Tempered fractional order compartment models and applications in biology

Yejuan Wang; Lijuan Zhang; Yuan Yuan

doi:10.3934/dcdsb.2021275

Article Contents

2022, Volume 27, Issue 9: 5297-5316. Doi: 10.3934/dcdsb.2021275

This issue Previous Article Stability and applications of multi-order fractional systems Next Article Pullback attractors via quasi-stability for non-autonomous lattice dynamical 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

^* Corresponding author
^* Corresponding author

Received: January 2021

Revised: October 2021

Early access: November 2021

Published: September 2022

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

Abstract Full Text(HTML) Figure(10) / Table(2) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 34A08, 60G22, 60K40, 92D30.

Citation:

Full Text(HTML)

Figure 1. Flux flow of tempered fractional SEIR model

Download: Full-size image PowerPoint slide

Figure 2. The endemic steady state plotted as the functions of $ \lambda_2 $ $ ( $Left$ ) $ and $ \lambda_3 $ $ ( $Right$ ) $ for the tfrSEIR model

Download: Full-size image PowerPoint slide

Figure 3. The endemic steady state plotted as the functions of $ \alpha_2 $ $ ( $Left$ ) $ and $ \alpha_3 $ $ ( $Right$ ) $ for the tfrSEIR model

Download: Full-size image PowerPoint slide

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 $ $ ( $Right$ ) $

Download: Full-size image PowerPoint slide

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 $ $ ( $Right$ ) $

Download: Full-size image PowerPoint slide

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 $ $ ( $Right$ ) $

Download: Full-size image PowerPoint slide

Figure 7. The variation of Infected $ I(t) $ in the tfrSEIR model for $ \beta = 2.69e-3,2.79e-3,2.89e-3. $

Download: Full-size image PowerPoint slide

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 $ $ ( $Right$ ) $

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.