Phase Transitions of an Epidemic Spreading Model in Small-World Networks

Transcription

1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 6, June 15, 2011 Phase Transitions of an Epidemic Spreading Model in Small-World Networks HUA Da-Yin (Ù ) and GAO Ke (Ô ) Department of Physics, Ningbo University, Ningbo , China (Received October 8, 2010; revised manuscript received November 15, 2010) Abstract We propose a modified susceptible-infected-refractory-susceptible (SIRS) model to investigate the global oscillations of the epidemic spreading in Watts Strogatz (WS) small-world networks. It is found that when an individual immunity does not change or decays slowly in an immune period, the system can exhibit complex transition from an infecting stationary state to a large amplitude sustained oscillation or an absorbing state with no infection. When the immunity decays rapidly in the immune period, the transition to the global oscillation disappears and there is no oscillation. Furthermore, based on the spatio-temporal evolution patterns and the phase diagram, it is disclosed that a long immunity period takes an important role in the emergence of the global oscillation in small-world networks. PACS numbers: k, Cn, Uu, q Key words: susceptible-infected-refractory-susceptible (SIRS) model, small-world networks, oscillation 1 Introduction Recently, the epidemic spreading behavior has been widely investigated, such as the spread of computer viruses and human diseases. [1 13] In particular, the sustained periodic behaviors of disease spreading in realistic systems and theoretical models have attracted a great deal of attention. Kuperman and Abramson have investigated an SIRS model based on small-world networks [14] introduced by Watts and Strogatz, [15] where a link between two sites in one dimension is rewired with a probability p. They have observed that, with p increasing, the system exhibits a transition from a stationary state with fluctuation to a large amplitude self-sustained oscillation. On the other hand, the periodic behaviors of epidemic spreading on the community network and scale free networks are observed. [16 17] Kuperman and Abramson have supposed that the emergence of the oscillation state should be due to the variation of the clustering coefficient. [14] But Szolnoki and Szabó have suggested that the clustering coefficient could not play a significant role in this phase transition. [18] On the other hand, McKane and Newman proposed that the resonant amplification of demographic fluctuation may take an important role in the real (finite) populations for the oscillatory state. [19] Furthermore, Rozhnova and Nunes suggested that the oscillation could persist in infinite populations with spatial correlations in which confer a long lasting immunity in a pair approximation and mean field approximation. [20] In this paper, we investigate the effect of the immunity decay on the global oscillation. It is found that the immunity persistence and its gradual decay will take important effects on the transition to the global oscillation. Our results will be helpful to understand the mechanism of the global oscillation in the epidemic spreading process deeply. We introduce our epidemic spreading model and simulation proceeds in Sec. 2. The simulation results are presented in Sec. 3 and the conclusion is presented in Sec Model and Simulation The system consists of N individuals and each individual is in one of three states: susceptible (S), infected (I) or refractory (R). In a previous work, [14] a refractory individual can not be infected and its immunity jumps to zero in the end of the immune period. In fact, an individual immunity can decay gradually in many realistic conditions. We consider an individual immunity is in a function R i (t 1 ) in an immune period, where t 1 is Monte Carlo time after an individual enters into the immune period, Obviously, t 1 is different for different individuals. Therefore, a refractory individual can be infected with a probability 1 R i (t 1 ). In each Monte Carlo attempt for an individual i: (i) if it is in a susceptible state, its one of nearest neighbors j is randomly chosen. If j is an infected individual, i is infected at once; (ii) if it is in an infected state, it will recover and enter into a refractory state with a probability z; (iii) if it is in a refractory state, one of its neighbors j is randomly chosen, if j is an infected individual, i is infected again with a probability λ = 1 R i (t 1 ), otherwise it keeps its refractory state and t 1 increases. Once an infected individual recovers, t 1 = 0. A refractory individual will become to a susceptible individual when t 1 = T c. In our simulation, a synchronized update is employed. Supported by National Natural Science Foundation of China under Grand No and Sponsored by K.C. Wong Magna Fund in Ningbo University huadayin@nbu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 1128 Communications in Theoretical Physics Vol. 55 The system has an initial density of infected individuals ρ 0 and the others are susceptible individuals and a periodic boundary condition is used. The formation of the spatial structure is described as small-world networks. [15] The spatial structure restores to a regular lattice for p = 0. When p > 0, long range links are added and a small world network is formed. When p tends to 1, most of links have been rewired and the structure is similar to a random network. In this paper, we focus on the effects of immunity R i (t 1 ) and the network structure in epidemic spreading. So we fix the parameters z = 0.15, the average number of neighbors k = 4, N = 5000, and ρ 0 = 0.1 in the following sections. 3 Simulation Results The immunity for different infection can decay in different rates. We have performed extensive numerical simulations to discuss the effect of varying rate of immunity. As a compare, we first study the epidemic spreading behavior while the immunity does not change in an immune period as the previous work. [14] It is found that, with an increase of p, the spreading behavior changes from a stationary state with fluctuations to a low-amplitude irregular oscillation of the fraction ρ (t) of the infected individuals, and then transits to a large amplitude self-sustained oscillation as shown in Fig. 1(a). These results are in excellent accordance with the previous simulation results. [14] On the other hand, as shown in Fig. 1(b), with T c increasing, the system also exhibits a transition from a stationary state to a large amplitude global oscillation. But when T c > 40, the long immune period will suppress the epidemic spreading. The results denote clearly that the long immunity time takes an important effect on emergence of the global oscillatory behavior. domain and ρ (t,l) is the average value in a long time window t = 1, 2,..., T steps after the system enters into a stationary state. As shown in Fig. 2, it is found that σ almost keeps a constant for the condition of global oscillation while L increasing but it decays quickly for the cases of stationary and small amplitude oscillation. Fig. 2 Simulation results of variance σ for different L. (a) R i(t 1) = 1.0, T c = 40; (b) R i(t 1) = 1.0, p = 0.4. Compared to condition R i (t 1 ) = 1, we consider some more realistic conditions where immunity can decay gradually. We assume that immunity can decay slowly as an exponent function R i (t 1 ) = exp( t 1 /3T c ) in an immune period. It is found that with an increase of p, the spreading behavior changes from a stationary state to a large self-sustained oscillation as shown in Fig. 3(a). Fig. 3 Infected individual evolutionary behavior when R i(t 1) = exp( t 1/3T c) (0 < t 1 < T c). (a) For different values of parameter p (T c = 50); (b) For different values of immunity time T c (p = 0.2). Fig. 1 Infected individual evolutionary behavior when R i(t 1) = 1.0 (0< t 1 < T c). (a) For different values of parameter p (T c = 40); (b) For different values of immunity time T c (p = 0.4); (c) The damped oscillation for T c = 45 (p = 0.4). All individuals take an updating trial as a Monte Carlo step (MCS). We calculate a variance σ in a local domain with size L. σ is defined as σ = (1/T) t (ρ (t,l) ρ (t,l) ) 2, where ρ (t,l) is fraction of the infected individual in the local However, it is surprised that the large amplitude oscillation transits to a low-amplitude random oscillation again when p increases to 0.9. Furthermore, it is found that, with T c varying, the spreading behavior also transits from a stationary state to a large self-sustained oscillation as shown in Fig. 3(b). We have changed the decay function as R i (t 1 ) = exp( t 1 /(2T c )), there is a transition to a global oscillation also. When the decay function changes to R i (t 1 ) = exp( t 1 /T c ), the transition disappears and there is no global oscillation. We also have considered many other kinds of functions to discuss the effect of the

3 No. 6 Communications in Theoretical Physics 1129 decay rate on the oscillation behavior. It is found that the transition to the global oscillation can disappear when the immunity can decay rapidly. These results are very important to understand the oscillatory epidemic spreading behavior in many realistic systems. According to Ref. [14], we define a synchronization parameter ǫ(t) = N im j=1 eiφj(t) /N im, where φ j (t) = 2πt 1 /T c and the individuals in the immune period are included when calculating ǫ(t) and N im is the number of the individuals in the immune period. In a stationary infecting state, the individuals are in a different immune states randomly, ǫ(t) is very small. However, in a global oscillation, the synchronization increases and then ǫ(t) will increase obviously. As shown in Fig. 4(a), we calculate the average of ǫ(t) when the system enters into a stable state. It is found that the average of ǫ(t) changes with T c varying and discloses the phase transition behavior clearly. [14] Therefore, we can estimate the transition point roughly from a stationary infection state to a global oscillation. The phase diagram in p-t c plane is obtained for two cases as shown in Fig. 4(b) and 4(c). From the phase diagram, it is shown there is a phase transition from an infecting state to an absorbing state with no infection when p is small. With p increasing, there are two phase transitions from an infecting state to a global oscillation state and then to an absorbing state with no infection. For a global oscillation in region II, it is found that much long immune time T c is needed for the transition to the global oscillation when the immunity can decay. On the other hand, in Fig. 4(c), the system can transit from an oscillation state into a non-oscillation state for an increasing of p when T c is small, this simulation is in consistence with the dynamical behavior in Fig. 3. Furthermore, from Fig. 5, it is shown that the oscillation period increases with T c. Fig. 4 (a) The change of ǫ with T c varying when p = 0.4 and R i(t 1) = 1. (b) The phase diagram p-t c plane for R i(t 1) = 1; (c) The phase diagram p-t c plane for R i(t 1) = exp( t 1/(3T c)). In region I, the system is in a stationary state with infected individuals, in region II, the system is in a global oscillation state and the system enters a state with no infection in region III. R i (t 1 ) = exp( t 1 /3T c ). The rapid infecting rate of a susceptible individual will take an important role in the global dynamical behavior. Fig. 5 The oscillation period T with immunity time T c varying for R i(t 1) = 1.0. In order to understand the emergence of the global oscillation, we firstly calculate the effect of the small world network on the infecting rate of the susceptible individual. As shown in Fig. 6, we define V 1 = inf s /N to describe the infecting rate of a susceptible individual, where inf s is the infected number from the susceptible individual in a Monte Carlo step. It is found that V 1 increases rapidly with p increasing to 0.2 for R i (t 1 ) = 1.0 and Fig. 6 The average infecting rate of susceptible individuals for R i(t 1) = 1.0, T c = 40; R i(t 1) = exp( t 1/3T c), T c = 50; and R i(t 1) = exp( t 1/T c), T c = 100, respectively. Secondly, we simulate the spatial-temporal patterns when the immunity function is in R i (t 1 ) = exp( t 1 /3T c ) as shown in Fig. 7. When p = 0.02, the infection spreads

4 1130 Communications in Theoretical Physics Vol. 55 in some local areas independently and then the global oscillation can not be induced. With p increasing to 0.4, as shown in Fig. 7(b), the system exhibits a sustained oscillation. We can understand the mechanism of the global oscillation deeply from the spatial-temporal pattern and the phase diagram in Fig. 4. With a long immune time T c, when the immunity does not decay or decays slowly in the immune period, most of refractory individuals are kept away from the infection. More infected individuals can be recovered gradually and enter into a refractory state and then into a susceptible state. When most of individuals enters into the susceptible state, with a rapid infecting rate for a large p as shown in Fig. 6, the infection begins to spread rapidly again in the small world network and then a global oscillation is induced. However, with a further increase of p, the infecting rate of a refractory individual increases also, much more refractory individuals are infected, and then the effect of long immunity time T c is inhibited and the global oscillation disappears as shown in Fig. 7(c). Fig. 7 Spatial-temporal evolution patterns for R i(t 1) = exp( t 1/3T c). Dark areas denote infected individuals, gray areas denote refractory individuals and light gray areas denote susceptible individuals. (a) T c = 50, p = 0.02; (b) T c = 50, p = 0.4; (c) T c = 50, p = 0.9. When the immunity decays rapidly following R i (t 1 ) = exp( t 1 /T c ), the spatial-temporal patterns are shown in Fig. 8. We can see that, in Fig. 8(a), there is no sustained oscillation because the epidemic spreads in small local areas as the same as in Fig. 7(a). On the other hand, when p increases, because the infecting rate of refractory individual increases also, we can predict that the oscillation disappears also as shown in Fig. 8(b). Fig. 8 (a) Spatial-temporal evolution patterns for R i(t 1) = exp( t 1/T c). Dark areas denote infected individuals, gray areas denote refractory individuals and light gray areas denote susceptible individuals. (a) T c = 100, p = 0.02; (b) T c = 100, p = Conclusion We investigate the effects of immunity decay on the global oscillation behavior of SIRS model in small-world networks. It is found that when the immunity of an individual does not change or decays slowly in the immune period, the system can exhibit complex transition from an infecting stationary state to a large amplitude sustained oscillation or an absorbing state with no infection. When the immunity decays quickly in an immune period, the transition disappears and there is no oscillation. Furthermore, based on the spatio-temporal evolution patterns, it is disclosed that, with an adjustment of a long immune period, most of individuals can enter into a susceptible state synchronously from a random initial condition and then the infection spreads periodically in small world networks. When the immunity decays rapidly, the infecting probability of a refractory individual increases and the effect of the long immunity time is inhibited and then the transition disappears. References [1] M. Boguna and R. Pastor-Satorras, Phys. Rev. E 66 (2002) [2] R.M. May and A.L. Lloyd, Phys. Rev. E 64 (2001) [3] Y. Moreno, J.B. Gomez, and A.F. Pacheco, Phys. Rev. E 68 (2003) (R). [4] N.H. Fefferman and K.L. Ng, Phys. Rev. E 76 (2007)

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