American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Change detection in the dynamics of an intracellular protein synthesis model using nonlinear Kalman filtering
  • MBE Home
  • This Issue
  • Next Article
    Multi-host transmission dynamics of schistosomiasis and its optimal control
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), 1265-1277. 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), 143-154. 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), 11-20. 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), 260-263. doi: 10.1109/TAC.1987.1104564.  Google Scholar

[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.  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), 225-239. Google Scholar

[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.  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), 919-927. 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), 1271-1292. 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, Philadelphia, 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), 629-637. 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, New York, 2000.  Google Scholar

[13]

A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens, J. Theor. Biol., 168 (1994), 291-308. 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), 245-255. 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), 285-301. 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), 4440-4443. 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), 65-72. 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, Philadelphia, 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), 194-200. 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), 1265-1277. 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), 143-154. 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), 11-20. 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), 260-263. doi: 10.1109/TAC.1987.1104564.  Google Scholar

[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.  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), 225-239. Google Scholar

[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.  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), 919-927. 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), 1271-1292. 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, Philadelphia, 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), 629-637. 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, New York, 2000.  Google Scholar

[13]

A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens, J. Theor. Biol., 168 (1994), 291-308. 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), 245-255. 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), 285-301. 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), 4440-4443. 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), 65-72. 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, Philadelphia, 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), 194-200. doi: 10.1016/S1471-4906(02)02189-0.  Google Scholar

[1]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[2]

Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure & Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005
[3]

Aiping Wang, Michael Y. Li. Viral dynamics of HIV-1 with CTL immune response. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2257-2272. doi: 10.3934/dcdsb.2020212
[4]

Andrei Korobeinikov, Conor Dempsey. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences & Engineering, 2014, 11 (4) : 919-927. doi: 10.3934/mbe.2014.11.919
[5]

Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021097
[6]

Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545
[7]

Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. doi: 10.3934/dcdsb.2020271
[8]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237
[9]

J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1017-1044. doi: 10.3934/mbe.2013.10.1017
[10]

Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021
[11]

Qi Deng, Zhipeng Qiu, Ting Guo, Libin Rong. Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3543-3562. doi: 10.3934/dcdsb.2020245
[12]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[13]

Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1
[14]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[15]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166
[16]

Stephen Pankavich, Deborah Shutt. An in-host model of HIV incorporating latent infection and viral mutation. Conference Publications, 2015, 2015 (special) : 913-922. doi: 10.3934/proc.2015.0913
[17]

Shingo Iwami, Shinji Nakaoka, Yasuhiro Takeuchi. Mathematical analysis of a HIV model with frequency dependence and viral diversity. Mathematical Biosciences & Engineering, 2008, 5 (3) : 457-476. doi: 10.3934/mbe.2008.5.457
[18]

Sukhitha W. Vidurupola, Linda J. S. Allen. Basic stochastic models for viral infection within a host. Mathematical Biosciences & Engineering, 2012, 9 (4) : 915-935. doi: 10.3934/mbe.2012.9.915
[19]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215
[20]

Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy