Spread of virus infections

V. L. de Rioja¹*, J. Fort¹, N. Isern¹

¹Universitat de Girona, Girona, Spain.
victor.lopezd@udg.edu

We propose a new reaction-diffusion set of equations to explain the spread of viruses on bacterial populations. Although the process

$$V + B \xrightarrow{k_1} I \xrightarrow{k_2} Y \cdot V, \qquad (1)$$

was first described years ago [1] and several modifications have been proposed since then [2-5], in this new model we introduce improvements. The two main features of the model are: (i) the second-order correction in time that takes care of the time delay due to the fact that viruses spend a time τ inside the cells before the new generation disperses away [4, 5], and (ii) a new mathematical formulation for the death of infected cells, also dependent on the time delay τ, and which is biologically more reasonable than other approaches using a logistic equation [4, 5].

We apply our model to experimental data for the front propagation speed of several T7 virus strains [6], calculating the necessary parameters from independent experiments (as opposed to other authors [2, 3, 7, 8]). We find good agreement between the model and experimental data. We compare our results to a classical model with no time delay by Yin and McCaskill [2], as well as to two models that describe the death process similarly to our approach but without the including the diffusive delay [7, 8]. All the other models overestimate the measured speed, therefore showing that our improvements are not only biologically sound, but also necessary to properly describe the processes of virus infection.

References

[1] A.L. Koch: *The growth of viral plaques during the enlargement phase*. J. Theor. Biol., **6** 413 (1964).

[2] J. Yin, J.S. McCaskill: *Replication of viruses in a growing plaque: a reaction-diffusion model*. Biophys. J., **61** 1540-1549 (1992).

[3] L. You, J. Yin: *Amplification and spread of viruses in a growing plaque*. J. Theor. Biol., **200** 365-373 (1999).

[4] J. Fort, V. Méndez: *Time-delayed spread of viruses in growing plaques*. Phys. Rev. Lett., **89** 178101 1-4 (2002).

[5] D.R. Amor, J. Fort: *Virus infection speeds: theory versus experiment*. Phys. Rev. E, **82** 061905 1-9 (2010).

[6] J. Yin: *Evolution of bacteriophage T7 in a growing plaque*. J. Bacteriol., **175(5)** 1272-1277 (1993).

[7] A. Jones, H.L. Smith, H.R. Thieme, G. Röst: *On spread of phage infection of bacteria in a petri dish*. SIAM J. Appl. Math., **72 (2)** 670-688 (2012).

[8] S.A. Gourley, Y. Kuang: *A delay reaction-diffusion model of the spread of bacteriophage infection*. SIAM J. Appl. Math., **65 (2)** 550-566 (2005).