Market and home production earnings gaps in Russia

Vladimir Hlasny; Vladimir Hlasny

doi:10.3934/NAR.2023007

National Accounting Review

2023, Volume 5, Issue 2: 108-124. doi: 10.3934/NAR.2023007

Previous Article Next Article

Research article Special Issues

Market and home production earnings gaps in Russia

Vladimir Hlasny ^,

Ewha Womans University, Economics Department, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, South Korea

Received: 27 February 2023 Revised: 28 March 2023 Accepted: 02 April 2023 Published: 10 April 2023
JEL Codes: F16, J31, D31

This study assesses the evolution of earnings across different groups of workers during Russia's 2000–2013 oil boom and amidst the 2014–2015 oil bust and a trade war. Unconditional quantile regressions and growth incidence curves are applied to nine household surveys for 2000–2016 to estimate earnings gaps across urban/rural, farming/non-farming and gender divides at various earnings quantiles. Gaps in pre-fiscal formal labor market earnings and informal non-market home production are assessed, distinguishing the roles of workers' endowments differentials and returns differentials in production markets. Earning gaps are found to be pervasive, with rural and some female-headed households receiving lower returns on their human capital in part because they lack employment opportunities. Rural Russian households face mobility barriers and lack decent employment opportunities, thus lacking incentives for skill investment. Rural households tend to be less educated and face low returns on their various marketable characteristics. Gender gaps were particularly high historically, particularly among lower quantile groups, but have been steadily falling over the past decade. Growth incidence curves reveal that the 2014 shocks affected particularly harshly the farming and urban households, both immediately and over the following years. We conclude that Russia should strengthen its rural assistance programs and lower the mobility and resettlement barriers to improve rural households' access to education and employment.

Keywords:

Citation: Vladimir Hlasny. Market and home production earnings gaps in Russia[J]. National Accounting Review, 2023, 5(2): 108-124. doi: 10.3934/NAR.2023007

Related Papers:

[1]	Mdi Begum Jeelani, Kamal Shah, Hussam Alrabaiah, Abeer S. Alnahdi . On a SEIR-type model of COVID-19 using piecewise and stochastic differential operators undertaking management strategies. AIMS Mathematics, 2023, 8(11): 27268-27290. doi: 10.3934/math.20231395
[2]	Muhammad Farman, Ali Akgül, Kottakkaran Sooppy Nisar, Dilshad Ahmad, Aqeel Ahmad, Sarfaraz Kamangar, C Ahamed Saleel . Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel. AIMS Mathematics, 2022, 7(1): 756-783. doi: 10.3934/math.2022046
[3]	Rukhsar Ikram, Ghulam Hussain, Inayat Khan, Amir Khan, Gul Zaman, Aeshah A. Raezah . Stationary distribution of stochastic COVID-19 epidemic model with control strategies. AIMS Mathematics, 2024, 9(11): 30413-30442. doi: 10.3934/math.20241468
[4]	Murugesan Sivashankar, Sriramulu Sabarinathan, Vediyappan Govindan, Unai Fernandez-Gamiz, Samad Noeiaghdam . Stability analysis of COVID-19 outbreak using Caputo-Fabrizio fractional differential equation. AIMS Mathematics, 2023, 8(2): 2720-2735. doi: 10.3934/math.2023143
[5]	C. W. Chukwu, Fatmawati . Modelling fractional-order dynamics of COVID-19 with environmental transmission and vaccination: A case study of Indonesia. AIMS Mathematics, 2022, 7(3): 4416-4438. doi: 10.3934/math.2022246
[6]	Rahat Zarin, Amir Khan, Aurangzeb, Ali Akgül, Esra Karatas Akgül, Usa Wannasingha Humphries . Fractional modeling of COVID-19 pandemic model with real data from Pakistan under the ABC operator. AIMS Mathematics, 2022, 7(9): 15939-15964. doi: 10.3934/math.2022872
[7]	Maryam Amin, Muhammad Farman, Ali Akgül, Mohammad Partohaghighi, Fahd Jarad . Computational analysis of COVID-19 model outbreak with singular and nonlocal operator. AIMS Mathematics, 2022, 7(9): 16741-16759. doi: 10.3934/math.2022919
[8]	Aqeel Ahmad, Cicik Alfiniyah, Ali Akgül, Aeshah A. Raezah . Analysis of COVID-19 outbreak in Democratic Republic of the Congo using fractional operators. AIMS Mathematics, 2023, 8(11): 25654-25687. doi: 10.3934/math.20231309
[9]	Iqbal M. Batiha, Abeer A. Al-Nana, Ramzi B. Albadarneh, Adel Ouannas, Ahmad Al-Khasawneh, Shaher Momani . Fractional-order coronavirus models with vaccination strategies impacted on Saudi Arabia's infections. AIMS Mathematics, 2022, 7(7): 12842-12858. doi: 10.3934/math.2022711
[10]	Mohammad Sajid, Biplab Dhar, Ahmed S. Almohaimeed . Differential order analysis and sensitivity analysis of a CoVID-19 infection system with memory effect. AIMS Mathematics, 2022, 7(12): 20594-20614. doi: 10.3934/math.20221129

Abstract

1. Introduction

Despite significant advances in treatment and prevention technologies, contagious diseases continue to impact countless individuals globally. To limit the spread of these diseases, it is imperative to maintain strict control over factors such as transmission routes, population size, rates of human contact, duration of infection, and other critical parameters. Comprehending the behavior and management of pandemic diseases is particularly crucial during the initial infection phases or in the absence of immunization. In this context, mathematical models play an indispensable role. In recent years, scientists from various disciplines have delved into and researched numerous biological models, (see ^[1,2,3] and references therein).

The recent global health crisis, termed the COVID-19 pandemic, stems from the emergence of a novel coronavirus strain, SARS-CoV-2. This virus rapidly evolved into a significant worldwide concern. It has affected millions, placing an unparalleled burden on healthcare infrastructures and economies across the globe ^[4]. In the unyielding fight against this highly contagious and ever-evolving virus, mathematical modeling has proven indispensable. Offering quantitative tools to understand, simulate, and predict the virus's spread, these models have been instrumental in guiding public health measures and shaping policy decisions. Although mathematical modeling has been valuable in past epidemics, its significance has been magnified during the COVID-19 outbreak, where swift and informed decision-making has been crucial^[5,6].

Many systems pertinent to real-world scenarios exhibit crossover behavior. Modeling systems based on this behavior has historically posed significant challenges. The transition from Markovian to non-Markovian processes has unveiled a plethora of examples that highlight real-world complexities. A prime illustration of this is in epidemiology, where the propagation of contagious diseases and occasional chaos may be attributed to this transition. Numerous intricate real-world challenges, including those involving chaos, have been articulated using the frameworks of piecewise integral and differential operators ^[7]. The cited authors delineated three distinct scenarios to frame deterministic-stochastic chaotic models and undertook numerical explorations of these constructs. Additionally, piecewise differentiation has emerged as a potent tool in epidemiological modeling, especially when challenges are presented by crossover dynamics ^[8,9].

Piecewise operators represent a novel class of operators established by Atangana et al. ^[10]. Traditional Mittag-Leffler or exponential mappings in fractional calculus need to provide a precise way to determine crossover time. A unique approach to piecewise derivation addressing this challenge was proposed in ^[10]. Piecewise differential equations offer insights into the present and serve as tools for future planning. Researchers employ these models to simulate diverse scenarios, such as potential virus resurgence and implications of new policy decisions. Active exploration of crossover behaviors with these operators is underway, with studies like ^[11,12] utilizing them to investigate qualitative characteristics of differential equations (DEs) and various infectious disease models. Further applications of piecewise operators are showcased in ^{[13,14,15,16]}.

Atangana et al. ^[9] extended the concept of epidemiology modeling to portray waves with varying patterns, introducing a novel approach to ensure the existence and uniqueness of system solutions. They present a piecewise numerical approach for deriving solutions, with an illustrative example compared against data from Turkey, Spain, and Czechia, concluding that this method offers a new perspective for understanding natural phenomena. Zeb et al. ^[17] introduces a piecewise mathematical model for COVID-19 that incorporates a quarantine category and vaccination strategies. This model is based on the SEIQR framework for epidemic modeling. The authors in ^[18] developed a mathematical model using the Atangana–Baleanu fractional derivative to assess the Omicron variant of COVID-19, exploring strategies for reducing transmission risk. They expanded this model into a piecewise fractional stochastic Atangana–Baleanu differential equation, applying it to South African COVID-19 case data, and presented detailed numerical solutions and graphical results. The authors in ^[19] formulated a new mathematical model to investigate COVID-19 infection dynamics using Caputo fractional differential equations and piecewise stochastic differential equations. They present numerical solutions and graphical results for critical parameters, demonstrating that isolating healthy individuals from various stages of infected individuals can significantly reduce infection rates.

In this article, we introduce a COVID-19 model that comprises classes such as susceptible, exposed, symptomatic infectious, superspreaders (infectious but asymptomatic), hospitalized, recovery, and fatality. The model incorporates hybrid fractional order and variable order operators, and extends the stochastic differential equations with fractional Brownian motion. We introduce two numerical methods: the nonstandard modified Euler Maruyama method (NEMM) for solving the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method (CPC-GLNFDM) for the fractional and variable order deterministic models. Numerical simulations are conducted to validate the theoretical results and demonstrate the effectiveness of the proposed method. These experiments further provide insights into various phenomena, including crossover and stochastic processes.

