On application of asymptotic generalized discrepancy principle to the analysis of epidemiology models

References

• Morozov VA. Choice of parameter for the solution of functional equations by the regularization methods. Sov. Math. Dokl. 1967;8:1000–1003.
   Google Scholar
• Morozov VA. The error principle in the solution of operational equations by the regularization method. USSR Comput. Math. Math. Phys. 1968;8:63–87.
   Google Scholar
• Morozov VA. Methods for solving incorrectly posed problems. Berlin: Springer-Verlag; 1984.
   Google Scholar
• Tikhonov AN, Goncharsky AV, Stepanov VV, Yagola AG. Numerical methods for the solution of ill-posed problems. Dordrecht: Kluwer Academic; 1995.
   Google Scholar
• Tikhonov AN, Leonov AS, Yagola AG. Nonlinear ill-posed problems. Vol. 1, 2. London: Chapman and Hall; 1998.
   Google Scholar
• Vogel CR. Computational methods for inverse problems, research monograph. Frontiers in applied mathematics. Philadelphia (PA): Society for Industrial and Applied Mathematics (SIAM); 2002.
   Google Scholar
• Bauer F, Lukas MA. Comparing parameter choice methods for regularization of ill-posed problems. Math. Comput. Simul. 2011;81:1795–1841.
   Web of Science ®Google Scholar
• Engl H, Neubauer A. Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind. Hammerlin G, Hoffmann KH,editors. Constructive methods for the practical treatment of integral equations. ISNM 73. Basel: Birkhauser Verlag; 1985. p. 120–141.
   Google Scholar
• Engl H. Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates. J. Optim. Theory Appl. 1987;52:209–215.
   Web of Science ®Google Scholar
• Gfrerer H. An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Math. Comput. 1987;49:507–522.
   Web of Science ®Google Scholar
• Neubauer A. An aposteriori parameter choice for Tikhonov regularization in Hilbert spaces leading to optimal convergence rates. SIAM J. Numer. Anal. 1988;25:1313–1326.
   Web of Science ®Google Scholar
• Kaltenbacher B, Neubauer A, Scherzer O. Iterative regularization methods for nonlinear ill-posed problems. Vol. 6, Radon series on computational and applied mathematics. Berlin: Walter de Gruyter; 2008.
   Google Scholar
• Hamarik U, Palm R, Raus T. A family of rules for the choice of the regularization parameter in the Larentiev method in the case of rough estimate of the noise level data. J. Inverse Ill-Posed Probl. 2012;20:831–854.
   Web of Science ®Google Scholar
• Jin Q. Further convergence results on the general iteratively regularized Gauss-Newton methods under the discrepancy principle. Math. Comp. 2013;82:1647–1665.
   Web of Science ®Google Scholar
• Goncharskii AV, Leonov AS, Yagola AG. Principle of discrepancy when solving nonlinear incorrect problems. Dokl. Akad. Nauk SSSR. 1974;214:499–500.
   Google Scholar
• Leonov AS. Choice of regularization parameter for non-linear ill-posed problems with approximately specified operator. USSR Comput. Math. Math. Phys. 1979;19:1–15.
   Google Scholar
• Leonov AS. Accuracy-order optimality of the generalized residual principle and some algorithms of solving nonlinear ill-posed problems with approximate data. Siberian Math. J. 1988;29:940–947.
   Web of Science ®Google Scholar
• Jin Q, Tautenhahn U. On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems. Numer. Math. 2009;111:509–558.
   Web of Science ®Google Scholar
• Bakushinsky AB, Smirnova A. On application of generalized discrepancy principle to iterative methods for nonlinear ill-posed problems. Numer. Funct. Anal. Optim. 2005;26:35–48.
   Web of Science ®Google Scholar
• Bakushinsky AB, Smirnova A. Discrepancy principal for generalized GN iterations combined with the reverse connection control. J. Inverse Ill-Posed Probl. 2010;18:421–432.
   Web of Science ®Google Scholar
• Goncharskii AV, Leonov AS, Yagola AG. A generalized discrepancy principle. USSR Comput. Math. Math. Phys. 1973;13:25–37.
   Google Scholar
• Leonov AS, Yagola AG. Special regularizing methods for ill-posed problems with sourcewise represented solutions. Inverse Probl. 1998;14:1539–1550.
   Web of Science ®Google Scholar
• Bakushinskii AB. Principle of the residual in the case of a perturbed operator for general regularizing algorithms. Zh. Vychisl. Mat. i Mat. Fiz. 1982;22:989–993. Russian.
   Google Scholar
• Bakushinskii AB, Goncharsky A. Ill-posed problems: theory and applications. Dordrecht: Kluwer Academic; 1994.
   Google Scholar
• Leonov AS. Elimination of accuracy saturation in regularizing algorithms. Numer. Anal. Appl. 2008;1:135–150.
