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 |
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form
(∂t+X⋅∇Y)u=∇X⋅(A(∇Xu,X,Y,t)).
The function A=A(ξ,X,Y,t):Rm×Rm×Rm×R→Rm is assumed to be continuous with respect to ξ, and measurable with respect to X,Y and t. A=A(ξ,X,Y,t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded X, Y and t dependent domains.
Citation: Prashanta Garain, Kaj Nyström. On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients[J]. Mathematics in Engineering, 2023, 5(2): 1-37. doi: 10.3934/mine.2023043
[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 |
We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form
(∂t+X⋅∇Y)u=∇X⋅(A(∇Xu,X,Y,t)).
The function A=A(ξ,X,Y,t):Rm×Rm×Rm×R→Rm is assumed to be continuous with respect to ξ, and measurable with respect to X,Y and t. A=A(ξ,X,Y,t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded X, Y and t dependent domains.
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.
The classical model for Ebola virus model is given in
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
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
Consider the fractional-order Ebola virus model (2.1), by using Caputo definition with Laplace transform, we have
We get the followings generalized form for analysis and numerical solution.
The results of fractional order model (2.1) is represented in followings tables and graphs.
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 |
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 |
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 |
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 |
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 |
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.
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.
[1] | D. Albritton, S. N. Armstrong, J. C. Mourrat, M. Novack, Variational methods for the kinetic Fokker-Planck equation, 2019, arXiv: 1902.04037. |
[2] |
F. Anceschi, S. Polidoro, M. A. Ragusa, Moser's estimates for degenerate Kolmogorov equations with non-negative divergence lower order coefficients, Nonlinear Anal., 189 (2019), 111568. https://doi.org/10.1016/j.na.2019.07.001 doi: 10.1016/j.na.2019.07.001
![]() |
[3] | F. Anceschi, A. Rebucci, A note on the weak regularity theory for degenerate Kolmogorov equations, 2021, arXiv: 2107.04441. |
[4] |
S. N. Armstrong, A. Bordas, J. C. Mourrat, Quantitative stochastic homogenization and regularity theory of parabolic equations, Anal. PDE, 11 (2018), 1945–2014. https://doi.org/10.2140/apde.2018.11.1945 doi: 10.2140/apde.2018.11.1945
![]() |
[5] |
S. N. Armstrong, J. C. Mourrat, Lipschitz regularity for elliptic equations with random coefficients, Arch. Rational Mech. Anal., 219 (2016), 255–348. https://doi.org/10.1007/s00205-015-0908-4 doi: 10.1007/s00205-015-0908-4
![]() |
[6] |
L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. Theor., 19 (1992), 581–597. https://doi.org/10.1016/0362-546X(92)90023-8 doi: 10.1016/0362-546X(92)90023-8
![]() |
[7] |
F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pure. Appl., 81 (2002), 1135–1159. https://doi.org/10.1016/S0021-7824(02)01264-3 doi: 10.1016/S0021-7824(02)01264-3
![]() |
[8] |
M. Bramanti, M. C. Cerutti, M. Manfredini, Lp estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl., 200 (1996), 332–354. https://doi.org/10.1006/jmaa.1996.0209 doi: 10.1006/jmaa.1996.0209
![]() |
[9] | H. Brézis, I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. Le cas dépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A1197–A1198. |
[10] | H. Brézis, I. Ekeland, Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A971–A974. |
[11] | C. Cercignani, H-theorem and trend to equilibrium in the kinetic theory of gases, Arch. Mech., 34 (1982), 231–241. |
[12] |
C. Cinti, A. Pascucci, S. Polidoro, Pointwise estimates for a class of non-homogeneous Kolmogorov equations, Math. Ann., 340 (2008), 237–264. https://doi.org/10.1007/s00208-007-0147-6 doi: 10.1007/s00208-007-0147-6
![]() |
[13] |
L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal., 269 (2015), 1359–1403. https://doi.org/10.1016/j.jfa.2015.05.009 doi: 10.1016/j.jfa.2015.05.009
![]() |
[14] |
L. Desvillettes, C. Mouhot, C. Villani, Celebrating Cercignani's conjecture for the Boltzmann equation, Kinet. Relat. Mod., 4 (2011), 277–294. https://doi.org/10.3934/krm.2011.4.277 doi: 10.3934/krm.2011.4.277
![]() |
[15] |
L. Desvillettes, C. Villani, On the spatially homogeneous Landau equation for hard potentials Ⅱ: H-theorem and applications, Commun. Part. Diff. Eq., 25 (2000), 261–298. https://doi.org/10.1080/03605300008821513 doi: 10.1080/03605300008821513