The paper is structured as follows: Section 2 provides some basic concepts and preliminaries, formulation of the proposed model using the piecewise calculus, the existence and uniqueness of the model, and a numerical method for approximating the model's solutions. Numerical simulations, which validate the results from earlier sections, can be found in Section 3. In the end, Section 4 offers concluding observations.

2. Proposed model

2.1. Preliminaries and notations

Here, we define key terms from fractional calculus that will be consistently used in this study.

Definition 1. ^[20] Assuming $\mathfrak{f}(\tau)$ is a continuous function, given $\Omega = [\mu, \nu]$ , $-\infty < \mu < \nu < +\infty$ , $\beta \in \mathbb{C}, \ \Re(\beta) > 0$ . Below, the definitions are provided for the Riemann-Liouville derivatives, encompassing both the left and right variants, of order $\beta$ :

$\begin{eqnarray} {}_{\mu}D^{^{\beta}}_{\tau}\mathfrak{f}(\tau) = \frac{1}{\Gamma(n-\beta)} (\frac{d}{d\tau})^n\int^{\tau}_{\mu}\frac{\mathfrak{f}(s)}{(\tau-s)^{1-n+\beta}}ds, \ \ \tau > \mu, \ \ \qquad \end{eqnarray}$

(2.1)

$\begin{eqnarray} {}_{\tau}D^{^{\beta}}_{\nu}\mathfrak{f}(\tau) = \frac{1}{\Gamma(n-\beta)}( \frac{-d}{d\tau})^n\int^{\nu}_{\tau}\frac{\mathfrak{f}(s)}{(s-\tau)^{1-n+\beta}}ds, \ \ \tau < \nu\qquad\quad \end{eqnarray}$

(2.2)

where $n = [\Re(\beta)]+1$ .

Definition 2. ^[20,21] Let $\Omega = [\mu, \nu]$ , where $-\infty < \mu < \nu < +\infty$ , and let $\beta \in \mathbb{C}$ with $\Re(\beta) > 0$ . The left and right Riemann-Liouville integrals of order $\beta$ for a continuous function $\mathfrak{f}(\tau)$ are defined as:

$\begin{eqnarray} ^{RL}_{\mu}I^{^{\beta}}_{\tau}\mathfrak{f}(\tau) = {}_{\mu}D^{^{-\beta}}_{\tau}\mathfrak{f}(\tau) = \frac{1}{\Gamma(\beta)}\Bigg[\int^{\tau}_{\mu}\mathfrak{f}(s)(\tau-s)^{\beta-1}ds \Bigg], \ \ \ \ \tau > \mu, \ \ \qquad \end{eqnarray}$

(2.3)

$\begin{eqnarray} ^{RL}_{\tau}I^{^{\beta}}_{\nu}\mathfrak{f}(\tau) = {}_{\tau}D^{^{-\beta}}_{\nu}\mathfrak{f}(\tau) = \frac{1}{\Gamma(\beta)}\Bigg[\int^{\nu}_{\tau}\mathfrak{f}(s)(\tau-s)^{\beta-1}ds \Bigg]\ \ \ \ \ \tau < \nu\qquad\quad \end{eqnarray}$

(2.4)

where $0 < \beta < 1$ .

Definition 3. ^[20] Within the field of complex numbers $\mathbb{C}$ , the descriptions of Caputo derivatives on the left and right sides of order $\beta$ for a function $\mathfrak{f}(\tau)$ are given by

$\begin{align} (^{C}D^{\beta}_{\mu+}\mathfrak{f})(\tau)& = (^{C}_{\mu}D_{\tau}^{\beta}\mathfrak{f})(\tau) = \frac{1}{\Gamma(n-\beta)}\int^{\tau}_{\mu}\frac{\mathfrak{f}^{n}(s)}{(\tau-\xi)^{1-n+\beta}}ds, \ \ \tau > \mu, \ \ \ \end{align}$

(2.5)

$\begin{align} (^{C}D^{\beta}_{\nu-}\mathfrak{f})(\tau)& = (^{C}_{\tau}D_{\nu}^{\beta}\mathfrak{f})(\tau) = \frac{(-1)^n}{\Gamma(n-\beta)}\int^{\nu}_{\tau}\frac{\mathfrak{f}^n(s)}{(s-\tau)^{1-n+\beta}}ds, \ \ \ \tau < \nu \end{align}$

(2.6)

where $n = [\Re(\beta)]+1$ , $\Re(\beta) \notin \mathbb{N}_0$ .

Definition 4. ^[22] In general, the operator known as the Caputo Proportional Fractional Hybrid (CP) operator is characterized as

$\begin{align} ^{CP}_{0}D^{\beta}_{\tau}y(\tau)& = \Bigg(\int^{\tau}_{0} (y(s)K_{1}(s, \beta) + y'(s) K_{0}(s, \beta)) (\tau-s)^{-\beta} ds\Bigg ) \frac{1}{\Gamma(1-\beta)}, \\ & = (K_{1}( \tau, \beta) y(\tau)+ K_{0}(\tau, \beta)y'(\tau)) \Bigg(\frac{\tau^{-\beta}}{\Gamma(1-\beta)}\Bigg) \end{align}$

(2.7)

where $K_0(\beta, \tau) = \beta \tau^{(1-\beta)}$ , $K_1(\beta, \tau) = (1-\beta) \tau^{\beta}$ , and $0 < \beta < 1$ .

Definition 5. ^[22] Fractional hybrid operators with a Caputo proportional constant (CPC) are described as

$\begin{align} ^{CPC}_{0}D^{\beta}_{\tau}y(\tau)& = \Bigg(\int^{\tau}_{0}(\tau-s)^{-\beta}(y(s)K_{1}(\beta) +y'(s) K_{0}(\beta)) ds\Bigg)\frac{1}{\Gamma(1-\beta)} \\ & = K_{1}(\beta) ^{RL}_{0}I^{1-\beta}_{\tau}y(\tau)+ K_{0}(\beta) ^{C}_{0}D^{\beta}_{\tau}y(\tau) \end{align}$

(2.8)

where $K_0(\beta) = \beta Q^{(1-\beta)}$ , $K_1(\beta) = (1-\beta) Q^{\beta}$ are kernels $Q$ is a constant, and $0 < \beta < 1$ .

Definition 6. ^[22] The variable-order fractional Caputo proportional operator $(C P)$ is given as follows:

$\begin{aligned} { }_0^{C P} D_\tau^{ \beta (\tau)} y (\tau) & = \int_0^t(\Gamma(1- \beta (\tau)))^{-1}(\tau-s)^{- \beta (\tau)}\left(y^{\prime}(s) K_0(s, \beta (\tau))+y(s) K_1(s, \beta (\tau))\right) d s, \\ & = \left(\frac{\Gamma(1- \beta (\tau))^{-1}}{\tau^{ \beta (\tau)}}\right)\left(y^{\prime} (\tau) K_0(t, \beta (\tau))+y (\tau) K_1(t, \beta (\tau))\right) . \end{aligned}$

Here, $K_1(\beta (\tau), \tau) = (- \beta (\tau)+1) \tau^{ \beta (\tau)}, K_0(\beta (\tau), \tau) = \tau^{(1- \beta (\tau))} \beta (\tau),$ and $1 > \beta (\tau) > 0.$ Alternatively, the constant proportional Caputo $({CPC})$ variable-order fractional hybrid operator can be formulated as follows:

$\begin{aligned} { }_0^{C P C} D_\tau ^{ \beta (\tau)} y (\tau) & = \left(\int_0^t(\tau-s)^{- \beta (\tau)} \frac{1}{\Gamma(1- \beta (\tau))}\left(K_1( \beta (\tau)) y(s)+y^{\prime}(s) K_0( \beta (\tau))\right) d s\right) \\ & = K_1( \beta (\tau))_0^{R L} I_\tau^{1- \beta (\tau)} y (\tau)+K_0( \beta (\tau))_0^C D_\tau^{ \beta (\tau)} y (\tau), \end{aligned}$

where $K_0(\beta (\tau)) = Q^{(- \beta (\tau)+1)} \beta (\tau),$ $K_1(\beta (\tau)) = Q^{ \beta (\tau)}(- \beta (\tau)+1)$ , $Q$ is a constant, and $1 > \beta (\tau) > 0.$ Moreover, its inverse operator is:

${ }_0^{C P C} I_t^{ \beta (\tau)} y(t) = \left(\int_0^\tau \exp \left[\frac{K_1( \beta (\tau))}{K_0( \beta (\tau))}(\tau-s)\right]_0^{R L} D_\tau^{1- \beta (\tau)} y(s) d s\right) \frac{1}{K_0( \beta (\tau))} .$

2.2. The Hybrid piecewise mathematical model

The concept of a piecewise differential equation is utilized to expand upon the COVID-19 pandemic model initially introduced in ^[23]. We address potential dimensional incompatibilities by introducing an auxiliary parameter, $\vartheta$ , to the fractional model. Consequently, the proposed model segments the population into eight classes: susceptible $(\mathcal{S}),$ exposed $(\mathcal{E}),$ symptomatic and infectious $(\mathcal{I}),$ superspreaders $(\mathcal{P}),$ infectious but asymptomatic $(\mathcal{A}),$ hospitalized $(\mathcal{H}),$ recovery $(\mathcal{R}),$ and fatality $(\mathcal{F})$ . The corresponding piecewise (fractional-variable order-stochastic) differential equations for the COVID-19 epidemic are presented as follows:

$\begin{eqnarray} \begin{cases} \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{S}(\tau)}& = -a \mathcal{I}\mathcal{S}-b\mathcal{H}\mathcal{S}-c\mathcal{P}\mathcal{S}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{E}(\tau)}& = a\mathcal{I}\mathcal{S}+b\mathcal{H}\mathcal{S}+c\mathcal{P}\mathcal{S}-d\mathcal{E}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{I}(\tau)}& = -d\mathcal{E}-f\mathcal{I}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{P}(\tau)}& = -g\mathcal{E}-\lambda \mathcal{P}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{A}(\tau)}& = \mu \mathcal{E}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{H}(\tau)}& = v(\mathcal{P}+\mathcal{I})-\mathcal{H}+u\mathcal{H}, \ \ \ \ \ \ \ 0 < \tau\leq T_{1}, \, \, \, 0 < \beta < 1, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{R}(\tau)}& = f(\mathcal{P}+\mathcal{I})+\mathcal{H}, \\ \frac{1}{\vartheta^{1-\beta}}{^{CPC}_{0}D^{\beta}_{\tau}\mathcal{F}(\tau)}& = f\mathcal{I}+\mathcal{P}+u\mathcal{H}\\ \end{cases} \end{eqnarray}$

(2.9)

with initial conditions

$\begin{align} \mathcal{I}(0)& = \mathcal{I}_{0}\geq0, \ \mathcal{S}(0) = \mathcal{S}_{0}\geq0, \mathcal{E}(0) = \mathcal{E}_{0}\geq0, \mathcal{A}(0) = \mathcal{A}_{0}\geq0, \\ & \mathcal{P}(0) = \mathcal{P}_{0}\geq0, \mathcal{R}(0) = \mathcal{R}_{0}\geq0, \mathcal{F}(0) = \mathfrak{F}_{0}\geq0, \mathcal{H}(0) = \mathcal{H}_{0}\geq0. \end{align}$

(2.10)

$\begin{eqnarray} \begin{cases} \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{S}(\tau)}& = -a \mathcal{I}\mathcal{S}-b\mathcal{H}\mathcal{S}-c\mathcal{P}\mathcal{S}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{E}(\tau)}& = a\mathcal{I}\mathcal{S}+b\mathcal{H}\mathcal{S}+c\mathcal{P}\mathcal{S}-d\mathcal{E}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{I}(\tau)}& = -d\mathcal{E}-f\mathcal{I}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{P}(\tau)}& = -g\mathcal{E}-\lambda \mathcal{P}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{A}(\tau)}& = \mu \mathcal{E}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{H}(\tau)}& = v(\mathcal{P}+\mathcal{I})-\mathcal{H}+u\mathcal{H}, \ \ \ \ \ \ \ T_{1} < \tau\leq T_{2}, , \, \, \, 0 < \beta(\tau) < 1\\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{R}(\tau)}& = f(\mathcal{P}+\mathcal{I})+\mathcal{H}, \\ \frac{1}{\vartheta^{1-\beta(\tau)}}{^{CPC}_{0}D^{\beta(\tau)}_{\tau}\mathcal{F}(\tau)}& = f\mathcal{I}+\mathcal{P}+u\mathcal{H}, \\ \end{cases} \end{eqnarray}$

(2.11)

$\begin{align} \mathcal{I}(T_{1})& = \mathcal{I}_{1}\geq0, \ \mathcal{S}(T_{1}) = \mathcal{S}_{1}\geq0, \mathcal{E}(T_{1}) = \mathcal{E}_{1}\geq0, \mathcal{A}(T_{1}) = \mathcal{A}_{1}\geq0, \\ & \mathcal{P}(T_{1}) = \mathcal{P}_{1}\geq0, \mathcal{R}(T_{1}) = \mathcal{R}_{1}\geq0, \mathcal{F}(T_{1}) = \mathfrak{F}_{1}\geq0, \mathcal{H}(T_{1}) = \mathcal{H}_{1}\geq0. \end{align}$

(2.12)

$\begin{eqnarray} \begin{cases} d\mathcal{S}& = (-a \mathcal{I}\mathcal{S}-b\mathcal{H}\mathcal{S}-c\mathcal{P}\mathcal{S})dt+\sigma_{1} \mathcal{S}(\tau)dB_{1}^{H^{\ast}}, \\ d\mathcal{E}& = (a\mathcal{I}\mathcal{S}+b\mathcal{H}\mathcal{S}+c\mathcal{P}\mathcal{S}-d\mathcal{E})dt+\sigma_{2} \mathcal{E}(\tau)dB_{2}^{H^{\ast}}, \\ d\mathcal{I}& = (-d\mathcal{E}-f\mathcal{I})dt+\sigma_{3} \mathcal{I}(\tau)dB_{3}^{H^{\ast}}, \\ d\mathcal{P}& = (-g\mathcal{E}-\lambda \mathcal{P})dt+\sigma_{4} \mathcal{P}(\tau)dB_{4}^{H^{\ast}}, \\ d\mathcal{A}& = (\mu \mathcal{E})dt+\sigma_{5} \mathcal{A}(\tau)dB_{5}^{H^{\ast}}, \\ d\mathcal{H}& = (v(\mathcal{P}+\mathcal{I})-\mathcal{H}+u\mathcal{H})dt+\sigma_{6} \mathcal{H}(\tau)dB_{6}^{H^{\ast}}, \ \ \ \ \ \ \ T_{2} < \tau\leq T_{f}, \\ d\mathcal{R}& = (f(\mathcal{P}+\mathcal{I})+\mathcal{H})dt+\sigma_{7} \mathcal{R}(\tau)dB_{7}^{H^{\ast}}, \\ d\mathcal{F}(\tau)& = (f\mathcal{I}+\mathcal{P}+u\mathcal{H})dt+\sigma_{8} \mathcal{F}(\tau)dB_{8}^{H^{\ast}}, \\ \end{cases} \end{eqnarray}$

(2.13)

$\begin{align} \mathcal{I}(T_{2})& = \mathcal{I}_{2}\geq0, \ \mathcal{S}(T_{2}) = \mathcal{S}_{2}\geq0, \mathcal{E}(T_{2}) = \mathcal{E}_{2}\geq0, \mathcal{A}(T_{2}) = \mathcal{A}_{2}\geq0, \\ & \mathcal{P}(T_{2}) = \mathcal{P}_{2}\geq0, \mathcal{R}(T_{2}) = \mathcal{R}_{2}\geq0, \mathcal{F}(T_{2}) = {F}_{2}\geq0, \mathcal{H}(T_{2}) = \mathcal{H}_{2}\geq0 \end{align}$

(2.14)

where $\sigma_{i}$ and $B_{i}(t),$ $i = 1, 2, 3, ..., 8$ are the density of randomness and environmental noise, respectively, and $H^{\ast}$ is the Hurst index.

2.3. Solution's existence and uniqueness

In the following, we demonstrate the existence and uniqueness of the system given by (2.9) and (2.10). However, before proceeding, it is essential to verify the conditions of Perov's theorem ^[24]. To do this, we first present preliminary results related to the Perov fixed point theorem and generalized Banach spaces. We refer the reader to ^[24] for a more detailed discussion.

Definition 7. Consider $E$ as a vector space with $\mathbb{K}$ as its field, which could be $\mathbb{R}$ or $\mathbb{C}$ . In such a scenario, one can define a generalized norm as a function acting on $E$

$\begin{aligned} \|\cdot\|_{G}: \ &E \longrightarrow [0, +\infty)^{n} \\& \phi\mapsto\| \phi\|_{G} = \begin{pmatrix} \| \phi\|_{1} \\ \vdots \\ \| \phi\|_{n} \\ \end{pmatrix} \end{aligned}$

characterized by the subsequent properties

$(i)$ For all $\phi\in E$ ; if $\| \phi\|_{G} = 0_{\mathbb{R}_{+}^{n}}$ , then $\phi = 0_{\mathcal{E}}$ ,

$(ii)$ $\|a \|_{G} = |a|\| \phi\|_{G}$ for all $\phi\in \mathcal{E}$ and $a \in \mathbb{K}$ , and

$(iii)$ $\| \phi+ \omega \|_{G}\preccurlyeq \| \phi\|_{G} + \|\omega\|_{G}$ for all $\phi, \omega \in E.$

The pair $(E, \|\cdot\|_{G})$ defines a generalized normed space. If the vector-valued metric space is complete, the space $(E, \|\cdot\|_{G}) $ is termed a generalized Banach space (GBS), characterized by $\delta_{G}(\phi_{1}, \phi_{2}) = \| \phi_{1}-\phi_{2}\|_{G}$.

Definition 8. Given a matrix $\varDelta\in \mathcal{M}_{n\times n}(\mathbb{R_{+}})$ , it is considered to converge to zero when

$\varDelta^{m} \longrightarrow \mathcal{O}_{n}, \quad \text{ as }\quad m \longrightarrow \infty$

where $\mathcal{O}_{n}$ is denoted as the zero $n \times n$ matrix.

Definition 9. Consider a generalized metric space $(E, \delta_{G}) $ and an operator $N $ mapping $E $ to itself. The operator $N $ is termed a $\varDelta $-contraction with matrix $\varDelta $ from $\mathcal{M}_{n\times n}(\mathbb{R}_{+}) $ that tends towards $\mathcal{O}_{n} $, provided that, for any $\varrho, v \in E $, the following holds:

$\delta_{G}(N(\varrho), N(v)) \preccurlyeq \varDelta \delta_{G}(\varrho, v).$

The following result is Perov's extension of the Banach contraction principle.