   Google Scholar
• Lavrent’ev MM, Romanov VG. Ill-posed problems of mathematical physics and analysis. Translated from the Russian by J. R. Schulenberger. Translation edited by Lev J. Leifman. Translations of Mathematical Monographs. Vol. 64. Providence (RI): American Mathematical Society; 1986.
   Google Scholar
• Lamm Patricia K. Full convergence of sequential local regularization methods for Volterra inverse problems. Inverse Probl. 2005;21:785–803.
   Web of Science ®Google Scholar
• Dolgopolova TF, Ivanov VK. On numerical differentiation. Zh. Vycisl. Mat. i Mat. Fiz. 1966;6:570–576. Russian.
   Google Scholar
• Zhao Z. A truncated Legendre spectral method for solving numerical differentiation. Int. J. Comput. Math. 2010;87:3209–3217.
   Web of Science ®Google Scholar
• Lu S, Naumova V, Pereverzev SV. Legendre polynomials as a recommended basis for numerical differentiation in the presence of stochastic white noise. J. Inverse Ill-Posed Probl. 2013;21:193–216.
   Web of Science ®Google Scholar
• Brooks CD, Lamm PK. A discrepancy principle for generalized local regularization of linear inverse problems. J. Inverse Ill-Posed Probl. 2013. doi: 10.1515/jip-2012-0056
   Web of Science ®Google Scholar
• Martcheva M. Avian influenza: modeling and implications for control. Math. Mod. Nat. Phenom. Forthcoming.
   Google Scholar
• World Health Organization, Confirmed Human Cases of Avian Influenza A(H5N1). Available from: http://www.who.int/influenza/human%20animal%20interface/H5N1%20cumulative%20table%20archives/en/.
   Google Scholar
• Smirnova AB, Tuncer N. Estimating time dependent transmission rate of avian influenza via stable numerical algorithm. J. Inverse Ill-Posed Probl. 2013. doi: 10.1515/jip-2012-0097
   Web of Science ®Google Scholar
• Iwami S, Takeuchi Y, Liu X. Avian-human influenza epidemic model. Math. Biosci. 2007;207:1–25.
   PubMed Web of Science ®Google Scholar
• Iwami S, Takeuchi Y, Korobeiniov A, Liu X. Prevention of avian influenza epidemic: what policy should we choose? JTB. 2008;252:732–741.
   PubMed Web of Science ®Google Scholar
• Iwami S, Takeuchi Y, Liu X. Avian flu pandemic: can we prevent it? JTB. 2009;257:181–190.
   PubMed Web of Science ®Google Scholar
• Kim K, Lin Z, Zhang L. Avian-human influenza epidemic model with diffusion. Nonlinear Anal. Real World Appl. 2010;11:313–322.
   Web of Science ®Google Scholar
• Lucchetti J, Roy M, Martcheva M. An avian influenza model and its fit to human avian influenza cases. In: Tchuenche JM, Mukandavire Z, editors. Advances in disease epidemiology. New York (NY): Nova Science; 2009. p. 1–30.
   Google Scholar
• Tuncer N, Martcheva M. Modeling seasonality in avian influenza H5N1, submitted.
   Google Scholar
• Engl H, Hanke M, Neubauer A. Regularization of inverse problems. Dordecht: Kluwer Academic; 1996.
   Google Scholar
• Apartsin AS. Multilinear Volterra equations of the first kind and some control problems. Avtomat. i Telemekh. 2008;2008:3–16. Russian; translation Autom. Remote Control. 69:545–558.
   Google Scholar
• Apartsin AS. On the investigation of the stability of a solution of the polynomial Volterra equation of the first kind. Avtomat. i Telemekh. 2011;2011:95–102. Russian; translation Autom. Remote Control. 72:1229–1236.
   Google Scholar
• Groetsch CW. The theory of Tikhonov regularization for Fredholm equations of the first kind. Boston (MA): Pitman; 1984.
   Google Scholar
• Vasin VV, Ageev AL. Ill-posed problems with a priori information. Inverse and Ill-posed problems series. Utrecht: VNU; 1995.
   Google Scholar
• Yagola AG. Ill-posed problems and methods for their numerical solution. Optimization and regularization for computational inverse problems and applications. Heidelberg: Springer; 2010. p. 17–34.
   Google Scholar
• Brooks CD, Lamm PK. A generalized approach to local regularization of linear Volterra problems in Lp spaces. Inverse Probl. 2011;27: 055010. 26 p.
   Web of Science ®Google Scholar
• Flemming J, Hofmann B, Mathé P. Sharp converse results for the regularization error using distance functions. Inverse Probl. 2011;27:025006.
   Web of Science ®Google Scholar
• Food and Agriculture Organization of the United Nations, FAO Statistics, Country- STAT. Available from: http://www.fao.org.
   Google Scholar