Virus infection speeds : Theory versus experiment

In order to explain the speed of Vesicular Stomatitis Virus VSV infections, we develop a simple model that improves previous approaches to the propagation of virus infections. For VSV infections, we find that the delay time elapsed between the adsorption of a viral particle into a cell and the release of its progeny has a very important effect. Moreover, this delay time makes the adsorption rate essentially irrelevant in order to predict VSV infection speeds. Numerical simulations are in agreement with the analytical results. Our model satisfactorily explains the experimentally measured speeds of VSV infections.


I. INTRODUCTION
Mathematical models of infectious diseases are a field that has advanced substantially during the last decades ͓1͔.It makes use of many methods from Physics, e.g., reactiondiffusion equations and front speed computations.On the other hand, in the last century vaccination has become one of the best tools against infection spreading, and some models including vaccination strategies have been developed recently ͓2,3͔.Moreover, new computer technology advances have played an important role on infection spreading research, since they have made it possible to introduce novel methods such as complex networks ͓4-6͔.However, in addition with macroscopic epidemiology studies ͓1-3,5͔, the effects of infectious diseases at a cellular levels ͑as well as microscopic viral infection front propagation͒ are also of scientific interest ͓7-12͔.
When a virus infects a cell, some time elapses before the new generation of viruses is released from the cell.Some years ago, it was shown that this delay time has an important effect and can satisfactory explain the infection front speeds of T7 viruses infecting E. Coli bacteria ͓13-15͔.In Ref. ͓15͔, we mentioned that it should be possible to apply the same model to Vesicular Stomatitis Viruses ͑VSV͒, which replicate on mammalian or insect cells ͑not on bacte-ria͒, because there is again a delay due to the time elapsed between virus adsorption into a cell and the release of its progeny.Detailed measurements of infection front speeds for VSV were not available when we developed our models ͓8,13͔, but such measurements have been performed recently ͓16,17͔.
Some very recent models for the spread of VSV infections have been proposed by Haseltine et al. ͓17͔.Their models include the reaction-diffusion process, as well as some additional relevant biological phenomena, in order to explain the experimental images of this phenomenon.In this paper we revisit the approach by Haseltine et al. ͓17͔ to reduce the complexity of the models, and to better account for the experimental speeds.We will present a simple model, using as few unknown parameters as possible, and compare its predicted front speeds to VSV experimental data ͓16͔.We will show that it is necessary to take it into account the effect of the delay time in order to explain properly the front speeds of VSV infections.
The plan of this paper is as follows.In Sec.II we present a time-delayed model using as few free or adjustable parameters as possible.In Sec.III we derive some approximate explicit equations for the front speed.In Sec.IV, we perform numerical simulations to check the validity of our theoretical model.In Sec.V we compare to experimental data.Section VI is devoted to concluding remarks, especially the importance of the delay time for VSV infections.We will also find that this delay-time effect leads to the front speed being approximately independent of the adsorption constant k 1 over four orders of magnitude.This is very important in order to compare predicted speeds to experimental ones because the value of k 1 is very uncertain.

II. MODEL
The infection and virus replication processes can be summarized by the reactions

YV. ͑1͒
In Eq. ͑1͒, k 1 is the adsorption rate of viruses V into uninfected cells C, k 2 is the rate constant for the death of infected cells I, and the yield Y is the number of new viruses produced per infected cell.In contrast to some previous papers ͓8,13-15͔, here we use the symbol C instead of B because VSV infections propagate on mammalian or insect cells ͑not on bacteria, in contrast to T7 infections͒.
The experiments reported in Refs.where D is the diffusion coefficient, ͓ ...͔ denotes concentration and t is the time.The set of Eqs.͑2͒-͑4͒ can be further improved for two reasons, which we now discuss.First, Eq. ͑4͒ implies a virus dynamics that does not agree with experimental data in homogeneous media ͓see Eq. ͑A4͒ and the text below it in Appendix A͔.In order to improve this point, it has been proposed ͓13͔ that k 2 ͓I͔͑r , t͒ in Eqs.͑2͒ and ͑4͒ should be replaced by a logistic-type growth function, namely, k 2 ͓I͔͑r , t͕͒1−͓I͔͑r , t͒ / I MAX ͖, which leads to a virus dynamics that agrees very well with one-step growth experimental data ͑see Appendix A͒.Such an improved model is therefore described, instead of by Eqs.͑2͒-͑4͒, by the following set: ‫͓ץ‬I͔͑r,t͒ ‫ץ‬t = k 1 ͓V͔͑r,t͓͒C͔͑r,t͒ − k 2 ͓I͔͑r,t͒ͩ1 − ͓I͔͑r,t͒