Theorem 2.1. ^[24] Let $E$ be a full generalized metric space and $N: E \longrightarrow E$ be an operator that is an $M$ -contraction. Then, $N$ has a single fixed point in $E.$

Now, given that the systems (2.9) and (2.10) can be rewritten in the following classical form

$\begin{equation} \begin{cases} ^{CPC}D^{\beta}_{\tau}{X}(\tau) = \vartheta^{1-\beta} \mathfrak{F}({X}(\tau)), \\ {X}(0) = {X}_0, \end{cases}\quad 0 < \tau < T_{1} < \infty, \end{equation}$

(2.15)

where, the vector ${X}(\tau) = (\mathcal{S}_0, \mathcal{E}_0, \mathcal{I}_0, \mathcal{P}_0, \mathcal{A}_0, \mathcal{H}_0, \mathcal{R}_0, \mathcal{F}_0)^{\tau}$ and the operator $\mathfrak{F}$ is defined as follows

$\begin{eqnarray} \mathfrak{F}(X) = \begin{pmatrix} \mathfrak{F}_{1}(X)\\ \mathfrak{F}_{2}(X)\\ \mathfrak{F}_{3}(X)\\ \mathfrak{F}_{4}(X)\\ \mathfrak{F}_{5}(X)\\ \mathfrak{F}_{6}(X)\\ \mathfrak{F}_{7}(X)\\ \mathfrak{F}_{8}(X) \end{pmatrix} = \begin{pmatrix} -a \mathcal{I}\mathcal{S}-b\mathcal{H}\mathcal{S}-c\mathcal{P}\mathcal{S}\\ a\mathcal{I}\mathcal{S}+b\mathcal{H}\mathcal{S}+c\mathcal{P}\mathcal{S}-d\mathcal{E}\\ -d\mathcal{E}-f\mathcal{I}\\ -g\mathcal{E}-\lambda \mathcal{P}\\\mu \mathcal{E}\\v(\mathcal{P}+\mathcal{I})-\mathcal{H}+u\mathcal{H}\\ f(\mathcal{P}+\mathcal{I})+\mathcal{H}\\f\mathcal{I}+\mathcal{P}+u\mathcal{H}\\ \end{pmatrix}. \end{eqnarray}$

(2.16)

Consider $E = \prod_{i = 1}^{8}\mathcal{C}([0, \tau], \mathbb{R})$ to be a generalized Banach space when we endow it with the following generalized norm:

$\begin{aligned} \|.\|_{G} : &{E} \longrightarrow \mathbb{R}^{8}_{+} \\& {X} \mapsto\|{X}\|_{G} = \begin{pmatrix} \|{\mathcal{S}}\|_{\infty} \\ \|\mathcal{E}\|_{\infty} \\ \|\mathcal{I}\|_{\infty} \\ \|\mathcal{P}\|_{\infty} \\ \|\mathcal{A}\|_{\infty} \\ \|\mathcal{H}\|_{\infty} \\ \|\mathcal{R}\|_{\infty} \\ \|\mathcal{F}\|_{\infty} \\ \end{pmatrix}. \end{aligned}$

To show that systems (2.9) and (2.10) have a unique solution, we transform them into a fixed point problem for the operator $\mathfrak{N}$ , defined by $\mathfrak{N}:\prod_{i = 1}^{8}\mathcal{C}([0, \tau], \mathbb{R})\to :\prod_{i = 1}^{8}\mathcal{C}([0, \tau], \mathbb{R}),$

$\begin{equation} \mathfrak{N}( X(\tau)) = {X}(0)+ \frac{\vartheta^{1-\beta}}{K_0(\beta)} \int_0^\tau \exp \left(-\frac{K_1(\beta)}{K_0(\beta(\tau))}(\tau-s)\right) {}^{RL}_{0} D_\tau^{1-\beta} \mathfrak{F}( X(s)) d s . \end{equation}$

(2.17)

The next lemma is needed.

Lemma 2.1. Assume a vector $\varDelta\in \mathbb{R}^8$ satisfies the given conditions

$\begin{equation} \varDelta = \begin{pmatrix} \varDelta_{1}\\[ 6mm]\varDelta_{2}\\[ 6mm]\varDelta_{3}\\[ 6mm] \varDelta_{4}\\[ 6mm]\varDelta_{5}\\[ 6mm]\varDelta_{6} \\[ 6mm]\varDelta_{7}\\[ 6mm]\varDelta_{8} \end{pmatrix}\succcurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|} \begin{pmatrix} a \varDelta_{3}\varDelta_{1}+b\varDelta_{6}\varDelta_{1}+c\varDelta_{4}\varDelta_{1}\\ | a \varDelta_{3}\varDelta_{1}+b\varDelta_{6}\varDelta_{1}+c\varDelta_{4}\varDelta_{1}-d\varDelta_{2}|\\ d\varDelta_{2}f\varDelta_{3}\\ g\varDelta_{2}\lambda \varDelta_{4}\\ \mu \varDelta_{2}\\|v(\varDelta_{4}+\varDelta_{3})-\varDelta_{6}+u\varDelta_{6}|\\ f(\varDelta_{4}+\varDelta_{3})+\varDelta_{6}\\f\varDelta_{3}+\varDelta_{4}+u\varDelta_{6}\\ \end{pmatrix}. \end{equation}$

(2.18)

Then, the operator $\mathfrak{N}$ maps the generalized ball $\bar{B}(X_0, \varDelta)\subset \mathcal{E}$ into itself.

Next, we make sure that the operator $\mathfrak{F}$ is G-Lipschitz.

Lemma 2.2. The operator $\mathfrak{F}$ defined in (2.16) is a G-Lipschitz operator; i.e., there exists a square matrix $\mho \in \mathcal{M}_{6}(\mathbb{R}_{+})$ such that

$\|\mathfrak{F}(X)-\mathfrak{F}(\bar{X})\|_{G}\preccurlyeq \mho \|X-\bar{X}\|_{G}.$

The proofs of the above two lemmas are stated in the Appendix. The next theorem ensures the existence and the uniqueness of the proposed model.

Theorem 2.2. Assuming that the matrix

$\begin{equation} \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|}\mho, \end{equation}$

(2.19)

converges to $O_{8},$ then there is a unique solution for the initial value problem given by (2.9) and (2.10), and the solution holds for all $\tau > 0$ in $\bar{B}(X_0, \varDelta)$ .

Proof. Further, we have for any $X, \, \bar{ X} \in \bar{B}(X_0, \varDelta)$ , and by using Lemma 2.2,

$\begin{align*} \left\| \mathfrak{N}( X)-\mathfrak{N}(\bar{ X})\right\|_G& = \left\| \frac{\vartheta^{1-\beta}}{K_0(\beta)} \int_0^\tau \exp \left(-\frac{K_1(\beta)}{K_0(\beta)}(\tau-s)\right)\Big({ }_0^{R L} D_\tau^{1-\beta} \mathfrak{F}( X(s))-{ }_0^{R L} D_\tau^{1-\beta}\mathfrak{F}(\bar{ X}(s))\Big)ds \right\|_G\\ &\preccurlyeq \frac{\vartheta^{1-\beta}}{|K_0(\beta)|} \int_0^\tau \Big|\exp \left(-\frac{K_1(\beta)}{K_0(\beta)}(\tau-s)\right)\Big|\left\| { }_0^{R L} D_\tau^{1-\beta} \mathfrak{F}( X(s))-{ }_0^{R L} D_\tau^{1-\beta}\mathfrak{F}(\bar{ X}(s))\right\|_Gds \\ &\preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)}}{\Gamma(\beta-1)|K_0(\beta)|}\left\|\int_0^\tau(\tau-s)^{\beta-2}\left(\mathfrak{F}({ X}(s))-\mathfrak{F}(\bar{ X}(s))\right) d s\right\|_{G} \\ & \preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|}\left\|\mathfrak{F}({ X})-\mathfrak{F}(\bar{ X})\right\|_{G} \\ & \preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|}\mho\left\|{ X}-\bar{ X}\right\|_{G} . \end{align*}$

Since the matrix $\frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|} \mho$ converges to zero, the operator $\mathfrak{N}$ is a G-contraction, and hence by applying Perov's fixed point theorem (2.1), systems (2.9) and (2.10) have only one solution in $\bar{B}(X_0, \varDelta)$ . □

2.4. Numerical method for crossover model

We propose an effective discretization for the examined cross-over system using the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method (CPC-GLNFDM). The approach we take to addressing the subsequent linear cross-over model (including fractional and variable order, both deterministic and stochastic) is outlined as follows:

$\begin{eqnarray} \begin{split} (^{CPC}_{0}D_{\tau}^{\beta}y)(\tau) = &\rho y(\tau), \ \ 0 < \tau\leq T_{1}, \ \ 0 < \beta\leq 1, \ \rho < 0 \\ \ \ \ \ \ \ \ \ y(0) = y_0 \end{split} \end{eqnarray}$

(2.20)

$\begin{eqnarray} \begin{split} (^{CPC}_{0}D_{\tau}^{\beta(\tau)}y)(\tau) = &\rho y(\tau), \ \ T_{1} < \tau\leq T_{2}, \ \ 0 < \beta(\tau)\leq 1, \ \rho < 0 \\ \ \ \ \ \ \ \ \ \ \ \ \ y(T_{1}) = y_1 \end{split} \end{eqnarray}$

(2.21)

$\begin{eqnarray} \begin{split} y(\tau) = &(\rho y(\tau)+\sigma y(\tau)dB^{H^{\ast}}(\tau) , \ \ T_{2} < \tau\leq T_{f}, \\ \ \ \ \ \ \ \ \ \ \ \ \ y(T_{2}) = y_2, \end{split} \end{eqnarray}$

(2.22)

where $B(\tau)$ represents the standard Brownian motion, $\sigma$ , denotes the real constants or the intensity of the stochastic environment, and $H^{*}$ is the Hurst index.

The relation given by (2.8) can be expressed as follows:

$\begin{align} ^{CPC}_{0}D^{\beta}_{\tau}y(\tau)& = \frac{1}{\Gamma(1-\beta)}\int^{\tau}_{0}(\tau-s)^{-\beta}(K_{1}(\beta) y(s)+ K_{0}(\beta)y'(s)) ds, \\ & = K_{1}(\beta) ^{RL}_{0}I^{1-\beta}_{\tau}y(\tau)+ K_{0}(\beta) ^{C}_{0}D^{\beta}_{\tau}y(\tau), \\ & = K_{1}(\beta) ^{RL}_{0}D^{\beta-1}_{\tau}y(\tau)+ K_{0}(\beta) ^{C}_{0}D^{\beta}_{\tau}y(\tau), \end{align}$

(2.23)

where $K_{1}(\beta)$ and $K_{0}(\beta)$ are kernels depending solely on $\beta$ and $K_0(\beta) = \beta Q^{(1-\beta)}$ , $K_1(\beta) = (1-\beta) Q^{\beta},$ $Q$ is a constant, and $\omega_{0} = 1.$ Using the GLNFDM approximation, (2.23) can be discretized as follows:

$\begin{align} ^{CPC}_{0}D^{\beta}_{\tau}y(\tau)|_{\tau = \tau^{n}} = \frac{K_{1}(\beta)}{(\Theta(\Delta \tau))^{\beta-1}}\Bigg(y_{n+1}+\sum\limits_{i = 1}^{n+1} \omega_{i}y_{n+1-i}\Bigg)+\frac{K_{0}(\beta)}{(\Theta(\Delta \tau))^{\beta}}\Bigg(y_{n+1}-\sum\limits_{i = 1}^{n+1} \mu_{i}y_{n+1-i}-q_{n+1}y_{0}\Bigg) \end{align}$

(2.24)

where

$\Theta(\Delta \tau) = \Delta \tau+ O(\Delta \tau^{2}), ..., 0 < \Theta(\Delta \tau) < 1, \Delta \tau \longrightarrow 0.$

Subsequently, (2.20) can be discretized as follows:

$\begin{align} \frac{K_{1}(\beta)}{(\Theta(\Delta \tau))^{\beta-1}}\Bigg(y_{n+1}+\sum\limits_{i = 1}^{n+1} \omega_{i}y_{n+1-i}\Bigg)+\frac{K_{0}(\beta)}{(\Theta(\Delta \tau))^{\beta}}\Bigg(y_{n+1}-\sum\limits_{i = 1}^{n+1} \mu_{i}y_{n+1-i}-q_{n+1}y_{0}\Bigg) = \rho y(t_{n}) \end{align}$

(2.25)

where $\omega_{0} = 1,$ $\omega_{i} = (1-\frac{\beta}{i})\omega_{i-1},$ $\tau^{n} = n(\Theta(\Delta \tau)), \ \ \ \ \, \Delta \tau = \frac{T_{f}}{N_{n}},$ $N_{n}$ is a natural number. $\mu_{i} = (-1)^{i-1}\Big(\ ^{\beta}_{i} \Big),$ $\mu_{1} = \beta,$ $q_{i} = \frac{i^{\beta}}{\Gamma(1-\beta)}$ , and $i = 1, 2, ..., n+1.$

In addition, we will proceed with the assumption that ^[25]

$0 < \mu_{i+1} < \mu_{i} < ... < \mu_{1} = \beta < 1,$

$0 < q_{i+1} < q_{i} < ... < q_{1} = \frac{1}{\Gamma(-\beta+1)}.$

Hence, if $K_1(\beta) = 0$ and $K_0(\beta) = 1$ in (2.25), then one has the discretization of the Caputo operator (C- GLNFDM) using the finite difference method.

Additionally, (2.21) can be discretized as follows:

$\begin{align} \frac{K_{1}(\beta(t_{i}))}{(\Theta(\Delta \tau))^{\beta(t_{i})-1}}\Bigg(y_{n_{2}+1}+\sum\limits_{i = n+1}^{n_{2}+1} \omega_{i}y_{n_{2}+1-i}\Bigg)+\frac{K_{0}\beta(t_i)}{(\Theta(\Delta \tau))^{\beta(t_i)}}\Bigg(y_{n_{2}+1}-\sum\limits_{i = n+1}^{n_{2}+1} \mu_{i}y_{n_{2}+1-i}-q_{n_{2}+1}y_{0}\Bigg) = \rho y(t_{n_{2}}) \end{align}$

(2.26)

where $\omega_{i} = (1-\frac{\beta(t_{i})}{i})\omega_{i-1},$ $\mu_{i} = (-1)^{i-1}\Big(\ ^{\beta(t_{i})}_{i} \Big),$ $\mu_{1} = \beta(t_{i}),$ $q_{i} = \frac{i^{\beta(t_{i})}}{\Gamma(1-\beta(t_{i}))}$ , and we can discretize (2.22) using the nonstandard modified Euler-Maruyama method (NMEMM) ^[26] as follows:

$\begin{eqnarray} \begin{split} y(t_{n_{3}+1}) = & y(t_{n_{3}})+\rho y(t_{n_{3}})\Theta(\Delta \tau)+\sigma y(t_{n_{3}})\Delta B_{n_{3}}+0.5 y(t_{n_{3}})\Theta(\Delta \tau)^{2H^{*}}, \notag\\& \ H^{*} > 0.5, \ \ \ \ n_{3} = n_2, ..., \kappa, \ \ \ \ \ T_{2} < \tau\leq T_{f}. \end{split} \end{eqnarray}$

3. Numerical simulations

To obtain the numerical solutions for the proposed piecewise models (2.9)–(2.14), we employ the following parameters: $a = 6.82621 10^{-5},$ $b = 1.0648 10^{-4},$ $c = 2.0478 10^{-4},$ $d = 0.25,$ $f = 3.5,$ $g = 2.5 10^{-4},$ $\lambda = 2.21,$ $\mu = 0.10475,$ $v = 0.94,$ $u = 0.3$ . The initial conditions are set as follows: $\mathcal{S}(0) = 59659,$ $\mathcal{I}(0) = 1,$ $\mathcal{E}(0) = 0,$ $\mathcal{P}(0) = 5,$ $\mathcal{A}(0) = 0,$ $\mathcal{R}(0) = 0,$ $\mathcal{H}(0) = 0,$ $\mathcal{R}(0) = 0,$ $\mathcal{F}(0) = 0,$ and $T_{1} = 25, T_{2} = 50,$ and $T_{f} = 100,$ $\Theta(\Delta \tau) = 1-e^{-\Delta \tau}$ . The numerical results for (2.9)-(2.13) are graphically represented at various values of $\beta $, $\beta(\tau) $, $\sigma_{i} $ (where $i $ ranges from 1 to 8), and $H^{\ast}(\tau) $. The simulations are conducted using the nonstandard Euler-Maruyama method (NEMM) (2.27) and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method CPC-GLNFDM (2.26). The dynamical behavior of the systems given by (2.9)–(2.13) is illustrated in through 5. It is observed that the crossover behavior is prominent around the values $T_{1} = 25 $, $T_{2} = 50 $, and $T_{f} = 100 $. After these points, the dynamics exhibit multiplicity in their behavior. illustrates the simulation for (2.9)–(2.13) with $\beta(\tau) = 0.80-0.003t$ and $H^{\ast} = 0.7$ , considering various values of $\beta$ and $\sigma_{i}$ . demonstrates how the solutions behave under various scenarios, with different values of $\beta (\tau)$ and for $\beta = 0.8$ , and $H^{\ast} = 0.7$ . illustrates the change in the behavior of the solutions (2.9)–(2.13) when employing different values of $H^{\ast},$ $\beta(\tau) = 0.75-0.001(cos^2(\tau/10))$ , and $\beta = 0.80, \sigma_{i} = 0.05$ . illustrates the behavior of the solution of (2.9)–(2.13) for different values of $H^{\ast}$ and $\sigma_{i}$ $\beta(\tau) = 0.75-0.001(cos^2(\tau/10))$ and $\beta = 0.80$ . To illustrate how parameter values $\sigma_{i}$ and $\mathcal{H}$ influence the results, we refer to Figure 5. From the simulation, we discerned several nuanced insights of the model, which proved invaluable insights from both biological and mathematical perspectives.

Figure 1. Simulation for (2.11–2.13) at different values of

$\beta$ and

$\beta(\tau) = 0.80-0.0003\tau,$

$H^{\ast} = 0.7,$

$\sigma_{1} = 0.05, \sigma_{2} = 0.05, \sigma_{3} = 0.05,$

$\sigma_{4} = 0.05, \sigma_{5} = 0.05, \sigma_{6} = 0.05,$

$\sigma_{7} = 0.05, \sigma_{8} = 0.05$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Simulation for (2.11–2.13) at different values of

$\beta(\tau)$ and

$\beta = 0.80$

$H^{\ast} = 0.7,$

$\sigma_{i} = 0.05, \ i = 1, 2, 3, 4, 5, 6, 7, 8$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Simulation for (2.11–2.13) at different values of

$H^{\ast},$

$\beta(\tau) = 0.75-0.001(cos^2(\tau/10))$ , and

$\beta = 0.80$

$\sigma_{i} = 0.05, \ i = 1, 2, 3, 4, 5, 6, 7, 8$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. Simulation for (2.11–2.13) at different values of

$H^{\ast},$

$\beta(\tau) = 0.75-0.001(cos^2(\tau/10))$ , and

$\beta = 0.80,$

$\sigma_{1} = 0.5, \sigma_{2} = 0.04, \sigma_{3} = 0.01, \sigma_{4} = 0.5, \sigma_{5} = 0.04, \sigma_{6} = 0.01, \sigma_{7} = 0.5, \sigma_{8} = 0.04$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Simulation for (2.11–2.13) at different values of

$H^{\ast},$

$\beta(\tau) = 0.7-0.001\tau$ , and

$\beta = 0.80$

$\sigma_{1} = 0.5, \sigma_{2} = 0.04, \sigma_{3} = 0.01, \sigma_{4} = 0.5, \sigma_{5} = 0.04, \sigma_{6} = 0.01, \sigma_{7} = 0.5, \sigma_{8} = 0.04$ .

