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doi: 10.3934/dcdsb.2021275
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Tempered fractional order compartment models and applications in biology

Yejuan Wang 1,, , Lijuan Zhang 1, and  Yuan Yuan 2,

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

* Corresponding author

Received  January 2021 Revised  October 2021 Early access November 2021

Fund Project: This work was supported by the National Natural Science Foundation of China under grant 41875084

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. [2]. The divergent first moment makes the power-law waiting time distribution less physical because of the finite lifespan of the particles. In this work, we take the tempered power-law function as the waiting time distribution, which has finite first moment while keeping the power-law properties. From the underlying physical stochastic process with the exponentially truncated power-law waiting time distribution, we build the tempered fractional compartment model. As an application, the tempered fractional SEIR epidemic model is proposed to simulate the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. (Colombia). Some analysis and numerical simulations are carried out around the equilibrium behavior.

Keywords: Riemann-Liouville, compartment models, tempered fractional derivative, epidemic models, influenza A(H1N1), exponentially truncated power-law, waiting time distribution.
Mathematics Subject Classification: Primary: 34A08, 60G22, 60K40, 92D30.
Citation: Yejuan Wang, Lijuan Zhang, Yuan Yuan. Tempered fractional order compartment models and applications in biology. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021275
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

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M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403.  doi: 10.1029/2008GL034899.  Google Scholar

[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.  Google Scholar

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E. A. Novikov, Infinitely divisible distributions in turbulence, Phys. Rev. E, 50 (1994), R3303.  doi: 10.1103/PhysRevE.50.R3303.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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J. Rosiński, Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.  doi: 10.1016/j.spa.2006.10.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8] H. Bateman, Higher Transcendental Functions, McGraw-Hill Book Company, New York, 1953.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, Walter de Gruyter & Co., Berlin, 2012.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[37]

E. A. Novikov, Infinitely divisible distributions in turbulence, Phys. Rev. E, 50 (1994), R3303.  doi: 10.1103/PhysRevE.50.R3303.  Google Scholar

[38] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, UK, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[41]

J. Rosiński, Tempering stable processes, Stochastic Process. Appl., 117 (2007), 677-707.  doi: 10.1016/j.spa.2006.10.003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Flux flow of tempered fractional SEIR model
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Figure 2.  The endemic steady state plotted as the functions of $ \lambda_2 $ $ ( $Left$ ) $ and $ \lambda_3 $ $ ( $Right$ ) $ for the tfrSEIR model
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Figure 3.  The endemic steady state plotted as the functions of $ \alpha_2 $ $ ( $Left$ ) $ and $ \alpha_3 $ $ ( $Right$ ) $ for the tfrSEIR model
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Figure 4.  Plots of Infected $ I(t) $ with respect to time in the tfrSEIR model with $ \alpha_2 = 0.5, 0.7, 0.9, \alpha_3 = 0.7, \lambda_2 = \lambda_3 = 0.0001 $ $ ( $Left$ ) $ and $ \alpha_3 = 0.5, 0.7, 0.9, \alpha_2 = 0.7, \lambda_2 = \lambda_3 = 0.0001 $ $ ( $Right$ ) $
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Figure 5.  Plots of Infected $ I(t) $ with respect to time in the tfrSEIR model with $ \lambda_2 = 0.0001, 0.001, 0.01, \lambda_3 = 0.0001, \alpha_2 = \alpha_3 = 0.7 $ $ ( $Left$ ) $ and $ \lambda_3 = 0.0001, 0.001, 0.01, \lambda_2 = 0.0001, \alpha_2 = \alpha_3 = 0.7 $ $ ( $Right$ ) $
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Figure 6.  The variation of Infected $ I(t) $ in the tfrSEIR model for $ \alpha_2 = 0.6,0.8,1 $ $ ( $Left$ ) $ and $ \alpha_3 = 0.8,0.9,1 $ $ ( $Right$ ) $
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Figure 7.  The variation of Infected $ I(t) $ in the tfrSEIR model for $ \beta = 2.69e-3,2.79e-3,2.89e-3. $
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Figure 8.  The variation of Infected $ I(t) $ in the tfrSEIR model for $ \lambda_2 = 0.2,0.05,0.001 $ $ ( $Left$ ) $ and $ \lambda_3 = 0.09,0.07,0.05 $ $ ( $Right$ ) $
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Figure 9.  Comparing with the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. $ ( $Colombia$ ) $, the numerical solutions of the classical SEIR model $ ( $Left$ ) $ and the frSEIR model $ ( $Right$ ) $ with $ \alpha_2 = 0.6 $, $ \alpha_3 = 0.88 $
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Figure 10.  Numerical solution of the tfrSEIR model adjusted to real data of people infected with influenza AH1N1/09 in Bogotá D.C. $ ( $Colombia$ ) $ with $ \alpha_2 = 0.6 $, $ \alpha_3 = 0.88 $, $ \lambda_2 = 0.015 $, $ \lambda_3 = 0.09 $
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Table 1.  Data provided by the Secretaria be Salud Distrital de Bogotá D.C. First row corresponds to the week number of year 2009. Second row presents the number of infectious individuals by AH1N1/09 detected in each week
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
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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
Table 2.  Values of various parameters and initial conditions
Model $ \mu $ $ \mu_1 $ $ \beta $ $ \Omega_2 $ $ \Omega_3 $ $ \alpha_2 $ $ \alpha_3 $
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 $ \lambda_2 $ $ \lambda_3 $ $ S_0 $ $ E_0 $ $ I_0 $ $ R_0 $
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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Model $ \mu $ $ \mu_1 $ $ \beta $ $ \Omega_2 $ $ \Omega_3 $ $ \alpha_2 $ $ \alpha_3 $
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 $ \lambda_2 $ $ \lambda_3 $ $ S_0 $ $ E_0 $ $ I_0 $ $ R_0 $
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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