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A SPATIAL SIRS BOOLEAN NETWORK MODEL FOR THE SPREAD OF H5N1 AVIAN INFLUENZA VIRUS AMONG POULTRY FARMS Alexander Kasyanov1 , Leona Kirkland2 , and Mihaela Teodora Matache3 1 Laboratory for Avian Inﬂuenza and Poultry Disease Epidemiology, Federal Centre for Animal Health, 600901 Yur’evets, Vladimir, Russia, 2 GR Exypnos, Inc., 6426 S 164th Ave., Omaha, NE 68135, USA, 3 Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182, USA kasjanov@arriah.ru, lkirkland@exypnos.org, dmatache@mail.unomaha.edu ABSTRACT propose a new model in which each farm is considered a To predict the spread of Avian Inﬂuenza we propose a node of a Boolean network and can be in one of two states, synchronous Susceptible-Infected-Recovered-Susceptible ”infected” or ”not infected” by the disease. A node can (SIRS) Boolean network of poultry farms, using proba- become infected based on the number of other infected bilistic Boolean rules. Gravity models from transportation nodes in its neighborhood, their distance from the node theory are used for the probability of infection of a node in under consideration (small distances allow for an easier one time step, taking into account farm sizes, distances be- spread of infection through wild bird or workers inter- tween farms, and mean distance travelled by birds. Basic action of neighboring farms through the common market reproduction numbers are computed analytically and nu- places), the size of the nodes (large farms have a bigger merically. The dynamics of the network are analyzed and chance being infected through the synanthropic birds in- various statistics considered such as number of infected teraction and humans and equipment movement), and the nodes or time until eradication of the epidemic. We con- distance travelled by birds in one time step. To deﬁne the clude that mostly when large farms (eventually) become probability of infection in one time step we use an ap- infected the epidemic is more encompassing, but for a proach similar to Xia et. al. [9] who have implemented a farm that does not have a very large poultry population, gravity model from transportation theory [10] to epidemi- the epidemic could be contained. ological coupling and dynamics using a transient force of infection exerted by infecteds in one location on suscep- 1. INTRODUCTION tibles in a different location, proportional to the number The spread of Highly Pathogenic Avian Inﬂuenza (HPAI) of susceptibles and the number of infecteds, and inverse H5N1 viruses across Asian and European countries has proportional to the distance between the locations. This is devastated domestic poultry industries. The development similar to Newton’s gravitational law. of strategies to moderate the spread of inﬂuenza among 2. THE BOOLEAN NETWORK MODEL poultry ﬂocks and humans is a top government priority. To investigate the spread of HPAI between poultry farms we In this section we describe the SIRS Boolean model. Con- propose a Susceptible-Infected-Recovered-Susceptible sider a network with N nodes (farms). Each node cn can (SIRS) Boolean network model. take on two values 0 (not infected) or 1 (infected). The Various individual-based models have been success- synchronous evolution of the nodes from time t to time ful in modelling real-world epidemics and understanding t + 1 is given by a Boolean rule which is considered the mechanisms of epidemic outbreaks [1]. The ﬁeld of com- same for all nodes, but depends on varying parameters plex networks has now been recognized as an important from one node to another. Initially all the nodes are con- line of study for epidemiology. For example, Barth´lemy e sidered susceptible (S). If a node is infected (I), it under- et.al. [2], or May and Lloyd [3], have published a num- goes a period of cleaning and quarantine during which it ber of papers on epidemics in scale-free networks. A large could spread the disease to other nodes in its neighbor- class of physical, biological, chemical networks have been hood; however the force of infection decreases with time, modelled as Boolean networks in recent years (e.g. [4], and the node recovers (R) completely eventually. After the [5], [6], [7]). General interest in Boolean networks and quarantine the node becomes again susceptible (S), unless their applications started much earlier with publications it goes out of business. such as the one by Kauffman [8], whose work on the Let cn (t) be the value of the node cn at time t. Deﬁne self-organization and adaptation in complex systems has the Boolean rule inspired many other research studies. cn (t + 1) = X(t) · χ{0} (cn (t)) + Y (t) · χ{1} (cn (t)) (1) The spread of HPAI among poultry farms has not yet been investigated in the context of Boolean networks. We where X(t) is a Bernoulli random variable X with param- eter pn (t) representing the probability that the susceptible Network of Farms node cn becomes infected at time t, and Y (t) = 1 if the node is infectious at time t + 1, and Y (t) = 