͑7͒
This set is more reasonable than Eqs.͑2͒-͑4͒ because it agrees better with experimental data in homogeneous media ͑see Appendix A͒, but we would like to stress that this point does not affect the analytical results in the present paper because the infection speeds that we will derive would be the same if the last term in Eqs.͑5͒ and ͑7͒ were neglected ͑see Appendix A͒.
A second, much more important improvement can be made.The diffusion dynamics in both Eqs.͑5͒ and ͑2͒ ͑left side and first term in the right side͒ is Fickian or nondelayed.This means that it does not take into account the effect of the time interval during which a virus does not move appreciably because it is inside a cell.The parameter is the time interval elapsed from the adsorption of a virus into a cell until the virus has reproduced inside it and the virus progeny have left the cell.Obviously, this effect will slow down the propagation of virus fronts.Indeed, it has been previously shown that taking into account this delay time is essential to find good agreement with the experimental observations in other kinds of virus infections ͓13͔.For this reason, Eq. ͑5͒ must be replaced by a time-delayed diffusion equation ͑9͒ Note that Eq. ͑8͒ simplifies to Eq. ͑5͒ if the effect of the delay time is neglected ͑ =0͒.
The symbol . . .͉ g indicates that the corresponding time derivatives in Eqs.͑8͒ and ͑9͒ take into account exclusively growth ͑i.e., reactive͒ but not diffusive processes.In a recent paper ͓21͔ it has been shown that taking this subindex properly into account improves previous results ͓22͔ for the speed of fronts ͑albeit Refs.͓21,22͔ consider a simpler system with a single species, whereas here we have three species͒.For this very same reason, Eq. ͑8͒ here improves Eq. ͑6͒ in Ref.
͓13͔, and yields a different infection speed that will be derived below.In order to do so, let us further develop the last term in Eq. ͑8͒,

͑13͒
Our model will be based on the set of Eqs.͑11͒-͑13͒.Before going ahead, however, we can see better the importance of the symbol . . .͉ g by noting that, if it had not been included in Eq. ͑8͒, Eq. ͑10͒ would be the same but without the symbols . . .͉ g , so its first term on the right would be − 2 k 1 ͓͑V͔ ‫ץ‬ ͓C͔ / ‫ץ‬t + ͓C͔ ‫ץ‬ ͓V͔ / ‫ץ‬t͒ and, since the symbol . . .͉ g does not appear, we would be unable to substitute Eq. ͑9͒.Therefore, instead of Eq. ͑11͒ we would have obtained

͑14͒
This corresponds to the model in Refs.͓13,14͔.The fourth term in the right of Eq. ͑14͒ is improved by that in Eq. ͑11͒.
The difference is that this term in Eq. ͑11͒ corresponds to the variation of ͓V͔͑r , t͒ due to adsorption and replication, whereas in Eq. ͑14͒ it also includes its variation due to diffusion ͓i.e., Eq. ͑14͒ uses ‫͓ץ‬V͔͑r,t͒ ‫ץ‬t instead of͉ ‫͓ץ‬V͔͑r,t͒ ‫ץ‬t ͉ g = F͑r , t͔͒.For details, see the derivation of Eq. ͑8͒ in Appendix B in the present paper.
Our model will be based on the set of Eqs.͑11͒-͑13͒.Their front speed will be derived below, and they will be integrated numerically in Sec.IV.
Equations ͑11͒-͑13͒ can be written in terms of dimensionless variables C ¯ϵ͓C͔ / C 0 , V ¯ϵ͓V͔ / C 0 , I ¯ϵ͓I͔ / C 0 , t ¯ϵ k 2 t, and r ¯ϵ r ͱ k 2 / D and dimensionless parameters where F ¯is the dimensionless growth function defined as

͑18͒
For simplicity, in Eqs.͑15͒-͑18͒ we have used de notation V ¯t ¯, C ¯t We look for solutions depending only on the new variable z ¯ϵ r ¯− c ¯t ¯, where c ¯Ͼ 0 is the dimensionless wave front speed, which is related to dimensional speed c by c ¯= c / ͱ k 2 D. As usual, we linearize our Eqs.͑15͒-͑18͒ around the unstable steady state ͓͑V͔ , ͓C͔ , ͓I͔͒ = ͑0,C 0 ,0͒, i.e., ͑V ¯, C ¯, For nontrivial solutions ͑⑀ V , ⑀ C , ⑀ I ͒ ͑0,0,0͒ to exist, the determinant of the matrix corresponding to the linearized set of three evolution equations must vanish.Therefore, the following characteristic equation must be satisfied: For simplicity, we have introduced the parameter ␤ = ¯/ 2. According to marginal stability analysis ͓23͔, the wave front speed can be calculated numerically from where c ¯͑͒ is given implicitly by Eq. ͑19͒.

III. APPROXIMATE EXPLICIT EQUATIONS FOR THE SPEED
In this section we derive explicit expressions for the dimensionless front speed c ¯. Its exact value is given by the implicit Eq. ͑20͒.Since an exact explicit equation for the speed would be very complicated, some assumptions and approximations will be made.
In order to avoid nonpositive values for concentrations, we must impose that the three solutions for in Eq. ͑19͒ are real, so it must be satisfied that where C 1 , C 2 , and C 3 are the coefficients of second, first, and zeroth powers of , respectively.We rewrite condition ͑21͒ in terms of = c ¯2 and then we get where the coefficients a i are easily derived from condition ͑21͒.However, the exact expressions for a i are rather long.Fortunately, they can be simplified under some simple approximations that take into account typical values of the parameters in the present paper, namely, a high value of the yield ͑e.g., Y = 2.77ϫ 10 5 ͒ and a small value of ͑e.g., = 2.15ϫ 10 −3 ͒.In Sec.V we will give a detailed discussion of these and other parameter values.Then the coefficients a i can be written as where =1+␤, = ␤ − 1, we have assumed the typical values mentioned above and made the approximation ͑Y −1͒ Ӎ Y.Even at this point, an exact solution for the front speed would be extremely cumbersome.Therefore, below we present two special cases where a manageable expression for the front speed can be found.First, we deal with the case ␤ ӷ 1.Then, Eqs.͑23͒-͑26͒ can be simplified as where we have just kept the highest-order power of ␤ for every coefficient a i .The critical condition to obtain the propagation speed is given by Eq. ͑22͒ when the equality holds.Then, substituting Eqs.͑27͒-͑30͒ into Eq.͑22͒, it is easy to see that the only real, positive solution if ␤ ӷ 1 is simply Note the interesting fact that the velocity is seen to be independent of ͑hence, independent of k 1 ͒ and Y in this limit.In Sec.V we will check this point by means of numerical simulations and discuss it in more detail.
On the other hand, we can easily a solution for the nondelayed case ͑ =0͒.Then ␤ = 0 and all of the terms containing ␤ in Eqs.͑23͒-͑26͒ vanish.We substitute the resulting coefficients a i in Eq. ͑22͒ and obtain where ␦ 3 =4, ␦ 2 = 6, and ␦ 1 = −27.

IV. NUMERICAL SIMULATIONS
We have integrated numerically the set of Eqs.͑15͒-͑17͒, in order to check the exact speeds obtained from Eqs. ͑19͒ and ͑20͒ and the approximate speeds from Eqs. ͑31͒ and ͑32͒.Numerical simulations are also interesting because they make it possible to obtain not only the front speed, but also profiles for the concentrations of the three species, namely, ͓V͔, ͓C͔, and ͓I͔.
In the experiments that we want to explain, the virus concentration of the initial inoculum is ͓17͔ where ͓V͔ 0 is the concentration of viruses at t = 0 and v 0 = 9.3ϫ 10 8 / ml.The initial condition ͑33͒, together with ͓C͔ = C 0 everywhere at t = 0 and ͓I͔ = 0 everywhere at t = 0, was used in all simulations.We used finite differences to approximate the partial derivatives in Eqs.͑15͒-͑17͒.Typical values for the space and time steps used in our simulations ͓e.g., in Fig. 1 and most of the simulations in Fig. 2͑b͔͒ were 1.5 s and 1 ϫ 10 −5 cm, respectively ͓dimensionless values for these steps were computed in order to use Eqs.͑15͒-͑17͔͒.However, in some cases a higher resolution was required in order to find good agreement with the theoretical front speeds.When high values of the parameters ͑Y, k, or ͒ were explored, the steps of space and time had to be reduced to 1 ϫ 10 −6 cm and below 0.3 s, respectively.Moreover, the same high resolution was required when very low values of the same parameters were used.Such step reductions lead to a substantial increase in computing time.Such higher reso-lution was needed in some of the simulations in Figs.2͑a͒  and 3. Figure 1 depicts an example of the concentration profiles ͓V͔, ͓C͔, and ͓I͔ after 48 h of infection.In Fig. 1 we observe that the profile of the uninfected cells C has the shape of a front ͑full curve͒, whereas that of the infected cells I has the shape of a pulse.This was to be expected intuitively, because the infection ͓first reaction in Eq. ͑1͔͒ decreases the number of uninfected cells C, which transform into infected cells I and eventually die.On the other hand, their death releases new viruses V ͓second reaction in Eq. ͑1͔͒ after some delay time , which explains the fact that the virus profile lags substantially behind that of infected cells I ͑in contrast to what is observed in numerical simulations of nondelayed models ͓24͔͒.Finally, let us compute the distance traveled by the infection front during 48 h.This should be 48 h times 0.053 cm/h ͓from Fig. 2͑a͒, rhombus with Ӎ 8 h͔, i.e., 0.25 cm.Adding 0.10 cm from the initial condition ͑33͒ we obtain 0.35 cm for the front position, in agreement with Fig. 1.

V. COMPARISON TO EXPERIMENT
In this section we apply our model to the specific case of a VSV focal infection spreading in Baby Hamster Kidney ͑BHK͒ cells.Experimental results of this biophysical system, including the observed front speeds, have been published in Ref. ͓16͔.Below we compare the observed speed range to both simulations and analytical solutions of our model.
First, we briefly discuss the parameter values used in our computations.Since we have not found any value for I MAX in the VSV experiments ͓16͔, we assume that I MAX Ӎ C 0 ͑in agreement with Fig. 1͒, i.e., that viruses are able to infect almost all of the cells at large enough times after the infection.The value of I MAX is thus I MAX Ӎ C 0 = 3.8ϫ 10 7 / ml ͓17͔.
The rate constant for the death of infected cells, k 2 , and the virus yield, Y, can be easily derived from the one-step growth data of VSV on BHK cells ͑Fig.3͑a͒ in ͓17͔͒ by fitting the logistic function ͓Eq.͑A11͒ in Appendix A͔.This The parameter values used are C 0 = 3.8ϫ 10 7 / ml, Y = 2.77ϫ 10 5 , k 1 = 1.4ϫ 10 −10 cm 3 / h, k 2 = 2.47/ h, = 8.1 h and D = 8.37 ϫ 10 −5 cm 2 / h ͑see Sec.V for a discussion on these values͒.Since the diffusion coefficient D of VSV in agar solutions has not been measured, we performed our analytical and numerical computations using two different values of D, namely, that in agar for other viruses ͓25,13͔ ͓D = 1.44 ϫ 10 −4 cm 2 / h, full curves in Figs.2͑a͒ and 2͑b͔͒ and the only value of D for VSV available, which refers to a very specific water solution ͓26͔ ͓D = 8.37ϫ 10 −5 cm 2 / h, dashed curves in Figs.2͑a͒ and 2͑b͔͒.
After VSV viruses infect BHK cells, their progeny leave the cells between Ӎ 2 h and Ӎ 10 h after infection ͑see Fig. 3͑a͒ in ͓17͔͒.Within this range, the predicted speeds in our Fig.2͑a͒ are consistent ͑for 3 Ͻ Ͻ 7 h͒ with the observed range, namely, ͑6.6-8.0͒ϫ 10 −3 cm/ h ͓27͔, in spite of the fact that the value of the diffusivity for the VSV virus is rather uncertain because ͑as mentioned above͒ it has not been measured through agar ͓29͔.Essentially the same agreement between theory and experiment is also obtained for the value of Y proposed in Ref. ͓17͔ ͓Fig.2͑b͔͒.In contrast, the nondelayed model ͓ = 0, as assumed in the model in Ref. ͓17͔ corresponding to Eq. ͑2͒ in the present paper͔ yields a very large error, as it is clear from its mismatch with the experimental data ͓compare the dotted horizontal lines to the shaded rectangle in Figs.2͑a͒ and 2͑b͔͒.This comparison with the experiments clearly shows the need to take into account the delay-time effect in order to explain the infection speed.This conclusion had not been reached before for VSV infections.
Our approximate solutions for the speed ͓Eqs.͑31͒ and ͑32͔͒ make it possible to understand better the differences between Figs. 2͑a͒ and 2͑b͒.If the delay time is large enough, according to Eq. ͑31͒ the front speed does not depend on Y neither .Both simulations and exact analytical results agree with the nondependence of the speed on Y, since for Ͼ 2 h the curves in Figs.2͑a͒ and 2͑b͒ are approximately the same.Note also that the approximate Eq. ͑31͒ has been derived assuming a high value of Y.Because of this, for Ͻ 2 h some differences between the exact and the approximate results arise ͓specially in Fig. 2͑b͒, because the value of Y is two orders of magnitude smaller than in Fig. 2͑a͔͒.Similarly, the approximate speed for =0 ͓Eq.͑32͔͒ is much more accurate for high values of Y ͓Fig.2͑a͔͒.
In Figs. 1 and 2 we have used the value k 1 = 1.4 ϫ 10 −10 cm 3 / h proposed in Ref. ͓17͔ for the adsorption rate because the value of k 1 has not been measured experimentally for VSV infecting BHK cells.However, the former value k 1 = 1.4ϫ 10 −10 cm 3 / h was obtained by fitting the observations to an extracellular model in Ref. ͓17͔.Because that model does not take into account the role of the delay time ͓compare Eqs.͑2͒-͑11͒ above͔, this value of k 1 should be regarded as highly uncertain.Therefore, we analyzed the dependence of the front speed on k 1 .We found the very interesting result, shown in Fig. 3, that the value of the front speed is approximately independent of k 1 over several orders of magnitude ͑provided that the delay time is taken into account, as first done in the present paper for VSV infec-tions͒.Indeed, in Fig. 3, the speed remains almost independent of the adsorption rate k 1 for large enough values of the delay time , namely, Ͼ 2 h.In contrast, k 1 becomes a relevant parameter for the case = 0.This is reasonable intuitively, because a long enough delay in the release of the new generation of viruses from the infected cells will substantially slow down the infection process.Then we should expect the time needed for the parent virus to cross the cell membrane ͑which is measured by k 1 ͒ to become irrelevant.This is indeed observed in Fig. 3 for Ͼ 2 h.The approximate solution given by Eqs.͑31͒ and ͑32͒ also supports this conclusion, because for large enough values of ͑␤ ӷ 1͒ according to Eq. ͑31͒ the speed does not depend on , hence neither in k 1 ͓whereas for the nondelayed case ͑ =0͒, Eq. ͑32͒ shows a nonlinear dependence on and therefore on k 1 ͔.This also illustrates that the conclusions obtained from delayed and nondelayed models can be very different.Furthermore, this makes it possible to make quantitative predictions of the front speed in spite of the uncertainty in the value of k 1 , and strongly supports the validity of our Figs.2͑a͒ and 2͑b͒, because it shows that they would remain almost the same even if the value of k 1 differed several orders of magnitude from the value used to obtain those figures ͑namely, k 1 = 1.4ϫ 10 −10 cm 3 / h, from Ref. ͓17͔͒.
We can also see that there is good agreement between the analytical results from Eqs. ͑19͒ and ͑20͒ ͑curves in Figs. 2 and 3͒ and the numerical simulations of Eqs.͑15͒-͑18͒ ͑full symbols͒.
FIG. 3. Front speed versus adsorption rate k 1 for several values of the delay time.The hatched rectangle corresponds to the experimental range, namely, ͑6.6-8.0͒ϫ 10 −3 cm/ h ͓27͔.The curves correspond to the analytical results from Eqs. ͑19͒ and ͑20͒ and the full symbols to the numerical simulations.The empty symbols indicate the approximate analytical speed results.The parameter values used are C 0 = 3.8ϫ 10 7 / ml, D = 8.37ϫ 10 −5 cm 2 / h, k 2 = 2.47/ h, and Y = 2.77ϫ 10 5 .
Relative differences between the speeds in Fig. 3 would not change if a different value of the virus diffusivity D were used.This is due to the fact that in the dimensionless Eq. ͑19͒, the parameter D does not appear, thus it is used only when computing the dimensional speed c from the dimensionless one c ¯as c = c ¯ͱk 2 D. Thus, using another diffusivity value D ˜would only change the speeds by the factor ͱ D ˜/ D.
Hence, we have shown that the only two parameters that have not been measured experimentally ͑namely, D and k 1 ͒ do not affect our conclusion that the role of the delay time cannot be neglected in order to understand VSV infection speeds.
In this section we have shown that the delay time is a strongly significant parameter in models of VSV infection spread.The authors of Ref. ͓17͔ fitted a large number of parameters ͑up to 9͒.In some cases, this can lead to overfitting the observed phenomena.We also think that it can sometimes lead to unjustified conclusions ͑e.g., to neglect the role of the delay time, which here we have found to be of utmost importance͒.In this paper, we have only used two unknown parameters ͑D and k 1 ͒ and shown that they are not strongly relevant to our main conclusions.

