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2014, 11(4): 919-927. doi: 10.3934/mbe.2014.11.919

A continuous phenotype space model of RNA virus evolution within a host

Andrei Korobeinikov 1, and  Conor Dempsey 2,

1. 

Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona
2. 

Department of Neuroscience, Columbia University, 40 Haven Avenue, New York, NY 10032, United States

Received  August 2013 Revised  October 2013 Published  March 2014

Due to their very high replication and mutation rates, RNA viruses can serve as an excellent testing model for verifying hypothesis and addressing questions in evolutionary biology. In this paper, we suggest a simple deterministic mathematical model of the within-host viral dynamics, where a possibility for random mutations incorporates. This model assumes a continuous distribution of viral strains in a one-dimensional phenotype space where random mutations are modelled by Brownian motion (that is, by diffusion). Numerical simulations show that random mutations combined with competition for a resource result in evolution towards higher Darwinian fitness: a stable pulse traveling wave of evolution, moving towards higher levels of fitness, is formed in the phenotype space. The advantage of this model, compared with the previously constructed models, is that this model is mechanistic and is based on commonly accepted model of virus dynamics within a host, and thus it allows an incorporation of features of the real-life host-virus system such as immune response, antiviral therapy, etc.
Keywords: HIV, Evolution, traveling wave., random mutation, mathematical modelling.
Mathematics Subject Classification: 92D3.
Citation: 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
References:
[1]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. Ser. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005.  Google Scholar

[2]

V. Andreasen, Dynamics of annual influenza A epidemics with immuno-selection, J. Math. Biol., 46 (2003), 504-536. doi: 10.1007/s00285-002-0186-2.  Google Scholar

[3]

V. Andreasen, S. Levin and J. Lin, A model of influenza A drift evolution, Z. Angew. Math. Mech., 76 (1996), 421-424. Google Scholar

[4]

M. F. Boni, J. R. Gog, V. Andreasen and M. W. Feldman, Epidemic dynamics and antigenic evolution in a single season of influenza A, Proc. R. Soc. B, 273 (2006), 1307-1316. doi: 10.1098/rspb.2006.3466.  Google Scholar

[5]

V. Calvez, A. Korobeinikov and P. K. Maini, Cluster formation for multi-strain infections with cross-immunity, J. Theor. Biol., 233 (2005), 75-83. doi: 10.1016/j.jtbi.2004.09.016.  Google Scholar

[6]

J. R. Gog and B. T. Grenfell, Dynamics and selection of many-strain pathogens, Proc. Natl Acad. Sci. USA, 99 (2002), 17209-17214. doi: 10.1073/pnas.252512799.  Google Scholar

[7]

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

[8]

T. Inoue, T. Kajiwara and T. Sasaki, Global stability of models of humoral immunity against multiple viral strains, Journal of Biological Dynamics, 4 (2010), 282-295. doi: 10.1080/17513750903180275.  Google Scholar

[9]

S. Iwami, T. Miura, S. Nakaoka and Y. Takeuchi, Immune impairment in HIV infection: Existence of risky and immunodeficiency thresholds, J. Theor. Biol., 260 (2009), 490-501. doi: 10.1016/j.jtbi.2009.06.023.  Google Scholar

[10]

Y. Iwasa, F. Michor and M. A. Nowak, Virus evolution within patients increases pathogenicity, J. Theor. Biol., 232 (2005), 17-26. doi: 10.1016/j.jtbi.2004.07.016.  Google Scholar

[11]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. also in Selected Works of A.N. Kolmogorov: Mathematics and Mechanics, Kluwer, Dordrecht, (1991), 1-25. Google Scholar

[12]

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. doi: 10.1093/imammb/dqp009.  Google Scholar

[13]

A. Korobeinikov, 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

[14]

J. Lin, V. Andreasen, R. Casagrandi and S. A. Levin, Traveling waves in a model of influenza A drift, J. Theor. Biol., 222 (2003), 437-445. doi: 10.1016/S0022-5193(03)00056-0.  Google Scholar

[15]

L. M. Mansky and H. M. Temin, Lower in vivo mutation rate of human immunodeficiency virus type 1 than that predicted from the fidelity of purified reverse transcriptase, Journal of Virology, 69 (1995), 5087-5094. Google Scholar

[16]

M. A. Nowak, R. M. Anderson, A. R. McLean, T. F. W. Wolfs, J. Goudsmit and R. M. May, Antigenic diversity threshold and the development of AIDS, Science, 254 (1991), 963-969. doi: 10.1126/science.1683006.  Google Scholar

[17]

M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, 2000.  Google Scholar

[18]

A. Rambaut, D. Posada, K. A. Crandall and E. C. Holmes, The causes and consequences of HIV evolution, Nature Reviews, 5 (2004), 52-61. http://tree.bio.ed.ac.uk/downloadPaper.php?id=242. doi: 10.1038/nrg1246.  Google Scholar

[19]

