doi: 10.3934/dcdss.2021025

A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay

a. 

Department of Mathematics, University of Mazandaran, Babolsar, Iran

b. 

Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa

c. 

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan

d. 

Department of Mathematics and Informatics, Azerbaijan University, Jeyhun Hajibeyli, 71, AZ1007, Baku, Azerbaijan

* Corresponding author: Hossein Jafari

Received  July 2020 Revised  November 2020 Published  March 2021

In present work, a step-by-step Legendre collocation method is employed to solve a class of nonlinear fractional stochastic delay differential equations (FSDDEs). The step-by-step method converts the nonlinear FSDDE into a non-delay nonlinear fractional stochastic differential equation (FSDE). Then, a Legendre collocation approach is considered to obtain the numerical solution in each step. By using a collocation scheme, the non-delay nonlinear FSDE is reduced to a nonlinear system. Moreover, the error analysis of this numerical approach is investigated and convergence rate is examined. The accuracy and reliability of this method is shown on three test examples and the effect of different noise measures is investigated. Finally, as an useful application, the proposed scheme is applied to obtain the numerical solution of a stochastic SIRS model.

Citation: Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021025
References:
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G. CottoneM. D. Paola and S. Butera, Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach, Probabilistic Engineering Mechanics, 26 (2011), 101-108.  doi: 10.1016/j.probengmech.2010.06.010.  Google Scholar

[2]

N. Bellomo, Z. Brzezniak and L. M. de Socio, Nonlinear Stochastic Evolution Problems in Applied Sciences, Kluwer Academic Publishers, Springer, Dordrecht, 1992. doi: 10.1007/978-94-011-1820-0.  Google Scholar

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R. AboulaichA. DarouichiI. Elmouki and A. Jraifi, A stochastic optimal control model for BCG immunotherapy in superficial bladder cancer, Math. Model. Nat. Phenom., 12 (2017), 99-119.  doi: 10.1051/mmnp/201712507.  Google Scholar

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J. YangY. Tan and R. A. Cheke, Thresholds for extinction and proliferation in a stochastic tumour-immune model with pulsed comprehensive therapy, Commun. Nonlinear. Sci. Numer. Simulat., 73 (2019), 363-378.  doi: 10.1016/j.cnsns.2019.02.025.  Google Scholar

[5]

S. JerezS. Diaz-Infante and B. Chen, Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process, Mathematical Biosciences, 299 (2018), 153-164.  doi: 10.1016/j.mbs.2018.03.006.  Google Scholar

[6]

S. Singh and S. S. Ray, Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion, Int. J. Biomath., 10 (2017), 1750103. doi: 10.1142/S1793524517501030.  Google Scholar

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W. Padgett and C. Tsokos, A new stochastic formulation of a population growth problem, Mathematical Biosciences, 17 (1973), 105-120.  doi: 10.1016/0025-5564(73)90064-3.  Google Scholar

[8]

G. I. Zmievskaya, A. L. Bondareva, T. V. Levchenko and G. Maino, Computational stochastic model of ions implantation, AIP Conf. Proc., (2015), 1648: 230003. doi: 10.1063/1.4912495.  Google Scholar

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B. Oksendal, Stochastic Differential Equations, An Introduction with Applications, 5$^th$ edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar

[10]

X. ChenP. HuS. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Oper. Res., 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524.  Google Scholar

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A. N. Huu and B. Costa-Lima, Orbits in a stochastic Goodwin-Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 419 (2014), 48-67.  doi: 10.1016/j.jmaa.2014.04.035.  Google Scholar

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D. Henderson and P. Plaschko, Differential Equation in Science and Engineering, Provo Utah, USA, Mexico CityDF, 2006. doi: 10.1142/9789812774798.  Google Scholar

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G. Chen and T. Li, Stability of stochastic delayed SIR model, Stochastics and Dynamics, 22 (2009), 231-252.  doi: 10.1142/S0219493709002658.  Google Scholar

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G. Shevchenko, Mixed stochastic delay differential equations, Theory of Probability and Mathematical Statistics, 89 (2014), 181-195.  doi: 10.1090/S0094-9000-2015-00944-3.  Google Scholar

