Approximate solution to the speed of spreading viruses

Grup de Fı ́sica Estadı ́stica, Departamento de Fı ́sica, Universitat Auto `noma de Barcelona, 08193 Bellaterrra, Spain Departamento de Fı ́sica, Universitat de Girona, Campus de Montilivi, 17071 Girona, Catalonia, Spain Departamento de Medicina, Facultat de Cie `ncies de la Salut, Universitat Internacional de Catalunya, Gomera s/n, 08190 Sant Cugat del Valle `s, Barcelona, Spain ~Received 20 November 2003; published 24 March 2004 !


I. INTRODUCTION
The role of the delay time in the spread of viruses in a plaque has been recently analyzed ͓1͔ by considering a delay time for virus diffusion.It has been shown that the delay time plays a crucial role in the dynamics of the advancing virus front, because it substantially reduces the value for the predicted speed as compared to the classical, parabolic model ͓2͔.In this sense, is the time that a virus particle spends, from the moment it is adsorbed into a host bacterium, to take control of it, replicate its proper genetic material, reproduce, and kill the cell.We consider a model of three species, the virus particles V, the host bacteria B, and the infected host bacteria I.There are two reactions involved in the virus expansion over the bacterial colony: ͑i͒ the adsorption process, during which a virus particle couples to a host bacterium through its membrane and the cell becomes infected and ͑ii͒ the lysis process, at the end of which the cell is killed and the virus progeny outbreak takes place.Thereafter, ͑iii͒ the phages disperse and ͑iv͒ they infect new hosts, so the process begins again.Let k 1 be the rate constant of adsorption.The virus particle introduces its genetic material in the infected bacteria and begins the reproduction.After a certain delay time ͑latent or lag time͒, the virus particle is completely reproduced and the infected bacterium dies ͑ly-sis͒.
In this work we obtain an analytic expression for the speed of the growth of virus plaques and compare them with the numerical solution of the complete system and with the experimental data.Comparison with the classical or nondelay time models ͓2,3͔ are not included because it was already done in ͓1͔.

A. Virus spreading dynamics
The process of infection, virus replication, and bacterium death can be summarized by a three species reaction as follows: Parameter Y ͑the ''yield''͒ is the production of new viruses per infected bacterium and k 2 is the rate constant of lysis of infected bacteria.
In order to find a good quantitative agreement with the experimental observations, it has been previously shown ͓1͔ that a better way to model the virus diffusion process is by taking into account the delay time between virus adsorption and bacteria death and the spreading of the newborn viruses.In practice, it implies that parabolic or classical reactiondiffusion equation must be replaced by its hyperbolic generalization ͓4,5͔, where the mentioned delay time appears explicitly.Assuming logistic dynamics for the growth process, the equations for our models are 2 In these equations ͓¯͔ denotes concentration and subindices ͓¯͔ tt ,͓¯͔ t , and ͓¯͔ rr stand for second time derivative, time derivative, and second spatial derivative in the radial direction from the plaque center, respectively.In Eq. ͑2͒ D e f f appears instead of the usual diffusion coefficient D. The reason is that the diffusing particles, i.e., viruses, do not move in a homogeneous continuous medium ͑agar in our case͒ but in the presence of a suspension of ellipsoids ͑host bacteria͒ which adsorb them.This is known as hindered diffusion, and the effective diffusion coefficient D e f f for this type of diffusion is related to the usual one, D, according to Fricke's equation ͓6͔: where f ϭB 0 /B max is the ratio of bacteria concentration to its maximum possible value and x takes care of the bacterium shape.
Equations ͑2͒-͑4͒ can be written in terms of dimensionless variables B ¯ϵ͓B͔/B 0 ,V ¯ϵ͓V͔/B 0 , I ¯ϵ͓I͔/B 0 , t ¯ϵk 2 t, and r ¯ϵrͱk 2 /D e f f and dimensionless parameters ¯ϵk 2 and ϵk 1 B 0 /k 2 , where B 0 is the initial bacterium concentration.We look for solutions depending only on new variable z ¯ϵr Ϫc ¯t ¯where c ¯Ͼ0 is the dimensionless wave front speed, which is related to dimensional speed c by c ¯ϵc/ͱD e f f k 2 .As usual, we linearize our equations around the unstable steady state (͓V͔,͓B͔,͓I͔)ϭ(0,B 0 ,0), i.e., (V ¯,B ¯, I ¯)ϭ( V ,1 Ϫ B , I ), where ϵ( V , B , I )Ӷ1.Then solutions to the linearized version of Eqs.͑2-4͒ are given by ϳexp(Ϫ z ¯) where, in order to avoid trivial solutions, the following characteristic equation must be satisfied: For simplicity, we have introduced the parameters ␤ ϵ ¯/2,␥ϵY Ϫ1, and ␦ϵkϩ1.

