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Critical properties of a vector-mediated epidemic process
Physica A 533 (2019) 122085
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Critical properties of a vector-mediated epidemic process F.L. Santos a , M.L. Almeida a , E.L. Albuquerque a , A. Macedo-Filho b , M.L. Lyra c , ∗ U.L. Fulco a , a
Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil Campus Prof. Antonio Geovanne Alves de Sousa, Universidade Estadual do Piauí, 64260-000, Piripiri-PI, Brazil c Instituto de Física, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil b
highlights • • • •
Vector-mediated epidemic propagation model was performed in a linear chain. Monte Carlo simulations and finite-time scaling was used to obtain the critical exponents. The critical properties from the statistically stationary state are estimated. The transition from the inactive vacuum absorbing state to the epidemic statistically stationary state belongs to the diffusive epidemic universality class.
article
info
Article history: Received 17 April 2019 Received in revised form 5 June 2019 Available online 16 July 2019 Keywords: Critical properties Epidemic process Vector-mediated model Monte Carlo simulations Dynamical critical exponents Universality class
a b s t r a c t We study the critical behavior of an epidemic propagation model with interacting static individuals and diffusive vectors. The model presents a non-equilibrium phase transition from an absorbing vacuum state to an epidemic state at a critical vector density which depends on the recovery rates of infected individuals and vectors. The simulation was performed in a linear chain of the proposed model and the finite time scale hypothesis was explored to estimate the vector critical density and dynamic critical exponents. Our results show that the absorbing-state phase transition belongs to the universality class of the symmetric diffusive epidemic process irrespective to the relative values of the recovery rates. On the other hand, the critical vector density shows a much stronger dependence on the recovery rate of vectors than on the corresponding recovery rate of individuals. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Accordingly to the World Health Organization, 17% of all infectious diseases in the world are transmitted by vectors. This problem draws the attention of the scientific community which has devoted a great deal of effort to study these epidemics processes [1–4]. One focus of these investigations is the universal characteristics of epidemic outbreaks, which is a problem not only in medical science, but also in public health. As examples, an outbreak of the dengue fever or malaria disease can kill many people worldwide justifying, by itself, the importance of a deeper understanding of the underlying dynamics that can provide insights in what public actions can be more effective in preventing or stopping the spreading of this disease. ∗ Corresponding author. E-mail address: [email protected] (U.L. Fulco). https://doi.org/10.1016/j.physa.2019.122085 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
Epidemic spreading as a kind of a diffusive process model was theoretically investigated by Kree et al. [5], with the aim to model the effect of pollution on a population on the brink of extinction. This process was also characterized by the coupling between the fluctuations in the total density with respect to the density of the species that is trying to survive [6], being afterwards extended to consider also distinct diffusion rates [7,8]. Although very simple, this model presents an extraordinary richness. This process belongs to the general category of dynamical complex systems, which exhibits a continuous phasetransition from the active stationary state to an absorbing one [8–11]. An important fact is that the critical behavior falls in a very specific universality class with exponents, as defined by renormalization group techniques, 1/z = 1/2 and 1/z ν = 1 − ϵ/4 where, ϵ = 4 − d, d being the dimension of the system, z and ν the dynamic and correlation length exponents, respectively [5,7,12]. In this paper we consider a diffusive epidemic model that simulates the transmission strategies mediated by a vector and analyze its universal characteristics at the vicinity of the transition to the vacuum state on which the epidemic spreading stops. Its main focus is the investigation on the statistical properties of a model for vector-mediated transmission of a disease in a population consisting of two species, namely hosts and vectors, which contract the disease from one another. It captures some relevant issues of tropical diseases, like malaria, which are transmitted by mosquitoes. The hosts are sedentary, meaning that each individual is kept fixed in a site of the network, while the vectors diffuse within the network at a constant rate. The spreading of vector-mediated diseases have also been investigated using alternative models, as for example a coupled susceptible–infected–recovered (SIR) dynamics for the humans and a susceptible– infected–susceptible (SIS) dynamics for the vectors [13]. A mean-field approach as well as simulation results considering static populations of humans and vectors occupying sites of a bipartite lattice while interacting by contact unveiled that spreading of the disease is impossible for any death rate of vectors if the infection rate is low [13]. To investigate the critical behavior of the model, we use Monte Carlo simulations and finite-time scaling to obtain the critical exponents β/z ν , 1/z and 1/z ν , where β is the critical parameter associated with the decay of the order parameter as the critical point is approached. A detailed study of the critical behavior considering distinct scenarios for the average lifetime of both infected hosts and vectors controlled by their recovery rates λ and φ will be presented, and its main features discussed. This work is organized as follows. In Section 2 we present in some detail the proposed model and the algorithm used in the numerical simulations. Section 3 reports our main numerical results, leaving the summary and conclusions of this work to be depicted in Section 4. 2. Vector–mediated epidemic propagation model In our simulation, the description of a vector-borne disease is based on the life cycle of the parasite, like dengue. Similar models of coupled populations of hosts and vectors have been previously introduced in the literature showing that it can capture the transition from the vacuum state with no infected individuals to the active state with a finite population of infected hosts and infective vectors [14,15]. Initially, we consider a randomly distribution of equally number of infected (Vi ) and non-infected (V ) vectors between the N sites of a linear chain with periodic boundary conditions, with each site being a locus, where an arbitrary number of vectors can be located, which means that they do not have a hardcore repulsive potential. Vectors diffuse over the chain with a density ρ (the number of vectors divided by the number of sites) kept fixed in time. However, each site can either be free of vectors or occupied by infected and/or non-infected vectors. There is a host at each lattice site, which can be in one of two states, namely the infected (Hi ) or non-infected (H) ones, respectively. The contamination process takes place only when vectors and hosts are found on the same site. The whole dynamics of this reaction–diffusion model is done in three stages: (a) The recovering process: an infected vector (hosts) is replaced by a non-infective one at rate φ (λ). As such, φ (λ) controls the average time a vector (host) stay in the infected state. (b) The contamination process: active states are transmitted only between different species. Each non-infected host in the same site of at least one infected vector will be transformed in a infected host with probability k1 . A noninfective vector becomes infected at the rate k2 if it shares the same site with an infected host. Our simulations are performed using k1 = k2 = 1. (c) The diffusive motion: the vectors jump to a neighbor at a rate D = 0.5 with equal probability to each one of its neighboring sites. In the above contamination process, we are considering the regime of maximal infectivity ki = 1 for simplicity because the universality class of the underlying absorbing-state phase transition is not expected to depend on the specific value of these rate parameters. For finite infection rates, one should consider that a non-infected host would be transformed into an infected host at rate k1 by each infected vector occupying the same site. In this way, the effective contamination would depend on the number of infected vector at a given site. The above model can be summarized by the set of reaction-rate equations represented by λ
Hi ↦ − → H;
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
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Fig. 1. Log–Log plot of the time-evolution of the density of infected individuals Ψ (ρ, t) in the close vicinity of the critical point ρ = [1.590, 1.600] for the case φ = 0.50 and λ = 0.90. Notice that for ρ < 1.595 (ρ > 1.595) the curves clearly veer downward (upward), leading to the estimate of ρc = 1.595. This procedure was used to locate the critical vector density for all pairs of recovery rates explored in this work.
φ
→ V; Vi ↦ − k1
H + Vi ↦ − → Hi ; k2
V + Hi ↦ − → Vi . The mechanism of dissemination of the infection among these coupled populations is made by the diffusion of the vectors, which are infected only by the hosts. Therefore, any infected host is also infective. A detailed study will be now be performed by varying the recovery rates of vectors and hosts. Our main goal is to investigate how the critical density of vectors depends on these factors and to unveil the robustness of the corresponding universality class. 3. Simulation results and discussions To estimate the critical properties from the statistically stationary state, we explore finite-time effects on Monte Carlo numerical simulation data. Our aim here is the finding of the critical density of vectors below which the epidemic process does not spread and the system evolves towards the absorbing state free of infected individuals. Furthermore, we should get precise estimations of the dynamical critical exponents β/z ν , 1/z and 1/z ν , where β is the critical exponent associated with the density of infected individuals that behaves as an order parameter of this non-equilibrium phase-transition. In order to explore the critical relaxation dynamics of the system, we start considering all active hosts Hi and measure how the average density of active hosts (Ψ = ⟨n(Hi )⟩/L) decays as a function of time t, where n(Hi ) is the number of infected hosts at site i. When the system is driven to the vacuum inactive (epidemic active) state, the density of active (infected) individuals decays exponentially in a fast way (evolves towards a statistically stationary value). This is usually the case achieved when the density of vectors ρ is small (large). At the critical point ρc (total density of vectors), the timeevolution towards the vacuum state becomes slower and power-law dynamic scaling laws emerge [16–19]. In particular, the order parameter (density of infected individuals) is expected to decay in time at ρc as:
Ψ (ρc , t) ∼ t −β/z ν .