VI. CONCLUDING REMARKS
We have built a simple time-delayed model of VSV infections.In order to do so, we have improved a model previously applied to other kinds of viruses ͓13͔.Our analytical speeds agree well with those from numerical simulations.They are also in agreement with experimentally observed VSV infection speeds ͓Figs.2͑a͒ and 2͑b͔͒.Moreover, we have derived approximate front speeds that have lead us to additional understanding on the results and made our conclusions more intuitive.
As expected intuitively, the front speed decreases with increasing values of the delay time ͓Figs.2͑a͒ and 2͑b͔͒.We have obtained a strong decrease from =0 to Ӎ 1 h, and a smoother decrease for larger values of .We have shown that the introduction of the delay time is critically important to account for the front speeds of VSV infections.Indeed, neglecting the delay-time effect ͑i.e., =0͒ leads to an infection front speed one order of magnitude faster than the observed range ͓Figs.2͑a͒, 2͑b͒, and 3͒.
Moreover, we have shown that the infection speed is almost independent of the adsorption rate k 1 over four orders of magnitude for large enough values of the delay time ͑Fig.3͒, a conclusion not previously reached for any kind of virus infections.
As mentioned in Sec.II, Eqs.͑2͒-͑4͒ do not agree with the experimental data in the so-called one-step growth experiments.To see this, consider the very simple case of a homogeneous medium ͓so that ‫ץ‬ 2 ͓V͔͑r , t͒ / ‫ץ‬r 2 =0 ͔, composed initially only of infected cells ͓͓C͔͑t͒Ӎ0 and ͓I͔ ͑t =0͒ = ͓I͔ 0 ͔ and free viruses ͓͓V͔͑t =0͒ = ͓V͔ 0 ͔.In such a situation, Eq. ͑4͒ becomes simply and Eq.͑2͒ simplifies to However, this exponential behavior does not agree with the experimental data in the so-called one-step growth experiments, because in those experiments a logistic or S-shape curve is always observed instead ͑see, e.g., Fig. 1 in Ref.

