Order reduction for an RNA virus evolution model

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

Order reduction for an RNA virus evolution model

Andrei Korobeinikov ^1, , Aleksei Archibasov ^2, and Vladimir Sobolev ^3,

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

Abstract
Related Papers

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.

Keywords: viral dynamics, phenotype space, basic reproduction number., Nowak--May model, singularly perturbed system, Darwinian fitness, variant space, HIV, slow-fast dynamics, viral evolution, integro--differential equations.

Mathematics Subject Classification: Primary: 92D15; Secondary: 35Q92, 34E1.

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.

show all references

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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