Journal of Applied Probability

A stochastic model for virus growth in a cell population

J. E. Björnberg, T. Britton, E. I. Broman, and E. Natan

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In this work we introduce a stochastic model for the spread of a virus in a cell population where the virus has two ways of spreading: either by allowing its host cell to live and duplicate, or by multiplying in large numbers within the host cell, causing the host cell to burst and thereby let the virus enter new uninfected cells. The model is a kind of interacting Markov branching process. We focus in particular on the probability that the virus population survives and how this depends on a certain parameter λ which quantifies the `aggressiveness' of the virus. Our main goal is to determine the optimal balance between aggressive growth and long-term success. Our analysis shows that the optimal strategy of the virus (in terms of survival) is obtained when the virus has no effect on the host cell's life cycle, corresponding to λ = 0. This is in agreement with experimental data about real viruses.

Article information

J. Appl. Probab., Volume 51, Number 3 (2014), 599-612.

First available in Project Euclid: 5 September 2014

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx]
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D15: Problems related to evolution

Branching process interacting branching process model for virus growth


Björnberg, J. E.; Britton, T.; Broman, E. I.; Natan, E. A stochastic model for virus growth in a cell population. J. Appl. Probab. 51 (2014), no. 3, 599--612. doi:10.1239/jap/1409932661.

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  • Ackers, G. K., Johnson, A. D. and Shea, M. A. (1982). Quantitative model for gene regulation by lambda phage repressor. Proc. Nat. Acad. Sci. USA 79, 1129–1133.
  • Arkin, A., Ross, J. and McAdams, H. H. (1998). Stochastic kinetic analysis of developmental pathway bifurcation in phage $\lambda$-infected Escherichia coli cells. Genetics 149, 1633–1648.
  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • Aurell, E. and Sneppen, K. (2002). Epigenetics as a first exit problem. Phys. Rev. Lett. 88, 048101.
  • Björnberg, J. E. and Broman, E. I. (2014). Coexistence and noncoexistence of Markovian viruses and their hosts. J. Appl. Prob. 51, 191–208.
  • Haccou, P., Jagers, P. and Vatutin, V. A. (2007). Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge University Press.
  • Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
  • Joh, R. I. and Weitz, J. S. (2011). To lyse or not to lyse: transient-mediated stochastic fate determination in cells infected by bacteriophages. PLoS Comput. Biol. 7, 1002006.
  • Johnson, A. D. \et (1981). $\lambda$ repressor and cro-components of an efficient molecular switch. Nature 294, 217–223.
  • Kendall, W. S. and Saunders, I. W. (1983). Epidemics in competition II: the general epidemic. J. R. Statist. Soc. B 45, 238–244.
  • Lieb, M. (1953). The establishment of lysogenicity in Escherichia coli. J. Bacteriology 65, 642–651.
  • Little, J. W., Shepley, D. P. and Wert, D. W. (1999). Robustness of a gene regulatory circuit. EMBO J. 18, 4299–4307.
  • Lwoff, A. (1953). Lysogeny. Bacteriological Rev. 17, 269–337.
  • McAdams, H. H. and Shapiro, L. (1995). Circuit simulation of genetic networks. Science 269, 650–656.
  • Nowak, M. A. and May, R. M. (2000). Virus Dynamics. Mathematical Principles of Immunology and Virology. Oxford University Press.
  • Oppenheim, A. B. \et (2005). Switches in bacteriophage lambda development. Ann. Rev. Genetics 39, 409–429.
  • Reinitz, J. and Vaisnys, J. R. (1990). Theoretical and experimental analysis of the phage lambda genetic switch implies missing levels of co-operativity. J. Theoret. Biol. 145, 295–318.
  • Renshaw, E. (1991). Modelling Biological Populations in Space and Time. Cambridge University Press.
  • Santillán, M. and Mackey, M. C. (2004). Why the lysogenic state of phage $\lambda$ is so stable: a mathematical modeling approach. Biophysical J. 86, 75–84.
  • Shea, M. A. and Ackers, G. K. (1985). The OR control system of bacteriophage lambda: a physical-chemical model for gene regulation. J. Molec. Biol. 181, 211–230.
  • St-Perre, F. and Endy, D. (2008). Determination of cell fate selection during phage lambda infection. Proc. Nat. Acad. Sci. USA 105, 20705–20710.
  • Zeng, L. \et (2010). Decision making at a subcellular level determines the outcome of bacteriophage infection. Cell 141, 682–691.