How do green bonds affect green technology innovation? Firm evidence from China

Tao Lin; Mingyue Du; Siyu Ren; Tao Lin; Mingyue Du; Siyu Ren

doi:10.3934/GF.2022024

Green Finance

2022, Volume 4, Issue 4: 492-511. doi: 10.3934/GF.2022024

Previous Article Next Article

Research article

How do green bonds affect green technology innovation? Firm evidence from China

1.
Finance and Economics College, Jimei University, Xiamen 361021, China
2.
School of Economics, Beijing Technology and Business University, Beijing 100048, China
3.
School of Economics, Nankai University, Tianjin 300071, China
4.
Center for Transnationals' Studies of Nankai University, Tianjin 300071, China

Received: 04 October 2022 Revised: 10 November 2022 Accepted: 28 November 2022 Published: 07 December 2022
JEL Codes: G38, G12

As an emerging financial tool, green bonds can broaden the financing channels of enterprises and stimulate the green innovation of enterprises. Based on the A-share data of Chinese listed companies from 2012 to 2020, this paper analyzes the impact of green bonds on green technology innovation by using a method of Difference in Difference with Propensity Score Matching (PSM-DID). We found that green bonds can significantly improve enterprise green technology innovation. Its positive impact is attributed to increases in media attention and R&D capital investment and a reduction in financing constraints. Green bonds play a greater role in the green innovation of strong financial constraints enterprises, non-SOEs and large-scale enterprises. Our findings have important reference significance for the improvement of the resource allocation role of green bonds and achievement of sustainable growth.

Keywords:

Citation: Tao Lin, Mingyue Du, Siyu Ren. How do green bonds affect green technology innovation? Firm evidence from China[J]. Green Finance, 2022, 4(4): 492-511. doi: 10.3934/GF.2022024

Related Papers:

[1]	Norliyana Nor Hisham Shah, Rashid Jan, Hassan Ahmad, Normy Norfiza Abdul Razak, Imtiaz Ahmad, Hijaz Ahmad . Enhancing public health strategies for tungiasis: A mathematical approach with fractional derivative. AIMS Bioengineering, 2023, 10(4): 384-405. doi: 10.3934/bioeng.2023023
[2]	Rashid Jan, Imtiaz Ahmad, Hijaz Ahmad, Narcisa Vrinceanu, Adrian Gheorghe Hasegan . Insights into dengue transmission modeling: Index of memory, carriers, and vaccination dynamics explored via non-integer derivative. AIMS Bioengineering, 2024, 11(1): 44-65. doi: 10.3934/bioeng.2024004
[3]	Honar J. Hamad, Sarbaz H. A. Khoshnaw, Muhammad Shahzad . Model analysis for an HIV infectious disease using elasticity and sensitivity techniques. AIMS Bioengineering, 2024, 11(3): 281-300. doi: 10.3934/bioeng.2024015
[4]	Atta Ullah, Hamzah Sakidin, Shehza Gul, Kamal Shah, Yaman Hamed, Maggie Aphane, Thabet Abdeljawad . Sensitivity analysis-based control strategies of a mathematical model for reducing marijuana smoking. AIMS Bioengineering, 2023, 10(4): 491-510. doi: 10.3934/bioeng.2023028
[5]	Mehmet Yavuz, Waled Yavız Ahmed Haydar . A new mathematical modelling and parameter estimation of COVID-19: a case study in Iraq. AIMS Bioengineering, 2022, 9(4): 420-446. doi: 10.3934/bioeng.2022030
[6]	Fırat Evirgen, Fatma Özköse, Mehmet Yavuz, Necati Özdemir . Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 2023, 10(3): 218-239. doi: 10.3934/bioeng.2023015
[7]	Ayub Ahmed, Bashdar Salam, Mahmud Mohammad, Ali Akgül, Sarbaz H. A. Khoshnaw . Analysis coronavirus disease (COVID-19) model using numerical approaches and logistic model. AIMS Bioengineering, 2020, 7(3): 130-146. doi: 10.3934/bioeng.2020013
[8]	Mariem Jelassi, Kayode Oshinubi, Mustapha Rachdi, Jacques Demongeot . Epidemic dynamics on social interaction networks. AIMS Bioengineering, 2022, 9(4): 348-361. doi: 10.3934/bioeng.2022025
[9]	Shital Hajare, Rajendra Rewatkar, K.T.V. Reddy . Design of an iterative method for enhanced early prediction of acute coronary syndrome using XAI analysis. AIMS Bioengineering, 2024, 11(3): 301-322. doi: 10.3934/bioeng.2024016
[10]	Adam B Fisher, Stephen S Fong . Lignin biodegradation and industrial implications. AIMS Bioengineering, 2014, 1(2): 92-112. doi: 10.3934/bioeng.2014.2.92

Abstract

1. Introduction

In 2014, an outbreak of Ebola virus (Ebola) decimated many people in Western Africa. With more than 16,000 clinically confirmed cases and approximately 70% mortality cases, this was the more deadly outbreak compared to 20 Ebola threats that occurred since 1976 [1]. In Africa, and particularly in the regions that were affected by Ebola outbreaks, people live close to the rain-forests, hunt bats and monkeys and harvest forest fruits for food [2], [3].

In [4] develop a SIR type model which, incorporates both the direct and indirect transmissions in such a manner that there is a provision of Ebola viruses with stability and numerical analysis is discussed. A number of mathematical models have been developed to understand the transmission dynamics of Ebola and other infectious diseases outbreak from various aspects [5], [6]. A commonly used model for characterising epidemics of diseases including Ebola is the susceptible-exposed-infectious-recovered (SEIR) model [7], and extensions to this basic model include explicit incorporation of transmission from Ebola deceased hosts [1], [8] or accounting for mismatches between symptoms and infectiousness [9], [10].

Many researchers and mathematicians have shown that fractional extensions of mathematical integer-order models are a very systematic representation of natural reality [11], [12], [13]. Recently, a non-integer-order idea is given by Caputo and Fabrizio [14]. The primary goal of this article is to use a fresh non-integer order derivative to study the model of diabetes and to present information about the diabetes model solution's uniqueness and existence using a fixed point theorem [15]. Atangana and Baleanu [16] then proposed another non-singular derivative version using the Mittag Leffler kernel function. In many apps in the actual globe, these operators have been successful [17], [18], [19]. The few existing works [4], [8], [9], [20] on the mathematical modeling tells transmission of the virus and spread of Ebola virus on the population of human. The classical settings of mathematical studies tells about spread of EVD, such as SI model, SIR model, SEIR model [4], SEIRD model, or SEIRHD model. World medical association invented medicines for Ebola virus. Quantitative approaches and obtaining an analysis of the reproduction number of Ebola outbreak were important modeling for EVD epidemics. Demographic data on Ebola risk factors and on the transmission of virus were studied through the household structured epidemic model [4], [21]. Predications, different valuable insights, personal and genomic data for EVD was reported and discovered through mathematical models [22], [23]. In [24], the authors observed spread that follows a fading memory process and also shows crossover behaviour for the EVD. They captured this kind of spread using differential operators that posses crossover properties and fading memory using the SIRDP model in [4]. They also analyzed the Ebola disease dynamic by considering the Caputo, Caputo-Fabrizio, and Atangana-Baleanu differential operators.

In this paper, we developed fractional order Ebola virus model by using the Caputo method of complex nonlinear differential equations. Caputo fractional derivative operator β ∈ (0,1] works to achieve the fractional differential equations. Laplace with Adomian Decomposition Methodsuccessfully solved the fractional differential equations. Ultimately, numerical simulations are also developed to evaluate the effects of the device parameter on spread of disease and effect of fractional parameter β on obtained solution which are also assessed by tabulated results.

2. Materials and method

2.1. Fractional order Ebola virus model

The classical model for Ebola virus model is given in [4], we developed the fractional order Ebola virus model in the followings equations

$\begin{array}{l} D^{φ_{1}} S (t) = Π - (β_{1} I + β_{2} D + λ P) S - µ S \\ D^{φ_{2}} I (t) = (β_{1} I + β_{2} D + λ P) S - (µ + δ + γ) I \\ D^{φ_{3}} R (t) = γ I - µ R \\ D^{φ_{4}} D (t) = (µ + δ) I - b D \\ D^{φ_{5}} P (t) = σ + ξ I + α D - η P \end{array}$ (2.1)

with initial conditions

$S (0) = N_{1}, I (0) = N_{2}, R (0) = N_{3}, D (0) = N_{4}, P (0) = N_{5}$ (2.2)

Where S(t) represent the susceptible individuals, I(t) the individuals infected, R(t) the individuals recovered from the EVD, D(t) the individual that died with the Ebola virus and P(t) in the virus concentration in the environment. The susceptible human population is replenished by a constant recruitment at rate Pi. susceptible individuals S may acquire infection after effective contacts β₁ with infectious and β₂ is effective contact rate of deceased human individuals. They can also catch the infection through contact with a contaminated environment at rate λ. Infectious individuals I experience an additional death due to the disease at rate δ and they are recovered at rate γ. Deceased human individuals can be buried directly during funerals at rate b. Susceptible, infectious and recovered individuals die naturally at rate µ. η, ξ, α, represent the decay rate, shedding rate of infected, and shedding rate of deceased, respectively. The recruitment rate of the Ebola virus in the environment expressed as σ.

2.2. Equilibrium point and stability

Here system (2.1) is analyzed qualitatively analyzed for feasibility and numerical solution at disease free and endemic equilibrium point. For this purpose, we used

$D^{φ_{1}} S (t) = D^{φ_{2}} I (t) = D^{φ_{3}} R (t) = D^{φ_{4}} D (t) = D^{φ_{5}} P (t) = 0$ (2.3)

in system (1). For disease free equilibrium, we have E = (π/µ,0,0,0,0) and endemic equilibrium is

$E^{*} = (S^{*}, I^{*}, R^{*}, D^{*}, P^{*}),$

where

$S^{*} = \frac{π}{µ R_{0}}; I^{*} = \frac{π (R_{0} - 1)}{(µ + δ + γ) R_{0}}; R^{*} = \frac{π γ (R_{0} - 1)}{µ (µ + δ + γ) R_{0}}; D^{*} = \frac{π (µ + δ) (R_{0} - 1)}{b (µ + δ + γ) R_{0}}$

$P^{*} = \frac{π (b ξ + α δ + α µ) (R_{0} - 1)}{b η (µ + δ + γ) R_{0}}$

is endemic equilibria of the system (1). Where reproductive number is

$R_{0} = \frac{η π (b β_{1} + β_{2} (µ + δ)) + λ π (b ξ + α δ + α µ)}{b η µ (µ + δ + γ)}$

Theorem. 1 There is a unique solution for the initial value problem given in system (2.1), and the solution remains in R⁵, x ≥ 0.

Proof: We need to show that the domain R⁵, x ≥ 0 is positively invariant. Since

$\begin{array}{l} D^{φ_{1}} S (t) |_{S = 0} = Π \geq 0 \\ D^{φ_{2}} I (t) |_{I = 0} = (β_{1} I + β_{2} D + λ P) S \geq 0 \\ D^{φ_{3}} R (t) |_{R = 0} = γ I \geq 0 \\ D^{φ_{4}} D (t) |_{D = 0} = (µ + δ) I \geq 0 \\ D^{φ_{5}} P (t) |_{P = 0} = σ + ξ I + α D \geq 0 \end{array}$

Hence the solution lies in feasible domain, so the uniqueness and solution of the system exists.