(1)
Fig. 1 depicts a log–log plot of Ψ (t) as a function of t for ρ ranging in the close vicinity of the critical density, i.e. ρ = [1.590, 1.600], for the case φ = 0.50 and λ = 0.90. This plot is obtained by using a lattice with L = 10240 sites and averaged over Nrep = 104 independent realizations, with a relaxation time trmax = 105 . The change in the concavity in the large run regime signals the critical vector density. The best estimation found for the critical vector density was
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F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
Fig. 2. Log–Log plot of the time-evolution of the order parameter at the critical density for φ = 0.50 and λ = 0.50. From the slope of this curve we estimate the critical exponent ratio β/z ν = 0.127(6).
Fig. 3. Log–Log plot of the time-evolution of the relative fluctuation of the order parameter at the critical density for φ = 0.80 and λ = 0.90, whose slope yields an estimation of the dynamical critical exponent 1/z = 0.49(3).
ρc = 1.595, where the curve shows the smallest curvature. Observe that from the data for ρ < 1.595 (ρ > 1.595) the curves clearly veer downward (upward). This procedure was performed for distinct sets of recovery rates to locate the corresponding critical vector density. In what follows, we will show the power-law dynamic scaling for some relevant quantities for some particular values of φ and λ. In Fig. 2 we report the time-evolution of the density of infected individuals for φ = λ = 0.5 for which we estimated ρc = 1.308(2). The best power-law fitting leads toβ/z ν = 0.127(6).
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
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Fig. 4. Same as in Fig. 3, but for φ = 0.20 and λ = 0.30. From the slope we estimate the critical exponent ratio 1/z ν = 0.24(4).
We can also obtain an independent estimate of the dynamic critical exponent z by exploring the time-evolution of the relative order parameter fluctuation at the critical point which shall scale as [20–23]: U(ρc , t) = (⟨Hi 2 ⟩/⟨Hi ⟩2 ) − 1 ∼ t 1/z .
(2)
Fig. 3 shows a typical plot of U(t) using as representative case the recovery rates φ = 0.80 and λ = 0.90. In this case we estimated ρc = 5.169(3). The best power-law fitting provided the estimate for the dynamic exponent 1/z = 0.49(3) for this case. We also got an independent estimate for the exponent ⏐ ratio⏐ 1/z ν . For this, we followed the critical relaxation of the
⏐ ∂ lnΨ (ρc ) ⏐ ⏐. According to the finite-time scaling hypothesis, it shall ∂ρ
logarithmic derivative of the order parameter D(t) = ⏐ scale at the critical point as D(t) ∼ t 1/z ν .
(3)
We illustrate such scaling at criticality in Fig. 4 using the parameters φ = 0.20 and λ = 0.30 for which we estimated ρc = 0.408(2). The asymptotic slope of the corresponding curve gives 1/z ν = 0.24(4). The full set of critical densities and exponents for all pairs of simulated recovery rates is summarized in Table 1. We notice that, within the error bars, the critical exponents are rather independent of the typical lifetime of either infected individuals or vectors. These exponents are compatible with this model belonging to the universality class of the symmetric diffusive epidemic process [6,8]. On the other hand, the critical vector density is strongly affected by the recovery rates. In order to explicitly show the relative role played by the recovery rate of either infected individuals or infective vectors, we plot in Fig. 5 the critical vector density as a function of φ and λ. Notice that ρc shows a stronger dependence on the recovery rate of vectors φ (Fig. 5a) than on the recovery rate of infected individuals λ (Fig. 5b). 4. Summary and conclusions In summary, we have investigated the critical behavior of an interacting two-species diffusion-limited reaction model which mimics the propagation of a disease in a static population mediated by a population of diffusive 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 the out-break of the epidemic processes. A critical short-time dynamics was used to provide some relevant critical exponents governing this non-equilibrium phase transition from simulations of this dynamical model in linear chains. The considered model contains three basic control parameters: The total density of vectors per site ρ , the recovery rates of individuals λ, and the vectors φ . These rates govern processes at which an individual or vector in the infected state
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F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
Fig. 5. The estimated vectors critical density ρc as a function of (a) the recovery rate of vectors φ and (b) the recovery rate of individuals λ. Notice that ρc depicts a much stronger dependence on φ .
changes to the non-infected state. They are related to the average lifetime an individual/vector remains in the infective state. The total density of vectors gives the background population through which the infected state can be transferred from one individual to the others. At very low vector densities the epidemic becomes extinct while it can become endemic above a critical density of vectors. The one-dimensional diffusive epidemic process has critical exponents given by z = ν = 2 [5,7] and β/z ν = 0.11(2) [8,10]. Our simulations suggested that the transition from the inactive vacuum absorbing state to the epidemic statistically stationary state of the investigated model belongs to such diffusive epidemic universality class irrespective to the relative recovery rates of individuals and vectors. Possible mechanisms that could influence the universal behavior include
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
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Table 1 The critical vector density ρc and the corresponding dynamical exponents β/z ν , 1/z and 1/z ν for the present reaction–diffusion model for distinct pairs of recovery rates φ (vectors) and λ (individuals). (φ, λ)
ρc
β/z ν
1/z
1/z ν
(0.20; 0.10) (0.20; 0.30) (0.20; 0.50) (0.20; 0.70) (0.20; 0.90) (0.50; 0.10) (0.50; 0.30) (0.50; 0.50) (0.50; 0.70) (0.50; 0.90) (0.80; 0.10) (0.80; 0.30) (0.80; 0.50) (0.80; 0.70) (0.80; 0.90)
0.210(2) 0.408(2) 0.509(2) 0.574(4) 0.619(1) 0.524(1) 1.037(2) 1.308(2) 1.477(3) 1.595(2) 1.645(3) 3.327(4) 4.228(3) 4.787(3) 5.169(3)
0.127(10) 0.115(12) 0.122(4) 0.118(9) 0.118(6) 0.125(6) 0.124(6) 0.127(6) 0.128(6) 0.118(10) 0.150(21) 0.136(11) 0.127(6) 0.128(4) 0.126(4)
0.54(4) 0.50(4) 0.50(4) 0.50(3) 0.50(2) 0.55(6) 0.51(4) 0.52(4) 0.52(4) 0.50(3) 0.56(5) 0.51(5) 0.50(4) 0.50(3) 0.49(3)
0.24(4) 0.24(4) 0.26(7) 0.24(8) 0.23(6) 0.26(4) 0.24(8) 0.27(5) 0.26(6) 0.22(9) 0.28(7) 0.29(7) 0.22(10) 0.29(5) 0.23(5)
anomalous diffusion [24] and stochastic resets of the diffusing particles to their initial positions [25–28] because these mechanisms introduce non-Gaussian fluctuations on the particles distribution. These aspects deserve future detailed investigations. On the other hand, the critical density of vectors needed to sustain the epidemic process depicts a stronger dependence (faster than linear growth) on the recovery rate of vectors φ than on the recovery rate of individuals λ (slower than linear growth). The above results shows clearly the relative effectiveness of interventions that can be put forward to drive an epidemic process mediated by diffusive vectors to extinction. The most effective actions are those directed to reduce the vector density. Usually this is performed by the use of pesticides or biological control of the population of vectors. Such population control demands strategies do reduce the reproduction rate by either interventions in the environment conditions needed for the vector reproduction, or by the introduction of genetically modified infertile vectors. On the other hand, the present results indicate that actions devoted to reduce the period over which an individual remains infective has a limited impact in the control of the epidemic process due to sub-linear dependence of the critical density of vectors on the recovery rate of hosts. Such control can be much more effective if the strategy is focused in reducing the lifetime of infected vectors. It would be interesting to have new epidemiological data to support this theoretical prediction. Acknowledgments This work was supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), Brazil, CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil, and FAPEAL (Fundação de Apoio à Pesquisa do Estado de Alagoas), Brazil. We would like to thank the Núcleo de Processamento de Alto Desempenho of the Universidade Federal do Rio Grande do Norte - NPAD/UFRN to allow us to access their computer facilities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
E. Alfinito, M. Beccaria, A. Fachechi, G. Macorini, Europhys. Lett. 117 (2017) 18002. A.A. Alemi, M. Bierbaum, C.R. Myers, J.P. Sethna, Phys. Rev. E 92 (2015) 052801. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, Rev. Modern Phys. 87 (2015) 925. R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65 (2002) 036104. R. Kree, B. Schaub, B. Schmittmann, Phys. Rev. A 39 (1989) 2214. U.L. Fulco, D.N. Messias, M.L. Lyra, Physica A 295 (2001) 49. F. van Wijland, K. Oerding, H.J. Hilhorst, Physica A 251 (1998) 179. U.L. Fulco, D.N. Messias, M.L. Lyra, Phys. Rev. E 63 (2001) 066118. A.M. Filho, G. Corso, M.L. Lyra, U.L. Fulco, J. Stat. Mech. 2010 (2010) P04027. D.S. Maia, R. Dickman, J. Phys.: Condens. Matter 19 (2007) 065143. J.E. Freitas, L.S. Lucena, L.R. Silva, H.J. Hilhorst, Phys. Rev. E 61 (2000) 6330. Hans-Karl Janssen, Phys. Rev. E 64 (2001) 058101. D.R. de Souza, T. Tomé, S.T.R. Pinho, F.R. Barreto, M.J. de Oliveira, Phys. Rev. E 87 (2013) 012709. E. Macnadbay, R. Bezerra, U.L. Fulco, M.L. Lyra, C. Argolo, Physica A 342 (2004) 249. A.G. Dickman, R. Dickman, Amer. J. Phys. 83 (2015) 468. H.K. Janssen, B. Shaub, B. Schmittmann, Z. Phys. B 73 (1989) 539. Z.B. Li, U. Ritschel, B. Zheng, J. Phys. A. 27 (1994) L837. Z.B. Li, L. SchMulke, B. Zheng, Phys. Rev. Lett. 74 (1995) 3396. N.V. Costa, U.L. Fulco, M.L. Lyra, I.M. Gléria, Phys. Rev. E 75 (2007) 031112. Z. Li, L. Schulke, B. Zheng, Phys. Rev. E 53 (1996) 2940. B. Zheng, Internat. J. Modern Phys. B 12 (1998) 1419. Guang-Ping Zheng, Mo Li, Phys. Rev. E 62 (2000) 6253.
8 [23] [24] [25] [26] [27] [28]
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085 Z.B. Li, L. Schülke, B. Zheng, Phys. Rev. E 53 (1996) 2940. F.A. Oliveira, R.M.S. Ferreira, L.C. Lapas, M.H. Vainstein, Front. Phys. 7 (2019) 18. M.R. Evans, S.N. Majumdar, Phys. Rev. Lett. 106 (2011) 160601. M.R. Evans, S.N. Majumdar, J. Phys. A 44 (2011) 435001. M.R. Evans, S.N. Majumdar, J. Phys. A 47 (2014) 285001. X. Durang, M. Henkel, H. Park, J. Phys. A 47 (2014) 045002.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Critical properties of a vector-mediated epidemic process F.L. Santos a , M.L. Almeida a , E.L. Albuquerque a , A. Macedo-Filho b , M.L. Lyra c , ∗ U.L. Fulco a , a
Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil Campus Prof. Antonio Geovanne Alves de Sousa, Universidade Estadual do Piauí, 64260-000, Piripiri-PI, Brazil c Instituto de Física, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil b
highlights • • • •
Vector-mediated epidemic propagation model was performed in a linear chain. Monte Carlo simulations and finite-time scaling was used to obtain the critical exponents. The critical properties from the statistically stationary state are estimated. The transition from the inactive vacuum absorbing state to the epidemic statistically stationary state belongs to the diffusive epidemic universality class.
article
info
Article history: Received 17 April 2019 Received in revised form 5 June 2019 Available online 16 July 2019 Keywords: Critical properties Epidemic process Vector-mediated model Monte Carlo simulations Dynamical critical exponents Universality class
a b s t r a c t We study the critical behavior of an epidemic propagation model with interacting static individuals and diffusive vectors. The model presents a non-equilibrium phase transition from an absorbing vacuum state to an epidemic state at a critical vector density which depends on the recovery rates of infected individuals and vectors. The simulation was performed in a linear chain of the proposed model and the finite time scale hypothesis was explored to estimate the vector critical density and dynamic critical exponents. Our results show that the absorbing-state phase transition belongs to the universality class of the symmetric diffusive epidemic process irrespective to the relative values of the recovery rates. On the other hand, the critical vector density shows a much stronger dependence on the recovery rate of vectors than on the corresponding recovery rate of individuals. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Accordingly to the World Health Organization, 17% of all infectious diseases in the world are transmitted by vectors. This problem draws the attention of the scientific community which has devoted a great deal of effort to study these epidemics processes [1–4]. One focus of these investigations is the universal characteristics of epidemic outbreaks, which is a problem not only in medical science, but also in public health. As examples, an outbreak of the dengue fever or malaria disease can kill many people worldwide justifying, by itself, the importance of a deeper understanding of the underlying dynamics that can provide insights in what public actions can be more effective in preventing or stopping the spreading of this disease. ∗ Corresponding author. E-mail address: [email protected] (U.L. Fulco). https://doi.org/10.1016/j.physa.2019.122085 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
Epidemic spreading as a kind of a diffusive process model was theoretically investigated by Kree et al. [5], with the aim to model the effect of pollution on a population on the brink of extinction. This process was also characterized by the coupling between the fluctuations in the total density with respect to the density of the species that is trying to survive [6], being afterwards extended to consider also distinct diffusion rates [7,8]. Although very simple, this model presents an extraordinary richness. This process belongs to the general category of dynamical complex systems, which exhibits a continuous phasetransition from the active stationary state to an absorbing one [8–11]. An important fact is that the critical behavior falls in a very specific universality class with exponents, as defined by renormalization group techniques, 1/z = 1/2 and 1/z ν = 1 − ϵ/4 where, ϵ = 4 − d, d being the dimension of the system, z and ν the dynamic and correlation length exponents, respectively [5,7,12]. In this paper we consider a diffusive epidemic model that simulates the transmission strategies mediated by a vector and analyze its universal characteristics at the vicinity of the transition to the vacuum state on which the epidemic spreading stops. Its main focus is the investigation on the statistical properties of a model for vector-mediated transmission of a disease in a population consisting of two species, namely hosts and vectors, which contract the disease from one another. It captures some relevant issues of tropical diseases, like malaria, which are transmitted by mosquitoes. The hosts are sedentary, meaning that each individual is kept fixed in a site of the network, while the vectors diffuse within the network at a constant rate. The spreading of vector-mediated diseases have also been investigated using alternative models, as for example a coupled susceptible–infected–recovered (SIR) dynamics for the humans and a susceptible– infected–susceptible (SIS) dynamics for the vectors [13]. A mean-field approach as well as simulation results considering static populations of humans and vectors occupying sites of a bipartite lattice while interacting by contact unveiled that spreading of the disease is impossible for any death rate of vectors if the infection rate is low [13]. To investigate the critical behavior of the model, we use Monte Carlo simulations and finite-time scaling to obtain the critical exponents β/z ν , 1/z and 1/z ν , where β is the critical parameter associated with the decay of the order parameter as the critical point is approached. A detailed study of the critical behavior considering distinct scenarios for the average lifetime of both infected hosts and vectors controlled by their recovery rates λ and φ will be presented, and its main features discussed. This work is organized as follows. In Section 2 we present in some detail the proposed model and the algorithm used in the numerical simulations. Section 3 reports our main numerical results, leaving the summary and conclusions of this work to be depicted in Section 4. 2. Vector–mediated epidemic propagation model In our simulation, the description of a vector-borne disease is based on the life cycle of the parasite, like dengue. Similar models of coupled populations of hosts and vectors have been previously introduced in the literature showing that it can capture the transition from the vacuum state with no infected individuals to the active state with a finite population of infected hosts and infective vectors [14,15]. Initially, we consider a randomly distribution of equally number of infected (Vi ) and non-infected (V ) vectors between the N sites of a linear chain with periodic boundary conditions, with each site being a locus, where an arbitrary number of vectors can be located, which means that they do not have a hardcore repulsive potential. Vectors diffuse over the chain with a density ρ (the number of vectors divided by the number of sites) kept fixed in time. However, each site can either be free of vectors or occupied by infected and/or non-infected vectors. There is a host at each lattice site, which can be in one of two states, namely the infected (Hi ) or non-infected (H) ones, respectively. The contamination process takes place only when vectors and hosts are found on the same site. The whole dynamics of this reaction–diffusion model is done in three stages: (a) The recovering process: an infected vector (hosts) is replaced by a non-infective one at rate φ (λ). As such, φ (λ) controls the average time a vector (host) stay in the infected state. (b) The contamination process: active states are transmitted only between different species. Each non-infected host in the same site of at least one infected vector will be transformed in a infected host with probability k1 . A noninfective vector becomes infected at the rate k2 if it shares the same site with an infected host. Our simulations are performed using k1 = k2 = 1. (c) The diffusive motion: the vectors jump to a neighbor at a rate D = 0.5 with equal probability to each one of its neighboring sites. In the above contamination process, we are considering the regime of maximal infectivity ki = 1 for simplicity because the universality class of the underlying absorbing-state phase transition is not expected to depend on the specific value of these rate parameters. For finite infection rates, one should consider that a non-infected host would be transformed into an infected host at rate k1 by each infected vector occupying the same site. In this way, the effective contamination would depend on the number of infected vector at a given site. The above model can be summarized by the set of reaction-rate equations represented by λ
Hi ↦ − → H;
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
3
Fig. 1. Log–Log plot of the time-evolution of the density of infected individuals Ψ (ρ, t) in the close vicinity of the critical point ρ = [1.590, 1.600] for the case φ = 0.50 and λ = 0.90. Notice that for ρ < 1.595 (ρ > 1.595) the curves clearly veer downward (upward), leading to the estimate of ρc = 1.595. This procedure was used to locate the critical vector density for all pairs of recovery rates explored in this work.
φ
→ V; Vi ↦ − k1
H + Vi ↦ − → Hi ; k2
V + Hi ↦ − → Vi . The mechanism of dissemination of the infection among these coupled populations is made by the diffusion of the vectors, which are infected only by the hosts. Therefore, any infected host is also infective. A detailed study will be now be performed by varying the recovery rates of vectors and hosts. Our main goal is to investigate how the critical density of vectors depends on these factors and to unveil the robustness of the corresponding universality class. 3. Simulation results and discussions To estimate the critical properties from the statistically stationary state, we explore finite-time effects on Monte Carlo numerical simulation data. Our aim here is the finding of the critical density of vectors below which the epidemic process does not spread and the system evolves towards the absorbing state free of infected individuals. Furthermore, we should get precise estimations of the dynamical critical exponents β/z ν , 1/z and 1/z ν , where β is the critical exponent associated with the density of infected individuals that behaves as an order parameter of this non-equilibrium phase-transition. In order to explore the critical relaxation dynamics of the system, we start considering all active hosts Hi and measure how the average density of active hosts (Ψ = ⟨n(Hi )⟩/L) decays as a function of time t, where n(Hi ) is the number of infected hosts at site i. When the system is driven to the vacuum inactive (epidemic active) state, the density of active (infected) individuals decays exponentially in a fast way (evolves towards a statistically stationary value). This is usually the case achieved when the density of vectors ρ is small (large). At the critical point ρc (total density of vectors), the timeevolution towards the vacuum state becomes slower and power-law dynamic scaling laws emerge [16–19]. In particular, the order parameter (density of infected individuals) is expected to decay in time at ρc as:
Ψ (ρc , t) ∼ t −β/z ν .
(1)
Fig. 1 depicts a log–log plot of Ψ (t) as a function of t for ρ ranging in the close vicinity of the critical density, i.e. ρ = [1.590, 1.600], for the case φ = 0.50 and λ = 0.90. This plot is obtained by using a lattice with L = 10240 sites and averaged over Nrep = 104 independent realizations, with a relaxation time trmax = 105 . The change in the concavity in the large run regime signals the critical vector density. The best estimation found for the critical vector density was
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Fig. 2. Log–Log plot of the time-evolution of the order parameter at the critical density for φ = 0.50 and λ = 0.50. From the slope of this curve we estimate the critical exponent ratio β/z ν = 0.127(6).
Fig. 3. Log–Log plot of the time-evolution of the relative fluctuation of the order parameter at the critical density for φ = 0.80 and λ = 0.90, whose slope yields an estimation of the dynamical critical exponent 1/z = 0.49(3).
ρc = 1.595, where the curve shows the smallest curvature. Observe that from the data for ρ < 1.595 (ρ > 1.595) the curves clearly veer downward (upward). This procedure was performed for distinct sets of recovery rates to locate the corresponding critical vector density. In what follows, we will show the power-law dynamic scaling for some relevant quantities for some particular values of φ and λ. In Fig. 2 we report the time-evolution of the density of infected individuals for φ = λ = 0.5 for which we estimated ρc = 1.308(2). The best power-law fitting leads toβ/z ν = 0.127(6).
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
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Fig. 4. Same as in Fig. 3, but for φ = 0.20 and λ = 0.30. From the slope we estimate the critical exponent ratio 1/z ν = 0.24(4).
We can also obtain an independent estimate of the dynamic critical exponent z by exploring the time-evolution of the relative order parameter fluctuation at the critical point which shall scale as [20–23]: U(ρc , t) = (⟨Hi 2 ⟩/⟨Hi ⟩2 ) − 1 ∼ t 1/z .
(2)
Fig. 3 shows a typical plot of U(t) using as representative case the recovery rates φ = 0.80 and λ = 0.90. In this case we estimated ρc = 5.169(3). The best power-law fitting provided the estimate for the dynamic exponent 1/z = 0.49(3) for this case. We also got an independent estimate for the exponent ⏐ ratio⏐ 1/z ν . For this, we followed the critical relaxation of the
⏐ ∂ lnΨ (ρc ) ⏐ ⏐. According to the finite-time scaling hypothesis, it shall ∂ρ
logarithmic derivative of the order parameter D(t) = ⏐ scale at the critical point as D(t) ∼ t 1/z ν .
(3)
We illustrate such scaling at criticality in Fig. 4 using the parameters φ = 0.20 and λ = 0.30 for which we estimated ρc = 0.408(2). The asymptotic slope of the corresponding curve gives 1/z ν = 0.24(4). The full set of critical densities and exponents for all pairs of simulated recovery rates is summarized in Table 1. We notice that, within the error bars, the critical exponents are rather independent of the typical lifetime of either infected individuals or vectors. These exponents are compatible with this model belonging to the universality class of the symmetric diffusive epidemic process [6,8]. On the other hand, the critical vector density is strongly affected by the recovery rates. In order to explicitly show the relative role played by the recovery rate of either infected individuals or infective vectors, we plot in Fig. 5 the critical vector density as a function of φ and λ. Notice that ρc shows a stronger dependence on the recovery rate of vectors φ (Fig. 5a) than on the recovery rate of infected individuals λ (Fig. 5b). 4. Summary and conclusions In summary, we have investigated the critical behavior of an interacting two-species diffusion-limited reaction model which mimics the propagation of a disease in a static population mediated by a population of diffusive 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 the out-break of the epidemic processes. A critical short-time dynamics was used to provide some relevant critical exponents governing this non-equilibrium phase transition from simulations of this dynamical model in linear chains. The considered model contains three basic control parameters: The total density of vectors per site ρ , the recovery rates of individuals λ, and the vectors φ . These rates govern processes at which an individual or vector in the infected state
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Fig. 5. The estimated vectors critical density ρc as a function of (a) the recovery rate of vectors φ and (b) the recovery rate of individuals λ. Notice that ρc depicts a much stronger dependence on φ .
