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Critical behavior of a vector-mediated propagation of an epidemic process
Physica A 342 (2004) 249 – 255
www.elsevier.com/locate/physa
Critical behavior of a vector-mediated propagation of an epidemic process E. Macnadbaya;∗ , R. Bezerraa , U.L. Fulcob , M.L. Lyrac , C. Argoloa a Departamento
de F sica, Centro Federal de Educaca˜ o Tecnol ogica de Alagoas, 57020-510 Macei o-AL, Brazil b Departamento de F sica, Universidade Federal do Piaui, 64049-550, Teresina-PI, Brazil c Departamento de F sica, Universidade Federal de Alagoas, 57072-970 Macei o-AL, Brazil Received 24 October 2003; received in revised form 19 January 2004 Available online 18 May 2004
Abstract We investigate the critical behavior of a model that mimics the propagation of an epidemic process over a population mediated by a density of di4usive individuals which can infect a static population upon contact. We simulate the above model on 5nite chains to determine the critical density of vectors above which the system achieves a stationary active state with a 5nite density of infected individuals. Further, we employ a scaling analysis to determine the order parameter, correlation length and critical relaxation exponents. We found evidences that this model does not belong to the usual direct percolation universality class. c 2004 Published by Elsevier B.V. PACS: 64.60.Fr; 64.60.Ak; 87.23.Cc Keywords: Absorbing state phase-transition; Directed percolation; Di4usion-limited reaction
The critical behavior of non-equilibrium systems describes relevant features of several phenomena in physics, chemistry and biology [1]. In general these systems present a second-order phase transition between a vacuum state, where the order parameter density vanishes, and a steady reactive state [2]. At high dimensions, where Auctuations can be neglected, these systems can be modelled by a set of mean-5eld-like di4erential ∗
Corresponding author.
c 2004 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.04.085
250
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
equations. On the other hand, microscopic stochastic models de5ned on a lattice have provided a more accurate description of low-dimensional di4usion-limited reactions [3]. Density Auctuations are known to strongly modify the mean-5eld picture of a large class of such critical low-dimensional non-equilibrium kinetic lattice models [4]. The directed percolation universality class has been observed to describe a large class of models presenting a dynamic transition into a single absorbing state [5]. The prototype model for this class of systems is the contact process [4,6]. In this model, sites of a regular lattice can be either active or inactive, with inactive sites becoming active upon contact with a neighboring active site having a 5nite lifetime. A short lifetime of active sites results in the whole system being driven to the absorbing state with only inactive sites. Above a critical lifetime, the system reaches a stationary active state with a Auctuating 5nite fraction of active sites. Recently, the propagation of an epidemic process in a population of di4using individuals was modelled as a contact-like reaction–di4usion decay process of two species [7–9]. The total density of individuals acts as the relevant control parameter. For increasing densities, the stationary state presents a transition between the absorbing and the active state. An interesting feature is that the dynamic transition does not belong to the usual directed percolation universality class. The critical exponents depend on the relative di4usive constants of active and inactive particles. Further, as it was conjectured in recent works [7,8], a Auctuation-induced 5rst-order transition can possibly emerge in a particular di4usive regime. In this work, we introduce a model that simulates the propagation of an epidemic process mediated by a density of di4usive individuals which can infect a static population upon contact. This model captures some relevant issues of tropical diseases, like malaria, which are transmitted by mosquitoes. For these kind of diseases, the mosquitoes population is the most relevant issue for public health control. We investigate the absorbing state phase-transition which takes place by varying the density of di4usive particles. We simulate this dynamic model in one dimension where density Auctuations are expected to be more pronounced. The critical behavior will be characterized through the measure of a set of relevant critical exponents related to the 5nite size scaling of the critical order parameter and the critical short-time dynamics of the order parameter and its relative Auctuation. We consider a model of two interacting species. One of the species corresponds to the individuals of a population that can be either in an inactive (non-infected) state Pi or in an active (infected) state Pa . These individuals occupy all sites of a linear chain and are not allowed to di4use. Sites in the active state have a 5nite lifetime, becoming inactive at a rate . There is no direct contamination by contact. Therefore, without any mechanism to spread the infection, the population naturally evolves towards the vacuum state with no active sites. The spread of the disease will be considered to be mediated by a population of vectors which can be either in an active (infective) state Va or in an inactive (non-infective) state Vi . Vectors di4use over the chain with a density of vectors (number of vectors divided by the number of sites). Infective vector become non-infective at a rate . Active states are transmitted only between di4erent species whenever an inactive individual, Pi or Vi , occupies the same site of an active individual, Pa or Va , of the other species. The dynamics of the above described
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
251
(a)
(b) Fig. 1. (a) Average stationary density of individuals in the active (infected) state versus the density of vector particles for distinct chain sizes. Data were obtained from simulations with = = 0:5. The stationary regime was considered to be achieved after 10L lattice sweeps. Average was performed considering 104 distinct con5gurations taken at each L lattice sweeps. (b) The relative order parameter Auctuation U as a function of the density of vectors . Data were obtained as in Fig. 1a but now averaged over 105 con5gurations. Its scale invariance at the critical point allowed us to precisely estimate the critical vector density as being c = 2:81.
model can be represented by reaction-rate equations as Pa + Vi → Pa + Va ; Pi + Va → Pa + Va ; P a → Pi ; Va → Vi :
(1)
In what follows, we are going to show results from simulations of the present reaction–di4usion model on 5nite chains of size L. In each lattice sweep (considered as
252
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
(a)
(b) Fig. 2. Finite size scaling of the critical order parameter (a) and its logarithmic derivative (b). From the best 5t to power-laws, we estimate the critical exponents as being = = 0:34 (1) and = 1:75 (5). These are distinct from those of the directed percolation universality class (= = 0:25 and = 1:09).
the time unit) each vector can di4use with equal probability to one of its neighboring sites. If it is inactive, it becomes active whenever such site has an individual in state Pa . On the other hand, an active vector becomes inactive with probability . Within the same lattice sweep, each inactive individual becomes active if its site is occupied by at least one active vector. Active individuals become inactive with probability . In our simulations, we started with active Pa and inactive Pi individuals randomly distributed and measured the average density of active individuals = n(Pa )=L in the stationary regime as a function of the total density of vectors . All vectors are initially in non-infective state Vi and randomly distributed over the lattice sites. The stationary state was considered to be reached after 10L lattice sweeps. Whenever the system becomes trapped in the vacuum state, we activate a vector chosen at random. This activation procedure is used essentially to compute stationary state properties. It
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
253
Fig. 3. Collapse of data from distinct chain sizes using the critical exponents obtained from the 5nite size scaling analysis. All data fall over a single universal curve, thus reinforcing the accuracy of the estimated critical exponents.
becomes unnecessary when studying the short-time dynamics once the absorbing state is never reached for the time scales considered. Results are for the particular case of = = 0:5. In Fig. 1a, we show the average density of active individuals as a function of the density of vectors as obtained from simulations on chains of distinct sizes. One can clearly note that, in the limit of L → ∞, a transition from the vacuum to the active state takes place by increasing the vectors concentration. In order to precisely locate the critical vectors concentration, we measured the relative Auctuation in the density of infected individuals de5ned as UL ()=[n(Pa )]2 =[n(Pa )]2 −1. The relative Auctuation is known to be independent of the system size at the critical point. In Fig. 1b, we plot UL () obtained from distinct chain sizes, which allowed us to estimate the critical concentration as c = 2:81. Once having located the critical concentration, 5nite size scaling relations were used to compute the critical exponents characterizing such non-equilibrium phase transition. In particular, the critical order parameter density shall scale as (c ) ˙ L−= and its 1= logarithmic derivative as d ln d (c ) ˙ L . These scaling laws are depicted in Fig. 2 from which we estimate = = 0:34 (1) and = 1:75 (5). These two exponents were used to collapse the order parameter data obtained from distinct chain sizes into a single universal curve, as shown in Fig. 3. The 5ne data collapse further supports the accuracy of the estimated exponents. We also investigated the critical relaxation dynamics by exploring the fact that, at the critical point, the physical quantities obey power-law dynamic scaling laws [10–12]. When the system evolves from an initial state with all individuals in the active state, the order parameter density is expected to decay in time as (c ; t) ˙ t −=z . Further, the relative order parameter Auctuation shall grow as U (c ; t) ˙ t 1=z [13–15]. In Fig. 4, we show the results from such study of the short-time dynamics as obtained
254
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
(a)
(b) Fig. 4. Temporal evolution of the order parameter density (a) and its relative Auctuation (b). Data correspond to averages over 1000 distinct runs in chains with L = 104 sites, starting from an initial con5guration with all individuals in the active state. From the best 5t to power-law scaling we estimate =z = 0:19 (1) and z = 1:92 (8). The dynamic exponent z is also distinct from that of the directed percolation universality class (z = 1:58). The exponent ratio = obtained from the short-time dynamic scaling agrees with the independent estimate based on the 5nite size scaling of the stationary state.