2.3. Numerical analysis of the model

Consider the fractional-order Ebola virus model (2.1), by using Caputo definition with Laplace transform, we have

$\begin{array}{l} ℒ {D^{φ_{1}} S (t)} = Π ℒ {1} - β_{1} ℒ {I S} - β_{2} ℒ {D S} - λ ℒ {P S} - µ ℒ {S} \\ ℒ {D^{φ_{2}} I (t)} = β_{1} ℒ {I S} + β_{2} ℒ {D S} + λ ℒ {P S} - (µ + δ + γ) ℒ {I} \\ ℒ {D^{φ_{3}} R (t)} = γ ℒ {I} - µ ℒ {R} \\ ℒ {D^{φ_{4}} D (t)} = (µ + δ) ℒ {I} - b ℒ {D} \\ ℒ {D^{φ_{5}} P (t)} = σ ℒ {1} + ξ ℒ {I} + α ℒ {D} - η ℒ {P} \end{array}$ (2.4)

$\begin{array}{l} S^{φ_{1}} ℒ {S (t)} - S^{φ_{1} - 1} S (0) = Π ℒ {1} - β_{1} ℒ {I S} - β_{2} ℒ {D S} - λ ℒ {P S} - µ ℒ {S} \\ S^{φ_{2}} ℒ {I (t)} - S^{φ_{2} - 1} I (0) = β_{1} ℒ {I S} + β_{2} ℒ {D S} + λ ℒ {P S} - (µ + δ + γ) ℒ {I} \\ S^{φ_{3}} ℒ {R (t)} - S^{φ_{3} - 1} R (0) = γ ℒ {I} - µ ℒ {R} \\ S^{φ_{4}} ℒ {D (t)} - S^{φ_{4} - 1} D (0) = {µ + δ} ℒ {I} - b ℒ {D} \\ S^{φ_{5}} ℒ {P (t)} - S^{φ_{5} - 1} P (0) = σ ℒ {1} + ξ ℒ {I} + α ℒ {D} - η ℒ {P} \end{array}$ (2.5)

by using the initial conditions (2.2), we get

$\begin{matrix} ℒ {S (t)} = \frac{N_{1}}{S} + \frac{Π}{S^{φ_{1} + 1}} - \frac{β_{1}}{S^{φ_{1}}} ℒ {I S} - \frac{β_{2}}{S^{φ_{1}}} ℒ {D S} - \frac{λ}{S^{φ_{1}}} ℒ {P S} - \frac{µ}{S^{φ_{1}}} ℒ {S} \\ ℒ {I (t)} = \frac{N_{2}}{S} + \frac{β_{1}}{S^{φ_{2}}} ℒ {I S} + \frac{β_{2}}{S^{φ_{2}}} ℒ {D S} + \frac{λ}{S^{φ_{2}}} ℒ {P S} - \frac{µ + δ + γ}{S^{φ_{2}}} ℒ {I} \\ ℒ {R (t)} = \frac{N_{3}}{S} + \frac{γ}{S^{φ_{3}}} ℒ {I} - \frac{µ}{S^{φ_{3}}} ℒ {R} \\ ℒ {D (t)} = \frac{N_{4}}{S} + \frac{µ + δ}{S^{φ_{4}}} ℒ {I} - \frac{b}{S^{φ_{4}}} ℒ {D} \\ ℒ {P (t)} = \frac{N_{5}}{S} + \frac{σ}{S^{φ_{5} + 1}} + \frac{ξ}{S^{φ_{5}}} ℒ {I} + \frac{α}{S^{φ_{5}}} ℒ {D} - \frac{η}{S^{φ_{5}}} ℒ {P} \end{matrix}$ (2.6)

We have followings infinite series solution

$S = \sum_{k = 0}^{\infty} S_{k}, I = \sum_{k = 0}^{\infty} I_{k}, R = \sum_{k = 0}^{\infty} R_{k}, D = \sum_{k = 0}^{\infty} D_{k}, P = \sum_{k = 0}^{\infty} P_{k}$ (2.7)

The nonlinearity IS, DS and PS can be written as

$I S = \sum_{k = 0}^{\infty} A_{k}, D S = \sum_{k = 0}^{\infty} B_{k}, P S = \sum_{k = 0}^{\infty} C_{k}$ (2.8)

where A_k, B_k and C_k is called the Adomian polynomials. We have the followings results

$ℒ {S_{0}} = \frac{N_{1}}{S} + \frac{Π}{S^{φ_{1} + 1}}, ℒ {I_{0}} = \frac{N_{2}}{S}, ℒ {R_{0}} = \frac{N_{3}}{S}, ℒ {D_{0}} = \frac{N_{4}}{S}, ℒ {P_{0}} = \frac{N_{5}}{S} + \frac{σ}{S^{φ_{5} + 1}}$ (2.9)

Similarly, we have

$\begin{array}{l} ℒ {S_{1}} = - \frac{β_{1}}{S^{φ_{1}}} ℒ {A_{0}} - \frac{β_{2}}{S^{φ_{1}}} ℒ {B_{0}} - \frac{λ}{S^{φ_{1}}} ℒ {C_{0}} - \frac{µ}{S^{φ_{1}}} ℒ {S_{0}},... \\ ℒ {S_{k + 1}} = - \frac{β_{1}}{S^{φ_{1}}} ℒ {A_{k}} - \frac{β_{2}}{S^{φ_{1}}} ℒ {B_{k}} - \frac{λ}{S^{φ_{1}}} ℒ {C_{k}} - \frac{µ}{S^{φ_{1}}} ℒ {S_{k}} \end{array}$ (2.10)

$\begin{array}{l} ℒ {I_{1}} = \frac{β_{1}}{S^{φ_{2}}} ℒ {A_{0}} + \frac{β_{2}}{S^{φ_{2}}} ℒ {B_{0}} + \frac{λ}{S^{φ_{2}}} ℒ {C_{0}} - \frac{µ + δ + γ}{S^{φ_{2}}} ℒ {I_{0}},... \\ ℒ {I_{k + 1}} = \frac{β_{1}}{S^{φ_{2}}} ℒ {A_{k}} + \frac{β_{2}}{S^{φ_{2}}} ℒ {B_{k}} + \frac{λ}{S^{φ_{2}}} ℒ {C_{k}} - \frac{µ + δ + γ}{S^{φ_{2}}} ℒ {I_{k}} \end{array}$ (2.11)

$\begin{array}{l} ℒ {R_{1}} = \frac{γ}{S^{φ_{3}}} ℒ {I_{0}} - \frac{µ}{S^{φ_{3}}} ℒ {R_{0}},... \\ ℒ {R_{k + 1}} = \frac{γ}{S^{φ_{3}}} ℒ {I_{k}} - \frac{µ}{S^{φ_{3}}} ℒ {R_{k}} \end{array}$ (2.12)

$\begin{array}{l} ℒ {D_{1}} = \frac{µ + δ}{S^{φ_{4}}} ℒ {I_{0}} - \frac{b}{S^{φ_{4}}} ℒ {D_{0}},... \\ ℒ {D_{k + 1}} = \frac{µ + δ}{S^{φ_{4}}} ℒ {I_{k}} - \frac{b}{S^{φ_{4}}} ℒ {D_{k}} \end{array}$ (2.13)

$\begin{array}{l} ℒ {P_{1}} = \frac{ξ}{S^{φ_{5}}} ℒ {I_{0}} + \frac{α}{S^{φ_{5}}} ℒ {D_{0}} - \frac{η}{S^{φ_{5}}} ℒ {P_{0}},... \\ ℒ {P_{k + 1}} = \frac{ξ}{S^{φ_{5}}} ℒ {I_{k}} + \frac{α}{S^{φ_{5}}} ℒ {D_{k}} - \frac{η}{S^{φ_{5}}} ℒ {P_{k}} \end{array}$ (2.14)

We get the followings generalized form for analysis and numerical solution.

$ℒ {S_{k + 1}} = - \frac{β_{1}}{S^{φ_{1}}} ℒ {A_{k}} - \frac{β_{2}}{S^{φ_{1}}} ℒ {B_{k}} - \frac{λ}{S^{φ_{1}}} ℒ {C_{k}} - \frac{µ}{S^{φ_{1}}} ℒ {S_{k}}$ (2.15)

$ℒ {I_{k + 1}} = \frac{β_{1}}{S^{φ_{2}}} ℒ {A_{k}} + \frac{β_{2}}{S^{φ_{2}}} ℒ {B_{k}} + \frac{λ}{S^{φ_{2}}} ℒ {C_{k}} - \frac{µ + δ + γ}{S^{φ_{2}}} ℒ {I_{k}}$ (2.16)

$ℒ {R_{k + 1}} = \frac{γ}{S^{φ_{3}}} ℒ {I_{k}} - \frac{µ}{S^{φ_{3}}} ℒ {R_{k}}$ (2.17)

$ℒ {D_{k + 1}} = \frac{µ + δ}{S^{φ_{4}}} ℒ {I_{k}} - \frac{b}{S^{φ_{4}}} ℒ {D_{k}}$ (2.18)

$ℒ {P_{k + 1}} = \frac{ξ}{S^{φ_{5}}} ℒ {I_{k}} + \frac{α}{S^{φ_{5}}} ℒ {D_{k}} - \frac{η}{S^{φ_{5}}} ℒ {P_{k}}$ (2.19)

3. Results

The results of fractional order model (2.1) is represented in followings tables and graphs.

Table 1. Numerical solution of S(t) with at different fractional values ϕ.

t	ϕ = 1	ϕ = 0.9	ϕ = 0.8	ϕ = 0.5
1	39.6912	39.5848	39.51	39.4383
1.5	39.3299	39.1117	39.0235	39.1181
3	37.2519	36.2268	36.264	37.5584
4.5	32.0652	29.2568	30.0105	34.514
6	19.0984	14.0068	17.2162	29.3074

| Show Table

DownLoad: CSV

Table 2. Numerical solution of I(t) with at different fractional values ϕ.

t	ϕ = 1	ϕ = 0.9	ϕ = 0.8	ϕ = 0.7
1	10.4879	10.6026	10.6542	10.7698
2	12.052	11.9378	11.8579	11.5689
4	13.2768	12.5387	12.2499	11.8581
6	8.7256	9.97166	10.5704	12.7973
8	5.0464	11.4457	13.7013	19.4353

| Show Table

DownLoad: CSV

Table 3. Numerical solution of R(t) with at different fractional values ϕ.

t	ϕ = 1	ϕ = 0.9	ϕ = 0.8	ϕ = 0.7
2	20.504	20.5195	20.5331	20.5414
4	21.952	21.8288	21.6815	21.509
6	25.448	24.6081	23.8154	23.076
8	32.096	29.4003	27.1489	25.2921
10	43	36.6889	31.8489	28.1877

| Show Table

DownLoad: CSV

Table 4. Numerical solution of D(t) with at different fractional values ϕ.

t	ϕ = 1	ϕ = 0.9	ϕ = 0.8	ϕ = 0.7
0.5	10.5931	10.6768	10.7803	10.9131
1	11.3486	11.5116	11.7127	11.9551
1.5	12.531	12.8083	13.1141	13.4308
2	14.4048	14.7773	15.124	15.4064

| Show Table

DownLoad: CSV

Table 5. Numerical solution of P(t) with at different fractional values ϕ.

t	ϕ = 1	ϕ = 0.95	ϕ = 0.9	ϕ = 0.85
1	5.67835	5.6959	5.68235	5.7302
2	6.4746	6.46707	6.45834	6.44629
4	8.788	8.62982	8.47227	8.30553
6	12.6746	12.1038	11.562	11.0225
8	18.8688	17.417	16.0958	14.8435
10	28.105	25.0658	22.3972	19.9683

| Show Table

DownLoad: CSV

Figure 1. Simulation of S(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 2. Simulation of I(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 3. Simulation of R(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 4. Simulation of D(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 5. Simulation of P(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 6. Simulation of S(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 7. Simulation of I(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 8. Simulation of R(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 9. Simulation of D(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

Figure 10. Simulation of P(t) at different fractional values in time t.

DownLoad: Full-Size Img PowerPoint

4. Discussion

The objective of our work is to develop a scheme of epidemic fractional Ebola virus model with Caputo fractional derivative also numerical solutions have been obtained by using the Laplace with the Adomian Decomposition Method. The results of fractional order Ebola virus model is presented and convergence results of fractional-order model are also presented to demonstrate the efficacy of the process. The analytical solution of the fractional-order Ebola virus model consisting of the non-linear system of the fractional differential equation has been presented by using the Caputo derivative. To observe the effects of the fractional parameter on the dynamics of the fractional-order model (2.1), we conclude several numerical simulations varying the values of parameter given in [4]. These simulations reveal that a change in the value affects the dynamics of the model. The numerical solutions at classical as well as different fractional values by using Caputo fractional derivative can be seen in Figures 1–5 for disease free equilibrium. The rate of susceptible individuals and pathogens decreases by reducing the fractional values to acquire the desired value, whereas the other compartment starts decreasing by increasing the fractional values. The fractional-order model shows the convergence with theoretical contribution and numerical results. The fractional-order parameter values show the impact of increasing or decreasing the disease. Also, we can fix the parameter values where the rate of infection is decrease and the recover rate will increase for some values which are representing in figures and tables. These results can be used for disease outbreak treatment and analysis without defining the control parameters in the model based on fractional values. In general, approaches to fractional-order modeling in situations with large refined data sets and good numerical algorithms may be worth it. The simulation and numerical solutions at classical as well as different fractional values by using Caputo fractional derivative can be seen in Figures 6–10 for endemic equilibrium as well as in Tables 1–5. Results in both cases are reliable at fractional values to overcome the outbreak of this epidemic and meet our desired accuracy. Results discuss in [1], [5] for classical model, but our results are on fractional order model, fractional parameters easily use to adjust the control strategy without defining others parameters in the model. Another important feature that plays a critical role in the 2014 EVD outbreaks is traditional/cultural belief systems and customs. For instance, while some individuals in the three Ebola-stricken nations believe that there is no Ebola, control the population or harvest human organs. We conclude that depending on the specific data set, the fractional order model either converges to the ordinary differential equation model and fits data similarly, or fits the data better and outperforms the ODE model.

5. Conclusion

We develop a scheme of epidemic fractional Ebola virus model with Caputo fractional derivative for numerical solutions that have been obtained by using the Laplace with the Adomian Decomposition Method. In [24] the use of three different fractional operators on the Ebola disease model suggests that the fractional-order parameter greatly affects disease elimination for the non-integer case when decreasing α. We constructed a numerical solution for the Ebola virus model to show a good agreement to control the bad impact of the Ebola virus for the different period for diseases free and endemic equilibrium point as well. However, in this work, we introduced the qualitative properties for solutions as well as the non-negative unique solution for a fractional-order nonlinear system. It is important to note that the Laplace Adomian Decomposition Method is used for the Ebola virus fractional-order model differential equation framework is a more efficient approach to computing convergent solutions that are represented through figures and tables for endemic and disease-free equilibrium point. Convergence results of the fractional-order model are also presented to demonstrate the efficacy of the process. The techniques developed to provide good results which are useful for understanding the Zika Virus outbreak in our community. It is worthy to observe that fractional derivative shows significant changes and memory effects as compare to ordinary derivatives. This model will assist the public health planar in framing an Ebola virus disease control policy. Also, we will expand the model incorporating determinist and stochastic model comparisons with fractional technique, as well as using optimal control theory for new outcomes.

