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2015, 12(5): 1007-1016. doi: 10.3934/mbe.2015.12.1007

Order reduction for an RNA virus evolution model

 1 Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain 2 Department of Applied Mathematics, Samara State Aerospace University (SSAU), 443086 Samara, 34, Moskovskoye shosse, Russian Federation 3 Department of Technical Cybernetics, Samara State Aerospace University (SSAU), 443086 Samara, 34, Moskovskoye shosse, Russian Federation

Received  August 2014 Revised  April 2015 Published  June 2015

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.
Citation: Andrei Korobeinikov, Aleksei Archibasov, Vladimir Sobolev. Order reduction for an RNA virus evolution model. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1007-1016. doi: 10.3934/mbe.2015.12.1007
References:
 [1] V. F. Butuzov, N. N. Nefedov, L. Recke and K. R. Schnieder, Global region of attraction of a periodic solution to a singularly perturbed parabolic problem, Applicable Analysis, 91 (2012), 1265-1277. doi: 10.1080/00036811.2011.567192. [2] L. H. Erbe and D. J. Guo, Method of upper and lower solutions for nonlinear integro-differential equations of mixed type in Banach spaces, Applicable Analysis, 52 (1994), 143-154. doi: 10.1080/00036819408840230. [3] Y. Haraguchi and A. Sasaki, Evolutionary pattern of intra-host pathogen antigenic drift: effect of crossreactivity in immune response, Phil. Trans. R. Soc. B, 352 (1997), 11-20. doi: 10.1098/rstb.1997.0002. [4] H. K. Khalil, Stability analysis of nonlinear multiparameter singularly perturbed systems, IEEE Trans. Aut. Control, 32 (1987), 260-263. doi: 10.1109/TAC.1987.1104564. [5] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [6] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. [7] A. Korobeinokov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [8] A. Korobeinikov and C. Dempsey, A continuous phenotype space model of RNA virus evolution within a host, Math. Biosci. Eng., 11 (2014), 919-927. doi: 10.3934/mbe.2014.11.919. [9] X. Lai, Sh. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect, Applicable Analysis, 89 (2010), 1271-1292. doi: 10.1080/00036811.2010.483557. [10] M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, Singular Perturbation and Hysteresis, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717860. [11] N. N. Nefedov and A. G. Nikitin, The Cauchy problem for a singularly perturbed integro-differential Fredholm equation, Computational Mathematics and Mathematical Physics, 47 (2007), 629-637. doi: 10.1134/S0965542507040082. [12] M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of Immunology and Virology, Oxford University Press, New York, 2000. [13] A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens, J. Theor. Biol., 168 (1994), 291-308. doi: 10.1006/jtbi.1994.1110. [14] A. Sasaki and Y. Haraguchi, Antigenic drift of viruses within a host: A finite site model with demographic stochasticity, J. Mol. Evol, 51 (2000), 245-255. [15] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perlson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. [16] L. S. Tsimring, H. Levine and D. A. Kessler, RNA virus evolution via a fitness-space model, Phys. Rev. Lett., 76 (1996), 4440-4443. doi: 10.1103/PhysRevLett.76.4440. [17] C. Vargas-De-León and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30 (2013), 65-72. doi: 10.1093/imammb/dqr027. [18] A. B. Vasilieva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970784. [19] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, TRENDS in Immunology, 23 (2002), 194-200. doi: 10.1016/S1471-4906(02)02189-0.

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References:
 [1] V. F. Butuzov, N. N. Nefedov, L. Recke and K. R. Schnieder, Global region of attraction of a periodic solution to a singularly perturbed parabolic problem, Applicable Analysis, 91 (2012), 1265-1277. doi: 10.1080/00036811.2011.567192. [2] L. H. Erbe and D. J. Guo, Method of upper and lower solutions for nonlinear integro-differential equations of mixed type in Banach spaces, Applicable Analysis, 52 (1994), 143-154. doi: 10.1080/00036819408840230. [3] Y. Haraguchi and A. Sasaki, Evolutionary pattern of intra-host pathogen antigenic drift: effect of crossreactivity in immune response, Phil. Trans. R. Soc. B, 352 (1997), 11-20. doi: 10.1098/rstb.1997.0002. [4] H. K. Khalil, Stability analysis of nonlinear multiparameter singularly perturbed systems, IEEE Trans. Aut. Control, 32 (1987), 260-263. doi: 10.1109/TAC.1987.1104564. [5] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [6] A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239. [7] A. Korobeinokov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321. doi: 10.1093/imammb/dqp009. [8] A. Korobeinikov and C. Dempsey, A continuous phenotype space model of RNA virus evolution within a host, Math. Biosci. Eng., 11 (2014), 919-927. doi: 10.3934/mbe.2014.11.919. [9] X. Lai, Sh. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect, Applicable Analysis, 89 (2010), 1271-1292. doi: 10.1080/00036811.2010.483557. [10] M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, Singular Perturbation and Hysteresis, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717860. [11] N. N. Nefedov and A. G. Nikitin, The Cauchy problem for a singularly perturbed integro-differential Fredholm equation, Computational Mathematics and Mathematical Physics, 47 (2007), 629-637. doi: 10.1134/S0965542507040082. [12] M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of Immunology and Virology, Oxford University Press, New York, 2000. [13] A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens, J. Theor. Biol., 168 (1994), 291-308. doi: 10.1006/jtbi.1994.1110. [14] A. Sasaki and Y. Haraguchi, Antigenic drift of viruses within a host: A finite site model with demographic stochasticity, J. Mol. Evol, 51 (2000), 245-255. [15] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perlson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076. [16] L. S. Tsimring, H. Levine and D. A. Kessler, RNA virus evolution via a fitness-space model, Phys. Rev. Lett., 76 (1996), 4440-4443. doi: 10.1103/PhysRevLett.76.4440. [17] C. Vargas-De-León and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Math. Med. Biol., 30 (2013), 65-72. doi: 10.1093/imammb/dqr027. [18] A. B. Vasilieva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970784. [19] D. Wodarz, J. P. Christensen and A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, TRENDS in Immunology, 23 (2002), 194-200. doi: 10.1016/S1471-4906(02)02189-0.
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