American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 826-834. doi: 10.3934/proc.2015.0826

Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation

Eleonora Messina 1,

1. 

Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples

Received  September 2014 Revised  July 2015 Published  November 2015

We consider a SIS epidemic model based on a Volterra integral equation and we compare the dynamical behavior of the analytical solution and its numerical approximation obtained by direct quadrature methods. We prove that, under suitable assumptions, the numerical scheme preserves the qualitative properties of the continuous equation and we show that, as the stepsize tends to zero, the numerical bifurcation points tend to the continuous ones.
Keywords: Volterra integral equations, bifurcations, numerical stability..
Mathematics Subject Classification: Primary: 45D05, 65R20; Secondary: 65L2.
Citation: Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826
References:
[1]

R. P.Agarwal and D. O'Regan, Integral and Integrodifferential Equations: Theory, Methods and Applications. Gordon and Breach Science Publishers, 2000. Google Scholar

[2]

M. Annunziato, H. Brunner and E. Messina, Asymptotic stability of solutions to Volterra-renewal integral equations with space maps, J. Math. Anal. Appl., 395 (2012), no. 2, 766-775. Google Scholar

[3]

C. T. H. Baker and M. S. Keech, Stability regions in the numerical treatment of Volterra integral equations, SIAM J. Numer. Anal., 15(2) (1978), 394-417. Google Scholar

[4]

F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal., 6 (1975), 312-317. Google Scholar

[5]

H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, 15. Cambridge University Press, Cambridge, 2004. Google Scholar

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, Vol. 3, North-Holland, Amsterdam, 1986. Google Scholar

[7]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19(4) (1987), 784-800. Google Scholar

[8]

K. L. Cooke, An epidemic equation with immigration, Math. Biosci., 29(1-2) (1976), 135-158. Google Scholar

[9]

O. Diekmann, Limiting behaviour in an epidemic model,, Nonlinear Anal., 1 (): 459.   Google Scholar

[10]

J. T. Edwards, N. J. Ford and J. A. Roberts, Bifurcations in numerical methods for Volterra integro-differential equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13(11) (2003), 3255-3271. Google Scholar

[11]

P. P. B. Eggermont and C. Lubich, Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line. Math. Comp., 56(193) (1991), 149-176. Google Scholar

[12]

S. Elaydi, An introduction to difference equations. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 2005. Google Scholar

[13]

N. J. Ford and C. T. H. Baker, Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type. Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), J. Comput. Appl. Math., 66(1-2) (1996), 213-225. Google Scholar

[14]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. Google Scholar

[15]

M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Biosci. 62 (1982), no. 1, 4574. Google Scholar

[16]

I. Győri and D.W. Reynolds, On admissibility of the resolvent of discrete Volterra equations, J. Difference Equ. Appl. 16 (2010), no. 12, 1393-1412. Google Scholar

[17]

H. W. Hethcote and P. van den Driessche, An SIS epidemic model with variable population size and a delay. J. Math. Biol. 34 (1995), no. 2, 177194. Google Scholar

[18]

V. B. Kolmanovskii, A. D. Myshkis and J.-P. Richard, Estimate of solutions for some Volterra difference equations. Lakshmikantham's legacy: a tribute on his 75th birthday, Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 345363. Google Scholar

[19]

P. Linz, Analytical and numerical methods for Volterra Equations, Philadelphia: S.I.A.M., 1985. Google Scholar

[20]

S.-O. Londen, On a nonlinear Volterra integral equation, J. Differential Equations 14 (1973), 106120. Google Scholar

[21]

Ch. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal., 3(4) (1983), 439-465. Google Scholar

[22]

E. Messina, Y. Muroya, E. Russo, and A. Vecchio, On the stability of numerical methods for nonlinear Volterra integral equations. Discrete Dyn. Nat. Soc., 2010, Art. ID 862538, 18 pp. Google Scholar

[23]

E. Messina, E. Russo, and A. Vecchio, Comparing analytical and numerical solution of a nonlinear two-delay integral equations. Math. Comput. Simulation 81(5) (2011), 1017-1026. Google Scholar

[24]

R. K. Miller, On the linearization of Volterra integral equations. J. Math. Anal. Appl., 23 (1968), 198-208. Google Scholar

[25]

Y. Song and C. T. H. Baker, Perturbation theory for discrete Volterra equations. J. Difference Equ. Appl., 9(10) (2003), 969-987. Google Scholar

[26]

P. van den Driessche, J. Watmough, A simple SIS epidemic model with a backward bifurcation. J. Math. Biol., 40(6) (2000), 525-540. Google Scholar

[27]

A. Vecchio, Stability of Direct Quadrature methods for systems of Volterra integral equations. J. of Comput. Meth. in Sci. and Eng., 3(2) (2003), 71-80. Google Scholar

[28]

A. Vecchio, On the resolvent kernel of Volterra discrete equations. Funct. Differ. Equ., 6(1-2) (1999), 191-201. Google Scholar

show all references

References:
[1]

R. P.Agarwal and D. O'Regan, Integral and Integrodifferential Equations: Theory, Methods and Applications. Gordon and Breach Science Publishers, 2000. Google Scholar

[2]

M. Annunziato, H. Brunner and E. Messina, Asymptotic stability of solutions to Volterra-renewal integral equations with space maps, J. Math. Anal. Appl., 395 (2012), no. 2, 766-775. Google Scholar

[3]

C. T. H. Baker and M. S. Keech, Stability regions in the numerical treatment of Volterra integral equations, SIAM J. Numer. Anal., 15(2) (1978), 394-417. Google Scholar

[4]

F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal., 6 (1975), 312-317. Google Scholar

[5]

H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, 15. Cambridge University Press, Cambridge, 2004. Google Scholar

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, Vol. 3, North-Holland, Amsterdam, 1986. Google Scholar

[7]

A. Buonocore, A. G. Nobile and L. M. Ricciardi, A new integral equation for the evaluation of first-passage-time probability densities, Adv. in Appl. Probab., 19(4) (1987), 784-800. Google Scholar

[8]

K. L. Cooke, An epidemic equation with immigration, Math. Biosci., 29(1-2) (1976), 135-158. Google Scholar

[9]

O. Diekmann, Limiting behaviour in an epidemic model,, Nonlinear Anal., 1 (): 459.   Google Scholar

[10]

J. T. Edwards, N. J. Ford and J. A. Roberts, Bifurcations in numerical methods for Volterra integro-differential equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13(11) (2003), 3255-3271. Google Scholar

[11]

P. P. B. Eggermont and C. Lubich, Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line. Math. Comp., 56(193) (1991), 149-176. Google Scholar

[12]

S. Elaydi, An introduction to difference equations. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 2005. Google Scholar

[13]

N. J. Ford and C. T. H. Baker, Qualitative behaviour and stability of solutions of discretised nonlinear Volterra integral equations of convolution type. Proceedings of the Sixth International Congress on Computational and Applied Mathematics (Leuven, 1994), J. Comput. Appl. Math., 66(1-2) (1996), 213-225. Google Scholar

[14]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations. Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. Google Scholar

[15]

M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Biosci. 62 (1982), no. 1, 4574. Google Scholar

[16]

I. Győri and D.W. Reynolds, On admissibility of the resolvent of discrete Volterra equations, J. Difference Equ. Appl. 16 (2010), no. 12, 1393-1412. Google Scholar

[17]

H. W. Hethcote and P. van den Driessche, An SIS epidemic model with variable population size and a delay. J. Math. Biol. 34 (1995), no. 2, 177194. Google Scholar

[18]

V. B. Kolmanovskii, A. D. Myshkis and J.-P. Richard, Estimate of solutions for some Volterra difference equations. Lakshmikantham's legacy: a tribute on his 75th birthday, Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 345363. Google Scholar

[19]

P. Linz, Analytical and numerical methods for Volterra Equations, Philadelphia: S.I.A.M., 1985. Google Scholar

[20]

S.-O. Londen, On a nonlinear Volterra integral equation, J. Differential Equations 14 (1973), 106120. Google Scholar

[21]

Ch. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal., 3(4) (1983), 439-465. Google Scholar

[22]

E. Messina, Y. Muroya, E. Russo, and A. Vecchio, On the stability of numerical methods for nonlinear Volterra integral equations. Discrete Dyn. Nat. Soc., 2010, Art. ID 862538, 18 pp. Google Scholar

[23]

E. Messina, E. Russo, and A. Vecchio, Comparing analytical and numerical solution of a nonlinear two-delay integral equations. Math. Comput. Simulation 81(5) (2011), 1017-1026. Google Scholar

[24]

R. K. Miller, On the linearization of Volterra integral equations. J. Math. Anal. Appl., 23 (1968), 198-208. Google Scholar

[25]

Y. Song and C. T. H. Baker, Perturbation theory for discrete Volterra equations. J. Difference Equ. Appl., 9(10) (2003), 969-987. Google Scholar

[26]

P. van den Driessche, J. Watmough, A simple SIS epidemic model with a backward bifurcation. J. Math. Biol., 40(6) (2000), 525-540. Google Scholar

[27]

A. Vecchio, Stability of Direct Quadrature methods for systems of Volterra integral equations. J. of Comput. Meth. in Sci. and Eng., 3(2) (2003), 71-80. Google Scholar

[28]

A. Vecchio, On the resolvent kernel of Volterra discrete equations. Funct. Differ. Equ., 6(1-2) (1999), 191-201. Google Scholar

[1]

Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087
[2]

T. Diogo, P. Lima, M. Rebelo. Numerical solution of a nonlinear Abel type Volterra integral equation. Communications on Pure & Applied Analysis, 2006, 5 (2) : 277-288. doi: 10.3934/cpaa.2006.5.277
[3]

M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42
[4]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[5]

Zhongying Chen, Bin Wu, Yuesheng Xu. Fast numerical collocation solutions of integral equations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 643-666. doi: 10.3934/cpaa.2007.6.643
[6]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
[7]

Onur Alp İlhan. Solvability of some volterra type integral equations in hilbert space. Conference Publications, 2007, 2007 (Special) : 28-34. doi: 10.3934/proc.2007.2007.28
[8]

Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251
[9]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[10]

Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004
[11]

Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397
[12]

Yushi Hamaguchi. Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems. Mathematical Control & Related Fields, 2021, 11 (2) : 433-478. doi: 10.3934/mcrf.2020043
[13]

Ludger Overbeck, Jasmin A. L. Röder. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 4-. doi: 10.1186/s41546-018-0030-2
[14]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[15]

Charles L. Epstein, Leslie Greengard, Thomas Hagstrom. On the stability of time-domain integral equations for acoustic wave propagation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4367-4382. doi: 10.3934/dcds.2016.36.4367
[16]

Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261
[17]

Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042
[18]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401
[19]

Hermann Brunner. On Volterra integral operators with highly oscillatory kernels. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 915-929. doi: 10.3934/dcds.2014.34.915
[20]

Sergiu Aizicovici, Yimin Ding, N. S. Papageorgiou. Time dependent Volterra integral inclusions in Banach spaces. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 53-63. doi: 10.3934/dcds.1996.2.53

 Impact Factor: 

Tools

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences