American Institute of Mathematical Sciences

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), no. 2, 766-775. [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. [4] F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal., 6 (1975), 312-317. [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. [6] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, Vol. 3, North-Holland, Amsterdam, 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(4) (1987), 784-800. [8] K. L. Cooke, An epidemic equation with immigration, Math. Biosci., 29(1-2) (1976), 135-158. [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(11) (2003), 3255-3271. [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. [12] S. Elaydi, An introduction to difference equations. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 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, 1994), J. Comput. Appl. Math., 66(1-2) (1996), 213-225. [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. [15] M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Biosci. 62 (1982), no. 1, 4574. [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. [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. [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. [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), 106120. [21] Ch. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal., 3(4) (1983), 439-465. [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. [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. [24] R. K. Miller, On the linearization of Volterra integral equations. J. Math. Anal. Appl., 23 (1968), 198-208. [25] Y. Song and C. T. H. Baker, Perturbation theory for discrete Volterra equations. J. Difference Equ. Appl., 9(10) (2003), 969-987. [26] P. van den Driessche, J. Watmough, A simple SIS epidemic model with a backward bifurcation. J. Math. Biol., 40(6) (2000), 525-540. [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. [28] A. Vecchio, On the resolvent kernel of Volterra discrete equations. Funct. Differ. Equ., 6(1-2) (1999), 191-201.

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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), no. 2, 766-775. [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. [4] F. Brauer, On a nonlinear integral equation for population growth problems, SIAM J. Math. Anal., 6 (1975), 312-317. [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. [6] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs, Vol. 3, North-Holland, Amsterdam, 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(4) (1987), 784-800. [8] K. L. Cooke, An epidemic equation with immigration, Math. Biosci., 29(1-2) (1976), 135-158. [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(11) (2003), 3255-3271. [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. [12] S. Elaydi, An introduction to difference equations. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 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, 1994), J. Comput. Appl. Math., 66(1-2) (1996), 213-225. [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. [15] M. Gyllenberg, Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures, Math. Biosci. 62 (1982), no. 1, 4574. [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. [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. [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. [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), 106120. [21] Ch. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal., 3(4) (1983), 439-465. [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. [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. [24] R. K. Miller, On the linearization of Volterra integral equations. J. Math. Anal. Appl., 23 (1968), 198-208. [25] Y. Song and C. T. H. Baker, Perturbation theory for discrete Volterra equations. J. Difference Equ. Appl., 9(10) (2003), 969-987. [26] P. van den Driessche, J. Watmough, A simple SIS epidemic model with a backward bifurcation. J. Math. Biol., 40(6) (2000), 525-540. [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. [28] A. Vecchio, On the resolvent kernel of Volterra discrete equations. Funct. Differ. Equ., 6(1-2) (1999), 191-201.
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