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Collocation of next-generation operators for computing the basic reproduction number of structured populations. (English) Zbl 07273331
Summary: We contribute a full analysis of theoretical and numerical aspects of the collocation approach recently proposed by some of the authors to compute the basic reproduction number of structured population dynamics as spectral radius of certain infinite-dimensional operators. On the one hand, we prove under mild regularity assumptions on the models coefficients that the concerned operators are compact, so that the problem can be properly recast as an eigenvalue problem thus allowing for numerical discretization. On the other hand, we prove through detailed and rigorous error and convergence analyses that the method performs the expected spectral accuracy. Several numerical tests validate the proposed analysis by highlighting diverse peculiarities of the investigated approach.
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
37N25 Dynamical systems in biology
47A75 Eigenvalue problems for linear operators
Full Text: DOI
[1] Barril, C.; Calsina, A.; Cuadrado, S.; Ripoll, J., On the basic reproduction number in continuously structured populations, Math. Meth. Appl. Sci. (2020)
[2] Barril, C.; Calsina, A.; Ripoll, J., On the reproduction number of a gut microbiota model, Bull. Math. Biol., 79, 2727-2746 (2017) · Zbl 1382.92214
[3] Barril, C.; Calsina, A.; Ripoll, J., A practical approach to \({R}_0\) in continuous-time ecological models, Math. Meth. Appl. Sci., 41, 18, 8432-8445 (2018) · Zbl 1406.92488
[4] Berrut, JP; Trefethen, LN, Barycentric Lagrange interpolation, SIAM Rev., 46, 3, 501-517 (2004) · Zbl 1061.65006
[5] Breda, D.; Florian, F.; Ripoll, J.; Vermiglio, R., Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384, 113165 (2021)
[6] Breda, D.; Maset, S.; Vermiglio, R., Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27, 2, 482-495 (2005) · Zbl 1092.65054
[7] Breda, D., Maset, S., Vermiglio, R.: Stability of linear delay differential equations—a numerical approach with MATLAB. In: SpringerBriefs in Control, Automation and Robotics. Springer, New York (2015) · Zbl 1315.65059
[8] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext (2011), New York: Springer, New York
[9] Calsina, A.; Palmada, JM; Ripoll, J., Optimal latent period in a bacteriophage population model structured by infection-age, Math. Mod. Meth. Appl. S, 21, 4, 693-718 (2011) · Zbl 1218.35136
[10] Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. In: International Series in Pure and Applied Mathematics. McGraw-Hill (1955) · Zbl 0064.33002
[11] Diekmann, O.; Heesterbeek, JAP; Britton, T., Mathematical Tools for Understanding Infectious Disease Dynamics. Theoretical and Computational Biology (2013), Princeton, Oxford: Princeton University Press, Princeton, Oxford · Zbl 1304.92009
[12] Diekmann, O.; Heesterbeek, JAP; Metz, JAJ, On the definition and the computation of the basic reproduction number \({R}_0\) in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018
[13] Diekmann, O.; Scarabel, F.; Vermiglio, R., Pseudospectral discretization of delay differential equations in sun-star formulation: results and conjectures, Discret. Contin. Dyn. S. - S, 8, 95-105 (2007)
[14] Erdös, P.; Turán, P., On interpolation, (I) quadrature and mean convergence in the Lagrange interpolation, Ann. Math., 38, 142-155 (1937) · Zbl 0016.10604
[15] Florian, F., Vermiglio, R.: PC-based sensitivity analysis of the basic reproduction number of population and epidemic models. In: Aguiar, M., Brauman, C., Kooi, B., Pugliese, A., Stollenwerk, N., Venturino, E. (eds.) Current Trends in Dynamical Systems in Biology and Natural Sciences, SEMA SIMAI Series. Springer, Berlin (to appear)
[16] Golub, G.; Van Loan, C., Matrix Computations. Johns Hopkins Studies in Mathematical Sciences (2013), Baltimore: Johns Hopkins University Press, Baltimore · Zbl 1268.65037
[17] Guo, W.; Ye, M.; Li, X.; Meyer-Baese, A.; Zhang, Q., A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon, Math. Biosci. Eng., 16, 5, 4107-4121 (2019)
[18] Heesterbeek, JAP, A brief history of \({R}_0\) and a recipe for its calculation, Acta Biother., 50, 189-204 (2002)
[19] Iannelli, M., Pugliese, A.: An Introduction to Mathematical Population Dynamics—Along the trail of Volterra and Lotka. No. 79 in La matematica per il 3+2. Springer, New York (2014) · Zbl 1305.92003
[20] Inaba, H., On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65, 309-348 (2012) · Zbl 1303.92117
[21] Inaba, H., Age-structured population dynamics in demography and epidemiology (2017), New York: Springer, New York · Zbl 1370.92010
[22] Inaba, H., The basic reproduction number \({R}_0\) in time-heterogeneous environments, J. Math. Biol., 79, 2, 731-764 (2019) · Zbl 1418.92170
[23] Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a banach space. Uspehi Matem. Nauk (N. S.) 3 1(23), 4-95 (1948). (in Russian). Amer. Math. Soc. Transl., 26:128pp, 1950 (in English) · Zbl 0030.12902
[24] Kress, R.: Linear integral equations. In: No. 82 in Applied Mathematical Sciences. Springer, New York (1989) · Zbl 0671.45001
[25] Kuniya, T., Numerical approximation of the basic reproduction number for a class of age-structured epidemic models, Appl. Math. Lett., 73, 106-112 (2017) · Zbl 1377.92090
[26] Kuniya, T., Prediction of the epidemic peak of Coronavirus disease in Japan, J. Clin. Med., 9, 3, 1-7 (2020)
[27] Liu, Z.; Magal, P.; Seydi, O.; Webb, G., A COVID-19 epidemic model with latency period, Infect. Dis. Model. (2020)
[28] Maset, S., The collocation method in the numerical solution of boundary value problems for neutral functional differential equations. Part I: convergence results, SIAM J. Numer. Anal., 53, 6, 2771-2793 (2015) · Zbl 1330.65122
[29] Mastroianni, G.; Milovanovic, G., Interpolation Processes—Basic Theory and Applications. Springer Monographs in Mathematics (2008), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 1154.41001
[30] van Neerven, J., The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory Advances and Applications (1996), Basel: Birkhäuser, Basel · Zbl 0905.47001
[31] Priestley, HA, Introduction to Complex Analysis (1990), New York: Oxford University Press, New York
[32] Pugliese, A., Sottile, S.: Inferring the COVID-19 infection curve in Italy (2020). arXiv:2004.09404
[33] Rivlin, T., An Introduction to the Approximation of Functions (1981), New York: Dover, New York
[34] Shaefer, HH, Banach Lattices and Positive Operators. Grundlehren der mathematischen Wissenschaften (1974), Berlin, Heidelberg: Springer, Berlin, Heidelberg
[35] Thieme, HR, Mathematics in Population Biology. Theoretical and Computational Biology (2003), Princeton, Oxford: Princeton University Press, Princeton, Oxford
[36] Thieme, HR, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70, 188-211 (2009) · Zbl 1191.47089
[37] Trefethen, L.N.: Spectral methods in MATLAB. Software—Environment—Tools series. SIAM, Philadelphia (2000)
[38] Trefethen, LN, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., 50, 1, 67-87 (2008) · Zbl 1141.65018
[39] Weideman, JA; Reddy, SC, A MATLAB differentiation matrix suite, ACM T. Math. Softw., 26, 4, 465-519 (2000)
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