Nonlocal finite difference discretization of a class of renewal equation models for epidemics

Eleonora Messina; Mario Pezzella; Antonia Vecchio; Eleonora Messina; Mario Pezzella; Antonia Vecchio

doi:10.3934/mbe.2023518

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 11656-11675. doi: 10.3934/mbe.2023518

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Nonlocal finite difference discretization of a class of renewal equation models for epidemics

1.
Department of Mathematics and Applications, University of Naples Federico II, Via Cintia, I-80126 Naples, 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 Naples, Italy

Received: 28 February 2023 Revised: 26 April 2023 Accepted: 28 April 2023 Published: 06 May 2023

In this paper we consider a non-standard discretization to a Volterra integro-differential system which includes a number of age-of-infection models in the literature. The aim is to provide a general framework to analyze the proposed scheme for the numerical solution of a class of problems whose continuous dynamic is well known in the literature and allow a deeper analysis in cases where the theory lacks.
- Volterra integral equations,
- epidemic models,
- non-standard difference scheme,
- asymptotic dynamics
Citation: Eleonora Messina, Mario Pezzella, Antonia Vecchio. Nonlocal finite difference discretization of a class of renewal equation models for epidemics[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 11656-11675. doi: 10.3934/mbe.2023518

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Abstract

In this paper we consider a non-standard discretization to a Volterra integro-differential system which includes a number of age-of-infection models in the literature. The aim is to provide a general framework to analyze the proposed scheme for the numerical solution of a class of problems whose continuous dynamic is well known in the literature and allow a deeper analysis in cases where the theory lacks.

References

[1]	J. Cresson, F. Pierret, Non standard finite difference scheme preserving dynamical properties, J. Comput. Appl. Math., 303 (2016), 15–30. https://doi.org/10.1016/j.cam.2016.02.007 doi: 10.1016/j.cam.2016.02.007
[2]	J. T. Edwards, N. J. Ford, J. A. Roberts, Bifurcations in numerical methods for Volterra integro-differential equations, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 3255–3271. https://doi.org/10.1142/S0218127403008570 doi: 10.1142/S0218127403008570
[3]	J. Lubuma, Y. Terefe, A nonstandard Volterra difference equation for the SIS epidemiological model, RACSAM, 109 (2015), 597–602. https://doi.org/10.1007/s13398-014-0203-5 doi: 10.1007/s13398-014-0203-5
[4]	E. Messina, Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation, Discrete Contin. Dyn. Syst., 2015 (2015), 826–834. https://doi.org/10.3934/proc.2015.0826 doi: 10.3934/proc.2015.0826
[5]	R. E. Mickens, A note on a discretization scheme for Volterra integro-differential equations that preserves stability and boundedness, J. Differ. Equ. Appl., 13 (2007), 547–550. https://doi.org/10.1080/10236190601143245 doi: 10.1080/10236190601143245
[6]	F. Milner, A. Pugliese, Periodic solutions: A robust numerical method for an S-I-R model of epidemics, J. Math. Biol., 39 (1999), 471–492. https://doi.org/10.1007/s002850050175 doi: 10.1007/s002850050175
[7]	S. Vaz, D. Torres, A dynamically-consistent nonstandard finite difference scheme for the SICA model, Math. Biosci. Eng., 18 (2021), 4552–4571. https://doi.org/10.3934/mbe.2021231 doi: 10.3934/mbe.2021231
[8]	B. Wacker, J. Schlüter, An age- and sex-structured SIR model: Theory and an explicit-implicit numerical solution algorithm, Math. Biosci. Eng., 17 (2020), 5752–5801. https://doi.org/10.3934/mbe.2020309 doi: 10.3934/mbe.2020309
[9]	F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. https://doi.org/10.1007/978-1-4939-9828-9
[10]	F. Bai, An age-of-infection model with both symptomatic and asymptomatic infections, J. Math. Biol., 86 (2023), 82. https://doi.org/10.1007/s00285-023-01920-w doi: 10.1007/s00285-023-01920-w
[11]	F. Brauer, J. Watmough, Age of infection epidemic models with heterogeneous mixing, J. Biol. Dyn., 3 (2009), 324–330. https://doi.org/10.1080/17513750802415822 doi: 10.1080/17513750802415822
[12]	E. Messina, M. Pezzella, A. Vecchio, A long-time behavior preserving numerical scheme for age-of-infection epidemic models with heterogeneous mixing, Appl. Numer. Math., (2023). https://doi.org/10.1016/j.apnum.2023.04.009
[13]	J. David, Epidemic models with heterogeneous mixing and indirect transmission, J. Biol. Dyn., 12 (2018), 375–399. https://doi.org/10.1080/17513758.2018.1467506 doi: 10.1080/17513758.2018.1467506
[14]	F. Brauer, A new epidemic model with indirect transmission, J. Biol. Dyn., 11 (2017), 285–293. https://doi.org/10.1080/17513758.2016.1207813 doi: 10.1080/17513758.2016.1207813
[15]	N. Ford, C. Baker, Preserving transient behaviour in numerical solutions of Volterra integral equations of convolution type, Integral and integrodifferential equations, Ser. Math. Anal. Appl., Gordon and Breach, Amsterdam, 2 (2000), 77–89.
[16]	E. Hairer, C. Lubich, On the stability of Volterra-Runge-Kutta methods, SIAM J. Numer. Anal., 21 (1984), 123–135. https://doi.org/10.1137/0721008 doi: 10.1137/0721008
[17]	C. B. Harris, R. D. Noren, Uniform $l^1$ behavior of a time discretization method for a Volterra integrodifferential equation with convex kernel; stability, SIAM J. Numer. Anal., 49 (2011), 1553–1571. https://doi.org/10.1137/100804656 doi: 10.1137/100804656
[18]	C. Lubich, On the stability of linear multistep methods for Volterra convolution equations, IMA J. Numer. Anal., 3 (1983), 439–465. https://doi.org/10.1093/imanum/3.4.439 doi: 10.1093/imanum/3.4.439
[19]	E. Messina, M. Pezzella, A. Vecchio, A non-standard numerical scheme for an age-of-infection epidemic model, J. Comput. Dyn., 9 (2022), 239–252. https://doi.org/10.3934/jcd.2021029 doi: 10.3934/jcd.2021029
[20]	E. Messina, M. Pezzella, A. Vecchio, Positive numerical approximation of integro-differential epidemic model, Axioms, 11 (2022), 69. https://doi.org/10.3390/axioms11020069 doi: 10.3390/axioms11020069
[21]	C. Zhang, S. Vandewalle, General linear methods for Volterra integro-differential equations with memory, SIAM J. Sci. Comput., 27 (2006), 2010–2031. https://doi.org/10.1137/040607058 doi: 10.1137/040607058
[22]	D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese, R. Vermiglio On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103–117. https://doi.org/10.1080/17513758.2012.716454 doi: 10.1080/17513758.2012.716454
[23]	O. Diekmann, J. Heesterbeek, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, Wiley series in mathematical and computational biology, John Wiley and Sons, United States, 2000.
[24]	P. Linz, Analytical and Numerical Methods for Volterra Equations, Society for Industrial and Applied Mathematics, 1985. https://doi.org/10.1137/1.9781611970852
[25]	P. Davis, P. Rabinowitz, Methods of numerical integration. Second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, 1984. https://doi.org/10.1016/C2013-0-10566-1
[26]	E. Messina, M. Pezzella, A. Vecchio, Asymptotic solutions of non-linear implicit Volterra discrete equations, J. Comput. Appl. Math., 425 (2023), 115068. https://doi.org/10.1016/j.cam.2023.115068 doi: 10.1016/j.cam.2023.115068
[27]	C. Barril, A. Calsina, J. Ripoll, A practical approach to $R_0$ in continuous-time ecological models, Math. Meth. Appl. Sci., 41 (2018), 8432–8445. https://doi.org/10.1002/mma.4673 doi: 10.1002/mma.4673
[28]	D. Breda, F. Florian, R. Vermiglio, J. Ripoll, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021), 113165. https://doi.org/10.1016/j.cam.2020.113165 doi: 10.1016/j.cam.2020.113165
[29]	G. Aldis, M. Roberts, An integral equation model for the control of a smallpox outbreak, Math. Biosci., 195 (2005), 1–22. https://doi.org/10.1016/j.mbs.2005.01.006 doi: 10.1016/j.mbs.2005.01.006
[30]	B. Wacker, J. Schlüter Time-continuous and time-discrete SIR models revisited: theory and applications, Adv. Differ. Equ., 556 (2020). https://doi.org/10.1186/s13662-020-02995-1

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