changes to the non-infected state. They are related to the average lifetime an individual/vector remains in the infective state. The total density of vectors gives the background population through which the infected state can be transferred from one individual to the others. At very low vector densities the epidemic becomes extinct while it can become endemic above a critical density of vectors. The one-dimensional diffusive epidemic process has critical exponents given by z = ν = 2 [5,7] and β/z ν = 0.11(2) [8,10]. Our simulations suggested that the transition from the inactive vacuum absorbing state to the epidemic statistically stationary state of the investigated model belongs to such diffusive epidemic universality class irrespective to the relative recovery rates of individuals and vectors. Possible mechanisms that could influence the universal behavior include
F.L. Santos, M.L. Almeida, E.L. Albuquerque et al. / Physica A 533 (2019) 122085
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Table 1 The critical vector density ρc and the corresponding dynamical exponents β/z ν , 1/z and 1/z ν for the present reaction–diffusion model for distinct pairs of recovery rates φ (vectors) and λ (individuals). (φ, λ)
ρc
β/z ν
1/z
1/z ν
(0.20; 0.10) (0.20; 0.30) (0.20; 0.50) (0.20; 0.70) (0.20; 0.90) (0.50; 0.10) (0.50; 0.30) (0.50; 0.50) (0.50; 0.70) (0.50; 0.90) (0.80; 0.10) (0.80; 0.30) (0.80; 0.50) (0.80; 0.70) (0.80; 0.90)
0.210(2) 0.408(2) 0.509(2) 0.574(4) 0.619(1) 0.524(1) 1.037(2) 1.308(2) 1.477(3) 1.595(2) 1.645(3) 3.327(4) 4.228(3) 4.787(3) 5.169(3)
0.127(10) 0.115(12) 0.122(4) 0.118(9) 0.118(6) 0.125(6) 0.124(6) 0.127(6) 0.128(6) 0.118(10) 0.150(21) 0.136(11) 0.127(6) 0.128(4) 0.126(4)
0.54(4) 0.50(4) 0.50(4) 0.50(3) 0.50(2) 0.55(6) 0.51(4) 0.52(4) 0.52(4) 0.50(3) 0.56(5) 0.51(5) 0.50(4) 0.50(3) 0.49(3)
0.24(4) 0.24(4) 0.26(7) 0.24(8) 0.23(6) 0.26(4) 0.24(8) 0.27(5) 0.26(6) 0.22(9) 0.28(7) 0.29(7) 0.22(10) 0.29(5) 0.23(5)
anomalous diffusion [24] and stochastic resets of the diffusing particles to their initial positions [25–28] because these mechanisms introduce non-Gaussian fluctuations on the particles distribution. These aspects deserve future detailed investigations. On the other hand, the critical density of vectors needed to sustain the epidemic process depicts a stronger dependence (faster than linear growth) on the recovery rate of vectors φ than on the recovery rate of individuals λ (slower than linear growth). The above results shows clearly the relative effectiveness of interventions that can be put forward to drive an epidemic process mediated by diffusive vectors to extinction. The most effective actions are those directed to reduce the vector density. Usually this is performed by the use of pesticides or biological control of the population of vectors. Such population control demands strategies do reduce the reproduction rate by either interventions in the environment conditions needed for the vector reproduction, or by the introduction of genetically modified infertile vectors. On the other hand, the present results indicate that actions devoted to reduce the period over which an individual remains infective has a limited impact in the control of the epidemic process due to sub-linear dependence of the critical density of vectors on the recovery rate of hosts. Such control can be much more effective if the strategy is focused in reducing the lifetime of infected vectors. It would be interesting to have new epidemiological data to support this theoretical prediction. Acknowledgments This work was supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), Brazil, CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil, and FAPEAL (Fundação de Apoio à Pesquisa do Estado de Alagoas), Brazil. We would like to thank the Núcleo de Processamento de Alto Desempenho of the Universidade Federal do Rio Grande do Norte - NPAD/UFRN to allow us to access their computer facilities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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