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Stability and applications of multi-order fractional systems
Tempered fractional order compartment models and applications in biology
1. | School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China |
2. | Department of Mathematics and Statistics, Memorial University of Newfoundland, St. Johns, NL, A1C 5S7, Canada |
Compartment models with classical derivatives have diverse applications and attracted a lot of interest among scientists. To model the dynamical behavior of the particles that existed in the system for a long period of time with little chance to be removed, a power-law waiting time technique was introduced in the most recent work of Angstmann et al. [
References:
[1] |
R. Almeida,
Analysis of a fractional SEIR model with treatment, Appl. Math. Lett., 84 (2018), 56-62.
doi: 10.1016/j.aml.2018.04.015. |
[2] |
C. N. Angstmann, A. M. Erickson, B. I. Henry, A. V. McGann, J. M. Murray and J. A. Nichols,
Fractional order compartment models, SIAM J. Appl. Math., 77 (2017), 430-446.
doi: 10.1137/16M1069249. |
[3] |
C. N. Angstmann, B. I. Henry and A. V. McGann,
A fractional order recovery SIR model from a stochastic process, Bull. Math. Biol., 78 (2016), 468-499.
doi: 10.1007/s11538-016-0151-7. |
[4] |
C. N. Angstmann, B. I. Henry and A. V. McGann,
A fractional-order infectivity SIR model, Phys. A, 452 (2016), 86-93.
doi: 10.1016/j.physa.2016.02.029. |
[5] |
A. A. M. Arafa, S. Z. Rida and M. Khalil,
Solutions of fractional order model of childhood diseases with constant vaccination strategy, Math. Sci. Lett., 1 (2012), 17-23.
|
[6] |
I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh and Á. Torres,
On a fractional order Ebola epidemic model, Adv. Difference Equ., 2015 (2015), 1-12.
doi: 10.1186/s13662-015-0613-5. |
[7] |
A. Atangana and R. T. Alqahtani,
Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative, Entropy, 18 (2016), 40.
|
[8] |
H. Bateman, Higher Transcendental Functions, McGraw-Hill Book Company, New York, 1953.
![]() |
[9] |
K. B. Bischoff, R. L. Dedrick, D. S. Zaharko and J. A. Longstreth,
Methotrexate pharmacokinetics, J. Pharm. Sci., 60 (1971), 1128-1133.
doi: 10.1002/jps.2600600803. |
[10] |
S. I. Boyarchenko and S. Z. Levendorskiǐ,
Option pricing for truncated Levy processes, Int. J. Theor. Appl. Finance, 3 (2000), 549-552.
doi: 10.1142/S0219024900000541. |
[11] |
P. Carr, H. Geman, D. B. Madan and M. Yor,
The fine structure of asset returns: An empirical investigation, J. Business, 75 (2002), 305-332.
doi: 10.1086/338705. |
[12] |
P. Carr, H. Geman, D. B. Madan and M. Yor,
Stochastic volatility for Lévy processes, Math. Finance, 13 (2003), 345-382.
doi: 10.1111/1467-9965.00020. |
[13] |
Á. Cartea and D. del-Castillo-Negrete,
Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105.
doi: 10.1103/PhysRevE.76.041105. |
[14] |
R. Casagrandi, L. Bolzoni, S. A. Levin and V. Andreasen,
The SIRC model and influenza A, Math. Biosci., 200 (2006), 152-169.
doi: 10.1016/j.mbs.2005.12.029. |
[15] |
M. H. Chen and W. H. Deng,
Discretized fractional substantial calculus, ESAIM Math. Model. Numer. Anal., 49 (2015), 373-394.
doi: 10.1051/m2an/2014037. |
[16] |
R. L. Dedrick, D. D. Forrester, J. N. Cannon, S. M. E. Dareer and L. B. Mellett,
Pharmacokinetics of 1-$\beta$-D-arabinofuranosylcytosine (Ara-C) deamination in several species, Biochem. Pharmacol., 22 (1973), 2405-2417.
doi: 10.1016/0006-2952(73)90342-0. |
[17] |
W. H. Deng, M. H. Chen and E. Barkai,
Numerical algorithms for the forward and backward fractional Feynman-Kac equations, J. Sci. Comput., 62 (2015), 718-746.
doi: 10.1007/s10915-014-9873-6. |
[18] |
W. H. Deng, R. Hou, W. L. Wang and P. B. Xu, Modelling Anomalous Diffusion: From Statistics to Mathematic, World Scientific Publishing Company, China, 2020.
![]() |
[19] |
K. Diethelm,
A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613-619.
doi: 10.1007/s11071-012-0475-2. |
[20] |
Y. S. Ding and H. P. Ye,
A fractional-order differential equation model of HIV infection of CD4+ T-cells, Math. Comput. Modelling, 50 (2009), 386-392.
doi: 10.1016/j.mcm.2009.04.019. |
[21] |
H. A. A. El-Saka,
The fractional-order SIS epidemic model with variable population size, J. Egyptian Math. Soc., 22 (2014), 50-54.
doi: 10.1016/j.joems.2013.06.006. |
[22] |
M. El-Shahed and A. Alsaedi,
The fractional SIRC model and influenza A, Math. Probl. Eng., 2011 (2011), 1-9.
doi: 10.1155/2011/480378. |
[23] |
M. El-Shahed and F. A. El-Naby,
Fractional calculus model for childhood diseases and vaccines, Appl. Math. Sci., 8 (2014), 4859-4866.
doi: 10.12988/ams.2014.4294. |
[24] |
G. González-Parra, A. J. Arenas and B. M. Chen-Charpentier,
A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1), Math. Methods Appl. Sci., 37 (2014), 2218-2226.
doi: 10.1002/mma.2968. |
[25] |
E. Hanert, E. Schumacher and E. Deleersnijder,
Front dynamics in fractional-order epidemic models, J. Theoret. Biol., 279 (2011), 9-16.
doi: 10.1016/j.jtbi.2011.03.012. |
[26] |
R. Hilfer and L. Anton,
Fractional master equations and fractal time random walks, Phys. Rev. E, 51 (1995), R848.
doi: 10.1103/PhysRevE.51.R848. |
[27] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London. Ser. A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[28] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. Roy. Soc. London. Ser. A, 138 (1932), 55-83.
doi: 10.1098/rspa.1932.0171. |
[29] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity, Proc. Roy. Soc. London. Ser. A, 141 (1933), 94-122.
doi: 10.1098/rspa.1933.0106. |
[30] |
H. Kleinert,
Option pricing from path integral for non-Gaussian fluctuations. Natural martingale and application to truncated Lévy distributions, Phys. A, 312 (2002), 217-242.
doi: 10.1016/S0378-4371(02)00839-7. |
[31] |
I. Koponen,
Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process, Phys. Rev. E, 52 (1995), 1197.
doi: 10.1103/PhysRevE.52.1197. |
[32] |
C. P. Li and F. H. Zeng,
Finite difference methods for fractional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1230014.
doi: 10.1142/S0218127412300145. |
[33] |
R. N. Mantegna and H. E. Stanley,
Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.
doi: 10.1103/PhysRevLett.73.2946. |
[34] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter & Co., Berlin, 2012.
![]() ![]() |
[35] |
M. M. Meerschaert, Y. Zhang and B. Baeumer,
Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403.
doi: 10.1029/2008GL034899. |
[36] |
H. Nakao,
Multi-scaling properties of truncated Lévy flights, Phys. Lett. A, 266 (2000), 282-289.
doi: 10.1016/S0375-9601(00)00059-1. |
[37] |
E. A. Novikov,
Infinitely divisible distributions in turbulence, Phys. Rev. E, 50 (1994), R3303.
doi: 10.1103/PhysRevE.50.R3303. |
[38] |
M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, UK, 2000.