͓13͔͒.
As a solution to avoid this inconsistency, it has been proposed ͓13͔ to replace Eqs.͑2͒-͑4͒ by Eqs.͑5͒-͑7͒.In order to explain this point, let us derive the virus dynamics predicted by the set ͓Eqs.͑5͒-͑7͔͒ in one-step growth experiments.Taking into account again that those experiments are performed in an homogeneous medium ͓so that the first term in the right-hand side of Eq. ͑5͒ is negligible͔ and in the absence of uninfected cells ͓so that ͓C͔͑r , t͒Ӎ0 ͔, Eqs.͑5͒-͑7͒ simplify to Integration of Eq. ͑A6͒ yields where ͓I͔ 0 is the value of ͓I͔͑t͒ at time t = 0. Note that this equation leads to ͓I͔͑t͒ → 0 for t → ϱ, as it should ͑because after a sufficiently long time, all infected cells will have died due to the second of reactions ͓Eq.͑1͔͒͒.
On the other hand, integrating Eq. ͑A5͒ from t =0 to t → ϱ we come to ͓V͔ ϱ = ͓V͔ 0 + Y͓I͔ 0 , ͑A8͒ so the increase in the concentration of viruses is limited by the initial concentration of infected cells ͓I͔ 0 and the yield Y, as it should ͑because Y viruses per infected cell are produced, according to the second of reactions ͓Eq.͑1͔͒͒.
If we integrate Eq. ͑A5͒ from t =0 to t we obtain, instead of Eq. ͑A8͒, ͓V͔͑t͒ = ͓V͔ 0 + Y †͓I͔ 0 − ͓I͔͑t͒ ‡, ͑A9͒ so, again, the increase in virus particles is equal to the decrease in infected cells multiplied by the yield.This is much more reasonable intuitively than Eq.͑A4͒.
Using Eq. ͑A7͒ into Eq.͑A9͒ leads us to

͑A10͒
If only a few viruses are present initially ͓͑V͔ 0 Ӷ Y͓I͔ 0 or ͓V͔ 0 Ӷ ͓V͔ ϱ ͒ and the initial concentration of infected cells is close to saturation ͓͑I͔ 0 Ӎ I MAX ͒, this simplifies to where c 1 ϵ͑ Equation ͑A11͒ is the typical logistic or S-shaped curve which ͑in contrast to the exponential curve ͓Eq.͑A4͔͒͒ agrees very well with experimental data of onegrowth experiments ͓see Eq. ͑3͒ and Fig. 1 in Ref. ͓13͔, and Fig. 2 in Ref. ͓16͔͔.We conclude that, both intuitively ͓see the text below Eq. ͑A9͔͒ and from the perspective of providing good fits to experimental data, Eqs.͑5͒-͑7͒ are more reasonable than Eqs.͑2͒-͑4͒.
Notwithstanding the former two strong arguments, the question arises if there is some physical mechanism that can lead to a dynamics with a quadratic saturation term ͓last term in Eq. ͑A6͔͒ or, equivalently, to its analytical solution ͓the logistic expression ͑A11͔͒.The answer is that there is at least one such mechanism, namely, to consider that in practice not all viruses spend exactly the same time inside a cell before releasing their progeny ͑i.e., that the value of the delay time is not the same for all virus particles͒.Then, if we choose the time origin t = 0 when the viruses enter the cells, the number of viruses at time t will be where ͑͒ is the probability distribution of the delay time .
Note that in the special case that all viruses have exactly the same value of , say 0 , ͑͒ is a Dirac delta centered at = 0 and Eq.͑A12͒ yields ͓V͔͑t͒ = ͓V͔ 0 if t Ͻ 0 and ͓V͔͑t͒ = Y͓V͔ 0 if t Ͼ 0 , as it should.If we consider the following delay-time distribution, which is bell-shaped ͑see the full curve in the inset to Fig. 1 in Ref. ͓15͔͒, using it into Eq.͑A12͒ and integrating yields the logistic Eq. ͑A11͒, i.e., the solution to the quadratic expression ͑A6͒.This gives a possible physical reason for the logistic solution ͓Eq.͑A11͔͒, and thus for the quadratic evolution Eq. ͑A6͒.However, we stress that there are other reasons why such a quadratic evolution equation is more reasonable than a linear one, both intuitively ͓see the text below Eq. ͑A9͔͒ and from the perspective of providing better fits to experimental data ͓see the text below Eq. ͑A11͔͒.In any case, the inclusion or not of a quadratic saturation term does not affect the analytical results in this paper, since the infection speeds derived would be the same if the quadratic term ͓last term in Eqs.͑5͒, ͑7͒, and ͑A6͔͒ were neglected ͓because after linearization, the last term in Eqs.͑15͒, ͑17͒, and ͑18͒ do not yield any term in Eq. ͑19͔͒.However, let us emphasize an important practical advantage of including this quadratic term, namely, that since Eq.͑A11͒ instead Eq. ͑A4͒ gives a much better fit to data, it leads to a more accurate estimate of the value of k 2 ͑as we have done in the Comparison to experiment section͒.Finally, it is worth to note that Y appears multiplying k 2 in all reaction-diffusion equations for ͓V͔ discussed in this paper ͓i.e., Eqs.͑2͒, ͑5͒, ͑9͒, and ͑11͔͒.Therefore, the question arises if it is possible to absorb Y into k 2 and, in this way, get rid of one parameter.The problem is that one-step growth experiments measure ͓V͔͑t͒ but not ͓I͔͑t͒ ͑see, e.g., p. 1736 in Ref. ͓17͔͒.Thus in order to estimate the parameter values, it is necessary to fit Eq. ͑A11͒ to the experimental data.But in Eq. ͑A11͒ Y no longer multiplies k 2 .For this reason, Y cannot be absorbed into k 2 .

APPENDIX B: TIME-DELAYED REACTION-DIFFUSION EQUATION
In order to make this paper as much self-contained as possible, in this appendix we give a derivation of the timedelayed reaction-diffusion Eq. ͑8͒.Variations in the population number density of viruses are due to two processes: population growth ͑replication minus adsorption͒ and dispersal.As usual ͓21͔, we Taylor-expand the variation due to population growth, where ͑x , y͒ are Cartesian space coordinates, is the generation time, the subindex g means growth, and we have introduced the growth function as F͑r , t͒ϵ͉ ‫͓ץ‬V͔ ‫ץ‬t ͉ g .On the other hand, the variation due to dispersal is ͓22͔ ͉͓V͔͑x,y,t + ͒ − ͓V͔͑x,y,t͉͒ d = ͵͵ ͓V͔͑x + ⌬ x ,y + ⌬ y ,t͒͑⌬ x ,⌬ y ͒d⌬ x d⌬ y − ͓V͔ ϫ͑x,y,t͒, ͑B2͒ where we have introduced the dispersal kernel ͑⌬ x , ⌬ y ͒, defined as the probability per unit area that a virus particle initially placed at ͑x + ⌬ x , y + ⌬ y ͒ has moved to ͑x , y͒ after a generation time .In a system involving both dispersal and population growth, the total variation is the sum of both contributions, ͓V͔͑x,y,t + ͒ − ͓V͔͑x,y,t͒ = ͵͵ ͓V͔͑x + ⌬ x ,y + ⌬ y ,t͒͑⌬ x ,⌬ y ͒d⌬ x d⌬ y − ͓V͔͑x,y,t͒ + F͑r,t͒ + 2 2 ͯ ‫ץ‬F͑r,t͒ ‫ץ‬t ͯ g + ¯͑B3͒ Finally, we Taylor-expand Eq. ͑B3͒ up to second order in time and space and assume an isotropic kernel ͓i.e., ͑⌬ x , ⌬ y ͒ = ͑⌬͒, where D is the diffusion coefficient 2 .Equation ͑B4͒ is the time-delayed Eq. ͑8͒ in our model.Note that the last term reads͉ ‫ץ‬F͑r,t͒ ‫ץ‬t ͉ g instead of ‫ץ‬F͑r,t͒ ‫ץ‬t .This corrects an error in a previous derivation ͓22͔ that was later applied to T7 virus infection fronts infecting E. Coli bacteria ͓13,14͔.That error was due to not including the symbol . . .͉ g in the last term in Eq. ͑B1͒.

FIG. 2 .
FIG. 2. Predicted front speed versus the delay time , for ͑a͒ the value Y = 2.77ϫ 10 5 proposed in this paper and ͑b͒ the value Y = 4389 proposed in Ref. ͓17͔.The dotted horizontal lines correspond to the nondelayed model ͑ =0͒ and the hatched rectangle to the experimental range, namely, ͑6.6-8.0͒ϫ 10 −3 cm/ h ͓27͔.The curves are the analytical results from Eqs. ͑19͒ and ͑20͒ and the full symbols are the numerical simulations.The empty symbols indicate the approximate analytical speed results.The parameter values used are C 0 = 3.8ϫ 10 7 / ml, k 1 = 1.4ϫ 10 −10 cm 3 / h, and k 2 = 2.47/ h.