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April  2022, 9(2): 239-252. doi: 10.3934/jcd.2021029

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 Published  April 2022 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, 2022, 9 (2) : 239-252. 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.

[2]

F. Brauer, Age of infection epidemic models, Mathematical and Statistical Modeling for Emerging and Re-Emerging Infectious Diseases, Springer, [Cham], (2016), 207–220.

[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.

[4]

F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Biosci., 198 (2005), 119-131.  doi: 10.1016/j.mbs.2005.07.006.

[5]

F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285-293.  doi: 10.1080/17513758.2016.1207813.

[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.

[7]

F. BrauerY. 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.

[8]

D. BredaO. DiekmannW. F. de GraafA. 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.

[9] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.
[10]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, The Netherlands, 1986.

[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.

[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.

[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.

[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.

[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.

[16]

P. Linz, Analytical and Numerical Methods for Volterra Equations, Studies in Applied and Numerical Mathematics, Philadelphia, 1985. doi: 10.1137/1.9781611970852.

[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.

[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.

[19]

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Pub Co. Inc., River Edge, NJ, 1994. doi: 10.1142/2081.

[20]

R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Difference Equ. Appl., 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.

[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.

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.

[2]

F. Brauer, Age of infection epidemic models, Mathematical and Statistical Modeling for Emerging and Re-Emerging Infectious Diseases, Springer, [Cham], (2016), 207–220.

[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.

[4]

F. Brauer, The Kermack-McKendrick epidemic model revisited, Math. Biosci., 198 (2005), 119-131.  doi: 10.1016/j.mbs.2005.07.006.

[5]

F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285-293.  doi: 10.1080/17513758.2016.1207813.

[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.

[7]

F. BrauerY. 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.

[8]

D. BredaO. DiekmannW. F. de GraafA. 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.

[9] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations,, Cambridge University Press, Cambridge, UK, 2004.  doi: 10.1017/CBO9780511543234.
[10]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, North-Holland, Amsterdam, The Netherlands, 1986.

[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.

[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.

[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.

[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.

[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.

[16]

P. Linz, Analytical and Numerical Methods for Volterra Equations, Studies in Applied and Numerical Mathematics, Philadelphia, 1985. doi: 10.1137/1.9781611970852.

[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.

[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.

[19]

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific Pub Co. Inc., River Edge, NJ, 1994. doi: 10.1142/2081.

[20]

R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Difference Equ. Appl., 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.

[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.

Figure 1.  Problem (2)-(28): norm of the relative errors (solid line) as functions of the stepsize, compared to the slope of order one (dotted line)
Figure 2.  Problem (2)-(29): numerical solution with $ h = 0.1. $
Figure 3.  Problem (2)-(29): comparison of numerical solutions with $ h = 0.5. $
Table 1.  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 $
Table 2.  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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