Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads

Wen Shen; Wen Shen

doi:10.3934/nhm.2019028

Networks and Heterogeneous Media

2019, Volume 14, Issue 4: 709-732. doi: 10.3934/nhm.2019028

Previous Article Next Article

Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads

Wen Shen ^,

Mathematics Department, Pennsylvania State University, University Park, PA 16802, USA

Received: 01 September 2018 Revised: 01 March 2019
Primary: 65M20, 35L02, 35L65; Secondary: 34B99, 35Q99

We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model one considers an averaged downstream velocity. The road condition is piecewise constant with a jump at $ x = 0 $. We study stationary traveling wave profiles cross $ x = 0 $, for all possible cases. We show that, depending on the case, there could exit infinitely many profiles, a unique profile, or no profiles at all. Furthermore, some of the profiles are time asymptotic solutions for the Cauchy problem of the conservation laws under mild assumption on the initial data, while other profiles are unstable.

Keywords:

Citation: Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads[J]. Networks and Heterogeneous Media, 2019, 14(4): 709-732. doi: 10.3934/nhm.2019028

Related Papers:

[1]	Humaira Kalsoom, Muhammad Amer Latif, Muhammad Idrees, Muhammad Arif, Zabidin Salleh . Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions. AIMS Mathematics, 2021, 6(12): 13291-13310. doi: 10.3934/math.2021769
[2]	Xue-Xiao You, Muhammad Aamir Ali, Ghulam Murtaza, Saowaluck Chasreechai, Sotiris K. Ntouyas, Thanin Sitthiwirattham . Post-quantum Simpson's type inequalities for coordinated convex functions. AIMS Mathematics, 2022, 7(2): 3097-3132. doi: 10.3934/math.2022172
[3]	Chunhong Li, Dandan Yang, Chuanzhi Bai . Some Opial type inequalities in (p, q)-calculus. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377
[4]	Muhammad Uzair Awan, Sadia Talib, Artion Kashuri, Ibrahim Slimane, Kamsing Nonlaopon, Y. S. Hamed . Some new (p, q)-Dragomir–Agarwal and Iyengar type integral inequalities and their applications. AIMS Mathematics, 2022, 7(4): 5728-5751. doi: 10.3934/math.2022317
[5]	Maimoona Karim, Aliya Fahmi, Shahid Qaisar, Zafar Ullah, Ather Qayyum . New developments in fractional integral inequalities via convexity with applications. AIMS Mathematics, 2023, 8(7): 15950-15968. doi: 10.3934/math.2023814
[6]	Da Shi, Ghulam Farid, Abd Elmotaleb A. M. A. Elamin, Wajida Akram, Abdullah A. Alahmari, B. A. Younis . Generalizations of some $ q $-integral inequalities of Hölder, Ostrowski and Grüss type. AIMS Mathematics, 2023, 8(10): 23459-23471. doi: 10.3934/math.20231192
[7]	Kamsing Nonlaopon, Muhammad Uzair Awan, Sadia Talib, Hüseyin Budak . Parametric generalized $ (p, q) $-integral inequalities and applications. AIMS Mathematics, 2022, 7(7): 12437-12457. doi: 10.3934/math.2022690
[8]	Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu . Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831
[9]	Muhammad Tariq, Asif Ali Shaikh, Sotiris K. Ntouyas, Jessada Tariboon . Some novel refinements of Hermite-Hadamard and Pachpatte type integral inequalities involving a generalized preinvex function pertaining to Caputo-Fabrizio fractional integral operator. AIMS Mathematics, 2023, 8(11): 25572-25610. doi: 10.3934/math.20231306
[10]	Muhammad Umar, Saad Ihsan Butt, Youngsoo Seol . Milne and Hermite-Hadamard's type inequalities for strongly multiplicative convex function via multiplicative calculus. AIMS Mathematics, 2024, 9(12): 34090-34108. doi: 10.3934/math.20241625

Abstract

Colon cancer is one of the leading causes of mortality and morbidity worldwide [1], with a dramatic rise in incidence and mortality rates over the past 3 decades, both worldwide and in China [2]. Surgery is the principal therapy for early-stage colon cancer because it offers the only method for complete tumor removal, thereby chance for cure [3]. However, there is still proportions of colon cancer patients that develop recurrence even after radical resection [4]. Unfortunately, long term control of the recurrent colon cancer has been a difficult dilemma to tackle and the clinical outcomes of these patients are poor [5],[6]. Adjuvant chemotherapy in colon cancer is able to complement curative surgery to reduce the risk of recurrence and death from relapsed or metastatic disease [7],[8]. However, patient willingness to be adjuvantly treated has been limited due to its associated financial burden, treatment toxicity and debatable prolongation of survival that need to be carefully balanced against the associated-toxicity. Therefore, the identification of robust biomarkers to assess the risk of postoperative recurrence in colon cancer has been a longing need.

Circular RNAs (circRNAs), a recently discovered type of noncoding RNA, have a special circular structure formed by 3′- and 5′-ends linking covalently [9]. Because they do not have 5′ or 3′ ends, they are resistant to exonuclease-mediated degradation [10]. Therefore, circRNAs are characterized by their high stability, expression level, and evolutionary conservation. Recent literatures [11]–[13] have confirmed circRNAs as playing an important role in cancer initiation and progression. Additionally, they are differentially expressed in cancer tissues and circulation of cancer patients. Due to these characteristics, circRNAs have gained recognition as a promising novel biomarker for cancer [14]. However, till now little is known about circRNAs and their relationship with colon cancer. In a recent study published in EMBO Molecular Medicine, entitled “A circRNA signature predicts postoperative recurrence in stage II/III colon cancer”, Ju et al. [15] identified and validated a circRNA-based signature that could improve postoperative prognostic stratification of patients with stage II/III colon cancer.

In that study, 437 colon cancer circRNAs were examined in tumoral and adjacent normal tissues. The authors found that 103 circRNAs were differentially expressed in recurrent colon cancer patients as compared to non-recurrent counterparts. Among them, 100 up-regulated circRNAs were validated in patients with stage II/III colon cancer to test whether they could be used as prognostic biomarkers. Four circRNAs, including hsa_circ_0122319, hsa_circ_0087391, hsa_circ_0079480, and hsa_circ_0008039, showed strong predictive values for disease-free survival (DFS) in the validation cohort. The following circRNA-based prognostic model was thereby generated:

$c i r S c o r e = 0. 46 \times {Exp}_{h s a_c i r c_0122319} + (- 0. 386 \times {Exp}_{h s a_c i r c_0083791}) + 0. 293 \times {Exp}_{h s a_c i r c_0079480} + 0. 439 \times {Exp}_{h s a_c i r c_0008039}$

After adjusting for baseline clinicopathological factors, the cirScore was found suitable for serving as predictive biomarkers of postoperative recurrence in stage II/III colon cancer. Patients in the high-risk group (cirScore ≥ −0.323) had poorer DFS (hazard ratio [HR], 2.89, 95% confidence interval [CI], 1.37–6.09, P < 0.001) and overall survival (OS; HR, 4.22, 95% CI, 1.61–11.03, P < 0.001) than those in the low-risk group (cirScore < −0.323). Based on this circRNA-based prognostic model, the authors further established a nomogram which exhibited good performance for estimating the 3- or 5-year DFS and OS.

To further explore the biological functions of hsa_circ_0122319, hsa_circ_0087391, and hsa_circ_0079480 (the abundance of hsa_circ_0008039 was found to be low in colon cancer cell line was therefore not further explored), the authors performed a series of functional in vitro and in vivo experiments. They found that suppression of these three circRNAs in colon cancer cells could significantly alter in vitro migration capacity. Additionally, the knockdown of hsa_circ_0079480 in colon cancer cells could both inhibit in vivo lung and liver metastasis.

The strengths of this study include the well-designed identification approaches, mature clinical data with clinical outcome and long-term follow-up, and well-characterized tissue samples for colon cancer. Limitations included that the prognostic model was only applicable for the patients with stage II/III colon cancer. In summary, this study identified a 4-circRNA-based expression signature as highly predictive for cancer recurrence in stage II/III colon cancer, with the potential of identification of high-risk early-stage colon cancer individuals who would benefit most from adjuvant chemotherapy.

References

[1]	Nonlocal systems of conservation laws in several space dimensions. SIAM J. Numer. Anal. (2015) 53: 963-983.
[2]	On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation. J. Differential Equations (2017) 262: 978-1022.
[3]	Front tracking approximations for slow erosion. Dicrete Contin. Dyn. Syst. (2012) 32: 1481-1502.
[4]	On the numerical integration of scalar nonlocal conservation laws. ESAIM Math. Model. Numer. Anal. (2015) 49: 19-37.
[5]	On nonlocal conservation laws modelling sedimentation. Nonlinearity (2011) 27: 855-885.
[6]	Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math. (2016) 132: 217-241.
[7]	Solutions for a nonlocal conservation law with fading memory. Proc. Amer. Math. Soc. (2007) 135: 3905-3915.
[8]	J. Chien and W. Shen, Traveling Waves for nonlocal particle models of traffic flow on rough roads, Discrete Contin. Dyn. Syst., 39 (2019), 4001—4040, arXiv: 1902.08537.
[9]	M. Colombo, G. Crippa and L. V. Spinolo, On the singular local limit for conservation laws with nonlocal fluxes, Arch. Ration. Mech. Anal., 233 (2019), 1131–1167, arXiv: 1710.04547.
[10]	M. Colombo, G. Crippa and L. V. Spinolo, Blow-up of the total variation in the local limit of a nonlocal traffic model, Preprint, arXiv: 1808.03529.
[11]	Nonlocal crowd dynamics models for several populations. Acta Math. Sci. (2012) 32: 177-196.
[12]	Existence and stability of solutions of a delay-differential system. Arch. Rational Mech. Anal. (1962) 10: 401-426.
[13]	R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20. Springer-Verlag, New York-Heidelberg, 1977.
[14]	A new approach for a nonlocal, nonlinear conservation law. SIAM J. Appl. Math. (2012) 72: 464-487.
[15]	J. Friedrich, O. Kolb and S. Göttlich, A Godunov type scheme for a class of LWR traffic flow models with non-local flux, Netw. Heterog. Media, 13 (2018), 531–547, arXiv: 1802.07484.
[16]	Existence and stability of traveling waves for an integro-differential equation for slow erosion. J. Differential Equations (2014) 256: 253-282.
[17]	On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A (1955) 229: 317-345.
[18]	J. Ridder and W. Shen, Traveling waves for nonlocal models of traffic flow, Discrete Contin. Dyn. Syst., 39 (2019), 4001–4040, arXiv: 1808.03734.
[19]	Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition. Netw. Heterog. Media (2018) 13: 449-478.
[20]	Traveling waves for a microscopic model of traffic flow. Discrete Contin. Dyn. Syst. (2018) 38: 2571-2589.
[21]	Erosion profile by a global model for granular flow. Arch. Rational Mech. Anal. (2012) 204: 837-879.
[22]	On a nonlocal dispersive equation modeling particle suspensions. Q. Appl. Math. (1999) 57: 573-600.

This article has been cited by:

1.	Thanin Sitthiwirattham, Muhammad Aamir Ali, Hüseyin Budak, Sina Etemad, Shahram Rezapour, A new version of $( p,q ) $-Hermite–Hadamard’s midpoint and trapezoidal inequalities via special operators in $( p,q ) $-calculus, 2022, 2022, 1687-2770, 10.1186/s13661-022-01665-3
2.	Waewta Luangboon, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas, Hüseyin Budak, Some (p, q)-Integral Inequalities of Hermite–Hadamard Inequalities for (p, q)-Differentiable Convex Functions, 2022, 10, 2227-7390, 826, 10.3390/math10050826
3.	Yonghong Liu, Ghulam Farid, Dina Abuzaid, Kamsing Nonlaopon, On q-Hermite-Hadamard Inequalities via q − h-Integrals, 2022, 14, 2073-8994, 2648, 10.3390/sym14122648

Reader Comments

Your name:*

Email:*
© 2019 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)