Thomson, E. M. Harrison, et al., SARS-CoV-2 variants, spike mutations and immune escape, Nat. Rev. Microbiol., 19 (2021), 409–424. https://doi.org/10.1038/s41579-021-00573-0 doi: 10.1038/s41579-021-00573-0
[14]	Hawaii Department of Health, Disease Outbreak Control Division: COVID-19 Dashboard. Available from: https://health.hawaii.gov/coronavirusdisease2019
[15]	Hawaii Population Model. Hawai'i Data Collaborative, Department of Health, Disease Outbreak Control Division: COVID-19 Data Reports, 2022. Available from: https://health.hawaii.gov/coronavirusdisease2019/what-you-should-know/covid-19-data-reports
[16]	Hawai'i Department of Health: Vaccination Distribution Plan, 2022. Available from: www.hawaiicovid19.com/wp-content/uploads/2021/01/Executive-Summary_Final1_010721.pdf
[17]	Hawaii News Now: Ventilators from Hawaii's emergency stockpile now in use as more relief nurses are on their way, By Allyson Blair, Aug. 25, 2021. Available from: https://www.hawaiinewsnow.com/2021/08/26/ventilators-emergency-stockpile-now-use-morerelief-nurses-are-their-way/
[18]	Hawai'i Pandemic Applied Modeling Work Group, 2022. Available from: www.hipam.org
[19]	H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
[20]	Honolulu Star-Advertiser: Hawaii's daily COVID cases are forecast to peak at 3,700 in October, By Nina Wu, Aug. 23, 2021. Available from: https://www.staradvertiser.com/2021/08/23/hawaiinews/hawaiis-daily-covid-cases-are-forecast-to-peak-at-3700-in-october/
[21]	Honolulu Star-Advertiser: Kaiser postpones elective surgeries and procedures on Oahu, Maui due to COVID surge, By Nina Wu, Aug. 27, 2021. Available from: https://www.staradvertiser.com/2021/08/27/breaking-news/kaiser-postpones-elective-surgeriesand-procedures-on-oahu-maui-due-to-covid-surge/
[22]	Honolulu Star-Advertiser: No ICU beds available at Queen's medical facilities as COVID cases surge in Hawaii, By Sophie Cocke, Aug. 17, 2021. Available from: https://www.staradvertiser.com/2021/08/17/hawaii-news/no-icu-beds-available-at-queensmedical-facilities-as-covid-cases-surge-in-hawaii/
[23]	A. D. Iuliano, J. M. Brunkard, T. K. Boehmer, E. Peterson, S. Adjei, A. M. Binder, et al., Trends in Disease Severity and Health Care Utilization During the Early Omicron Variant Period Compared with Previous SARS-CoV-2 High Transmission Periods — United States, December 2020–January 2022, MMWR Morb. Mortal., 71 (2022), 146–152. https://doi.org/10.15585/mmwr.mm7104e4 doi: 10.15585/mmwr.mm7104e4
[24]	V. Kala, K. Guo, E. Swantek, A. Tong, M. Chyba, Y. Mileyko, et al., Pandemics in Hawai‘i: 1918 Influenza and COVID-19, The Ninth International Conference on Global Health Challenges, (2020), 26–31.
[25]	D. G. Kendall, Deterministic and stochastic epidemics in closed populations, University of California Press, (2020), 149–166. https://doi.org/10.1525/9780520350717-011
[26]	W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, R. Soc. Lond. A, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
[27]	W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. Ⅱ.—The problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55–83. https://doi.org/10.1098/rspa.1932.0171 doi: 10.1098/rspa.1932.0171
[28]	P. Kunwar, O. Markovichenko, M. Chyba, Y. Mileyko, A. Koniges, T. Lee, A study of computational and conceptual complexities of compartment and agent based models, Netw. Heterog. Media, 17 (2022), 359–384. https://doi.org/10.3934/nhm.2022011 doi: 10.3934/nhm.2022011
[29]	J. O. Lloyd-Smith, A. P. Galvani, W. M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Proceedings of the Royal Society of London. Series B: Biological Sciences, 270 (2003), 1979–1989. https://doi.org/10.1098/rspb.2003.2481 doi: 10.1098/rspb.2003.2481
[30]	N. Ortega, M. Ribes, M. Vidal, R. Rubio, R. Aguilar, S. Williams, et al, Seven-month kinetics of SARS-CoV-2 antibodies and role of pre-existing antibodies to human coronaviruses, Nat. Commun., 12 (2021), 1–10. https://doi.org/10.1038/s41467-021-24979-9 doi: 10.1038/s41467-021-24979-9
[31]	M. Park, A. R. Cook, J. T. Lim, Y. Sun, B. L. Dickens, A Systematic Review of COVID-19 Epidemiology Based on Current Evidence, J. Clin. Med., 9 (2020), 967. https://doi.org/10.3390/jcm9040967 doi: 10.3390/jcm9040967
[32]	P. Qu, J. N. Faraone, J. P. Evans, Y. M. Zheng, L. Yu, Q. Ma, et al., Durability of Booster mRNA Vaccine against SARS- CoV-2 BA.2.12.1, BA.4, and BA.5 Subvariants, New Engl. J. Med., 387 (2022), 1329–1331. https://doi.org/10.1056/NEJMc2210546 doi: 10.1056/NEJMc2210546
[33]	A. Richterman, E. A. Meyerowitz, M. Cevik, Indirect Protection by Reducing Transmission: Ending the Pandemic With Severe Acute Respiratory Syndrome Coronavirus 2 Vaccination, Open Forum Infectious Diseases Open Forum Infectious Diseases, 9(2) (2022), ofab259. https://doi.org/10.1093/ofid/ofab259
[34]	R. Ross, H. P. Hudson, An Application of the Theory of Probabilities to the Study of a priori Pathometry. Part Ⅱ, Proceedings of the Royal Society of London Series A, 93 (1917), 212–225. https://doi.org/10.1098/rspa.1917.0014 doi: 10.1098/rspa.1917.0014
[35]	R. Ross, An Application of the Theory of Probabilities to the Study of a priori Pathometry. Part Ⅰ, Proceedings of the Royal Society of London Series A, 92 (1916), 204–230. https://doi.org/10.1098/rspa.1916.0007 doi: 10.1098/rspa.1916.0007
[36]	Y. B. Ruhomally, M. Mungur, A. A. H. Khoodaruth, V. Oree, M. Z. Dauhoo, Assessing the Impact of Contact Tracing, Quarantine and Red Zone on the Dynamical Evolution of the Covid-19 Pandemic using the Cellular Automata Approach and the Resulting Mean Field System: A Case study in Mauritius, Appl. Math. Model., 111 (2022), 567–589. https://doi.org/10.1016/j.apm.2022.07.008 doi: 10.1016/j.apm.2022.07.008
[37]	Science, Science Insider, Health: More people are getting COVID-19 twice, suggesting immunity wanes quickly in some, By Jop De Vrieze, Nov. 18, 2020. Available from: https://www.science.org/content/article/more-people-are-getting-covid-19-twice-suggestingimmunity-wanes-quickly-some
[38]	M. T. Sofonea, B. Roquebert, V. Foulongne, L. Verdurme, S. Trombert-Paolantoni, M. Roussel, et al., From Delta to Omicron: analysing the SARS-CoV-2 epidemic in France using variant-specific screening tests (September 1 to December 18, 2021) Cold Spring Harbor Laboratory Press, 2022. https://www.medrxiv.org/content/10.1101/2021.12.31.21268583v1
[39]	S. Sturniolo, W. Waites, T. Colbourn, D. Manheim, J. Panovska-Griffiths, Testing, tracing and isolation in compartmental models, PLoS Comput. Biol., 17 (2021), e1008633. https://doi.org/10.1371/journal.pcbi.1008633 doi: 10.1371/journal.pcbi.1008633
[40]	State of Hawai'i Portal: Travel Data. Available from: https://hawaiicovid19.com/travel/data/
[41]	S. T. Tan, A. T. Kwan, I. Rodríguez-Barraquer, B. J. Singer, H. J. Park, J. A. Lewnard, et al., Infectiousness of SARS-CoV-2 breakthrough infections and reinfections during the Omicron wave, medRxiv 2022.08.08.22278547.
[42]	U.S. Bureau of Labor Statistics: Healthcare Population, May 2020, State Occupational Employment and Wage Estimates Hawaii. Available from: https://www.bls.gov/oes/current/oeshi.htm29-0000
[43]	U.S. Census Bureau. Available from: https://data.census.gov/cedsci/
[44]	F. Zhou, T. Yu, R. Du, G. Fan, Y. Liu, Z. Liu, et al., Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study, The Lancet, 395 (2020), 1054–1062. https://doi.org/10.1016/S0140-6736(20)30566-3 doi: 10.1016/S0140-6736(20)30566-3

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