References

[1]	Barbieri N, Marzucchi A, Rizzo U (2020) Knowledge sources and impacts on subsequent inventions: Do green technologies differ from non-green ones? Res Policy 49: 103901. https://doi.org/10.1016/j.respol.2019.103901 doi: 10.1016/j.respol.2019.103901
[2]	Baulkaran V (2019) Stock market reaction to green bond issuance. J Asset Manage 20: 331–340. https://doi.org/10.1057/s41260-018-00105-1 doi: 10.1057/s41260-018-00105-1
[3]	Borsatto JMLS, Bazani CL (2021) Green innovation and environmental regulations: A systematic review of international academic works. Environ Sci Pollution Res 28: 63751–63768. https://doi.org/10.1007/s11356-020-11379-7 doi: 10.1007/s11356-020-11379-7
[4]	Broadstock DC, Cheng LT (2019) Time-varying relation between black and green bond price benchmarks: Macroeconomic determinants for the first decade. Financ Res Lette 29: 17–22. https://doi.org/10.1016/j.frl.2019.02.006 doi: 10.1016/j.frl.2019.02.006
[5]	Brown JR, Fazzari SM, Petersen BC (2009) Financing innovation and growth: Cash flow, external equity, and the 1990s R&D boom. J Finance 64: 151–185. https://doi.org/10.1111/j.1540-6261.2008.01431.x doi: 10.1111/j.1540-6261.2008.01431.x
[6]	Dangelico RM, Pujari D (2010) Mainstreaming green product innovation: Why and how companies integrate environmental sustainability. J Bus Ethics 95: 471–486. https://doi.org/10.1007/s10551-010-0434-0 doi: 10.1007/s10551-010-0434-0
[7]	Du K, Cheng Y, Yao X (2021) Environmental regulation, green technology innovation, and industrial structure upgrading: The road to the green transformation of Chinese cities. Energy Econ 98: 105247. https://doi.org/10.1016/j.eneco.2021.105247 doi: 10.1016/j.eneco.2021.105247
[8]	El Ghoul S, Guedhami O, Kim H, et al. (2018) Corporate environmental responsibility and the cost of capital: International evidence. J Bus Ethics 149: 335–361. https://doi.org/10.1007/s10551-015-3005-6 doi: 10.1007/s10551-015-3005-6
[9]	Flammer C (2021) Corporate green bonds. J Financ Econ 142: 499–516. https://doi.org/10.1016/j.jfineco.2021.01.010 doi: 10.1016/j.jfineco.2021.01.010
[10]	Ghisetti C, Quatraro F (2017) Green technologies and environmental productivity: A cross-sectoral analysis of direct and indirect effects in Italian regions. Ecol Econ 132: 1–13. https://doi.org/10.1016/j.ecolecon.2016.10.003 doi: 10.1016/j.ecolecon.2016.10.003
[11]	Hachenberg B, Schiereck D (2018). Are green bonds priced differently from conventional bonds? J Asset Manag 19: 371–383. https://doi.org/10.1057/s41260-018-0088-5 doi: 10.1057/s41260-018-0088-5
[12]	Hao Y, Ba N, Ren S, et al. (2021) How does international technology spillover affect China's carbon emissions? A new perspective through intellectual property protection. Sustain Prod Consump 25: 577–590. https://doi.org/10.1016/j.spc.2020.12.008 doi: 10.1016/j.spc.2020.12.008
[13]	Hao Y, Huang J, Guo Y, et al. (2022) Does the legacy of state planning put pressure on ecological efficiency? Evidence from China. Bus Strateg Environ 5: 1–22. https://doi.org/10.1002/bse.3066 doi: 10.1002/bse.3066
[14]	Hu AG, Jefferson GH (2009) A great wall of patents: What is behind China's recent patent explosion?. J Dev Econ 90: 57–68. https://doi.org/10.1016/j.jdeveco.2008.11.004 doi: 10.1016/j.jdeveco.2008.11.004
[15]	Huang H, Mbanyele W, Wang F, et al. (2022) Climbing the quality ladder of green innovation: Does green finance matter? Technol Forecast Soc 184: 122007. https://doi.org/10.1016/j.techfore.2022.122007 doi: 10.1016/j.techfore.2022.122007
[16]	Huang Z, Liao G, Li Z (2019) Loaning scale and government subsidy for promoting green innovation. Technol Forecast Soc 144: 148–156. https://doi.org/10.1016/j.techfore.2019.04.023 doi: 10.1016/j.techfore.2019.04.023
[17]	Hyun S, Park D, Tian S (2020) The price of going green: the role of greenness in green bond markets. Account Financ 60: 73–95. https://doi.org/10.1111/acfi.12515 doi: 10.1111/acfi.12515
[18]	Jiang Z, Wang Z, Lan X (2021) How environmental regulations affect corporate innovation? The coupling mechanism of mandatory rules and voluntary management. Technol Soc 65: 101575. https://doi.org/10.1016/j.techsoc.2021.101575 doi: 10.1016/j.techsoc.2021.101575
[19]	Keohane NO, Olmstead SM (2016) Economic Efficiency and Environmental Protection. In Markets and the Environment. 11–34. Island Press, Washington, DC. https://doi.org/10.5822/978-1-61091-608-0_2
[20]	Larcker DF, Watts EM (2020) Where's the greenium? J Account Econ 69: 101312. https://doi.org/10.1016/j.jacceco.2020.101312 doi: 10.1016/j.jacceco.2020.101312
[21]	Li F, Xu X, Li Z, et al. (2021) Can low-carbon technological innovation truly improve enterprise performance? The case of Chinese manufacturing companies. J Clean Prod 293: 125949. https://doi.org/10.1016/j.jclepro.2021.125949 doi: 10.1016/j.jclepro.2021.125949
[22]	Li Z, Liao G, Albitar K (2020) Does corporate environmental responsibility engagement affect firm value? The mediating role of corporate innovation. Bus Strateg Environ 29: 1045–1055. https://doi.org/10.1002/bse.2416 doi: 10.1002/bse.2416
[23]	Lin B, Luan R (2020) Do government subsidies promote efficiency in technological innovation of China's photovoltaic enterprises? J Clean Prod 254: 120108. https://doi.org/10.1016/j.jclepro.2020.120108 doi: 10.1016/j.jclepro.2020.120108
[24]	Liu J, Zhao M, Wang Y (2020) Impacts of government subsidies and environmental regulations on green process innovation: A nonlinear approach. Technol Soc 63: 101417. https://doi.org/10.1016/j.techsoc.2020.101417 doi: 10.1016/j.techsoc.2020.101417
[25]	Liu P, Zhao Y, Zhu J, et al. (2022) Technological industry agglomeration, green innovation efficiency, and development quality of city cluster. Green Financ 4: 411–435. https://doi.org/10.3934/gf.2022020 doi: 10.3934/gf.2022020
[26]	Lv C, Shao C, Lee CC (2021) Green technology innovation and financial development: Do environmental regulation and innovation output matter? Energy Econ 98: 105237. https://doi.org/10.1016/j.eneco.2021.105237 doi: 10.1016/j.eneco.2021.105237
[27]	Managi S, Opaluch JJ, Jin D, et al. (2005) Environmental regulations and technological change in the offshore oil and gas industry. Land Econ 81: 303–319. https://doi.org/10.3368/le.81.2.303 doi: 10.3368/le.81.2.303
[28]	Mbanyele W, Huang H, Li Y, et al. (2022) Corporate social responsibility and green innovation: Evidence from mandatory CSR disclosure laws. Econ Lett 212: 110322. https://doi.org/10.1016/j.econlet.2022.110322 doi: 10.1016/j.econlet.2022.110322
[29]	Miao CL, Meng XN, Duan MM, et al. (2020) Energy consumption, environmental pollution, and technological innovation efficiency: taking industrial enterprises in China as empirical analysis object. Environ Sci Pollut Res 27: 34147–34157. https://doi.org/10.1007/s11356-020-09537-y doi: 10.1007/s11356-020-09537-y