0 if the node is noninfectious at time t + 1. Here χ{a} (b) = 1 if a = b and zero otherwise. If cn (t) becomes 0 at time t during quarantine, then we set Y (t) = 0 automatically until the end of quarantine. On the other hand, if a node goes out of business after an infection, then cn = 0 permanently. To deﬁne pn (t) let Bn denote the size of the node cn , that is ˆ Bn is the number of poultry at location cn . Let cn denote the collection of all the farms in the neighborhood of node Figure 1. Geographical spread of the network. cn (excluding the node itself). Then B(n) = ck ∈ˆn Bk c is the total number of poultry in the neighborhood of node (a) Boxplot − farm sizes (b) Frequencies − farm distances cn . Let dnk denote the physical distance between nodes 4000 0.006 cn and ck , with k ∈ {1, 2, 3, . . . , N }. We deﬁne the prob- Butler 967,344 ability pn (t) that the node cn becomes infected at time t as follows: Frequency Farm size Polk 514,038 τ 1 pn (t) = ck (t) (Bk /B(n)) ρ f (t). 1 + (dnk /d0 ) 1000 c ck ∈ˆn (2) 0 Here d0 represents the mean distance the infected wild 0 1152 farms 0 800 distance (km) birds are able to cover in one time step. The function f (t) ∈ [0, 1] is a random factor that accounts for a re- Figure 2. Boxplot of node sizes and distance frequencies. duction of the probability of infection from the infectious Most of the nodes have a rather small size, except farms node ck while cleaning and disinfection take place. The in Butler and Polk counties, provided separately. factor Bk /B(n) = Bn Bk / ck ∈ˆn Bn Bk is a version of c ρ the size terms and 1/ (1 + (dnk /d0 ) ) is a version of a distance kernel in the gravity model of [9]. Here τ deter- 4. THE PARAMETERS OF THE MODEL mines how the transient “emigration” probability scales with the donor population size, while ρ quantiﬁes how at- Recent studies have shown that the human inﬂuenza has traction decays with distance. an average time interval from infection of one individual to when their contacts are infected of about two days [11]. In the next sections we analyze the actual network of This number has been used by various authors in assessing farms and discuss the parameters of the model. Then we the potential impact of a human pandemic of HPAI. At the study the evolution of the disease in the network and we same time time two days is the minimum time needed to compute some related statistics. get preliminary results back from a diagnostic center for 3. THE NETWORK OF FARMS HPAI. Consequently we assume that the basic time step is two days, so the infections within a time step are sec- Information regarding the poultry farms are taken from the ondary cases from infected nodes in the previous time step National Agriculture Statistics Service USDA (www.nass. within a neighborhood. The basic neighborhood is consid- usda.gov) and topographic maps (1 : 100, 000 Digital Raster ered a circle of radius R km centered at each node. Given Graphics; Conservation and Survey Division; School of an infected node, the probability that it will infect nodes Natural Resources; University of Nebraska-Lincoln). To outside its neighborhood is equal to zero. We will use simulate a network of farms we identify the geographical mostly R = 100 km, but the impact of the value of R will center of each county and compute the distances between be considered in the analysis. The parameters τ and ρ will these centers. We approximate each county by a square be varied to understand the impact of how the transient centered at the county center. In each square we apply “emigration” probability scales with the donor population a uniform geographical spread of the farms. The size of size, and how attraction decays with distance. We do not each farm is obtained as a random number from a Pois- posses real data to estimate these parameters. The parame- son distribution with mean equal to the average number ter d0 is roughly estimated to 3.1 km from available infor- of poultry per farm in each county. In Figure 1, we pro- mation on home ranges for permanent resident birds and vide a network of 1198 poultry farms generated as above. migrating birds of Nebraska. However, due to incomplete This network is used further in the paper. We observe that data, we believe that this number underestimates the true Butler county accounts for about 63% and Polk county for value of d0 and therefore we use values of d0 ≥ 10 km. about 32% of the poultry population of Nebraska. The government quarantines an infected location for We provide a boxplot for the node sizes in Figure 2 Q = 21 time steps as speciﬁed in the USDA national (a). The frequencies of the distances between nodes are in response plan. During this process the disease can still Figure 2 (b). spread to other locations due to migration of synanthropic birds, rodents, humans and equipment movement, but the probability of infection decreases with time. To account for this, the random factor f (t) in formula 2 is set equal to 1 during the ﬁrst time step after infection, and is