from simulations of large chains with L=10,000 sites averaged over 103 independent realizations. From these, we could estimate =z = 0:19 (1) and z = 1:92 (8). It has been previously pointed out that the above short-time dynamics scaling may not work well for some speci5c models [14]. In order to verify the accuracy of the exponents derived from such scaling, we observed that it provides an independent estimate (although less precise) of = = 0:36 (3) in agreement with the 5nite size scaling analysis of the stationary state. This result suggests that the dynamic exponent z of the presently
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
255
investigated model is accurately determined by following the time evolution of a suitable Binder cumulant. In summary, we have investigated the critical behavior of an interacting two-species di4usion-limited reaction model which mimics the propagation of a disease in a population mediated by a population of di4usive vectors. We showed that the proposed model presents a transition from an absorbing to an active state at a critical density of vectors. The scaling analysis at the vicinity of this phase transition characterizes the universal behavior of out-break of epidemic processes. For example, the critical short-time dynamics can provide useful data concerning the rate at which the number of infected individuals decreases in time after the density of infective vectors is driven to the critical point. Using 5nite size and short-time dynamics scaling relations, we computed some relevant critical exponents governing this non-equilibrium phase transition from simulations of this dynamical model in linear chains. We obtained that, for the particular parameters set employed, the scaling relations provide = = 0:34, = 1:75 and z = 1:92. It is relevant to stress that these are quite distinct from the exponents characterizing the directed percolation universality class (= = 0:25, = 1:09 and z = 1:58). This result agrees with previous works which demonstrated that di4usion is indeed a relevant mechanism that can inAuence the critical behavior of absorbing state phase-transitions. It would be interesting to investigate the possible non-universality of the presently reported exponents with respect to the relative lifetime of active populations and the characteristic di4usion time. We expect to report on this point in future communications. This work was partially supported by the Brazilian research agencies CNPq and CAPES and by the Alagoas State agency FAPEAL. References [1] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992. [2] J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Cambridge University Press, Cambridge, 1999, and references therein. [3] D.C. Mattis, M.L. Glasser, Rev. Mod. Phys. 70 (1998) 979. [4] R. Dickman, in: V. Privman (Ed.), Nonequilibrium Statistical Mechanics in One Dimension, Cambridge University Press, Cambridge, 1996. [5] H. Hinrichsen, Adv. Phys. 49 (2000) 815. [6] T.E. Harris, Ann. Probab. 2 (1974) 969. [7] F. van Wijland, K. Oerding, H.J. Hilhorst, Physica A 251 (1998) 179. [8] K. Oerding, F. van Wijland, J.P. Leroy, H.J. Hilhorst, J. Stat. Phys. 99 (2000) 1365. [9] U.L. Fulco, D.N. Messias, M.L. Lyra, Phys. Rev. E 63 (2001) 066118. [10] H.K. Janssen, B. Shaub, B. Schmittmann, Z. Phys. B: Condens. Matter 73 (1989) 539. [11] Z.B. Li, U. Ritschel, B. Zheng, J. Phys. A 27 (1994) L837. [12] Z.B. Li, L. SchMulke, B. Zheng, Phys. Rev. Lett. 74 (1995) 3396. [13] Z.B. Li, L. SchMulke, B. Zheng, Phys. Rev. E 53 (1996) 2940. [14] B. Zheng, Int. J. Mod. Phys. B 12 (1998) 1419. [15] Guang-Ping Zheng, Mo Li, Phys. Rev. E 62 (2000) 6253.
www.elsevier.com/locate/physa
Critical behavior of a vector-mediated propagation of an epidemic process E. Macnadbaya;∗ , R. Bezerraa , U.L. Fulcob , M.L. Lyrac , C. Argoloa a Departamento
de F sica, Centro Federal de Educaca˜ o Tecnol ogica de Alagoas, 57020-510 Macei o-AL, Brazil b Departamento de F sica, Universidade Federal do Piaui, 64049-550, Teresina-PI, Brazil c Departamento de F sica, Universidade Federal de Alagoas, 57072-970 Macei o-AL, Brazil Received 24 October 2003; received in revised form 19 January 2004 Available online 18 May 2004
Abstract We investigate the critical behavior of a model that mimics the propagation of an epidemic process over a population mediated by a density of di4usive individuals which can infect a static population upon contact. We simulate the above model on 5nite chains to determine the critical density of vectors above which the system achieves a stationary active state with a 5nite density of infected individuals. Further, we employ a scaling analysis to determine the order parameter, correlation length and critical relaxation exponents. We found evidences that this model does not belong to the usual direct percolation universality class. c 2004 Published by Elsevier B.V. PACS: 64.60.Fr; 64.60.Ak; 87.23.Cc Keywords: Absorbing state phase-transition; Directed percolation; Di4usion-limited reaction
The critical behavior of non-equilibrium systems describes relevant features of several phenomena in physics, chemistry and biology [1]. In general these systems present a second-order phase transition between a vacuum state, where the order parameter density vanishes, and a steady reactive state [2]. At high dimensions, where Auctuations can be neglected, these systems can be modelled by a set of mean-5eld-like di4erential ∗
Corresponding author.
c 2004 Published by Elsevier B.V. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.04.085
250
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
equations. On the other hand, microscopic stochastic models de5ned on a lattice have provided a more accurate description of low-dimensional di4usion-limited reactions [3]. Density Auctuations are known to strongly modify the mean-5eld picture of a large class of such critical low-dimensional non-equilibrium kinetic lattice models [4]. The directed percolation universality class has been observed to describe a large class of models presenting a dynamic transition into a single absorbing state [5]. The prototype model for this class of systems is the contact process [4,6]. In this model, sites of a regular lattice can be either active or inactive, with inactive sites becoming active upon contact with a neighboring active site having a 5nite lifetime. A short lifetime of active sites results in the whole system being driven to the absorbing state with only inactive sites. Above a critical lifetime, the system reaches a stationary active state with a Auctuating 5nite fraction of active sites. Recently, the propagation of an epidemic process in a population of di4using individuals was modelled as a contact-like reaction–di4usion decay process of two species [7–9]. The total density of individuals acts as the relevant control parameter. For increasing densities, the stationary state presents a transition between the absorbing and the active state. An interesting feature is that the dynamic transition does not belong to the usual directed percolation universality class. The critical exponents depend on the relative di4usive constants of active and inactive particles. Further, as it was conjectured in recent works [7,8], a Auctuation-induced 5rst-order transition can possibly emerge in a particular di4usive regime. In this work, we introduce a model that simulates the propagation of an epidemic process mediated by a density of di4usive individuals which can infect a static population upon contact. This model captures some relevant issues of tropical diseases, like malaria, which are transmitted by mosquitoes. For these kind of diseases, the mosquitoes population is the most relevant issue for public health control. We investigate the absorbing state phase-transition which takes place by varying the density of di4usive particles. We simulate this dynamic model in one dimension where density Auctuations are expected to be more pronounced. The critical behavior will be characterized through the measure of a set of relevant critical exponents related to the 5nite size scaling of the critical order parameter and the critical short-time dynamics of the order parameter and its relative Auctuation. We consider a model of two interacting species. One of the species corresponds to the individuals of a population that can be either in an inactive (non-infected) state Pi or in an active (infected) state Pa . These individuals occupy all sites of a linear chain and are not allowed to di4use. Sites in the active state have a 5nite lifetime, becoming inactive at a rate . There is no direct contamination by contact. Therefore, without any mechanism to spread the infection, the population naturally evolves towards the vacuum state with no active sites. The spread of the disease will be considered to be mediated by a population of vectors which can be either in an active (infective) state Va or in an inactive (non-infective) state Vi . Vectors di4use over the chain with a density of vectors (number of vectors divided by the number of sites). Infective vector become non-infective at a rate . Active states are transmitted only between di4erent species whenever an inactive individual, Pi or Vi , occupies the same site of an active individual, Pa or Va , of the other species. The dynamics of the above described
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
251
(a)
(b) Fig. 1. (a) Average stationary density of individuals in the active (infected) state versus the density of vector particles for distinct chain sizes. Data were obtained from simulations with = = 0:5. The stationary regime was considered to be achieved after 10L lattice sweeps. Average was performed considering 104 distinct con5gurations taken at each L lattice sweeps. (b) The relative order parameter Auctuation U as a function of the density of vectors . Data were obtained as in Fig. 1a but now averaged over 105 con5gurations. Its scale invariance at the critical point allowed us to precisely estimate the critical vector density as being c = 2:81.
model can be represented by reaction-rate equations as Pa + Vi → Pa + Va ; Pi + Va → Pa + Va ; P a → Pi ; Va → Vi :
(1)
In what follows, we are going to show results from simulations of the present reaction–di4usion model on 5nite chains of size L. In each lattice sweep (considered as
252
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
(a)
(b) Fig. 2. Finite size scaling of the critical order parameter (a) and its logarithmic derivative (b). From the best 5t to power-laws, we estimate the critical exponents as being = = 0:34 (1) and = 1:75 (5). These are distinct from those of the directed percolation universality class (= = 0:25 and = 1:09).
the time unit) each vector can di4use with equal probability to one of its neighboring sites. If it is inactive, it becomes active whenever such site has an individual in state Pa . On the other hand, an active vector becomes inactive with probability . Within the same lattice sweep, each inactive individual becomes active if its site is occupied by at least one active vector. Active individuals become inactive with probability . In our simulations, we started with active Pa and inactive Pi individuals randomly distributed and measured the average density of active individuals = n(Pa )=L in the stationary regime as a function of the total density of vectors . All vectors are initially in non-infective state Vi and randomly distributed over the lattice sites. The stationary state was considered to be reached after 10L lattice sweeps. Whenever the system becomes trapped in the vacuum state, we activate a vector chosen at random. This activation procedure is used essentially to compute stationary state properties. It
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
253
Fig. 3. Collapse of data from distinct chain sizes using the critical exponents obtained from the 5nite size scaling analysis. All data fall over a single universal curve, thus reinforcing the accuracy of the estimated critical exponents.
becomes unnecessary when studying the short-time dynamics once the absorbing state is never reached for the time scales considered. Results are for the particular case of = = 0:5. In Fig. 1a, we show the average density of active individuals as a function of the density of vectors as obtained from simulations on chains of distinct sizes. One can clearly note that, in the limit of L → ∞, a transition from the vacuum to the active state takes place by increasing the vectors concentration. In order to precisely locate the critical vectors concentration, we measured the relative Auctuation in the density of infected individuals de5ned as UL ()=[n(Pa )]2 =[n(Pa )]2 −1. The relative Auctuation is known to be independent of the system size at the critical point. In Fig. 1b, we plot UL () obtained from distinct chain sizes, which allowed us to estimate the critical concentration as c = 2:81. Once having located the critical concentration, 5nite size scaling relations were used to compute the critical exponents characterizing such non-equilibrium phase transition. In particular, the critical order parameter density shall scale as (c ) ˙ L−= and its 1= logarithmic derivative as d ln d (c ) ˙ L . These scaling laws are depicted in Fig. 2 from which we estimate = = 0:34 (1) and = 1:75 (5). These two exponents were used to collapse the order parameter data obtained from distinct chain sizes into a single universal curve, as shown in Fig. 3. The 5ne data collapse further supports the accuracy of the estimated exponents. We also investigated the critical relaxation dynamics by exploring the fact that, at the critical point, the physical quantities obey power-law dynamic scaling laws [10–12]. When the system evolves from an initial state with all individuals in the active state, the order parameter density is expected to decay in time as (c ; t) ˙ t −=z . Further, the relative order parameter Auctuation shall grow as U (c ; t) ˙ t 1=z [13–15]. In Fig. 4, we show the results from such study of the short-time dynamics as obtained
254
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
(a)
(b) Fig. 4. Temporal evolution of the order parameter density (a) and its relative Auctuation (b). Data correspond to averages over 1000 distinct runs in chains with L = 104 sites, starting from an initial con5guration with all individuals in the active state. From the best 5t to power-law scaling we estimate =z = 0:19 (1) and z = 1:92 (8). The dynamic exponent z is also distinct from that of the directed percolation universality class (z = 1:58). The exponent ratio = obtained from the short-time dynamic scaling agrees with the independent estimate based on the 5nite size scaling of the stationary state.
