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Article Contents

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

• 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.
Mathematics Subject Classification: 92D30.

 Citation:

•  [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. [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. [3] V. Andreasen, S. Levin and J. Lin, A model of influenza A drift evolution, Z. Angew. Math. Mech., 76 (1996), 421-424. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, 2000. [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. [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. [20] A. Sasaki, Evolution of antigenic drift/switching: Continuously evading pathogens, J. Theor. Biol., 168 (1994), 291-308.doi: 10.1006/jtbi.1994.1110. [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. [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. [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. [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. [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. [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. [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.