B. Wave front speed
In order to avoid nonpositive values for concentrations, we must impose that the three solutions for in Eq. ͑6͒ 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 ͑7͒ in terms of ϵc ¯2 and then we get where coefficients a i are given by a 0 ϭϪ4␥, The speed of the wave front can be calculated numerically from c ¯ϭmin Ͼ0 ͓c ¯()͔, where c ¯() is given by Eq. ͑6͒ as it is done in Ref. ͓1͔, but now we shall try to obtain an approximated analytical expression for this minimum speed although, for this purpose, we shall make some approximations.On one hand, we define ⑀ϵk 1 /k 2 B max which implies that ϭ⑀ f .As we shall see in detail in the following section, when typical experimental values for the parameters are used, one observes that ⑀ is always a small parameter, i.e., ⑀Ӷ1.This fact allows us to expand the coefficients a i up to first order in ⑀, so we get Moreover, if ␤ӷ1 is verified, we can simplify Eqs.͑10͒ even further to get a 0 ϭϪ4 f ␥⑀ϵr 0 , Then Eq. ͑8͒ is reduced to r 3 3 ϩr 2 2 ϩr 1 ϩr 0 у0, ͑12͒ where coefficients r i are defined in Eq. ͑11͒.The condition critical to the propagation speed is given by Eq. ͑12͒ when equality holds, and then it is easy to show that positive solutions for the speed are or, in terms of the dimensional variables, According to the principle of marginal stability ͓7,8͔, from both expressions for the wave front speed, we must choose the minimal one.This will be confirmed in Sec.III below by means of numerical integrations of Eqs.͑2͒-͑4͒.It is easy to show the existence of a critical value of f, namely, f 0 , such that c 1 Ͻc 2 if f Ͻ f 0 and c 1 Ͼc 2 if f Ͼ f 0 .Figure 1 shows both c 1 and c 2 as functions of f for typical experimental values of parameters.Then we can write the minimal speed as follows: where f 0 is defined as

III. COMPARISON TO OBSERVATIONS
We compare in this section the results of Eq. ͑15͒ with the experimental values for virus T7 which spread in a medium containing agar-immobilized E. coli bacteria.We also compare the new results with numerical integrations performed on system ͑2͒-͑4͒.The values of the parameters are: B max ϭ10 7 Ϫ10 8 ml Ϫ1 , k 1 ϭ(1.29Ϯ0.59)ϫ10Ϫ9 ml/min, k 2 ϭ1.39 min Ϫ1 , ϭ18.4 min, Y ϭ34.5, Dϭ4ϫ10 Ϫ8 cm 2 /s, and xϭ1.67.To obtain Eq. ͑15͒ we have assumed that ␤ ӷ1 ͑in fact ␤ϭk 2 /2ϭ12.8)which basically implies that delay time is large enough, so comparison to nondelay time modes are out of place.The other assumption is ⑀Ӷ1 and from the experimental data we have that ⑀ ranges from 5 ϫ10 Ϫ3 to 0.135.
In Fig. 2͑a͒ we have taken B max ϭ10 7 ml Ϫ1 and the two extreme values for k 1 .We plot the analytic solution for the speed of the front ͑15͒ ͑lines͒ and the results from numerical solutions of the system ͑2͒-͑4͒ ͑symbols͒ and observe good agreement with the experimental results.In Fig. 2͑b͒ we take B max ϭ10 7 ml Ϫ1 for the same values of k 1 as before where good agreement with experimental results is also found.

IV. DISCUSSION
In the present paper we have found an explicit expression for the speed of the growth of virus plaques ͑15͒ which is valid only if the parameter values satisfy the specified conditions, i.e., ⑀Ӷ1 and ␤ӷ1.Moreover, we have performed numerical integrations on Eqs.͑2͒-͑4͒ in order to compare their results with predictions from Eq. ͑15͒.We can see this comparison in Fig. 2 and we note that both approaches are in FIG. 1. Solutions to Eq. ͑12͒ when equality holds as functions of the bacterial relative concentration f.The selected value for the speed is the minimal one, i.e., c 1 if f Ͻ f 0 and c 2 if f Ͼ f 0 .Both functions are drawn for B max ϭ10 7 ml Ϫ1 , k 1 ϭ0.7ϫ10Ϫ9 ml/min, k 2 ϭ1.39 min Ϫ1 , ϭ18.4 min, and Y ϭ34.5.FIG. 2. Curves: speed of the growth of T7 virus plaques on E. coli as a function of the bacterial relative concentration according to expression ͑15͒.Symbols: squares and triangles, numerical integrations of Eqs.͑2͒-͑4͒; open and closed circles, experimental data.In ͑a͒ ⑀ϭ5ϫ10 Ϫ3 for the dotted line and square symbols and ⑀ϭ0.013 for the dashed line and triangles.In ͑b͒ ⑀ϭ0.05 for the dotted line and square symbols and ⑀ϭ0.13 for the dashed line and triangles.For all cases ␤ϭ12.8.