![]() |
[16] | E. DiBenedetto, U. Gianazza, V. Vespri, Harnack's inequality for degenerate and singular parabolic equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-1584-8 |
[17] | I. Ekeland, R. Temam, Convex analysis and variational problems, Amsterdam-Oxford: North-Holland Publishing Co., 1976. |
[18] | N. Ghoussoub, Self-dual partial differential systems and their variational principles, New York: Springer, 2009. https://doi.org/10.1007/978-0-387-84897-6 |
[19] |
N. Ghoussoub, L. Tzou, A variational principle for gradient flows, Math. Ann., 330 (2004), 519–549. https://doi.org/10.1007/s00208-004-0558-6 doi: 10.1007/s00208-004-0558-6
![]() |
[20] |
F. Golse, C. Imbert, C. Mouhot, A. F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19 (2019), 253–295. https://doi.org/10.2422/2036-2145.201702_001 doi: 10.2422/2036-2145.201702_001
![]() |
[21] | J. Guerand, Quantitative regularity for parabolic De Giorgi classes, 2019, arXiv: 1903.07421. |
[22] | J. Guerand, C. Imbert, Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations, 2021, arXiv: 2102.04105. |
[23] | J. Guerand, C. Mouhot, Quantitative de Giorgi methods in kinetic theory, 2021, arXiv: 2103.09646. |
[24] |
L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147–171. https://doi.org/10.1007/BF02392081 doi: 10.1007/BF02392081
![]() |
[25] |
A. N. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. Math. (2), 35 (1934), 116–117. https://doi.org/10.2307/1968123 doi: 10.2307/1968123
![]() |
[26] |
E. Lanconelli, F. Lascialfari, A boundary value problem for a class of quasilinear operators of Fokker-Planck type, Ann. Univ. Ferrara, 41 (1996), 65–84. https://doi.org/10.1007/BF02825256 doi: 10.1007/BF02825256
![]() |
[27] |
F. Lascialfari, D. Morbidelli, A boundary value problem for a class of quasilinear ultraparabolic equations, Commun. Part. Diff. Eq., 23 (1998), 847–868. https://doi.org/10.1080/03605309808821369 doi: 10.1080/03605309808821369
![]() |
[28] |
P. L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 191–204. https://doi.org/10.1098/rsta.1994.0018 doi: 10.1098/rsta.1994.0018
![]() |
[29] |
M. Litsgård, K. Nyström, The Dirichlet problem for Kolmogorov-Fokker-Planck type equations with rough coefficients, J. Funct. Anal., 281 (2021), 109226. https://doi.org/10.1016/j.jfa.2021.109226 doi: 10.1016/j.jfa.2021.109226
![]() |
[30] |
M. Litsgård, K. Nyström, Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients, J. Math. Pure. Appl., 157 (2022), 45–100. https://doi.org/10.1016/j.matpur.2021.11.004 doi: 10.1016/j.matpur.2021.11.004
![]() |
[31] | M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831–866. |
[32] | M. Manfredini, S. Polidoro, Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients, Bollettino dell'Unione Matematica Italiana Serie 8, 1-B (1998), 651–675. |
[33] | C. Mouhot, De Giorgi–Nash–Moser and Hörmander theories: new interplays, In: Proceedings of the International Congress of Mathematicians (ICM 2018), Hackensack, NJ: World Sci. Publ., 2018, 2467–2493. https://doi.org/10.1142/9789813272880_0146 |
[34] |
A. Pascucci, S. Polidoro, The Moser's iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., 6 (2004), 395–417. https://doi.org/10.1142/S0219199704001355 doi: 10.1142/S0219199704001355
![]() |
[35] |
S. Polidoro, M. A. Ragusa, Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal., 14 (2001), 341–350. https://doi.org/10.1023/A:1011261019736 doi: 10.1023/A:1011261019736
![]() |
[36] | C. A. Truesdell, R. G. Muncaster, Fundamentals of Maxwell's kinetic theory of a simple monatomic gas, New York-London: Academic Press, Inc., 1980. |
[37] | C. Villani, A review of mathematical topics in collisional kinetic theory, In: Handbook of mathematical fluid dynamics, Vol I, Amsterdam: North-Holland, 2002, 71–74. https://doi.org/10.1016/S1874-5792(02)80004-0 |
[38] |
W. Wang, L. Zhang, The Cα regularity of a class of non-homogeneous ultraparabolic equations, Sci. China Ser. A-Math., 52 (2009), 1589–1606. https://doi.org/10.1007/s11425-009-0158-8 doi: 10.1007/s11425-009-0158-8
![]() |
[39] |
W. Wang, L. Zhang, The Cα regularity of weak solutions of ultraparabolic equations, Discrete Contin. Dyn. Syst., 29 (2011), 1261–1275. https://doi.org/10.3934/dcds.2011.29.1261 doi: 10.3934/dcds.2011.29.1261
![]() |
[40] | W. Wang, L. Zhang, Cα regularity of weak solutions of non-homogenous ultraparabolic equations with drift terms, 2017, arXiv: 1704.05323. |
[41] |
Y. Zhu, Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients, SIAM J. Math. Anal., 53 (2021), 2746–2775. https://doi.org/10.1137/20M1372147 doi: 10.1137/20M1372147
![]() |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |