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doi: 10.3934/jcd.2021029
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## A non-standard numerical scheme for an age-of-infection epidemic model

 1 Department of Mathematics and Applications, University of Naples Federico II, Via Cintia, I-80126 Napoli, Italy 2 Member of the Italian INdAM Research group GNCS 3 C.N.R. National Research Council of Italy, Institute for Computational Application "Mauro Picone", Via P. Castellino, 111 - 80131 Napoli, Italy

* Corresponding author: Eleonora Messina

Received  April 2021 Revised  November 2021 Early access December 2021

We propose a numerical method for approximating integro-differential equations arising in age-of-infection epidemic models. The method is based on a non-standard finite differences approximation of the integral term appearing in the equation. The study of convergence properties and the analysis of the qualitative behavior of the numerical solution show that it preserves all the basic properties of the continuous model with no restrictive conditions on the step-length $h$ of integration and that it recovers the continuous dynamic as $h$ tends to zero.

Citation: Eleonora Messina, Mario Pezzella, Antonia Vecchio. A non-standard numerical scheme for an age-of-infection epidemic model. Journal of Computational Dynamics, doi: 10.3934/jcd.2021029
##### References:
 [1] J. Arino and P. van den Driessche, Time delays in epidemic models, Delay Differential Equations and Applications, NATO Science Series, 205 (2006), 539–578. doi: 10.1007/1-4020-3647-7_13.  Google Scholar [2] F. Brauer, Age of infection epidemic models, Mathematical and Statistical Modeling for Emerging and Re-Emerging Infectious Diseases, Springer, [Cham], (2016), 207–220.  Google Scholar [3] F. Brauer, Age-of-infection and the final size relation, Math. Biosci. Eng., 5 (2008), 681-690.  doi: 10.3934/mbe.2008.5.681.  Google Scholar [4] F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Biosci., 198 (2005), 119-131.  doi: 10.1016/j.mbs.2005.07.006.  Google Scholar [5] F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285-293.  doi: 10.1080/17513758.2016.1207813.  Google Scholar [6] F. Brauer, C. Castillo-Chavez and Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. doi: 10.1007/978-1-4939-9828-9.  Google Scholar [7] F. Brauer, Y. Xiao and S. M. Moghadas, Drug resistance in an age-of-infection model, Math. Popul. Stud., 24 (2017), 64-78.  doi: 10.1080/08898480.2015.1054216.  Google Scholar [8] D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454.  Google Scholar [9] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.  Google Scholar [10] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, The Netherlands, 1986.  Google Scholar [11] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2$^{nd}$ edition, Computer Science and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1984.  Google Scholar [12] O. Diekmann, and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley series in mathematical and computational biology; Wiley, J.; New York, 2000.  Google Scholar [13] O. Diekmann, J. A. J. Metz and J. A. P. Heesterbeek, The legacy of Kermack and McKendrick, D. Mollison (ed.) Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, Cambridge, (1995), 95–115. Google Scholar [14] Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), no.3,803–833. doi: 10.1137/S0036139998347834.  Google Scholar [15] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar [16] P. Linz, Analytical and Numerical Methods for Volterra Equations, Studies in Applied and Numerical Mathematics, Philadelphia, 1985. doi: 10.1137/1.9781611970852.  Google Scholar [17] J. M. S. Lubuma and Y. A. Terefe, A nonstandard Volterra difference equation for the SIS epidemiological model, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 109 (2015), 597-602.  doi: 10.1007/s13398-014-0203-5.  Google Scholar [18] E. Messina and A. Vecchio, A sufficient condition for the stability of direct quadrature methods for Volterra integral equations, Numer. Algorithms, 74 (2017), 1223-1236.  doi: 10.1007/s11075-016-0193-9.  Google Scholar [19] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Pub Co. Inc., River Edge, NJ, 1994. doi: 10.1142/2081.  Google Scholar [20] R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Difference Equ. Appl., 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.  Google Scholar [21] R. E. Mickens, Numerical integration of population models satisfying conservation laws: NSFD methods, J. Biol. Dyn., 1 (2007), 427-436.  doi: 10.1080/17513750701605598.  Google Scholar

show all references

##### References:
 [1] J. Arino and P. van den Driessche, Time delays in epidemic models, Delay Differential Equations and Applications, NATO Science Series, 205 (2006), 539–578. doi: 10.1007/1-4020-3647-7_13.  Google Scholar [2] F. Brauer, Age of infection epidemic models, Mathematical and Statistical Modeling for Emerging and Re-Emerging Infectious Diseases, Springer, [Cham], (2016), 207–220.  Google Scholar [3] F. Brauer, Age-of-infection and the final size relation, Math. Biosci. Eng., 5 (2008), 681-690.  doi: 10.3934/mbe.2008.5.681.  Google Scholar [4] F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Biosci., 198 (2005), 119-131.  doi: 10.1016/j.mbs.2005.07.006.  Google Scholar [5] F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285-293.  doi: 10.1080/17513758.2016.1207813.  Google Scholar [6] F. Brauer, C. Castillo-Chavez and Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. doi: 10.1007/978-1-4939-9828-9.  Google Scholar [7] F. Brauer, Y. Xiao and S. M. Moghadas, Drug resistance in an age-of-infection model, Math. Popul. Stud., 24 (2017), 64-78.  doi: 10.1080/08898480.2015.1054216.  Google Scholar [8] D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454.  Google Scholar [9] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.  Google Scholar [10] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, The Netherlands, 1986.  Google Scholar [11] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2$^{nd}$ edition, Computer Science and Applied Mathematics. Academic Press, Inc., Orlando, FL, 1984.  Google Scholar [12] O. Diekmann, and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley series in mathematical and computational biology; Wiley, J.; New York, 2000.  Google Scholar [13] O. Diekmann, J. A. J. Metz and J. A. P. Heesterbeek, The legacy of Kermack and McKendrick, D. Mollison (ed.) Epidemic Models: Their Structure and Relation to Data, Cambridge University Press, Cambridge, (1995), 95–115. Google Scholar [14] Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), no.3,803–833. doi: 10.1137/S0036139998347834.  Google Scholar [15] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721.  doi: 10.1098/rspa.1927.0118.  Google Scholar [16] P. Linz, Analytical and Numerical Methods for Volterra Equations, Studies in Applied and Numerical Mathematics, Philadelphia, 1985. doi: 10.1137/1.9781611970852.  Google Scholar [17] J. M. S. Lubuma and Y. A. Terefe, A nonstandard Volterra difference equation for the SIS epidemiological model, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 109 (2015), 597-602.  doi: 10.1007/s13398-014-0203-5.  Google Scholar [18] E. Messina and A. Vecchio, A sufficient condition for the stability of direct quadrature methods for Volterra integral equations, Numer. Algorithms, 74 (2017), 1223-1236.  doi: 10.1007/s11075-016-0193-9.  Google Scholar [19] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Pub Co. Inc., River Edge, NJ, 1994. doi: 10.1142/2081.  Google Scholar [20] R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Difference Equ. Appl., 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.  Google Scholar [21] R. E. Mickens, Numerical integration of population models satisfying conservation laws: NSFD methods, J. Biol. Dyn., 1 (2007), 427-436.  doi: 10.1080/17513750701605598.  Google Scholar
Problem (2)-(28): norm of the relative errors (solid line) as functions of the stepsize, compared to the slope of order one (dotted line)
Problem (2)-(29): numerical solution with $h = 0.1.$
Problem (2)-(29): comparison of numerical solutions with $h = 0.5.$
Error values and experimental order of convergence for example (2)-(28)
 $h$ Error on $S$ Error on $\varphi$ Exp. ord. for $S$ Exp. ord. for $\varphi$ $10^{-1}$ $1.17\cdot10^{-1}$ $4.11\cdot10^{-1}$ $\setminus\setminus$ $\setminus\setminus$ $10^{-2}$ $1.46\cdot10^{-2}$ $5.02\cdot10^{-2}$ $0.90$ $0.91$ $10^{-3}$ $1.49\cdot10^{-3}$ $5.13\cdot10^{-3}$ $0.99$ $0.99$ $10^{-4}$ $1.48\cdot10^{-4}$ $5.09\cdot10^{-4}$ $1.00$ $1.00$
 $h$ Error on $S$ Error on $\varphi$ Exp. ord. for $S$ Exp. ord. for $\varphi$ $10^{-1}$ $1.17\cdot10^{-1}$ $4.11\cdot10^{-1}$ $\setminus\setminus$ $\setminus\setminus$ $10^{-2}$ $1.46\cdot10^{-2}$ $5.02\cdot10^{-2}$ $0.90$ $0.91$ $10^{-3}$ $1.49\cdot10^{-3}$ $5.13\cdot10^{-3}$ $0.99$ $0.99$ $10^{-4}$ $1.48\cdot10^{-4}$ $5.09\cdot10^{-4}$ $1.00$ $1.00$
Values of the final size $S_{\infty}(h)$ as function of $h$ for problem (2)-(29)
 $h$ $S_{\infty}(h)$ $10^{-1}$ $2.3211\cdot10^{4}$ $10^{-2}$ $1.8852\cdot10^{4}$ $10^{-3}$ $1.8435\cdot10^{4}$
 $h$ $S_{\infty}(h)$ $10^{-1}$ $2.3211\cdot10^{4}$ $10^{-2}$ $1.8852\cdot10^{4}$ $10^{-3}$ $1.8435\cdot10^{4}$
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