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

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.
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).

[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), 766.

[3]

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

[4]

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

[5]

H. Brunner, Collocation methods for Volterra integral and related functional differential equations,, Cambridge Monographs on Applied and Computational Mathematics, (2004).

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations,, CWI Monographs, (1986).

[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 (1987), 784.

[8]

K. L. Cooke, An epidemic equation with immigration,, Math. Biosci., 29 (1976), 1.

[9]

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

[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 (2003), 3255.

[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 (1991), 149.

[12]

S. Elaydi, An introduction to difference equations., Third edition. Undergraduate Texts in Mathematics. Springer, (2005).

[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, 66 (1996), 1.

[14]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations., Encyclopedia of Mathematics and its Applications, (1990).

[15]

M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures,, Math. Biosci. 62 (1982), 62 (1982).

[16]

I. Győri and D.W. Reynolds, On admissibility of the resolvent of discrete Volterra equations,, J. Difference Equ. Appl. 16 (2010), 16 (2010), 1393.

[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), 34 (1995).

[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), 40 (2000), 1.

[19]

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

[20]

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

[21]

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

[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).

[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), 81 (2011), 1017.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

A. Vecchio, On the resolvent kernel of Volterra discrete equations., Funct. Differ. Equ., 6 (1999), 1.

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).

[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), 766.

[3]

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

[4]

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

[5]

H. Brunner, Collocation methods for Volterra integral and related functional differential equations,, Cambridge Monographs on Applied and Computational Mathematics, (2004).

[6]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations,, CWI Monographs, (1986).

[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 (1987), 784.

[8]

K. L. Cooke, An epidemic equation with immigration,, Math. Biosci., 29 (1976), 1.

[9]

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

[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 (2003), 3255.

[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 (1991), 149.

[12]

S. Elaydi, An introduction to difference equations., Third edition. Undergraduate Texts in Mathematics. Springer, (2005).

[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, 66 (1996), 1.

[14]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra integral and functional equations., Encyclopedia of Mathematics and its Applications, (1990).

[15]

M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures,, Math. Biosci. 62 (1982), 62 (1982).

[16]

I. Győri and D.W. Reynolds, On admissibility of the resolvent of discrete Volterra equations,, J. Difference Equ. Appl. 16 (2010), 16 (2010), 1393.

[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), 34 (1995).

[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), 40 (2000), 1.

[19]

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

[20]

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

[21]

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

[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).

[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), 81 (2011), 1017.

[24]

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

[25]

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

[26]

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

[27]

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

[28]

A. Vecchio, On the resolvent kernel of Volterra discrete equations., Funct. Differ. Equ., 6 (1999), 1.

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177

[18]

Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897

[19]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559

[20]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118

 Impact Factor: 

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]