J. Saldaña, S. F. Elena and R. V. Solé, Coinfection and superinfection in RNA virus populations: A selection-mutation model, Math. Biosci., 183 (2003), 135-160. doi: 10.1016/S0025-5564(03)00038-5.  Google Scholar

[20]

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

[21]

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

[22]

M. O. Souza and J. P. Zubelli, Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625. doi: 10.1007/s11538-010-9543-2.  Google Scholar

[23]

M. A. Stafford et al., Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

D. Wodarz, P. Klenerman and M. A. Nowak, Dynamics of cytotoxic T-lymphocyte exhaustion, Proc. R. Soc. Lond. B 265 (1998), 191-203. doi: 10.1098/rspb.1998.0282.  Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Soc. Lond. Ser. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005.  Google Scholar

[2]

V. Andreasen, Dynamics of annual influenza A epidemics with immuno-selection, J. Math. Biol., 46 (2003), 504-536. doi: 10.1007/s00285-002-0186-2.  Google Scholar

[3]

V. Andreasen, S. Levin and J. Lin, A model of influenza A drift evolution, Z. Angew. Math. Mech., 76 (1996), 421-424. Google Scholar

[4]

M. F. Boni, J. R. Gog, V. Andreasen and M. W. Feldman, Epidemic dynamics and antigenic evolution in a single season of influenza A, Proc. R. Soc. B, 273 (2006), 1307-1316. doi: 10.1098/rspb.2006.3466.  Google Scholar

[5]

V. Calvez, A. Korobeinikov and P. K. Maini, Cluster formation for multi-strain infections with cross-immunity, J. Theor. Biol., 233 (2005), 75-83. doi: 10.1016/j.jtbi.2004.09.016.  Google Scholar

[6]

J. R. Gog and B. T. Grenfell, Dynamics and selection of many-strain pathogens, Proc. Natl Acad. Sci. USA, 99 (2002), 17209-17214. doi: 10.1073/pnas.252512799.  Google Scholar

[7]

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

[8]

T. Inoue, T. Kajiwara and T. Sasaki, Global stability of models of humoral immunity against multiple viral strains, Journal of Biological Dynamics, 4 (2010), 282-295. doi: 10.1080/17513750903180275.  Google Scholar

[9]

S. Iwami, T. Miura, S. Nakaoka and Y. Takeuchi, Immune impairment in HIV infection: Existence of risky and immunodeficiency thresholds, J. Theor. Biol., 260 (2009), 490-501. doi: 10.1016/j.jtbi.2009.06.023.  Google Scholar

[10]

Y. Iwasa, F. Michor and M. A. Nowak, Virus evolution within patients increases pathogenicity, J. Theor. Biol., 232 (2005), 17-26. doi: 10.1016/j.jtbi.2004.07.016.  Google Scholar

[11]

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. also in Selected Works of A.N. Kolmogorov: Mathematics and Mechanics, Kluwer, Dordrecht, (1991), 1-25. Google Scholar

[12]

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. doi: 10.1093/imammb/dqp009.  Google Scholar

[13]

A. Korobeinikov, 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

[14]

J. Lin, V. Andreasen, R. Casagrandi and S. A. Levin, Traveling waves in a model of influenza A drift, J. Theor. Biol., 222 (2003), 437-445. doi: 10.1016/S0022-5193(03)00056-0.  Google Scholar

[15]

L. M. Mansky and H. M. Temin, Lower in vivo mutation rate of human immunodeficiency virus type 1 than that predicted from the fidelity of purified reverse transcriptase, Journal of Virology, 69 (1995), 5087-5094. Google Scholar

[16]

M. A. Nowak, R. M. Anderson, A. R. McLean, T. F. W. Wolfs, J. Goudsmit and R. M. May, Antigenic diversity threshold and the development of AIDS, Science, 254 (1991), 963-969. doi: 10.1126/science.1683006.  Google Scholar

[17]

M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, 2000.  Google Scholar

[18]

A. Rambaut, D. Posada, K. A. Crandall and E. C. Holmes, The causes and consequences of HIV evolution, Nature Reviews, 5 (2004), 52-61. http://tree.bio.ed.ac.uk/downloadPaper.php?id=242. doi: 10.1038/nrg1246.  Google Scholar

[19]

J. Saldaña, S. F. Elena and R. V. Solé, Coinfection and superinfection in RNA virus populations: A selection-mutation model, Math. Biosci., 183 (2003), 135-160. doi: 10.1016/S0025-5564(03)00038-5.  Google Scholar

[20]

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

[21]

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

[22]

M. O. Souza and J. P. Zubelli, Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625. doi: 10.1007/s11538-010-9543-2.  Google Scholar

[23]

M. A. Stafford et al., Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. doi: 10.1006/jtbi.2000.1076.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

D. Wodarz, P. Klenerman and M. A. Nowak, Dynamics of cytotoxic T-lymphocyte exhaustion, Proc. R. Soc. Lond. B 265 (1998), 191-203. doi: 10.1098/rspb.1998.0282.  Google Scholar

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