DownLoad: Full-Size Img PowerPoint

4. Conclusions

In conclusion, a new mathematical model of piecewise fractional and variable-order differential equations and the fractional stochastic derivative of the COVID-19 epidemic has been developed. Moreover, two numerical techniques are constructed to solve the proposed model. These methods are the nonstandard modified Euler Maruyama method for the fractional stochastic model and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Utilizing this approach has enriched our understanding of the dynamics underlying this intricate health crisis. Furthermore, by integrating fractional Brownian motion into stochastic differential equations, we have gleaned valuable insights into the unpredictable nature of infectious disease propagation. The development of two specialized numerical methods, namely, Caputo's proportional constant-Grünwald-Letnikov nonstandard finite difference method and the nonstandard modified Euler Maruyama technique, enabled us to delve deeply into the behavior of our extended models. Through a series of numerical tests, we have showcased the efficiency of these methods and garnered robust empirical support for our theoretical findings. Perhaps most crucially, our research has paved new pathways for studying the COVID-19 pandemic. The extended models afford the flexibility to examine a broad spectrum of behaviors, from deterministic patterns to stochastic processes. This allows for more adaptive strategies and interventions as circumstances change. This study stands as a substantial contribution to epidemiology, offering indispensable tools and insights that can guide public health decisions and aid in addressing the ongoing global health crisis.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are thankful to the Deanship of Scientific Research at Najran University for funding this work under the Research Priorities and Najran Research funding program grant code (NU/NRP/SERC/12/3).

Conflicts of Interest:

The authors declare no conflicts of interest.

Appendix. Proofs of Lemmas 2.1 and 2.2

Proof of Lemma 2.1. Let $X \in \bar{B}(X_0, \varDelta)$ , then

$\begin{align*} \left\| \mathfrak{N}( X)-{{X}_0}\right\|_G& = \left\| \frac{\vartheta^{1-\beta}}{K_0(\beta)} \int_0^\tau \exp \left(-\frac{K_1(\beta)}{K_0(\beta)}(\tau-s)\right){ }_0^{R L} D_\tau^{1-\beta} \mathfrak{F}( X(s))ds \right\|_G\\ &\preccurlyeq \frac{\vartheta^{1-\beta}}{|K_0(\beta)|} \int_0^\tau \Big|\exp \left(-\frac{K_1(\beta)}{K_0(\beta)}(\tau-s)\right)\Big|\left\| { }_0^{R L} D_\tau^{1-\beta} \mathfrak{F}( X(s))\right\|_Gds \\ &\preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)}}{\Gamma(\beta-1)|K_0(\beta)|}\left\|\int_0^\tau(\tau-s)^{\beta-2}\mathfrak{F}({ X}(s)) d s\right\|_{G} \\ & \preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1) |K_0(\beta)|}C_\mathfrak{F} \end{align*}$

where

$C_\mathfrak{F} = \sup\limits_{X\in \bar{B}( X_0, \varDelta)} \Vert\mathfrak{F}(X)\Vert_{G}\preccurlyeq \frac{\vartheta^{1-\beta}\varXi^{max }_{(\beta)} \Psi^{max }_{(\beta)}}{\Gamma(\beta-1)|K_0(\beta)|} \begin{pmatrix} a \varDelta_{3}\varDelta_{1}+b\varDelta_{6}\varDelta_{1}+c\varDelta_{4}\varDelta_{1}\\ | a \varDelta_{3}\varDelta_{1}+b\varDelta_{6}\varDelta_{1}+c\varDelta_{4}\varDelta_{1}-d\varDelta_{2}|\\ d\varDelta_{2}f\varDelta_{3}\\ g\varDelta_{2}\lambda \varDelta_{4}\\ \mu \varDelta_{2}\\|v(\varDelta_{4}+\varDelta_{3})-\varDelta_{6}+u\varDelta_{6}|\\ f(\varDelta_{4}+\varDelta_{3})+\varDelta_{6}\\f\varDelta_{3}+\varDelta_{4}+u\varDelta_{6} \end{pmatrix}$

and this complete the proof. □

Proof of Lemma 2.2. Let $X = (\mathcal{S}, \mathcal{E}, \mathcal{I}, \mathcal{P}, \mathcal{A}, \mathcal{H}, \mathcal{R}, \mathcal{F})$ , $\bar{X} = (\bar{\mathcal{S}}, \bar{\mathcal{E}}, \bar{\mathcal{I}}, \bar{\mathcal{P}}, \bar{\mathcal{A}}, \bar{\mathcal{H}}, \bar{\mathcal{R}}, \bar{\mathcal{F}})\in \bar{B}(X_0, \varDelta)$ and we have

$\begin{aligned} | \mathfrak{F}_{1}(X)-\mathfrak{F}_{1}(\bar{X})|& = |-a \mathcal{I}\mathcal{S}-b\mathcal{H}\mathcal{S}-c\mathcal{P}\mathcal{S} +a \bar{\mathcal{I}}\bar{\mathcal{S}}+b\bar{\mathcal{H}}\bar{\mathcal{S}}+c\bar{\mathcal{P}}\bar{\mathcal{S}}|\\&\leq a|( \bar{\mathcal{I}}\bar{\mathcal{S}}-{\mathcal{I}}{\mathcal{S}})|+b|(\bar{\mathcal{H}}\bar{\mathcal{S}}-{\mathcal{H}}{\mathcal{S}})|+c|(\bar{\mathcal{P}}\bar{\mathcal{S}}-{\mathcal{P}}{\mathcal{S}})|. \end{aligned}$

Also, we have for all $\wp_{1}, \, \wp_{2}, \, \upsilon_{1}, \, \upsilon_{2} \in \mathbb{R}$

$| \wp_{1}\upsilon_{1}-\wp_{2}\upsilon_{2}| = \frac{1}{2} \Big[ (\wp_{1}-\wp_{2})(\upsilon_{1}+\upsilon_{2})+(\wp_{1}+\wp_{2})(\upsilon_{1}-\upsilon_{2})\Big]$

and combining that with Lemma 2.1 we find

$\begin{equation} \begin{aligned} \| \mathfrak{F}_{1}(X)-\mathfrak{F}_{1}(\bar{X})\|_\infty&\leq\frac{a}{2} \Big\| (\bar{\mathcal{S}}-{\mathcal{S}})(\bar{\mathcal{I}}+{\mathcal{I}})+(\bar{\mathcal{I}}-{\mathcal{I}})(\bar{\mathcal{S}}+{\mathcal{S}})\Big\|_\infty\\&+\frac{b}{2} \Big\| (\bar{\mathcal{S}}-{\mathcal{S}})(\bar{\mathcal{H}}+{\mathcal{H}})+(\bar{\mathcal{H}}-{\mathcal{H}})(\bar{\mathcal{S}}+{\mathcal{S}})\Big\|_\infty\\&+\frac{c}{2} \Big\| (\bar{\mathcal{S}}-{\mathcal{S}})(\bar{\mathcal{P}}+{\mathcal{P}})+(\bar{\mathcal{P}}-{\mathcal{P}})(\bar{\mathcal{S}}+{\mathcal{S}})\Big\|_\infty\\&\leq a \Big( \varDelta_{1}\| \bar{\mathcal{I}}-{ \mathcal{I}}\|_\infty+ \varDelta_{3}\|\bar{\mathcal{S}}-{\mathcal{S}}\|_\infty\Big)\\&+b \Big( \varDelta_{1}\|\bar{\mathcal{H}}-{\mathcal{H}}\|_\infty+ \varDelta_{6}\|\bar{\mathcal{S}}-{\mathcal{S}}\|_\infty\Big)\\&+c \Big( \varDelta_{1} \|\bar{\mathcal{P}}-{\mathcal{P}}\|_\infty+ \varDelta_{4}\|\bar{\mathcal{S}}-{\mathcal{S}}\|_\infty\Big)\\&\leq \varDelta_{1} \Big(a\|\bar{\mathcal{I}}-{ \mathcal{I}}\|_\infty+b\|\bar{\mathcal{H}}-{\mathcal{H}}\|_\infty+c \|\bar{\mathcal{P}}-{\mathcal{P}}\|_\infty\Big)\\&+ \Big(a \varDelta_{3}+b \varDelta_{6}+c \varDelta_{4}\Big)\|\bar{\mathcal{S}}-{\mathcal{S}}\|_\infty. \end{aligned} \end{equation}$

(4.1)

