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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

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.  doi: 10.1080/00036811.2011.567192.  Google Scholar

[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.  doi: 10.1080/00036819408840230.  Google Scholar

[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.  doi: 10.1098/rstb.1997.0002.  Google Scholar

[4]

H. K. Khalil, Stability analysis of nonlinear multiparameter singularly perturbed systems,, IEEE Trans. Aut. Control, 32 (1987), 260.  doi: 10.1109/TAC.1987.1104564.  Google Scholar

[5]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[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.   Google Scholar

[7]

A. Korobeinokov, Stability of ecosystem: Global properties of a general prey-predator model,, Math. Med. Biol., 26 (2009), 309.  doi: 10.1093/imammb/dqp009.  Google Scholar

[8]

A. Korobeinikov and C. Dempsey, A continuous phenotype space model of RNA virus evolution within a host,, Math. Biosci. Eng., 11 (2014), 919.  doi: 10.3934/mbe.2014.11.919.  Google Scholar

[9]

X. Lai, Sh. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect,, Applicable Analysis, 89 (2010), 1271.  doi: 10.1080/00036811.2010.483557.  Google Scholar

[10]

M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, Singular Perturbation and Hysteresis,, SIAM, (2005).  doi: 10.1137/1.9780898717860.  Google Scholar

[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.  doi: 10.1134/S0965542507040082.  Google Scholar

[12]

M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[13]

A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens,, J. Theor. Biol., 168 (1994), 291.  doi: 10.1006/jtbi.1994.1110.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[16]

L. S. Tsimring, H. Levine and D. A. Kessler, RNA virus evolution via a fitness-space model,, Phys. Rev. Lett., 76 (1996), 4440.  doi: 10.1103/PhysRevLett.76.4440.  Google Scholar

[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.  doi: 10.1093/imammb/dqr027.  Google Scholar

[18]

A. B. Vasilieva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems,, SIAM, (1995).  doi: 10.1137/1.9781611970784.  Google Scholar

[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.  doi: 10.1016/S1471-4906(02)02189-0.  Google Scholar

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.  doi: 10.1080/00036811.2011.567192.  Google Scholar

[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.  doi: 10.1080/00036819408840230.  Google Scholar

[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.  doi: 10.1098/rstb.1997.0002.  Google Scholar

[4]

H. K. Khalil, Stability analysis of nonlinear multiparameter singularly perturbed systems,, IEEE Trans. Aut. Control, 32 (1987), 260.  doi: 10.1109/TAC.1987.1104564.  Google Scholar

[5]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[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.   Google Scholar

[7]

A. Korobeinokov, Stability of ecosystem: Global properties of a general prey-predator model,, Math. Med. Biol., 26 (2009), 309.  doi: 10.1093/imammb/dqp009.  Google Scholar

[8]

A. Korobeinikov and C. Dempsey, A continuous phenotype space model of RNA virus evolution within a host,, Math. Biosci. Eng., 11 (2014), 919.  doi: 10.3934/mbe.2014.11.919.  Google Scholar

[9]

X. Lai, Sh. Liu and R. Lin, Rich dynamical behaviors for predator-prey model with weak Allee effect,, Applicable Analysis, 89 (2010), 1271.  doi: 10.1080/00036811.2010.483557.  Google Scholar

[10]

M. P. Mortell, R. E. O'Malley, A. Pokrovskii and V. A. Sobolev, Singular Perturbation and Hysteresis,, SIAM, (2005).  doi: 10.1137/1.9780898717860.  Google Scholar

[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.  doi: 10.1134/S0965542507040082.  Google Scholar

[12]

M. A. Nowak and R. M. May, Virus dynamics: Mathematical principles of Immunology and Virology,, Oxford University Press, (2000).   Google Scholar

[13]

A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens,, J. Theor. Biol., 168 (1994), 291.  doi: 10.1006/jtbi.1994.1110.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jtbi.2000.1076.  Google Scholar

[16]

L. S. Tsimring, H. Levine and D. A. Kessler, RNA virus evolution via a fitness-space model,, Phys. Rev. Lett., 76 (1996), 4440.  doi: 10.1103/PhysRevLett.76.4440.  Google Scholar

[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.  doi: 10.1093/imammb/dqr027.  Google Scholar

[18]

A. B. Vasilieva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems,, SIAM, (1995).  doi: 10.1137/1.9781611970784.  Google Scholar

[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.  doi: 10.1016/S1471-4906(02)02189-0.  Google Scholar

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