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M. Milosevic, An explicit analytic approximation of solutions for a class of neutral stochastic differential equations with time-dependent delay based on Taylor expansion, Applied Mathematics and Computation, 274 (2016), 745-761.  doi: 10.1016/j.amc.2015.11.026.  Google Scholar

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E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1.  Google Scholar

[20]

I. Podlubny, Fractional differential equations, Math. Sci. Eng., 198 (1998).  Google Scholar

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A. Babaei, H. Jafari and S. Banihashemi, Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, Journal of Computational and Applied Mathematics, 377 (2020), 112908. doi: 10.1016/j.cam.2020.112908.  Google Scholar

[22]

C. AngstmannA. M. EricksonB. I. HenryA. V. McGannJ. M. Murray and J. Nichols, Fractional order compartment models, SIAM Journal on Applied Mathematics, 77 (2017), 430-446.  doi: 10.1137/16M1069249.  Google Scholar

[23]

A. Babaei, B. Parsa Moghaddam, S. Banihashemi and J. A. Tenreiro Machado, Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations, Communications in Nonlinear Science and Numerical Simulation, 82 (2019), 104985. doi: 10.1016/j.cnsns.2019.104985.  Google Scholar

[24]

R. M. Ganji, H. Jafari and S. Nemati, A new approach for solving integro-differential equations of variable order, Journal of Computational and Applied Mathematics, 379 (2020), 112946. doi: 10.1016/j.cam.2020.112946.  Google Scholar

[25]

A. Babaei and S. Banihashemi, Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numerical Methods for Partial Differential Equations, 35 (2019), 976-992.  doi: 10.1002/num.22334.  Google Scholar

[26]

M. Izadi and C. Cattani, Generalized Bessel polynomial for multi-order fractional differential equations, Symmetry, 12 (2020), 1260. doi: 10.3390/sym12081260.  Google Scholar

[27]

D. N. Tien, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397 (2013), 334-348.  doi: 10.1016/j.jmaa.2012.07.062.  Google Scholar

[28]

Z. G. Yu, V. Anh, Y. Wang, D. Mao and J. Wanliss, Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition, J. Geophys. Res. Space Phys., 115 (2010). doi: 10.1029/2009JA015206.  Google Scholar

[29]

E. Abdel-Rehim, From the Ehrenfest model to time-fractional stochastic processes, J. Comput. Appl. Math., 233 (2009), 197-207.  doi: 10.1016/j.cam.2009.07.010.  Google Scholar

[30]

A. Babaei, H. Jafari and S. Banihashemi, A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise, Symmetry, 12 (2020), 904. doi: 10.3390/sym12060904.  Google Scholar

[31]

T. S. DoanP. T. HuongP. E. Kloeden and H. T. Tuana, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stochastic Analysis and Applications, 36 (2018), 1-11.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[32]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Physica D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[33]

B. P. MoghaddamL. ZhangA. M. LopesJ. A. Tenreiro Machado and Z. S. Mostaghim, Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, An International Journal of Probability and Stochastic Processes, 92 (2020), 379-396.  doi: 10.1080/17442508.2019.1625903.  Google Scholar

[34]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.  Google Scholar

[35]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.  Google Scholar

[36]

X. WangS. Gan and D. Wang, $ \theta $-Maruyama methods for nonlinear stochastic differential delay equations, Appl. Numer. Math., 98 (2015), 38-58.  doi: 10.1016/j.apnum.2015.08.004.  Google Scholar

[37]

B. P. MoghaddamL. ZhangA. M. LopesJ. A. T. Machado and Z. S. Mostaghim, Computational scheme for solving nonlinear fractional stochastic differential equations with delay, Stochastic Analysis and Applications, 37 (2019), 893-908.  doi: 10.1080/07362994.2019.1621182.  Google Scholar

[38]

I. J. Gyongy and T. Martinez, On numerical solution of stochastic partial differential equations of elliptic type, Stochastics: An International Journal of Probability and Stochastic Processes, 78 (2006), 213-231.  doi: 10.1080/17442500600805047.  Google Scholar

[39]

C. Roth, A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations, Stoch. Anal. Appl., 24 (2006), 221-240.  doi: 10.1080/07362990500397764.  Google Scholar

[40]

J. B.Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018.   Google Scholar

[41]

Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. Anal., 40 (2002), 1421-1445.  doi: 10.1137/S0036142901387956.  Google Scholar

[42]

M. H. HeydariM. R. HooshmandaslG. B. Loghmani and C. Cattani, Wavelets Galerkin method for solving stochastic heat equation, International Journal of Computer Mathematics, 93 (2016), 1579-1596.  doi: 10.1080/00207160.2015.1067311.  Google Scholar

[43]

F. Mirzaee and E. Hadadiyan, Solving system of linear Stratonovich Volterra integral equations via modification of hat functions, Applied Mathematics and Computation, 293 (2017), 254-264.  doi: 10.1016/j.amc.2016.08.016.  Google Scholar

[44]

Q. LiT. Kang and Q. Zhang, Mean-square dissipative methods for stochastic agedependent capital system with fractional Brownian motion and jumps, Appl. Math. Comput., 339 (2018), 81-92.  doi: 10.1016/j.amc.2018.07.018.  Google Scholar

[45]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006.  Google Scholar

[46]

X. ZhongS. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar

show all references

References:
[1]

G. CottoneM. D. Paola and S. Butera, Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach, Probabilistic Engineering Mechanics, 26 (2011), 101-108.  doi: 10.1016/j.probengmech.2010.06.010.  Google Scholar

[2]

N. Bellomo, Z. Brzezniak and L. M. de Socio, Nonlinear Stochastic Evolution Problems in Applied Sciences, Kluwer Academic Publishers, Springer, Dordrecht, 1992. doi: 10.1007/978-94-011-1820-0.  Google Scholar

[3]

R. AboulaichA. DarouichiI. Elmouki and A. Jraifi, A stochastic optimal control model for BCG immunotherapy in superficial bladder cancer, Math. Model. Nat. Phenom., 12 (2017), 99-119.  doi: 10.1051/mmnp/201712507.  Google Scholar

[4]

J. YangY. Tan and R. A. Cheke, Thresholds for extinction and proliferation in a stochastic tumour-immune model with pulsed comprehensive therapy, Commun. Nonlinear. Sci. Numer. Simulat., 73 (2019), 363-378.  doi: 10.1016/j.cnsns.2019.02.025.  Google Scholar

[5]

S. JerezS. Diaz-Infante and B. Chen, Fluctuating periodic solutions and moment boundedness of a stochastic model for the bone remodeling process, Mathematical Biosciences, 299 (2018), 153-164.  doi: 10.1016/j.mbs.2018.03.006.  Google Scholar

[6]

S. Singh and S. S. Ray, Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion, Int. J. Biomath., 10 (2017), 1750103. doi: 10.1142/S1793524517501030.  Google Scholar

[7]

W. Padgett and C. Tsokos, A new stochastic formulation of a population growth problem, Mathematical Biosciences, 17 (1973), 105-120.  doi: 10.1016/0025-5564(73)90064-3.  Google Scholar

[8]

G. I. Zmievskaya, A. L. Bondareva, T. V. Levchenko and G. Maino, Computational stochastic model of ions implantation, AIP Conf. Proc., (2015), 1648: 230003. doi: 10.1063/1.4912495.  Google Scholar

[9]

B. Oksendal, Stochastic Differential Equations, An Introduction with Applications, 5$^th$ edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar

[10]

X. ChenP. HuS. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Oper. Res., 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524.  Google Scholar

[11]

A. N. Huu and B. Costa-Lima, Orbits in a stochastic Goodwin-Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 419 (2014), 48-67.  doi: 10.1016/j.jmaa.2014.04.035.  Google Scholar

[12]

F. Klebaner, Introduction to Stochastic Calculus with Applications, 2nd edition, Imperial College Press, 2005. doi: 10.1142/p386.  Google Scholar

[13]

D. Henderson and P. Plaschko, Differential Equation in Science and Engineering, Provo Utah, USA, Mexico CityDF, 2006. doi: 10.1142/9789812774798.  Google Scholar

[14]

G. Chen and T. Li, Stability of stochastic delayed SIR model, Stochastics and Dynamics, 22 (2009), 231-252.  doi: 10.1142/S0219493709002658.  Google Scholar

[15]

B. Lian and S. Hu, Stochastic delay Gilpin-Ayala competition models, Stochastics and Dynamics, 6 (2006), 561-576.  doi: 10.1142/S0219493706001888.  Google Scholar

[16]

W. MaoS. YouX. Wu and X. Mao, On the averaging principle for stochastic delay differential equations with jumps, Advances in Difference Equations, 2015 (2015), 1-19.  doi: 10.1186/s13662-015-0411-0.  Google Scholar

[17]

G. Shevchenko, Mixed stochastic delay differential equations, Theory of Probability and Mathematical Statistics, 89 (2014), 181-195.  doi: 10.1090/S0094-9000-2015-00944-3.  Google Scholar

[18]

M. Milosevic, An explicit analytic approximation of solutions for a class of neutral stochastic differential equations with time-dependent delay based on Taylor expansion, Applied Mathematics and Computation, 274 (2016), 745-761.  doi: 10.1016/j.amc.2015.11.026.  Google Scholar

[19]

E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307.  doi: 10.1016/S0377-0427(00)00475-1.  Google Scholar

[20]

I. Podlubny, Fractional differential equations, Math. Sci. Eng., 198 (1998).  Google Scholar

[21]

A. Babaei, H. Jafari and S. Banihashemi, Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, Journal of Computational and Applied Mathematics, 377 (2020), 112908. doi: 10.1016/j.cam.2020.112908.  Google Scholar

[22]

C. AngstmannA. M. EricksonB. I. HenryA. V. McGannJ. M. Murray and J. Nichols, Fractional order compartment models, SIAM Journal on Applied Mathematics, 77 (2017), 430-446.  doi: 10.1137/16M1069249.  Google Scholar

[23]

A. Babaei, B. Parsa Moghaddam, S. Banihashemi and J. A. Tenreiro Machado, Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations, Communications in Nonlinear Science and Numerical Simulation, 82 (2019), 104985. doi: 10.1016/j.cnsns.2019.104985.  Google Scholar

[24]

R. M. Ganji, H. Jafari and S. Nemati, A new approach for solving integro-differential equations of variable order, Journal of Computational and Applied Mathematics, 379 (2020), 112946. doi: 10.1016/j.cam.2020.112946.  Google Scholar

[25]

A. Babaei and S. Banihashemi, Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numerical Methods for Partial Differential Equations, 35 (2019), 976-992.  doi: 10.1002/num.22334.  Google Scholar

[26]

M. Izadi and C. Cattani, Generalized Bessel polynomial for multi-order fractional differential equations, Symmetry, 12 (2020), 1260. doi: 10.3390/sym12081260.  Google Scholar

[27]

D. N. Tien, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397 (2013), 334-348.  doi: 10.1016/j.jmaa.2012.07.062.  Google Scholar

[28]

Z. G. Yu, V. Anh, Y. Wang, D. Mao and J. Wanliss, Modeling and simulation of the horizontal component of the geomagnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition, J. Geophys. Res. Space Phys., 115 (2010). doi: 10.1029/2009JA015206.  Google Scholar

[29]

E. Abdel-Rehim, From the Ehrenfest model to time-fractional stochastic processes, J. Comput. Appl. Math., 233 (2009), 197-207.  doi: 10.1016/j.cam.2009.07.010.  Google Scholar

[30]

A. Babaei, H. Jafari and S. Banihashemi, A collocation approach for solving time-fractional stochastic heat equation driven by an additive noise, Symmetry, 12 (2020), 904. doi: 10.3390/sym12060904.  Google Scholar

[31]

T. S. DoanP. T. HuongP. E. Kloeden and H. T. Tuana, Asymptotic separation between solutions of Caputo fractional stochastic differential equations, Stochastic Analysis and Applications, 36 (2018), 1-11.  doi: 10.1080/07362994.2018.1440243.  Google Scholar

[32]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Physica D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[33]

B. P. MoghaddamL. ZhangA. M. LopesJ. A. Tenreiro Machado and Z. S. Mostaghim, Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, An International Journal of Probability and Stochastic Processes, 92 (2020), 379-396.  doi: 10.1080/17442508.2019.1625903.  Google Scholar

[34]

E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, Int. J. Eng. Sci., 3 (1965), 213-229.  doi: 10.1016/0020-7225(65)90045-5.  Google Scholar

[35]

E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.  doi: 10.1214/aoms/1177699916.  Google Scholar

[36]

X. WangS. Gan and D. Wang, $ \theta $-Maruyama methods for nonlinear stochastic differential delay equations, Appl. Numer. Math., 98 (2015), 38-58.  doi: 10.1016/j.apnum.2015.08.004.  Google Scholar

[37]

B. P. MoghaddamL. ZhangA. M. LopesJ. A. T. Machado and Z. S. Mostaghim, Computational scheme for solving nonlinear fractional stochastic differential equations with delay, Stochastic Analysis and Applications, 37 (2019), 893-908.  doi: 10.1080/07362994.2019.1621182.  Google Scholar

[38]

I. J. Gyongy and T. Martinez, On numerical solution of stochastic partial differential equations of elliptic type, Stochastics: An International Journal of Probability and Stochastic Processes, 78 (2006), 213-231.  doi: 10.1080/17442500600805047.  Google Scholar

[39]

C. Roth, A combination of finite difference and Wong-Zakai methods for hyperbolic stochastic partial differential equations, Stoch. Anal. Appl., 24 (2006), 221-240.  doi: 10.1080/07362990500397764.  Google Scholar

[40]

J. B.Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018.   Google Scholar

[41]

Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises, SIAM J. Numer. Anal., 40 (2002), 1421-1445.  doi: 10.1137/S0036142901387956.  Google Scholar

[42]

M. H. HeydariM. R. HooshmandaslG. B. Loghmani and C. Cattani, Wavelets Galerkin method for solving stochastic heat equation, International Journal of Computer Mathematics, 93 (2016), 1579-1596.  doi: 10.1080/00207160.2015.1067311.  Google Scholar

[43]

F. Mirzaee and E. Hadadiyan, Solving system of linear Stratonovich Volterra integral equations via modification of hat functions, Applied Mathematics and Computation, 293 (2017), 254-264.  doi: 10.1016/j.amc.2016.08.016.  Google Scholar

[44]

Q. LiT. Kang and Q. Zhang, Mean-square dissipative methods for stochastic agedependent capital system with fractional Brownian motion and jumps, Appl. Math. Comput., 339 (2018), 81-92.  doi: 10.1016/j.amc.2018.07.018.  Google Scholar

[45]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006.  Google Scholar

[46]

X. ZhongS. Guo and M. Peng, Stability of stochastic SIRS epidemic models with saturated incidence rates and delay, Stochastic Analysis and Applications, 35 (2017), 1-26.  doi: 10.1080/07362994.2016.1244644.  Google Scholar