subse- quently given for all nodes by a Beta distribution β(1, h(T )) where T is the number of time steps since the beginning of the quarantine, and h(T ) is an increasing function of T (h(T ) = T in simulations). We set cn (t) = 0 after 15 time steps of quarantine. After the quarantine the node re-enters the normal process if the location is repopulated. Small farms are assumed to have a 50% chance of going out of business versus repopulation. Next we provide a formula for computing the basic reproduction numbers and generate simulations that allow us to understand the impact of a change in parameters on this quantity. Figure 3. Plot of the average number of infected nodes in one time step, AK , versus K, the index of the initial infected node. This is done in four different scenarios corresponding to the variation of one of the parameters ( (a) τ , (b) ρ, 5. BASIC REPRODUCTION NUMBERS (c) d0 , (d) R) while keeping the other ones ﬁxed as mentioned in the titles of the subplots. Observe that AK is decreasing as a function of τ , ρ or R, and increasing Consider now the infection probability given by formula as a function of d0 . The two peaks correspond to Butler and Polk counties. The values of AK are impacted most dramatically by changes in τ . When R increases 2, used to compute the basic reproduction numbers, or the there is an approximate threshold value beyond which the neighborhood size makes average amount of secondary infections generated by a no difference since far away farms will not be infected. primary infection. We assume that exactly one node, say cK , is infected at time t = 0, that is cK (0) = 1. We want to see what is the distribution of the number of infected nodes at time t = 1. j (1 − aj ) = a1 + a2 + · · · + ak , where the sum is over 1 ≤ i1 < i2 < · · · < il ≤ k, and j = 1, 2, . . . , k, j = Let cn be a node in the neighborhood of node cK . τ ρ in , n = 1, 2, . . . , l. Then pn = (BK /B(n)) / (1 + (dnK /d0 ) ). This is the Proof: The proof is by induction on k. Let Sk denote probability that cn (1) = 1 given that cK (0) = 1 and the sum in Remark 3. Observe that S1 = 0 · (1 − a1 ) + ck (0) = 0, for all nodes ck , k = K. Thus, at time t = 1 1 · a1 = a1 . Also Sk+1 = Sk · (1 − ak+1 ) + Sk · ak+1 + the number m of nodes that are turned ON can vary from 0 k to the number MK of nodes in the neighborhood of node ak+1 · l=0 ai1 ai2 . . . ail j (1−aj ) where the second cK (not including the node cK ). So if p1 , p2 , . . . , pMK sum is over 1 ≤ i1 < i2 < · · · < il ≤ k, and j = are the probabilities corresponding to the MK nodes of 1, 2, . . . , k, j = in , n = 1, 2, . . . , l. Thus, Sk+1 = Sk + the neighborhood of node cK , then the probability qK (m) ak+1 · 1 = a1 + a2 + · · · + ak+1 . ♦ that exactly m nodes are infected at time t = 1 is given So the average number of nodes infected at time t = 1 by qK (m) = pi1 pi2 . . . pim j (1 − pj ), where the or the basic reproduction numbers given cK (0) = 1 are sum is over all possible combinations of m nodes out of AK = p1 + p2 + · · · + pMK K = 1, 2, . . . N. (3) MK , 1 ≤ i1 < i2 < · · · < im ≤ MK , and j = 1, 2, . . . , MK , j = il , l = 1, . . . , m. Thus the random We graph AK versus K and one other parameter (τ , variable giving the number of nodes that are infected at ρ, d0 , and R respectively) in Figure 3. A modiﬁcation time t = 1 is: P (m infected nodes) = qK (m), m = of the ﬁxed parameters mentioned in the titles of the plots MK 0, 1, . . . , MK . Then m=0 qK (m) = 1 by the follow- does not change the shape of the graphs, only the values ing result. of AK . For example, when τ is varied, an increase in the Remark 2. For any integer k > 0 and any real num- ﬁxed ρ generates overall smaller values of AK due to the k bers a1 , a2 , . . . , ak , we have that l=0 ai1 ai2 . . . ail · fact that the distance kernel in formula 2 decreases. j (1 − aj ) = 1, where the sum is over 1 ≤ i1 < i2 < Now we can focus on one parameter combination and · · · < il ≤ k, and j = 1, 2, . . . , k, j = in , n = 1, 2, . . . , l. analyze the average number of infected nodes by time We make the convention that l = 0 means that there is steps and time until eradication of the epidemic. only one term in the inside sum and all the factors of this term are of the type (1 − aj ). 