Also, we have

$\begin{aligned} \| \mathfrak{F}_{2}(X)-\mathfrak{F}_{2}(\bar{X})\|_\infty& = \|a \mathcal{I}\mathcal{S}+b\mathcal{H}\mathcal{S}+c\mathcal{P}\mathcal{S} -d\mathcal{E} -a \bar{\mathcal{I}}\bar{\mathcal{S}}-b\bar{\mathcal{H}}\bar{\mathcal{S}}-c\bar{\mathcal{P}}\bar{\mathcal{S}} +d\bar{ \mathcal{E}}\|_\infty\\&\leq \varDelta_{1} \Big(a\|\bar{\mathcal{I}}-{ \mathcal{I}}\|_\infty+b\|\bar{\mathcal{H}}-{\mathcal{H}}\|_\infty+c \|\bar{\mathcal{P}}-{\mathcal{P}}\|_\infty\Big) +d\| \mathcal{E}-\bar{ \mathcal{E}}\|_\infty\\&+ \Big(a \varDelta_{3}+b \varDelta_{6}+c \varDelta_{4}\Big)\|\bar{\mathcal{S}}-{\mathcal{S}}\|_\infty. \end{aligned}$

The linearity of the operators $\mathfrak{F}_{3}, \, \mathfrak{F}_{4}, \, \mathfrak{F}_{5}, \ \ \mathfrak{F}_{6}, \, \mathfrak{F}_{7}, \ \ \mathfrak{F}_{8}$

$\begin{aligned} \| \mathfrak{F}_{3}(X)-\mathfrak{F}_{3}(\bar{X})\|_\infty&\leq d\| \mathcal{E}-\bar{ \mathcal{E}}\|_\infty+ f \|{\mathcal{I}}-\bar{ \mathcal{I}}\|_\infty \end{aligned}$

$\begin{aligned} \| \mathfrak{F}_{4}(X)-\mathfrak{F}_{4}(\bar{X})|&\leq g\| \mathcal{E}-\bar{\mathcal{ E}}\|_\infty+ \lambda \|{\mathcal{P}}-\bar{\mathcal{P}}\|_\infty \end{aligned}$

$\begin{aligned} \| \mathfrak{F}_{5}(X)-\mathfrak{F}_{5}(\bar{X})|&\leq \mu\| \mathcal{E}-\bar{ \mathcal{E}}\|_\infty, \end{aligned}$

$\begin{aligned} \| \mathfrak{F}_{6}(X)-\mathfrak{F}_{6}(\bar{X})\|_\infty&\leq v\| \mathcal{P}-\bar{\mathcal{P}}\|_\infty+v\| \mathcal{I}-\bar{ \mathcal{I}}\|_\infty+|1-u|\| \mathcal{H}-\bar{\mathcal{H}}\|_\infty, \end{aligned}$

$\begin{aligned} \| \mathfrak{F}_{7}(X)-\mathfrak{F}_{7}(\bar{X})\|_\infty&\leq f\| \mathcal{P}-\bar{\mathcal{P}}\|_\infty+f\| \mathcal{I}-\bar{ \mathcal{I}}\|_\infty+\| \mathcal{H}-\bar{\mathcal{H}}\|_\infty, \end{aligned}$

$\begin{aligned} \| \mathfrak{F}_{8}(X)-\mathfrak{F}_{8}(\bar{X})\|_\infty&\leq \| \mathcal{P}-\bar{\mathcal{P}}\|_\infty+f\| \mathcal{I}-\bar{ \mathcal{I}}\|_\infty+u\| \mathcal{H}-\bar{\mathcal{H}}\|_\infty. \end{aligned}$

By rewriting the above equations in matrix form we find

$\begin{equation*} \begin{pmatrix} \Vert \mathfrak{F}_{1}({X})- \mathfrak{F}_{1}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{2}({X})- \mathfrak{F}_{2}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{3}({X})- \mathfrak{F}_{3}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{4}({X})- \mathfrak{F}_{4}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{5}({X})- \mathfrak{F}_{5}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{6}({X})- \mathfrak{F}_{6}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{7}({X})- \mathfrak{F}_{7}(\bar{{X}})\Vert_{\infty}\\ \Vert \mathfrak{F}_{8}({X})- \mathfrak{F}_{8}(\bar{{X}})\Vert_{\infty}\\ \end{pmatrix} \preccurlyeq \mho\begin{pmatrix} \Vert {\mathcal{S}}-\bar{{\mathcal{S}}}\Vert_{\infty}\\ \Vert {\mathcal{E}}-\bar{{\mathcal{E}}}\Vert_{\infty}\\\Vert {I}-\bar{{I}}\Vert_{\infty}\\ \Vert {\mathcal{P}}-\bar{{\mathcal{P}}}\Vert_{\infty}\\ \Vert {\mathcal{A}}-\bar{\mathcal{{A}}}\Vert_{\infty}\\ \Vert {\mathcal{H}}-\bar{{\mathcal{H}}}\Vert_{\infty}\\ \Vert {\mathcal{R}}-\bar{{\mathcal{R}}}\Vert_{\infty}\\ \Vert {\mathcal{F}}-\bar{{\mathcal{F}}}\Vert_{\infty}\\ \end{pmatrix} \end{equation*}$

where

$\mho = \left(\begin{array}{cccccccc} \Big(a \varDelta_{3}+b \varDelta_{6}+c \varDelta_{4}\Big)&0&a \varDelta_{1}&c \varDelta_{1}&0&b \varDelta_{1} &0&0\\ \Big(a \varDelta_{3}+b \varDelta_{6}+c \varDelta_{4}\Big)&d&a \varDelta_{1}&c \varDelta_{1}&0&b \varDelta_{1} &0&0\\ 0&d&f&0&0&0 &0&0\\ 0&g&0&\lambda&0&0 &0&0\\ 0&\mu&0&0&0 &0&0&0\\ 0&0&v&v&0&|u-1| &0&0\\ 0&0&f&f&0&1&0&0\\ 0&0&f&1&0&u &0&0\\ \end{array}\right).$

□

References

[1]	Abanokova K, Dang HAH, Lokshin M (2022) Do Adjustments for Equivalence Scales Affect Poverty Dynamics? Evidence from the Russian Federation during 1994–2017. Rev Income Wealth 68: S167–S192. https://doi.org/10.1111/roiw.12559 doi: 10.1111/roiw.12559
[2]	Barseghyan, Gayane (2019) Sanctions and counter-sanctions: What did they do?, Available from: http://nbn-resolving.de/urn:NBN:fi:bof-201912101702.
[3]	Brainerd, Elizabeth (1998) Winners and losers in Russia's economic transition. Am Econ Rev 88:1094–1116.
[4]	Bukowski P, Novokmet F (2019) Between Communism and Capitalism: Long-Term Inequality in Poland, 1892–2015. J Econ Growth 26: 187–239 https://doi.org/10.1007/s10887-021-09190-1 doi: 10.1007/s10887-021-09190-1
[5]	Calvo PA, Lopez-Calva LF, Posadas J (2015) A decade of declining earnings inequality in the Russian Federation.
[6]	Clarke S (2005) Post‐socialist trade unions: China and Russia. Ind Relat J 36: 2–18. https://doi.org/10.1111/j.1468-2338.2005.00342.x doi: 10.1111/j.1468-2338.2005.00342.x
[7]	Dang HAH, Lokshin MM, Abanokova K (2019a) Did the poor adapt to their circumstances? Evidence from long-run Russian panel data. Econ Bull.
[8]	Dang, HAH, Lokshin MM, Abanokova K, et al. (2019b) Welfare and inequality dynamics in the Russian Federation during 1994–2015. Eur J Dev Res 32: 812–846. https://doi.org/10.1057/s41287-019-00241-3 doi: 10.1057/s41287-019-00241-3
[9]	DiNardo J, Fortin NM, Lemieux T (1996) Labor market institutions and the distribution of wages, 1973–1992: A semi-parametric approach. Econometrica 64: 1001–1044.
[10]	Drobyshevsky S, Pikulina E (2012) Analysis of a potential "bubble" on Russia's real estate market. http://dx.doi.org/10.2139/ssrn.2121244
[11]	Durand-Lasserve O, Blöchliger H (2018) The drivers of regional growth in Russia: a baseline model with application. https://doi.org/10.1787/9279f6c3-en
[12]	Fedorov L (2002) Regional inequality and regional polarization in Russia, 1990–99. World Dev 30: 443–456. https://doi.org/10.1016/S0305-750X(01)00124-3 doi: 10.1016/S0305-750X(01)00124-3
[13]	Firpo S, Fortin NM, Lemieux T (2009) Unconditional quantile regressions. Econometrica 77: 953–973. https://doi.org/10.3982/ECTA6822 doi: 10.3982/ECTA6822
[14]	Fournier JM, Koske I (2012) The determinants of earnings inequality: evidence from quantile regressions. https://doi.org/10.1787/eco_studies-2012-5k8zs3twbrd8
[15]	Gimpelson VY (2016) Structural change and inter-industry wage differentiation. J New Econ Assoc 3: 186–197. https://doi.org/10.31737/2221-2264-2016-31-3-9 doi: 10.31737/2221-2264-2016-31-3-9
[16]	Gimpelson VY, Lukiyanova A (2009) Are public sector workers underpaid in Russia? Estimating the public-private wage gap. https://doi.org/10.2139/ssrn.1329579
[17]	Gluschenko KP (2010) Methodologies of analyzing inter-regional income inequality and their applications to Russia. http://dx.doi.org/10.2139/ssrn.1649666
[18]	Gluschenko KP (2011a) Price convergence and market integration in Russia. Reg Sci Urban Econ 41: 160–172. https://doi.org/10.1016/j.regsciurbeco.2010.12.003 doi: 10.1016/j.regsciurbeco.2010.12.003
[19]	Gluschenko KP (2011b) Studies on income inequality among Russian regions: variations in social-economic development by region. Reg Res Russ 1: 319–330. https://doi.org/10.1134/S2079970511040034 doi: 10.1134/S2079970511040034
[20]	Gorodnichenko Y, Martinez-Vazquez J, Sabirianova-Peter K (2009) Myth and reality of flat tax reform: micro estimates of tax evasion response and welfare effects in Russia. J Polit Econ 117: 504–554. https://doi.org/10.1086/599760 doi: 10.1086/599760
[21]	Gorodnichenko Y, Sabirianova-Peter K, Stolyarov D (2010) Inequality and volatility moderation in Russia: evidence from micro-level panel data on consumption and income. Rev Econ Dynam 13: 209–237. https://doi.org/10.1016/j.red.2009.09.006 doi: 10.1016/j.red.2009.09.006
[22]	Götz L, Glauben T, Brümmer B (2013) Wheat export restrictions and domestic market effects in Russia and Ukraine during the food crisis. Food Policy 38: 214–226. https://doi.org/10.1016/j.foodpol.2012.12.001 doi: 10.1016/j.foodpol.2012.12.001
[23]	Guriev S, Rachinsky A (2005) The role of oligarchs in Russian capitalism. J Econ Perspect 19: 131–150.
[24]	Guriev, Sergei, and Andrei Rachinsky (2006) The evolution of personal wealth in the former Soviet Union and Central and Eastern Europe, UNU-Wider.
[25]	Guriev S, Vakulenko E (2012) Convergence between Russian regions.
[26]	Hlasny V (2019) Different faces of inequality across Asia: Decomposition of income gaps across demographic groups. In: Huang, B., Morgan, P.J., Yoshino, N., Demystifying Rising Inequality in Asia, 38–107, Tokyo: Asian Development Bank Institute. http://dx.doi.org/10.2139/ssrn.2881456
[27]	Hlasny V (2020a) Decomposition of income gaps across socioeconomic groups: the case of six Asian countries. J Income Distrib.
[28]	Hlasny V (2020b) Nonresponse bias in inequality measurement: cross-country analysis using Luxembourg Income Study surveys. Soc Sci Q 101: 712–731. https://doi.org/10.1111/ssqu.12762 doi: 10.1111/ssqu.12762
[29]	Hlasny V (2023) Les revenus des ménages dans la Russie de Poutine: changement distributionnel entre les groupes socioéconomiques, 2000–2016. Revue d'É conomie du Développement 30.
[30]	Ivanova A, Keen M, Klemm A (2005) The Russian 'flat tax' reform. Econ Policy 43: 397–444.
[31]	Johnson T (2013) Food price volatility and insecurity. Available from: https://www.cfr.org/backgrounder/food-price-volatility-and-insecurity.
[32]	Kolenikov S, Shorrocks A (2005) A decomposition analysis of regional poverty in Russia. Rev Dev Econ 9: 25–46. https://doi.org/10.1111/j.1467-9361.2005.00262.x doi: 10.1111/j.1467-9361.2005.00262.x
[33]	Kosareva N, Raymond JS, Tkachenko A (2000) Russia: dramatic shift to demand-side assistance, In: Stryuk, R., Homeownership and Housing Finance Policy in the Former Soviet Bloc: Costly Populism, Washington, DC: Urban Institute, 151–216.
[34]	Kozyreva P, Kosolapov M, Popkin BM (2015) Data resource profile: the Russian Longitudinal Monitoring Survey-Higher School of Economics phase II: monitoring the economic and health situation in Russia, 1994–2013. Int J Epidemiol 45: 395–401. https://doi.org/10.1093/ije/dyv357 doi: 10.1093/ije/dyv357
[35]	Leonard CS, Nazarov Z, Vakulenko ES (2016) The impact of sub‐national institutions: recentralization and regional growth in the Russian Federation (2001–2008). Econ Transit I Change 24: 421–446. https://doi.org/10.1111/ecot.12095 doi: 10.1111/ecot.12095
[36]	Lisina A, Kerm PV (2019) Understanding inequality and poverty trends in Russia. Available from: http://ecineq.org/ecineq_paris19/papers_EcineqPSE/paper_322.pdf.
[37]	Lukiyanova A (2015) Earnings inequality and informal employment in Russia. Econ Transit 23: 469–516. https://doi.org/10.1111/ecot.12069 doi: 10.1111/ecot.12069
[38]	Lukiyanova A, Vishnevskaya N (2016) Decentralization of Minimum Wage Setting in Russia: Causes and Consequences. Econ Labour Relat Rev 27: 98–117. https://doi.org/10.1177/1035304616629616 doi: 10.1177/1035304616629616
[39]	Luxembourg Income Study (LIS) (2019) Russia income reference years 2000–2016. Available from: http://www.lisdatacenter.org.
[40]	Machado JAF, Mata J (2005) Counterfactual decomposition of changes in wage distributions using quantile regression. J Appl Econ 20: 445–465. https://doi.org/10.1002/jae.788
[41]	Mahler C (2011) Diverging fortunes: recent developments in income inequality across Russian regions.
[42]	Malkina M (2017) Contribution of various income sources to interregional inequality of the per capita income in the Russian Federation. Equilibrium–Quart J Econ Econ Policy 12:3 99–416. https://doi.org/24136/eq.v12i3.21
[43]	Novokmet F, Piketty T, Zucman G (2018) From Soviets to oligarchs: inequality and property in Russia 1905–2016. J Econ Inequal 16:189–223. https://doi.org/10.1007/s10888-018-9383-0 doi: 10.1007/s10888-018-9383-0
[44]	Ovcharova LN, Popova DO, Rudberg AM (2016) Decomposition of income inequality in contemporary Russia. J New Econ Assoc 3: 170–186. https://doi.org/10.31737/2221-2264-2016-31-3-8 doi: 10.31737/2221-2264-2016-31-3-8
[45]	Popova D, Matytsin M, Sinnot E (2018) Distributional impact of taxes and social transfers in Russia over the downturn. J Eur Soc Policy 28: 535–548. https://doi.org/10.1177/0958928718767608 doi: 10.1177/0958928718767608
[46]	Ravallion M, Chen S (2003) Measuring pro-poor growth. Econ Lett 78: 93–99. https://doi.org/10.1016/S0165-1765(02)00205-7 doi: 10.1016/S0165-1765(02)00205-7
[47]	Remington TF (2011) The Politics of Inequality in Russia. Cambridge, UK: Cambridge University Press. https://doi.org/10.1017/CBO9780511973024
[48]	Wegren SK (2014) Rural Inequality in Post-Soviet Russia. Probl Post-Communism 61: 52–64. https://doi.org/10.2753/PPC1075-8216610104 doi: 10.2753/PPC1075-8216610104
[49]	World Bank (2015) Inflation: GDP Deflator (annual %), World Bank national accounts data, and OECD National Accounts data files. Available from: http://data.worldbank.org/indicator/NY.GDP.DEFL.KD.ZG.

NAR-05-02-007-s001.pdf

This article has been cited by:

1.	N.H. Sweilam, S.M. Al-Mekhlafi, W.S. Abdel Kareem, G. Alqurishi, A new crossover dynamics mathematical model of monkeypox disease based on fractional differential equations and the Ψ-Caputo derivative: Numerical treatments, 2025, 111, 11100168, 181, 10.1016/j.aej.2024.10.019
2.	Nasser H. Sweilam, Seham M. Al-Mekhlafi, Waleed S. Abdel Kareem, Ghader Alqurishi, Comparative Study of Crossover Mathematical Model of Breast Cancer Based on Ψ-Caputo Derivative and Mittag-Leffler Laws: Numerical Treatments, 2024, 16, 2073-8994, 1172, 10.3390/sym16091172
3.	N.H. Sweilam, Waleed Abdel Kareem, S.M. ALMekhlafi, Muner M. Abou Hasan, Taha H. El-Ghareeb, T.M. Soliman, New Crossover Lumpy Skin Disease: Numerical Treatments, 2024, 26668181, 100986, 10.1016/j.padiff.2024.100986
4.	Khaled Aldwoah, Hanen Louati, Nedal Eljaneid, Tariq Aljaaidi, Faez Alqarni, AbdelAziz Elsayed, Ahmad Al-Omari, Fractional and stochastic modeling of breast cancer progression with real data validation, 2025, 20, 1932-6203, e0313676, 10.1371/journal.pone.0313676

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

National Accounting Review

1.8 3.2

Metrics

Article views(1640) PDF downloads(73) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

National Accounting Review

Market and home production earnings gaps in Russia

Related Papers:

Abstract

1. Introduction

2. Proposed model

2.1. Preliminaries and notations

2.2. The Hybrid piecewise mathematical model

2.3. Solution's existence and uniqueness

2.4. Numerical method for crossover model

3. Numerical simulations

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflicts of Interest:

Appendix. Proofs of Lemmas 2.1 and 2.2

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Proposed model

2.1. Preliminaries and notations

2.2. The Hybrid piecewise mathematical model

2.3. Solution's existence and uniqueness

2.4. Numerical method for crossover model

3. Numerical simulations

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflicts of Interest:

Appendix. Proofs of Lemmas 2.1 and 2.2

References

National Accounting Review

Market and home production earnings gaps in Russia

Related Papers:

Abstract

1. Introduction

2. Proposed model

2.1. Preliminaries and notations

2.2. The Hybrid piecewise mathematical model

2.3. Solution's existence and uniqueness

2.4. Numerical method for crossover model

3. Numerical simulations

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflicts of Interest:

Appendix. Proofs of Lemmas 2.1 and 2.2

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Proposed model

2.1. Preliminaries and notations

2.2. The Hybrid piecewise mathematical model

2.3. Solution's existence and uniqueness

2.4. Numerical method for crossover model

3. Numerical simulations

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflicts of Interest:

Appendix. Proofs of Lemmas 2.1 and 2.2

References