Influence of infection rate and migration on extinction of disease in spatial epidemics

ARTICLE IN PRESS Journal of Theoretical Biology 264 (2010) 95–103

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Influence of infection rate and migration on extinction of disease in spatial epidemics Gui-Quan Sun a,b,c, Quan-Xing Liu d, Zhen Jin a,, Amit Chakraborty c, Bai-Lian Li a,c a

Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China School of Mechatronic Engineering, North University of China, Taiyuan, Shan’xi 030051, People’s Republic of China c Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA 92521-0124, USA d Spatial Ecology Department, the Netherlands Institute of Ecology (NIOO-KNAW), P.O. Box 140, 4400 AC Yerseke, the Netherlands b

a r t i c l e in fo

abstract

Article history: Received 18 September 2009 Received in revised form 23 December 2009 Accepted 7 January 2010 Available online 18 January 2010

Extinction of disease can be explained by the patterns of epidemic spreading, yet the underlying causes of extinction are far from being well understood. To reveal a mechanism of disease extinction, a cellular automata model with both birth, death rate and migration is presented. We find that, in single patch, when the infection rate is small or large enough, the disease will disappear for a long time. When the invasion form is in the coexistence of stable spiral and turbulent wave state, the disease will persist. Also, we find that the migration has dual effects on the epidemic spreading. On one hand, in the extinction region of single patch, if the migration rate is large enough, there is a phase transition from the disease free to endemic state in two patches. On the other hand, migration will induce extinction in the regime, which can ensure the persistence of the disease in single patch, due to emergence of antiphase synchrony. The results obtained well reveal the effect of infection rate and migration on the extinction of the disease, which enriches the finding in the filed of epidemiology and may provide some new ideas to control the disease in the real world. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Epidemic spreading Cellular automata Birth and death Patches Anti-phase synchrony

1. Introduction Epidemic spreading in space has been attracted considerable attention of epidemiologists as there are recurring threats posed by newly emerging and high profile infectious diseases, such as SARS (Guan et al., 2003; Marra et al., 2003; Riley et al., 2003), the H5N1 strain of avian influenza (Li et al., 2004; Shortridge et al., 1998; Ungchusak et al., 2005), HIV (Stremlau et al., 2004; Yeni et al., 2002), ebola (Chandran et al., 2005; Jones et al., 2005), whooping cough (Diavatopoulos et al., 2005; Soelaiman et al., 2003), dengue fever (Hales et al., 2002; Mongkolsapaya et al., 2003), influenza (Meleigy, 2009; Wu, 2009), and the spread of H1N1 (Garten et al., 2009; Liu et al., 2009a; Smith et al., 2009). Although obvious progress has been made during recent 10 years in understanding the invasion of the disease in space, many important issues have not been well addressed yet. We may ask that, what the main factors to enhance the disease spreading in space are (Liu et al., 2009b), how the emergence of traveling wave has effect on the invasion speed (Grenfell et al., 2001), how the disease spreading holds a power-law scaling distribution of the

 Corresponding author.

E-mail addresses: [email protected] (G.-Q. Sun), [email protected] (Q.-X. Liu), [email protected] (Z. Jin), [email protected] (A. Chakraborty), [email protected] (B.-L. Li). 0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.01.006

cluster sizes (Manor and Sherb, 2008) and what is the corresponding invasion form, whether dynamical resonance exists if the infection rate are not periodic (Dushoff et al., 2004), and so on. It is well-known that, many basic feature of the invasion of the disease in both the space and time can be well described by the interaction of the local birth, death, infection and local diffusion of the individual, from the biological point of view. In recent years, epidemic spreading modeled by reaction–diffusion models (Li et al., 2008; Sun et al., 2007, 2008), and patch-structured models, which include meta-populations (Rossi et al., 2007), coupled-map lattices (Dorogovtsev et al., 2008; Shirley and Rushton, 2005) and demestructured populations (Newman, 2002), are used. All these models are amenable to mathematical analysis, but these limit their applicability, in particular in evolutionary contexts. The advent of modern computing facilities has encouraged ecologists and epidemiologist to turn to individual-based spatial simulations, such as cellular automata (Durrett and Levin, 1994a, 1994b). In this model, space is represented by a network of sites, and the state of the network is updated through probabilistic rules representing stochastic demographic events (Lion and van Baalen, 2008). A basic issue in spatial epidemics is the relation between extinction and many different factors. One of the clearest manifestations of population threshold for extinction is the critical community size below which the disease does not persist, see measles for example (Keeling and Grenfell, 1997). Effect of vaccination on extinction was investigated and it showed that, in

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the absence of vaccination, the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point, with an unusual critical exponent. However, when random vaccination was added, even a very weak Poisson distributed random vaccination led to an exponential increase in the extinction rate (Dykman et al., 2008). Moreover, influence of separating the populations into sub-communities on the extinction time was studied by Mathias and Tom (Lindholm and Britton, 2007), and they found that the time to extinction increased with the degree of interaction between communities. In addition, some experiments were done to extinction risk in wild mammals with infectious diseases (Pedersen et al., 2007). On the other hand, the individual in each patch has its own dynamics, but connected by movement with another patch, which is called migration. Understanding how migration affects invasion, spread and persistence of the disease is very important for public health and wildlife management authorities (Grenfell and Harwood, 1997; Jesse et al., 2008; Ovaskainen and Hanski, 2004), and consequently the problems have been collected and continues to receive considerable attention by both the governments and the epidemiologists (Cross et al., 2005; Gog et al., 2002; Hagenaars et al., 2004; Keeling, 2000; Little, 2007; Rauch and BarYam, 2004; Zhang et al., 2009). Extinction may occur where the infection rate is so small that there is insufficient transmission to keep the disease in its endemic state (Dykman et al., 2008; Keeling, 2004). It is natural to ask, whether the disease will persist if the infection rate is large enough. To gain insight into disease extinction with respect to the infection rate, we will investigate how the disease varies as the infection rate increases. And for migration, several theoretical models have shown that migration of population is more stable (Rohani and Miramontes, 1995), which means that migration can promote the high density of the disease, and thus lead to the persistence of the disease. We will address the effect of the migration between two patches, especially to check that if migration may result in the extinction of the disease. As a result, modeling the spread of the disease using Cellular Automata and investigating the influence of infection rate and migration on extinction are the main work of this paper. The paper is organized according to the following orders: in Section 2, we present a SI model with birth and death using cellular automata, and also give the evolution rule. In Section 3, we study the effect of infection rate and migration on the epidemic spreading. The numerical results show that when the infection rate is small or large enough, the disease will disappear. Moreover, we find that the migration has dual effect: inducing the persistence of the disease in the extinction region of one single patch and the extinction of the disease in the persistence region of one single patch.

2. Method A spatial susceptible-infected model, with birth and death events, will be illustrated. Generally speaking, the population, in which a pathogenic agent is active, comprises two subgroups: the healthy individuals who are susceptible (S) to the infection and those individuals who are already infected (I) (Anderson and May, 1992). Also, there are another state in the cellular automation-an empty state (E) where none of the individuals is present. We use the Moore neighborhood, i.e., eight nearest neighbors reached by a chess-kings-move, for all ði; jÞ in the two-dimensional space. Detailed rules governing the dynamics of this epidemiological model are as follows: (i) Initially, the populations are randomly located in the spatial domain (note that the spatial structure at the equilibrium of

the system is insensitive to the initial condition) (Hiebeler and Morin, 2007). (ii) The susceptible (S) can be infected at a probability of b through the contact between the susceptible and infected individuals. As a result, a susceptible can become infected at a probability 1ð1bÞNI (Rhodes and Anderson, 1997; Tilman, 1994), where NI is the sum of infected in the neighborhood. (iii) There is only disease-related death, no natural death. When death events occur, the infected state (I) becomes the empty one (E). (iv) Only the susceptible (S) can colonize the empty suitable patches at a probability of b in each time step. Specifically, if the chosen patch is empty (E), it becomes occupied by the offspring reproduced from the neighboring 1ð1bÞNS (Bairagi et al., 2007; Deredec and Courchamp, 2006; Packer et al., 2003), where NS is the sum of susceptible in the neighborhood. Within a subpopulation, the dynamics for the local populations obey a basic reaction scheme conserving the number of population, which has been studied both in physics and mathematical epidemiology, namely the infection dynamics process identified by the following set of reactions: b

S þI!I þ I; d

I!E; b

Eþ S!S þ S:

ð1aÞ ð1bÞ ð1cÞ

In the cellular automation, S, I and E mean the state in one discrete lattice. The first reaction (Eq. (1a)) reflects the fact that an infected host (I) can infect susceptible (S) neighbors; the second reaction (Eq. (1c)) indicates the infected host (I) will die with death rate d; and the third reaction (Eq. (1c)) indicates the empty state becomes occupied by the offspring reproduced from the susceptible neighbors. The system is synchronously updated at each step, and the same time scale in individual infection and birth events.

3. Main results In this section, we reveal the effect of both infection rate and migration on the extinction of the disease. We fulfill our aim by two steps. In the first step, we investigate how the infection rate has effect on the extinction in one patch, including the spatial patterns. And then we show the influence of migration in two patches. Here, migration means the individual exhibit a motion from one patch to the other patch. Time step is set as 0.01, and 200  200 cells are used in the simulations. 3.1. Single patch To well reveal the effect of infection rate, we set b ¼ 0:5, d ¼ 0:5 and vary b. In Fig. 1(1), spatial patterns of infected population at t ¼ 0, 10, 20, 50, 320 and 710 with random initial conditions and small values of b are presented. One can see from this figure that, the disease number exhibits oscillation behavior and reaches its maximum value at t  50. And after this, the dominating trend of the density of the infected is decreasing and at last the disease disappear. To well show the rule of change in the density of the infected, time series of the infected is given in Fig. 1(2). A doubt of interest is when the disease will persist for the small values of the infection rate, i.e., what the critical values of b

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Fig. 1. (1) Spatial pattern of the susceptible (white), infected (red) and empty states (black) for different time. All panels are depicted on a 200  200 spatial grid. (A) t ¼ 0; (B) t ¼ 10; (C) t ¼ 20; (D) t ¼ 50; (E) t ¼ 320; (F) t ¼ 710; (2) the time series of the density of the infected. Parameter values used as: b ¼ 0:3, b ¼ 0:5, and d ¼ 0:5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

are. To address this issue, we performed extensive computer simulations for different parameter values (in total, more than 100 different values of b are examined). We find that, when b is more than bc  0:85, the density of the infected will be more than zero at last. In Fig. 2(1), it shows the evolution of the spatial pattern of infected population t ¼ 0, 50, 200, 500, 700 and 1000, with the same initial conditions in Fig. 1 and b ¼ 2. It can be seen from this figure that, stable target and spiral wave can coexist in the two-dimensional space, which means the infected are in the form of high density and can ensure the persistence of the disease. For the overall analysis, we show the density of the infected in Fig. 2(2). The question naturally arising here is whether the disease will persist for larger infection rate. In order to understand the mechanism, we need to check what will happen when infection rate further increases. By performing series of numerical simulations, we find that, when b 43:89, the disease will disappear. In Fig. 3(1), we show the evolutions of the infected corresponding to the cases t ¼ 0, 20, 30, 100, 220, 330, 420 and 570 with b ¼ 5. Also the same initial conditions are used with b ¼ 2. The spatial patterns exhibit spiral wave with a very high density. Conversely, with higher infection rate there will be a tendency for all the individuals in a cluster to be infected quickly (Liu et al., 2009b).

This in turn leads to relatively rapid local cluster extinction because of the death due to disease. In Fig. 3(2), the density of the infected is shown with respect to the time. At the beginning, the density increases with the increased time and has its maximum values at t ¼ 65. And then, it decreases when the time is further increased. After two jumps and downs, it reaches zero. In summary, the infection rate can induce the extinction of the disease, not only when it is small, but also for larger values. Compared with Figs. 1 and 3 shows much stronger oscillation behavior, and the extinction time is shorter. The reason is that, two cases of extinction have different mechanisms. For small infection rate, there is not large enough transmission to maintain the spread of the disease. However, for the larger values of infection rate, the disease is in the form of clustered group and easy to fade out because the infected will die. To well describe the effect of infection rate on the persistence of the disease, we show the density of the infected with respect to the parameter, b, in Fig. 4.

3.2. Two patches It is well known that, all the individuals can have the motion, such as migration. In that case, disease will spread from one place

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time Fig. 2. (1) Spatial pattern of the susceptible (white), infected (red) and empty states (black) for different time. All panels are depicted on a 200  200 spatial grid. (A) t ¼ 10; (B) t ¼ 50; (C) t ¼ 200; (D) t ¼ 500; (E) t ¼ 700; (F) t ¼ 1000; (2) the time series of the density of the infected. Parameter values used as: b ¼ 2, b ¼ 0:5, and d ¼ 0:5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to another. The invasion of disease between different patches has been much studied, particularly for pandemic invasion waves (Earn et al., 2000; Grassly et al., 2005; Hufnagel et al., 2004; Liu et al., 2009b; Xia et al., 2004). However, the key underlying relationship between migration and extinction of the disease has not been verified across wide spatial scales, especially when birth and death rates are added. To answer this question, we consider the linked dynamics of two patches, and each of size is N  N (in the simulations, N ¼ 200). For the discussion on effect of migration, we define that migration rate equals to the number of migrated individuals proportional to the size of community. In this subsection, we consider the case that there is migration from one patch to the other patch. We want to address three issues:

(1) Can migration promote the persistence of the disease, especially in the region, which is corresponding to the extinction in one patch?

(2) How the initial conditions of the other patch has influence on the epidemic spreading? In particular, are initial conditions more important than the migration rate? (3) Will migration induce the extinction of the disease, especially in the region, which is corresponding to the persistence in one patch?

Firstly, we give the time series of the density of the infected in two patches with b ¼ 0:3, which is corresponding to the extinction of the disease (cf. Fig. 1). We will investigate the effect of migration in four different cases: (1) the initial density of the infected in the second patch is zero, and the migration rate is small; (2) the initial density of the infected in the second patch is zero, and the migration rate is large; (3) the initial density of the infected in the second patch is large, and the migration rate is small; (4) the initial density of the infected in the second patch is large, and the migration rate is large.

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Fig. 3. (1) Spatial pattern of the susceptible (white), infected (red) and empty states (black) for different time. All panels are depicted on a 200  200 spatial grid. (A) t ¼ 10; (B) t ¼ 20; (C) t ¼ 30; (D) t ¼ 100; (E) t ¼ 220; (F) t ¼ 330; (G) t ¼ 420; (H) t ¼ 570; (2) the time series of the density of the infected. Parameter values used as: b ¼ 5, b ¼ 0:5, and d ¼ 0:5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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In Fig. 5(1), we show the four cases. One can see that, for zero initial condition and small migration rate, the disease is still extinct. However, when the migration rate is large, the disease will persist with a strong oscillation behavior. It can be found that there is a jump of the density at t  600 (also see Fig. 5(1)(b)). For the large initial condition, if the migrate rate is large, the disease will persist, otherwise, it will disappear. From the above analysis, it shows migration rate plays a more important role on the persistence than the initial condition in the patches. When b ¼ 5, b ¼ 0:5, d ¼ 0:5, which is also corresponding to the extinction region (see Fig. 3(2)), we do numerical simulations on the spatial pattern in two patches by choosing different initial conditions and migration rate (cf. Fig. 5(2)). We find that when the migration rate is just larger than a small value, the disease will persist. Compared with the case that b is small, it is easy to keep the disease spreading in the space when b is larger. Now, we pay attention to the case that the invasion form is coexistence of stable spiral and turbulent wave (see Fig. 2 for example). Firstly, we set the migration rate be small and find that the density of the infected in two patches is always more than zero regardless of the initial conditions. In other words, the small migration and initial conditions have no effect on the persistence of the disease, but just influence the values of the density.

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time Fig. 5. (1) The time series of the density of the infected in two patches with b ¼ 0:3: (a) initial density of the infected is 0, and migration rate is 0.2; (b) initial density of the infected is 0, and migration rate is 0.8; (c) initial density of the infected is 0.1, and migration rate is 0.2; (d) initial density of the infected is 0.1, and migration rate is 0.8. (2) The time series of the density of the infected with b ¼ 5 and the initial density of the infected of second patch is 0.1 and the migration rate is 0.2. (3) The time series of the density of the infective with b ¼ 2 and the initial density of the infected of second patch is 0.1 and the migration rate is 0.2. (4) The time series of the density of the infective with b ¼ 2 and the initial density of the infected of second patch is 0.15 and the migration rate is 0.8. (A) and (B) are corresponding to the first patch and second patch, respectively. Other parameter values are used as: b ¼ 0:5, and d ¼ 0:5.

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Moreover, the migration lead to the synchronous dynamics in the two patches. One typical time series of the infected are shown in Fig. 5(3). However, as migration rate is more than a critical value (about 0.765), we find that extinction of the disease in the first patch emerges. The main reason for this phenomenon is the existence of anti-phase synchronization1 (see Fig. 5(4)), and three stages may lead it. The first stage is that, for the migration rate, it will lead the infected well mixed in each patch, and are in a high density. The second is that when the number of the infected is large enough, for each single patch, the interaction in it has more important effect than outside it. The last stage is that when the interaction in each patch dominates, anti-phase synchrony emerges and lasts for some time, which causes the amplitude of the time series to become larger and larger and thus leads to the extinction of the disease in the first patch. In other words, the migration can induce the extinction of the disease. To well see the effect of the infection and migration rate on the persistence of the disease, we show the extinction and persistence regime of disease with respect to the two parameters in Fig. 6.

4. Discussion and conclusion Infection rate and migration between different patches are considered as one of the important factors in epidemic spreading. However, detail analysis of their effect on the extinction of the disease are still lacking. As a result, we investigate their influence on the extinction of the disease. We find some interesting results, i.e., if the infection rate is small or large enough, the disease will disappear. Furthermore, migration has dual effects on the epidemic spreading. More specifically, if the migration rate is large enough, there is a phase transition from the disease free to endemic state in the regime, which is corresponding to the 1 Anti-phase synchrony means that, for two patches, when the density of the infected in one patch is high, the density of the infected in the other patch is low (Earn et al., 1998; Grenfell et al., 2001; He and Stone, 2003; Lloyd and May, 1996). And the densities of the infected in the two patches oscillate in opposite direction.

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extinction of single patch. On the other hand, migration will induce extinction in the regime, which can ensure the persistence of the disease in single patch, due to emergence of anti-phase synchrony. In 1918, influenza out-broke in Spain firstly, called as ‘‘Spanish Flu’’ and spread all over the world (Corral-Corral and RodriguezNavarro, 2007; Llorca et al., 1998; Trilla et al., 2008). Until 1919, there were 50 million people died due to the disease. However, the obtained spatio-temporal data revealed that when the infected rate reached (or more than) 0.5, the immune barrier would emerge, and thus led to disease to be disappeared (Dory, 1977; Gregorio et al., 1993; Oxford et al., 2001, 2002; Schoenbaum, 2001). That is to say that, when the infection rate is large enough, the disease will disappear. Our results (cf. Figs. 3 and 4) are consistent with the finding in the real world. We show that when the migration is large, the disease will extinct in the region corresponding to the persistence of disease in one single patch. This finding contradicts some of known results about investigation on the persistence of infectious disease in different patches. In their papers, the results have revealed that persistence is always improved by increasing movement between patches (Hagenaars et al., 2004; Lindholm and Britton, 2007). Our results provide another explanation for the extinction of the disease: anti-phase synchronization between different patches can induce the disappearance of the disease. It should be noted that the mechanisms of the disease extinction for small and lager infection rate are different (cf. Section 3.1). And that, for different diseases, they have different infection rates. For example, for the influenza epidemics, the corresponding infection rate b equals to 500 per year (Dushoff et al., 2004), and for Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations, b equals to 0.2 per year (Courchamp et al., 1995; Hilker et al., 2007). Moreover, some disease exhibits seasonal oscillations in its infection rate, and b equals to b0 ½1 þ b1 cosð2ptÞ (Dushoff et al., 2004; Earn et al., 2002; Huntington, 1920). As a result, if we want to control the disease, we should pay attention to estimating its infection rate of the disease. Different methods are necessary to prevent the spreading of the diseases with different infection rates. Birth and dearth rate events are very important in the process of the invasion of the disease. However, it had been observed in the literature that the two events had been generally overlooked in epidemic models, despite their potential ecological reality and intrinsic theoretical interest (Green and Sadedin, 2005; Liu et al., 2006, 2009b; Santos et al., 2009; Su et al., 2008; Zhang and Atkinson, 2008). For such reason, birth and dearth rate events are included in our cellular automata model. If there is no birth and dearth rate events, when b 41, the disease always persists. However, when combined with these two events, it shows that for large infection rate, cluster extinction emerge (cf. Fig. 3). In this paper, we set birth and death rate as constant. Then one may ask, what transitions between different dynamical regimes arise as a result of perturbation of the birth and death rate, which need further investigation.

Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant no. 60771026, Program for New Century Excellent Talents in University (NCET050271), the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005), Graduate Students’ Excellent Innovative Item of Shanxi Province no. 20081018, the US National Science Foundation Biocomplexity Program and the University of California Agricultural Experiment Station.

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