Figure 1.  The exact and numerical solutions and absolute errors for Example 1 with $ \varepsilon = 1 $ when $ \alpha = 0.75 $ and $ n = 10 $
Figure 2.  The absolute error for $ t\in (0,\tau ) $ (left) and $ t\in (\tau , 1 ) $ (right) in Example 1 when $ \varepsilon = 1 $, $ \alpha = 0.75 $ and $ n = 10 $
Figure 3.  The exact and numerical solutions in Example 1 for several values of $ \bar{\mathrm{P}} $
Figure 4.  The exact and numerical solutions in Example 1 for several values of $ \varepsilon $
Figure 5.  The exact and numerical solutions (up) and absolute error (down) in Example 2 for $ \alpha = 0.75 $ on the domain $ [0,4] $
Figure 6.  The exact and numerical solutions in Example 2 for numerous values of $ \alpha $
Figure 7.  The phase-space diagram of Example 2 with $ \alpha = 0.99 $
40] (up) and the proposed method (down) with $ \alpha = 1 $ for Example 3">Figure 8.  The numerical solution obtained by $ \theta $-Maruyama methods [40] (up) and the proposed method (down) with $ \alpha = 1 $ for Example 3
Figure 9.  The phase-space diagram of Example 3 with $ \alpha = 0.75 $
Figure 10.  The numerical approximation of $ u(t) $ along $ \bar{\mathrm{P}} = 1 $ and $ \bar{\mathrm{P}} = 100 $ discretized Brownian paths in Example 3 with $ \alpha = 0.55 $ and CPU-time = 8.921
Figure 11.  The numerical solution for different values of $ \alpha $ with $ \varepsilon = 0.2 $ in Example 3
Figure 12.  Graphs of the trajectories of $ S(t) $, $ I(t) $ and $ R(t) $ for the deterministic SIRS model with $ \sigma = 0 $ (blue) and the stochastic SIRS model (red) with $ \sigma = 0.5 $ (up) and $ \sigma = 1.2 $ (down)
Figure 13.  The trajectories of $ S(t) $, $ I(t) $ and $ R(t) $ along different values of discretized Brownian paths
Algorithm.
Input: $ T, \tau\in \mathbb{R}^{+} $, $ n\in \mathbb{Z}^{+} $, $ \alpha\in (0,1) $, functions $ \mathrm{P} $, $ \mathrm{H} $, $ \eta $ and Brownian motion process $ \mathrm{B}(t) $.
Step 1: Compute the shifted Legendre polynomials $ \theta_{i}^{a,b}(t) $ from Definition 2.3.
Step 2: Compute the vector of shifted Legendre polynomials $ \Theta_{a,b}(t) $ from Eq.(13).
Step 3: Compute the collocation points $ t_{i}^{0,\tau} $ for $ i = 0, ..., n $ of the domain $ [0,\ \tau] $ from Eq.(20).
Step 4: Compute the matrices $ \mathbf{A}_{0,\tau} $ and $ \mathbf{B}_{0,\tau} $ from Eqs. (22) and (25).
Step 5: Compute the vector $ \mathbf{D}_{0,\tau}^{\alpha}(t) $ from Eq. (18) and the vectors $ \mathbf{P}_{0,\tau} $ and $ \mathbf{H}_{0,\tau} $ from Eqs. (23) and (24).
Step 6: Solve the nonlinear system $ \mathbf{A}_{0,\tau}^{T}\mathbf{C}_{0,\tau} = \mathbf{P}_{0,\tau}+\mathbf{B}_{0, \tau}\mathbf{H}_{0,\tau} $ and obtain the unknown vector $ \mathbf{C}_{0,\tau} $ by using Step 4 and Step 5.
Step 7: Let $ \mathfrak{U}_{n}^{1}(t) : = \mathbf{C}_{0,\tau}\Theta_{0,\tau}(t) $ on the interval $ [0, \tau] $.
Step 8: Start temporal loop for $ j = 2, ..., M $ where $ M = [\frac{T}{\tau}] $:
Step 8.1: Compute the collocation points $ t_{i}^{(j-1)\tau ,j\tau} $ for $ i = 0, ..., n $ of the domain $ [(j-1)\tau , j\tau] $ from Eqs. (35)-(36).
Step 8.2: Compute the matrices $ \mathbf{A}_{(j-1)\tau , j\tau} $ and $ \mathbf{B}_{(j-1)\tau , j\tau} $ from (30) and (34).
Step 8.3: Compute the vectors $ \mathbf{P}_{(j-1)\tau , j\tau} $, $ \mathbf{H}_{(j-1)\tau , j\tau} $ and $ \mathbf{D}_{(j-1)\tau , j\tau}^{\alpha}(t) $ from (31)-(33).
Step 8.4: Solve the nonlinear system
          $\mathbf{A}_{(j-1)\tau , j\tau}^{T} \mathbf{C}_{(j-1)\tau ,j\tau} = \mathbf{P}_{(j-1)\tau ,j\tau}+\mathbf{B}_{(j-1)\tau , j\tau}\mathbf{H}_{(j-1)\tau , j\tau} $
and obtain the unknown vector $ \mathbf{C}_{(j-1)\tau , j\tau} $ by using Step 8.2 and Step 8.3.
Step 8.5: Let $ \mathfrak{U}_{n}^{j}(t) : = \mathbf{C}_{(j-1)\tau , j\tau}^{T} \Theta _{(j-1)\tau , j\tau} (t) $ on $ [(j-1)\tau , j\tau] $.
Step 9: Post-processing the results.
Output: The approximate solution: $ u(t)\simeq \mathfrak{U}_{n}(t) $ from (37).
Algorithm.
Input: $ T, \tau\in \mathbb{R}^{+} $, $ n\in \mathbb{Z}^{+} $, $ \alpha\in (0,1) $, functions $ \mathrm{P} $, $ \mathrm{H} $, $ \eta $ and Brownian motion process $ \mathrm{B}(t) $.
Step 1: Compute the shifted Legendre polynomials $ \theta_{i}^{a,b}(t) $ from Definition 2.3.
Step 2: Compute the vector of shifted Legendre polynomials $ \Theta_{a,b}(t) $ from Eq.(13).
Step 3: Compute the collocation points $ t_{i}^{0,\tau} $ for $ i = 0, ..., n $ of the domain $ [0,\ \tau] $ from Eq.(20).
Step 4: Compute the matrices $ \mathbf{A}_{0,\tau} $ and $ \mathbf{B}_{0,\tau} $ from Eqs. (22) and (25).
Step 5: Compute the vector $ \mathbf{D}_{0,\tau}^{\alpha}(t) $ from Eq. (18) and the vectors $ \mathbf{P}_{0,\tau} $ and $ \mathbf{H}_{0,\tau} $ from Eqs. (23) and (24).
Step 6: Solve the nonlinear system $ \mathbf{A}_{0,\tau}^{T}\mathbf{C}_{0,\tau} = \mathbf{P}_{0,\tau}+\mathbf{B}_{0, \tau}\mathbf{H}_{0,\tau} $ and obtain the unknown vector $ \mathbf{C}_{0,\tau} $ by using Step 4 and Step 5.
Step 7: Let $ \mathfrak{U}_{n}^{1}(t) : = \mathbf{C}_{0,\tau}\Theta_{0,\tau}(t) $ on the interval $ [0, \tau] $.
Step 8: Start temporal loop for $ j = 2, ..., M $ where $ M = [\frac{T}{\tau}] $:
Step 8.1: Compute the collocation points $ t_{i}^{(j-1)\tau ,j\tau} $ for $ i = 0, ..., n $ of the domain $ [(j-1)\tau , j\tau] $ from Eqs. (35)-(36).
Step 8.2: Compute the matrices $ \mathbf{A}_{(j-1)\tau , j\tau} $ and $ \mathbf{B}_{(j-1)\tau , j\tau} $ from (30) and (34).
Step 8.3: Compute the vectors $ \mathbf{P}_{(j-1)\tau , j\tau} $, $ \mathbf{H}_{(j-1)\tau , j\tau} $ and $ \mathbf{D}_{(j-1)\tau , j\tau}^{\alpha}(t) $ from (31)-(33).
Step 8.4: Solve the nonlinear system