6. NETWORK EVOLUTION AND SOME Proof: The proof is by induction on k. Let Sk denote STATISTICS the sum in Remark 2. Clearly S1 = a1 + (1 − a1 ) = 1. We set the parameters as follows: τ = 1.5, ρ = 0.95, d0 = Also Sk+1 = Sk · ak+1 + Sk · (1 − ak+1 ) = Sk . ♦ 30 km, and R = 100 km which yields an average of 195 Then the average number of infected nodes given only farms per neighborhood. In the next graph we list the one infected node at time t = 0, cK (0) = 1, is AK = nodes horizontally (in the alphabetic order of the coun- MK m=0 m · qK (m). ties) and represent the infected ones by dots. We iterate Remark 3. For any integer k > 0 and any real num- formula 2 exactly 50 time steps. In Figure 4 we start with k bers a1 , a2 , . . . , ak , we have that l=0 l ai1 ai2 . . . ail · one infected node in Butler county and we plot dots for Initial infected county: Butler Butler Douglas Washington time evolving downwards 50 time steps Saline Thurston Howard Lancaster Washington Hamilton Frequency Washington Cuming Howard Lancaster 1 Burt Pierce 0 Howard 0 0 0 20 25 40 I(t) t 60 60 Boone NemahaPierce Sarpy 50 Butler Howard Lancaster Lancaster Richardson Average number of time steps until eradication of disease Douglas Nemaha Sarpy 40 Burt Merrick Stanton Cass average time until eradication per infected node until eradication Cedar overall average time until eradication = 18.358 average time Butler Lancaster 0 1200 farms (in alphabetic order of counties) Figure 4. Sample spread of the infection starting with one infected node in Butler county. The infected nodes are listed: Butler, Howard, Lancaster, and Pierce 10 counties, followed by the rest of the listed counties at various times. 0 40 80 infected node at time t=0 all the nodes infected at each time step listing their names. Figure 5. Frequency of I(t), t = 1, 2, . . . , 60 when the initial infected nodes are in Butler county. For small and large values of t less nodes are infected Note that Butler is followed by Howard, Lancaster, Pierce, as expected. Around t = 20 which is the time around which the ﬁrst infections are cured larger values of I(t) are more frequent. The second graph represents the and then the rest of the listed counties at various times. A average number of time steps for eradication of the network infection versus the total of 34 nodes are infected and the infection is con- initial infected node. The overall average is represented by a horizontal line. tained during the 50 time steps. N The quantity I(t) = n=1 cn (t) is the number of infected nodes at time t. We generate frequency plots of [4] R. Albert and A.-L. Barabasi, “Dynamics of com- I(t) for t = 1, 2, . . . 60, starting with one infected county. plex systems: scaling laws for the period of boolean For example the ﬁrst graph of Figure 5 corresponds to the networks,” Physical Review Letters, vol. 84(24), pp. infection of Butler. We observe that small and large values 5660–5663, 2000. of t correspond to mostly small values of I(t), while for [5] M. T. Matache and J. Heidel, “Asynchronous ran- medium values of t there are higher frequencies of larger dom boolean network model based on elementary values of I(t). The epidemic may not be contained. For cellular automata rule 126,” Physical Review E, vol. smaller counties, the plots are concentrated around small 71, pp. 026232, 2005. values of I(t) for all t. Now consider the time until the eradication of the dis- [6] C. Goodrich and M. T. Matache, “The stabilizing ef- ease starting with one infection, averaged over multiple fect of noise on the dynamics of a boolean network,” sample evolutions. The results are in the second graph of Physica A, vol. 379, pp. 334–356, 2007. Figure 5. The two peaks are for Butler and Polk counties. The overall network average is about 18 time steps. [7] H. Lahdesmaki, S. Hautaniemi, I. Shmulevich, and We note that the infection of small counties has little O. Yli-Harja, “Relationships between probabilistic impact on the network, unless they are close enough to boolean networks and dynamic bayesian networks one of the bigger nodes. When the spread of the disease as models of gene regulatory networks,” Signal Pro- is more encompassing, the bigger nodes are infected and cessing, vol. 86(4), pp. 814–834, 2006. spread the disease to other nodes faster and throughout a [8] S. A. Kauffman, The origins of order, Oxford Uni- wider area. However, for a medium node the disease could versity Press, Oxford, 1993. be contained rather fast. On the other hand, it could be that even small nodes spread the disease to bigger nodes [9] Y. Xia, O. N. Bjørnstad, and B. T. Grenfell, “Measels and produce an outbreak. However, for most cases the metapopulation dynamics: a gravity model for epi- infection spreads to only a few or no other nodes. demiological coupling and dynamics,” The Ameri- can Naturalist, vol. 164, pp. 267–281, 2004. 7. REFERENCES [10] S. Erlander and N. F. Stewart, The gravity model in [1] H. W. Hethcote, “The mathematics of infectious dis- transporation analysis: theory and extensions, In- eases,” SIAM Review, vol. 42(4), pp. 599–653, 2000. ternational Science, Netherlands, 1990. e [2] M. Barth´lemy, R. Pastor-Satorras, and A. Vespig- nani, “Dynamical patterns of epidemic outbreaks in [11] N. M. Ferguson, D. A. T. Cummings, S. Cauchemez, complex heterogeneous networks,” Journal of The- C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, oretical Biology, vol. 235, pp. 275–288, 2005. and D. S. Burke, “Strategies for containing an emerging inﬂuenza pandemic in southeast asia,” Na- [3] M. E. J. May and A. L. Lloyd, “Infection dynamics ture articles, vol. 437, pp. 209–214, 2005. on scale-free networks,” Physical Review E, vol. 64, pp. 066112, 2001.

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