from simulations of large chains with L=10,000 sites averaged over 103 independent realizations. From these, we could estimate =z = 0:19 (1) and z = 1:92 (8). It has been previously pointed out that the above short-time dynamics scaling may not work well for some speci5c models [14]. In order to verify the accuracy of the exponents derived from such scaling, we observed that it provides an independent estimate (although less precise) of = = 0:36 (3) in agreement with the 5nite size scaling analysis of the stationary state. This result suggests that the dynamic exponent z of the presently
E. Macnadbay et al. / Physica A 342 (2004) 249 – 255
255
investigated model is accurately determined by following the time evolution of a suitable Binder cumulant. In summary, we have investigated the critical behavior of an interacting two-species di4usion-limited reaction model which mimics the propagation of a disease in a population mediated by a population of di4usive vectors. We showed that the proposed model presents a transition from an absorbing to an active state at a critical density of vectors. The scaling analysis at the vicinity of this phase transition characterizes the universal behavior of out-break of epidemic processes. For example, the critical short-time dynamics can provide useful data concerning the rate at which the number of infected individuals decreases in time after the density of infective vectors is driven to the critical point. Using 5nite size and short-time dynamics scaling relations, we computed some relevant critical exponents governing this non-equilibrium phase transition from simulations of this dynamical model in linear chains. We obtained that, for the particular parameters set employed, the scaling relations provide = = 0:34, = 1:75 and z = 1:92. It is relevant to stress that these are quite distinct from the exponents characterizing the directed percolation universality class (= = 0:25, = 1:09 and z = 1:58). This result agrees with previous works which demonstrated that di4usion is indeed a relevant mechanism that can inAuence the critical behavior of absorbing state phase-transitions. It would be interesting to investigate the possible non-universality of the presently reported exponents with respect to the relative lifetime of active populations and the characteristic di4usion time. We expect to report on this point in future communications. This work was partially supported by the Brazilian research agencies CNPq and CAPES and by the Alagoas State agency FAPEAL. References [1] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992. [2] J. Marro, R. Dickman, Nonequilibrium Phase Transitions in Lattice Models, Cambridge University Press, Cambridge, 1999, and references therein. [3] D.C. Mattis, M.L. Glasser, Rev. Mod. Phys. 70 (1998) 979. [4] R. Dickman, in: V. Privman (Ed.), Nonequilibrium Statistical Mechanics in One Dimension, Cambridge University Press, Cambridge, 1996. [5] H. Hinrichsen, Adv. Phys. 49 (2000) 815. [6] T.E. Harris, Ann. Probab. 2 (1974) 969. [7] F. van Wijland, K. Oerding, H.J. Hilhorst, Physica A 251 (1998) 179. [8] K. Oerding, F. van Wijland, J.P. Leroy, H.J. Hilhorst, J. Stat. Phys. 99 (2000) 1365. [9] U.L. Fulco, D.N. Messias, M.L. Lyra, Phys. Rev. E 63 (2001) 066118. [10] H.K. Janssen, B. Shaub, B. Schmittmann, Z. Phys. B: Condens. Matter 73 (1989) 539. [11] Z.B. Li, U. Ritschel, B. Zheng, J. Phys. A 27 (1994) L837. [12] Z.B. Li, L. SchMulke, B. Zheng, Phys. Rev. Lett. 74 (1995) 3396. [13] Z.B. Li, L. SchMulke, B. Zheng, Phys. Rev. E 53 (1996) 2940. [14] B. Zheng, Int. J. Mod. Phys. B 12 (1998) 1419. [15] Guang-Ping Zheng, Mo Li, Phys. Rev. E 62 (2000) 6253.