# American Institute of Mathematical Sciences

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2014, 11(5): 1091-1113. doi: 10.3934/mbe.2014.11.1091

## Dynamics of evolutionary competition between budding and lytic viral release strategies

 1 Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada 2 Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  January 2014 Revised  April 2014 Published  June 2014

In this paper, we consider the evolutionary competition between budding and lytic viral release strategies, using a delay differential equation model with distributed delay. When antibody is not established, the dynamics of competition depends on the respective basic reproductive ratios of the two viruses. If the basic reproductive ratio of budding virus is greater than that of lytic virus and one, budding virus can survive. When antibody is established for both strains but the neutralization capacities are the same for both strains, consequence of the competition also depends only on the basic reproductive ratios of the budding and lytic viruses. Using two concrete forms of the viral production functions, we are also able to conclude that budding virus will outcompete if the rates of viral production, death rates of infected cells and neutralizing capacities of the antibodies are the same for budding and lytic viruses. In this case, budding strategy would have an evolutionary advantage. However, if the antibody neutralization capacity for the budding virus is larger than that for the lytic virus, the lytic virus can outcompete the budding virus provided that its reproductive ratio is very high. An explicit threshold is derived.
Citation: Xiulan Lai, Xingfu Zou. Dynamics of evolutionary competition between budding and lytic viral release strategies. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1091-1113. doi: 10.3934/mbe.2014.11.1091
##### References:
 [1] A. Brännstr$\ddot o$m and D. J. T. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B., 272 (2005), 2065-2072. [2] J. Carter and V. Saunders, Virology: Principles and Application, John Wiley and Sons, Ltd, 2007. [3] C. Castillo-Chaves and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino, et al.), Wuerz, Winnnipeg, 1995, 33-50. [4] D. Coombs, Optimal viral production, Bull. Math. Biol., 65 (2003), 1003-1023. doi: 10.1016/S0092-8240(03)00056-9. [5] H. Garoff, R. Hewson and D. Opstelten, Virus maturation by budding, Microbiology and Moleculer Biology Reviews, 62 (1998), 1171-1190. [6] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theor. Biol. 229 (2004), 281-288. doi: 10.1016/j.jtbi.2004.04.015. [7] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [8] N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive? J. Theor. Biol., 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013. [9] D. P. Nayak, Assembly and budding of influenza virus, Virus Research, 106 (2004), 147-165. doi: 10.1016/j.virusres.2004.08.012. [10] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variation in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267. [11] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. [12] H. L. Smith, Monotone Dynamical Systems. An Introduction To The Theory Of Competitive And Cooperative Systems, Mathematical Surveys and Monographs, 41, AMS, Providence, 1995. [13] I. N. Wang, D. E. Dykhuizen and L. B. Slobodkin, The evolution of phage lysis timing, Evolutionary Ecology, 10 (1996), 545-558. doi: 10.1007/BF01237884. [14] I. N. Wang, Lysis timing and bacteriophage fitness, Genetics, 172 (2006), 17-26. doi: 10.1534/genetics.105.045922.

show all references

##### References:
 [1] A. Brännstr$\ddot o$m and D. J. T. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B., 272 (2005), 2065-2072. [2] J. Carter and V. Saunders, Virology: Principles and Application, John Wiley and Sons, Ltd, 2007. [3] C. Castillo-Chaves and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino, et al.), Wuerz, Winnnipeg, 1995, 33-50. [4] D. Coombs, Optimal viral production, Bull. Math. Biol., 65 (2003), 1003-1023. doi: 10.1016/S0092-8240(03)00056-9. [5] H. Garoff, R. Hewson and D. Opstelten, Virus maturation by budding, Microbiology and Moleculer Biology Reviews, 62 (1998), 1171-1190. [6] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theor. Biol. 229 (2004), 281-288. doi: 10.1016/j.jtbi.2004.04.015. [7] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [8] N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive? J. Theor. Biol., 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013. [9] D. P. Nayak, Assembly and budding of influenza virus, Virus Research, 106 (2004), 147-165. doi: 10.1016/j.virusres.2004.08.012. [10] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variation in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267. [11] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. [12] H. L. Smith, Monotone Dynamical Systems. An Introduction To The Theory Of Competitive And Cooperative Systems, Mathematical Surveys and Monographs, 41, AMS, Providence, 1995. [13] I. N. Wang, D. E. Dykhuizen and L. B. Slobodkin, The evolution of phage lysis timing, Evolutionary Ecology, 10 (1996), 545-558. doi: 10.1007/BF01237884. [14] I. N. Wang, Lysis timing and bacteriophage fitness, Genetics, 172 (2006), 17-26. doi: 10.1534/genetics.105.045922.
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