![]() ![]() |
[39] |
N. ÖZalp and E. Demirci,
A fractional order SEIR model with vertical transmission, Math. Comput. Modelling, 54 (2011), 1-6.
doi: 10.1016/j.mcm.2010.12.051. |
[40] |
J. Pitman and M. Yor,
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab., 25 (1997), 855-900.
doi: 10.1214/aop/1024404422. |
[41] |
J. Rosiński,
Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.
doi: 10.1016/j.spa.2006.10.003. |
[42] |
F. Sabzikar, M. M. Meerschaert and J. H. Chen,
Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.
doi: 10.1016/j.jcp.2014.04.024. |
[43] |
S. M. Salman and A. M. Yousef,
On a fractional-order model for HBV infection with cure of infected cells, J. Egyptian Math. Soc., 25 (2017), 445-451.
doi: 10.1016/j.joems.2017.06.003. |
[44] |
T. Sardar, S. Rana and J. Chattopadhyay,
A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 511-525.
doi: 10.1016/j.cnsns.2014.08.009. |
[45] |
X. C. Wu, W. H. Deng and E. Barkai,
Tempered fractional Feynman-Kac equation: Theory and examples, Phys. Rev. E, 93 (2016), 032151.
doi: 10.1103/PhysRevE.93.032151. |
[46] |
A. Zeb, G. Zaman, S. Momani and V. S. Ertürk,
Solution of an SEIR epidemic model in fractional order, VFAST Trans. Math., 1 (2013), 7-15.
|
show all references
References:
[1] |
R. Almeida,
Analysis of a fractional SEIR model with treatment, Appl. Math. Lett., 84 (2018), 56-62.
doi: 10.1016/j.aml.2018.04.015. |
[2] |
C. N. Angstmann, A. M. Erickson, B. I. Henry, A. V. McGann, J. M. Murray and J. A. Nichols,
Fractional order compartment models, SIAM J. Appl. Math., 77 (2017), 430-446.
doi: 10.1137/16M1069249. |
[3] |
C. N. Angstmann, B. I. Henry and A. V. McGann,
A fractional order recovery SIR model from a stochastic process, Bull. Math. Biol., 78 (2016), 468-499.
doi: 10.1007/s11538-016-0151-7. |
[4] |
C. N. Angstmann, B. I. Henry and A. V. McGann,
A fractional-order infectivity SIR model, Phys. A, 452 (2016), 86-93.
doi: 10.1016/j.physa.2016.02.029. |
[5] |
A. A. M. Arafa, S. Z. Rida and M. Khalil,
Solutions of fractional order model of childhood diseases with constant vaccination strategy, Math. Sci. Lett., 1 (2012), 17-23.
|
[6] |
I. Area, H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh and Á. Torres,
On a fractional order Ebola epidemic model, Adv. Difference Equ., 2015 (2015), 1-12.
doi: 10.1186/s13662-015-0613-5. |
[7] |
A. Atangana and R. T. Alqahtani,
Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative, Entropy, 18 (2016), 40.
|
[8] |
H. Bateman, Higher Transcendental Functions, McGraw-Hill Book Company, New York, 1953.
![]() |
[9] |
K. B. Bischoff, R. L. Dedrick, D. S. Zaharko and J. A. Longstreth,
Methotrexate pharmacokinetics, J. Pharm. Sci., 60 (1971), 1128-1133.
doi: 10.1002/jps.2600600803. |
[10] |
S. I. Boyarchenko and S. Z. Levendorskiǐ,
Option pricing for truncated Levy processes, Int. J. Theor. Appl. Finance, 3 (2000), 549-552.
doi: 10.1142/S0219024900000541. |
[11] |
P. Carr, H. Geman, D. B. Madan and M. Yor,
The fine structure of asset returns: An empirical investigation, J. Business, 75 (2002), 305-332.
doi: 10.1086/338705. |
[12] |
P. Carr, H. Geman, D. B. Madan and M. Yor,
Stochastic volatility for Lévy processes, Math. Finance, 13 (2003), 345-382.
doi: 10.1111/1467-9965.00020. |
[13] |
Á. Cartea and D. del-Castillo-Negrete,
Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E, 76 (2007), 041105.
doi: 10.1103/PhysRevE.76.041105. |
[14] |
R. Casagrandi, L. Bolzoni, S. A. Levin and V. Andreasen,
The SIRC model and influenza A, Math. Biosci., 200 (2006), 152-169.
doi: 10.1016/j.mbs.2005.12.029. |
[15] |
M. H. Chen and W. H. Deng,
Discretized fractional substantial calculus, ESAIM Math. Model. Numer. Anal., 49 (2015), 373-394.
doi: 10.1051/m2an/2014037. |
[16] |
R. L. Dedrick, D. D. Forrester, J. N. Cannon, S. M. E. Dareer and L. B. Mellett,
Pharmacokinetics of 1-$\beta$-D-arabinofuranosylcytosine (Ara-C) deamination in several species, Biochem. Pharmacol., 22 (1973), 2405-2417.
doi: 10.1016/0006-2952(73)90342-0. |
[17] |
W. H. Deng, M. H. Chen and E. Barkai,
Numerical algorithms for the forward and backward fractional Feynman-Kac equations, J. Sci. Comput., 62 (2015), 718-746.
doi: 10.1007/s10915-014-9873-6. |
[18] |
W. H. Deng, R. Hou, W. L. Wang and P. B. Xu, Modelling Anomalous Diffusion: From Statistics to Mathematic, World Scientific Publishing Company, China, 2020.
![]() |
[19] |
K. Diethelm,
A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam., 71 (2013), 613-619.
doi: 10.1007/s11071-012-0475-2. |
[20] |
Y. S. Ding and H. P. Ye,
A fractional-order differential equation model of HIV infection of CD4+ T-cells, Math. Comput. Modelling, 50 (2009), 386-392.
doi: 10.1016/j.mcm.2009.04.019. |
[21] |
H. A. A. El-Saka,
The fractional-order SIS epidemic model with variable population size, J. Egyptian Math. Soc., 22 (2014), 50-54.
doi: 10.1016/j.joems.2013.06.006. |
[22] |
M. El-Shahed and A. Alsaedi,
The fractional SIRC model and influenza A, Math. Probl. Eng., 2011 (2011), 1-9.
doi: 10.1155/2011/480378. |
[23] |
M. El-Shahed and F. A. El-Naby,
Fractional calculus model for childhood diseases and vaccines, Appl. Math. Sci., 8 (2014), 4859-4866.
doi: 10.12988/ams.2014.4294. |
[24] |
G. González-Parra, A. J. Arenas and B. M. Chen-Charpentier,
A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1), Math. Methods Appl. Sci., 37 (2014), 2218-2226.
doi: 10.1002/mma.2968. |
[25] |
E. Hanert, E. Schumacher and E. Deleersnijder,
Front dynamics in fractional-order epidemic models, J. Theoret. Biol., 279 (2011), 9-16.
doi: 10.1016/j.jtbi.2011.03.012. |
[26] |
R. Hilfer and L. Anton,
Fractional master equations and fractal time random walks, Phys. Rev. E, 51 (1995), R848.
doi: 10.1103/PhysRevE.51.R848. |
[27] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London. Ser. A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[28] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. II. The problem of endemicity, Proc. Roy. Soc. London. Ser. A, 138 (1932), 55-83.
doi: 10.1098/rspa.1932.0171. |
[29] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity, Proc. Roy. Soc. London. Ser. A, 141 (1933), 94-122.
doi: 10.1098/rspa.1933.0106. |
[30] |
H. Kleinert,
Option pricing from path integral for non-Gaussian fluctuations. Natural martingale and application to truncated Lévy distributions, Phys. A, 312 (2002), 217-242.
doi: 10.1016/S0378-4371(02)00839-7. |
[31] |
I. Koponen,
Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process, Phys. Rev. E, 52 (1995), 1197.
doi: 10.1103/PhysRevE.52.1197. |
[32] |
C. P. Li and F. H. Zeng,
Finite difference methods for fractional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1230014.
doi: 10.1142/S0218127412300145. |
[33] |
R. N. Mantegna and H. E. Stanley,
Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight, Phys. Rev. Lett., 73 (1994), 2946-2949.
doi: 10.1103/PhysRevLett.73.2946. |
[34] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter & Co., Berlin, 2012.
![]() ![]() |
[35] |
M. M. Meerschaert, Y. Zhang and B. Baeumer,
Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403.
doi: 10.1029/2008GL034899. |
[36] |
H. Nakao,
Multi-scaling properties of truncated Lévy flights, Phys. Lett. A, 266 (2000), 282-289.
doi: 10.1016/S0375-9601(00)00059-1. |
[37] |
E. A. Novikov,
Infinitely divisible distributions in turbulence, Phys. Rev. E, 50 (1994), R3303.
doi: 10.1103/PhysRevE.50.R3303. |
[38] |
M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, UK, 2000.
![]() ![]() |
[39] |
N. ÖZalp and E. Demirci,
A fractional order SEIR model with vertical transmission, Math. Comput. Modelling, 54 (2011), 1-6.
doi: 10.1016/j.mcm.2010.12.051. |
[40] |
J. Pitman and M. Yor,
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab., 25 (1997), 855-900.
doi: 10.1214/aop/1024404422. |
[41] |
J. Rosiński,
Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.
doi: 10.1016/j.spa.2006.10.003. |
[42] |
F. Sabzikar, M. M. Meerschaert and J. H. Chen,
Tempered fractional calculus, J. Comput. Phys., 293 (2015), 14-28.
doi: 10.1016/j.jcp.2014.04.024. |
[43] |
S. M. Salman and A. M. Yousef,
On a fractional-order model for HBV infection with cure of infected cells, J. Egyptian Math. Soc., 25 (2017), 445-451.
doi: 10.1016/j.joems.2017.06.003. |
[44] |
T. Sardar, S. Rana and J. Chattopadhyay,
A mathematical model of dengue transmission with memory, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 511-525.
doi: 10.1016/j.cnsns.2014.08.009. |
[45] |
X. C. Wu, W. H. Deng and E. Barkai,
Tempered fractional Feynman-Kac equation: Theory and examples, Phys. Rev. E, 93 (2016), 032151.
doi: 10.1103/PhysRevE.93.032151. |
[46] |
A. Zeb, G. Zaman, S. Momani and V. S. Ertürk,
Solution of an SEIR epidemic model in fractional order, VFAST Trans. Math., 1 (2013), 7-15.
|










Week | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
Cases | 4 | 1 | 0 | 1 | 2 | 5 | 12 | 17 | 22 | 16 | 15 | 53 | 55 | 45 |
Week | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |
Cases | 84 | 102 | 127 | 261 | 210 | 155 | 109 | 116 | 125 | 76 | 52 | 19 | 15 | 1 |
Week | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
Cases | 4 | 1 | 0 | 1 | 2 | 5 | 12 | 17 | 22 | 16 | 15 | 53 | 55 | 45 |
Week | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |
Cases | 84 | 102 | 127 | 261 | 210 | 155 | 109 | 116 | 125 | 76 | 52 | 19 | 15 | 1 |
Model | |||||||
cSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 1 | 1 |
frSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 0.6 | 0.88 |
tfrSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 0.6 | 0.88 |
Model | |||||||
cSEIR | 0 | 0 | 100 | 0 | 1 | 0 | |
frSEIR | 0 | 0 | 100 | 0 | 1 | 0 | |
tfrSEIR | 0.015 | 0.09 | 100 | 0 | 1 | 0 |
Model | |||||||
cSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 1 | 1 |
frSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 0.6 | 0.88 |
tfrSEIR | 5.038e+01 | 1.0e-08 | 2.89e-03 | 2.25e-01 | 1.373e+00 | 0.6 | 0.88 |
Model | |||||||
cSEIR | 0 | 0 | 100 | 0 | 1 | 0 | |
frSEIR | 0 | 0 | 100 | 0 | 1 | 0 | |
tfrSEIR | 0.015 | 0.09 | 100 | 0 | 1 | 0 |
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