          $\mathbf{A}_{(j-1)\tau , j\tau}^{T} \mathbf{C}_{(j-1)\tau ,j\tau} = \mathbf{P}_{(j-1)\tau ,j\tau}+\mathbf{B}_{(j-1)\tau , j\tau}\mathbf{H}_{(j-1)\tau , j\tau} $
and obtain the unknown vector $ \mathbf{C}_{(j-1)\tau , j\tau} $ by using Step 8.2 and Step 8.3.
Step 8.5: Let $ \mathfrak{U}_{n}^{j}(t) : = \mathbf{C}_{(j-1)\tau , j\tau}^{T} \Theta _{(j-1)\tau , j\tau} (t) $ on $ [(j-1)\tau , j\tau] $.
Step 9: Post-processing the results.
Output: The approximate solution: $ u(t)\simeq \mathfrak{U}_{n}(t) $ from (37).
Table 1.  The $ l_{\infty } $-norm and $ l_{2 } $-norm errors, convergence orders and CPU-time for Example 1
$ {n} $ $ {\|\mathcal {E}_n\|_{_\infty}} $ CO $ {\|\mathcal {E}_n\|_{2}} $ CO CPU-time(s)
$ {6} $ $ {5.9498\times10^{-2}} $ $ {\; \; \; -} $ $ {1.1604\times10^{-2}} $ $ \; \; \; - $ $ 5.149 $
$ {9} $ $ { 1.1548 \times10^{-6}} $ $ {26.7585} $ $ 1.1768\times10^{-7} $ $ 28.3589 $ $ 7.982 $
$ {12} $ $ { 7.8465\times10^{-11}} $ $ { 33.3590} $ $ 4.2500\times10^{-12} $ $ 35.5559 $ $ 12.986 $
$ {n} $ $ {\|\mathcal {E}_n\|_{_\infty}} $ CO $ {\|\mathcal {E}_n\|_{2}} $ CO CPU-time(s)
$ {6} $ $ {5.9498\times10^{-2}} $ $ {\; \; \; -} $ $ {1.1604\times10^{-2}} $ $ \; \; \; - $ $ 5.149 $
$ {9} $ $ { 1.1548 \times10^{-6}} $ $ {26.7585} $ $ 1.1768\times10^{-7} $ $ 28.3589 $ $ 7.982 $
$ {12} $ $ { 7.8465\times10^{-11}} $ $ { 33.3590} $ $ 4.2500\times10^{-12} $ $ 35.5559 $ $ 12.986 $
Table 2.  Absolute errors, CPU-time and convergence orders for Example 2
$ {n} $ $ \alpha=0.25 $ $ \alpha =0.75 $
$ {\|\mathcal {E}_n\|_{_\infty}} $ CO $ {\|\mathcal {E}_n\|_{_\infty}} $ CO CPU-time(s)
$ {6} $ $ {5.7280\times10^{-6}} $ $ {\; \; \; -} $ $ {1.4068\times10^{-4}} $ $ \; \; \; - $ $ 77.011 $
$ {9} $ $ { 3.4219 \times10^{-9}} $ $ {18.3071} $ $ 1.9357\times10^{-7} $ $ 16.2495 $ $ 135.44 $
$ {12} $ $ { 1.1281\times10^{-11}} $ $ { 19.8650} $ $ 9.6833\times10^{-10} $ $ 18.4155 $ $ 178.156 $
$ {n} $ $ \alpha=0.25 $ $ \alpha =0.75 $
$ {\|\mathcal {E}_n\|_{_\infty}} $ CO $ {\|\mathcal {E}_n\|_{_\infty}} $ CO CPU-time(s)
$ {6} $ $ {5.7280\times10^{-6}} $ $ {\; \; \; -} $ $ {1.4068\times10^{-4}} $ $ \; \; \; - $ $ 77.011 $
$ {9} $ $ { 3.4219 \times10^{-9}} $ $ {18.3071} $ $ 1.9357\times10^{-7} $ $ 16.2495 $ $ 135.44 $
$ {12} $ $ { 1.1281\times10^{-11}} $ $ { 19.8650} $ $ 9.6833\times10^{-10} $ $ 18.4155 $ $ 178.156 $
Table 3.  The parameter values in the stochastic SIRS model
Parameter Value Parameter Value
$ \Lambda $ $ 1.8 $ $ \mu_{2} $ $ 0.5 $
$ \beta $ $ 0.2 $ $ \mu_{3} $ $ 0.5 $
$ \tilde{\beta} $ $ 0.1 $ $ \gamma $ $ 0.3 $
$ \mu_{1} $ $ 0.85 $ $ \tilde{\gamma} $ $ 0.25 $
Parameter Value Parameter Value
$ \Lambda $ $ 1.8 $ $ \mu_{2} $ $ 0.5 $
$ \beta $ $ 0.2 $ $ \mu_{3} $ $ 0.5 $
$ \tilde{\beta} $ $ 0.1 $ $ \gamma $ $ 0.3 $
$ \mu_{1} $ $ 0.85 $ $ \tilde{\gamma} $ $ 0.25 $
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