[30]	Mughal N, Arif A, Jain V, et al. (2022) The role of technological innovation in environmental pollution, energy consumption and sustainable economic growth: Evidence from South Asian economies. Energy Strateg Rev 39: 100745. https://doi.org/10.1016/j.esr.2021.100745 doi: 10.1016/j.esr.2021.100745
[31]	Rahman S, Moral IH, Hassan M, et al. (2022) A systematic review of green finance in the banking industry: perspectives from a developing country. Green Financ 4: 347–363. https://doi.org/10.3934/gf.2022017 doi: 10.3934/gf.2022017
[32]	Reboredo JC (2018) Green bond and financial markets: Co-movement, diversification and price spillover effects. Energy Econ 74: 38–50. https://doi.org/10.1016/j.eneco.2018.05.030 doi: 10.1016/j.eneco.2018.05.030
[33]	Ren S, Hao Y, Wu H (2021) Government corruption, market segmentation and renewable energy technology innovation: Evidence from China. J Environ Manage 300: 113686. https://doi.org/10.1016/j.jenvman.2021.113686 doi: 10.1016/j.jenvman.2021.113686
[34]	Ren S, Hao Y, Wu H (2022a) Digitalization and environment governance: does internet development reduce environmental pollution? J Environ Plann Manage 3: 1–30. https://doi.org/10.1080/09640568.2022.2033959 doi: 10.1080/09640568.2022.2033959
[35]	Ren S, Liu Z, Zhanbayev R, et al. (2022b) Does the internet development put pressure on energy-saving potential for environmental sustainability? Evidence from China. J Econ Anal 1: 81–101. https://doi.org/10.12410/jea.2811-0943.2022.01.004 doi: 10.12410/jea.2811-0943.2022.01.004
[36]	Sartzetakis ES (2021) Green bonds as an instrument to finance low carbon transition. Econ Chang Restruct 54: 755–779. https://doi.org/10.1007/s10644-020-09266-9 doi: 10.1007/s10644-020-09266-9
[37]	Singh MP, Chakraborty A, Roy M (2016) The link among innovation drivers, green innovation and business performance: empirical evidence from a developing economy. World Review of Science, Technol Sustainable Dev 12: 316–334. https://doi.org/10.1504/wrstsd.2016.10003088 doi: 10.1504/wrstsd.2016.10003088
[38]	Wang F, Wang R, He Z (2021a) The impact of environmental pollution and green finance on the high-quality development of energy based on spatial Dubin model. Resour Policy 74: 102451. https://doi.org/10.1016/j.resourpol.2021.102451 doi: 10.1016/j.resourpol.2021.102451
[39]	Wang J, Chen X, Li X, et al. (2020) The market reaction to green bond issuance: Evidence from China. Pacific-Basin Financ J 60: 101294. https://doi.org/10.1016/j.pacfin.2020.101294 doi: 10.1016/j.pacfin.2020.101294
[40]	Wang P, Dong C, Chen N, et al. (2021b) Environmental Regulation, Government Subsidies, and Green Technology Innovation—A Provincial Panel Data Analysis from China. Int J Environ Res Public Health 18: 11991. https://doi.org/10.3390/ijerph182211991 doi: 10.3390/ijerph182211991
[41]	Wu H, Hao Y, Ren S, et al. (2021a) Does internet development improve green total factor energy efficiency? Evidence from China. Energy Policy 153: 112247. https://doi.org/10.1016/j.enpol.2021.112247 doi: 10.1016/j.enpol.2021.112247
[42]	Wu H, Xue Y, Hao Y, et al. (2021b) How does internet development affect energy-saving and emission reduction? Evidence from China. Energy Econ 103: 105577. https://doi.org/10.1016/j.eneco.2021.105577 doi: 10.1016/j.eneco.2021.105577
[43]	Xie X, Huo J, Zou H (2019) Green process innovation, green product innovation, and corporate financial performance: A content analysis method. J Bus Res 101: 697–706. https://doi.org/10.1016/j.jbusres.2019.01.010 doi: 10.1016/j.jbusres.2019.01.010
[44]	Yang X, Wang W, Su X, et al. (2022) Analysis of the influence of land finance on haze pollution: An empirical study based on 269 prefecture‐level cities in China. Growth Chang 4: 1–22. https://doi.org/10.1016/j.strueco.2020.12.001 doi: 10.1016/j.strueco.2020.12.001
[45]	Yang X, Wu H, Ren S, et al. (2021) Does the development of the internet contribute to air pollution control in China? Mechanism discussion and empirical test. Struct Chang Econ Dyn 56: 207–224. https://doi.org/10.1016/j.strueco.2020.12.001 doi: 10.1016/j.strueco.2020.12.001
[46]	Yao Y, Hu D, Yang C, et al. (2021) The impact and mechanism of fintech on green total factor productivity. Green Financ 3: 198–221. https://doi.org/10.3934/gf.2021011 doi: 10.3934/gf.2021011
[47]	Yeow KE, Ng SH (2021) The impact of green bonds on corporate environmental and financial performance. Managerial Financ 1: 1–20. https://doi.org/10.1108/mf-09-2020-0481 doi: 10.1108/mf-09-2020-0481
[48]	Yii KJ, Geetha C (2017) The nexus between technology innovation and CO2 emissions in Malaysia: evidence from granger causality test. Energy Procedia 105: 3118–3124. https://doi.org/10.1016/j.egypro.2017.03.654 doi: 10.1016/j.egypro.2017.03.654
[49]	Yin S, Zhang N, Li B (2020) Enhancing the competitiveness of multi-agent cooperation for green manufacturing in China: An empirical study of the measure of green technology innovation capabilities and their influencing factors. Sustain Prod Consump 23: 63–76. https://doi.org/10.1016/j.spc.2020.05.003 doi: 10.1016/j.spc.2020.05.003
[50]	Zerbib OD (2019) The effect of pro-environmental preferences on bond prices: Evidence from green bonds. J Bank Financ 98: 39–60. https://doi.org/10.1016/j.jbankfin.2018.10.012 doi: 10.1016/j.jbankfin.2018.10.012
[51]	Zhang D, Zhang Z, Managi S (2019) A bibliometric analysis on green finance: Current status, development, and future directions. Financ Res Lett 29: 425–430. https://doi.org/10.1016/j.frl.2019.02.003 doi: 10.1016/j.frl.2019.02.003
[52]	Zhang W, Li G (2020) Environmental decentralization, environmental protection investment, and green technology innovation. Environ Sci Pollut Res 10: 1–16. https://doi.org/10.1007/s11356-020-09849-z doi: 10.1007/s11356-020-09849-z
[53]	Zhao L, Zhang L, Sun J, et al. (2022) Can public participation constraints promote green technological innovation of Chinese enterprises? The moderating role of government environmental regulatory enforcement. Technol Forecast Soc Chang 174: 121198. https://doi.org/10.1016/j.techfore.2021.121198 doi: 10.1016/j.techfore.2021.121198
[54]	Zheng C, Deng F, Zhuo C, et al. (2022) Green Credit Policy, Institution Supply and Enterprise Green Innovation. J Econ Anal 1: 28–51. https://doi.org/10.12410/jea.2811-0943.2022.01.002 doi: 10.12410/jea.2811-0943.2022.01.002
[55]	Zhou Q, Du M, Ren S (2022) How government corruption and market segmentation affect green total factor energy efficiency in the post-COVID-19 era: Evidence from China. Front Energy Res 10: 1–16. https://doi.org/10.3389/fenrg.2022.878065 doi: 10.3389/fenrg.2022.878065

This article has been cited by:

1.	Aqeel Ahmad, Muhammad Farman, Ali Akgül, Nabila Bukhari, Sumaiyah Imtiaz, Mathematical analysis and numerical simulation of co-infection of TB-HIV, 2020, 27, 2576-5299, 431, 10.1080/25765299.2020.1840771
2.	Rana Muhammad Zulqarnain, Imran Siddique, Fahd Jarad, Rifaqat Ali, Thabet Abdeljawad, Ahmed Mostafa Khalil, Development of TOPSIS Technique under Pythagorean Fuzzy Hypersoft Environment Based on Correlation Coefficient and Its Application towards the Selection of Antivirus Mask in COVID-19 Pandemic, 2021, 2021, 1099-0526, 1, 10.1155/2021/6634991
3.	Waheed Ahmad, Mujahid Abbas, Effect of quarantine on transmission dynamics of Ebola virus epidemic: a mathematical analysis, 2021, 136, 2190-5444, 10.1140/epjp/s13360-021-01360-9
4.	Muhammad Farman, Aqeel Ahmad, Ali Akg黮, Muhammad Umer Saleem, Muhammad Naeem, Dumitru Baleanu, Epidemiological Analysis of the Coronavirus Disease Outbreak with Random Effects, 2021, 67, 1546-2226, 3215, 10.32604/cmc.2021.014006
5.	SHAHER MOMANI, R. P. CHAUHAN, SUNIL KUMAR, SAMIR HADID, A THEORETICAL STUDY ON FRACTIONAL EBOLA HEMORRHAGIC FEVER MODEL, 2022, 30, 0218-348X, 10.1142/S0218348X22400321
6.	Maryam Amin, Muhammad Farman, Ali Akgül, Mohammad Partohaghighi, Fahd Jarad, Computational analysis of COVID-19 model outbreak with singular and nonlocal operator, 2022, 7, 2473-6988, 16741, 10.3934/math.2022919
7.	Muhammad Farman, Ali Akg黮, Aqeel Ahmad, Dumitru Baleanu, Muhammad Umer Saleem, Dynamical Transmission of Coronavirus Model with Analysis and Simulation, 2021, 127, 1526-1506, 753, 10.32604/cmes.2021.014882
8.	Jie Liu, Peng Zhang, Hailian Gui, Tong Xing, Hao Liu, Chen Zhang, Resonance study of fractional-order strongly nonlinear duffing systems, 2024, 98, 0973-1458, 3317, 10.1007/s12648-024-03080-z
9.	Isaac K. Adu, Fredrick A. Wireko, Mojeeb Al-R. El-N. Osman, Joshua Kiddy K. Asamoah, A fractional order Ebola transmission model for dogs and humans, 2024, 24, 24682276, e02230, 10.1016/j.sciaf.2024.e02230
10.	Mohammed A. Almalahi, Khaled Aldowah, Faez Alqarni, Manel Hleili, Kamal Shah, Fathea M. O. Birkea, On modified Mittag–Leffler coupled hybrid fractional system constrained by Dhage hybrid fixed point in Banach algebra, 2024, 14, 2045-2322, 10.1038/s41598-024-81568-8
11.	Kamel Guedri, Rahat Zarin, Mowffaq Oreijah, Samaher Khalaf Alharbi, Hamiden Abd El-Wahed Khalifa, Artificial neural network-driven modeling of Ebola transmission dynamics with delays and disability outcomes, 2025, 115, 14769271, 108350, 10.1016/j.compbiolchem.2025.108350

Reader Comments

Your name:*

Email:*
© 2022 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)