The Speed of Epidemic Waves in a One-Dimensional Lattice of SIR Models

Transcription

1 Math. Moel. Nat. Phenom. Vol. 3, No. 4, 2008, pp The Spee of Epiemic Waves in a One-Dimensional Lattice of SIR Moels Igor Sazonov a1, Mark Kelbert b an Michael B. Gravenor c a School of Engineering, Swansea University, Singleton Park, SA2 8PP, U.K. b Department of Mathematics, Swansea University b Institute of Life Science, School of Meicine, Swansea University Abstract. A one-imensional lattice of SIR (susceptible/infecte/remove) epiemic centres is consiere numerically an analytically. The limiting solutions escribing the behaviour of the stanar SIR moel with a small number of initially infecte iniviuals are erive, an expressions foun for the uration of an outbreak. We stuy a moel for a weakly mixe population istribute between the interacting centres. The centres are moelle as SIR noes with interaction between sites etermine by a iffusion-type migration process. Uner the assumption of fast migration, a one-imensional lattice of SIR noes is stuie numerically with eterministic an ranom coupling, an travelling wave-like solutions are foun in both cases. For weak coupling, the main part of the travelling wave is well approximate by the limiting SIR solution. Explicit formulae are foun for the spee of the travelling waves an compare with results of numerical simulation. Approximate formulae for the epiemic propagation spee are also erive when coupling coefficients are ranomly istribute, they allow us to estimate how the average spee in ranom meia is slowe own. Key wors: spatial epiemic moels; ynamic systems; travelling waves AMS subject classification: 92D30, 91D25 1. Introuction Mathematical moels have been long use to stuy the epiemiology of a wie range of infectious iseases [1]. The classical SIR (susceptible/infecte/remove) moel is one of the cornerstones of 1 Corresponing author. i.sazonov@swansea.ac.uk Article available at or 28

2 I. Sazonov et al. The spee of epiemic waves mathematical epiemiology, escribing the ynamics of infection in a single population [18, 5]. The analysis of infection sprea through linke systems of populations such as urban centres is of great importance [20] an attracts consierable interest, in particular for planning the response to emerging panemic iseases. An excellent review of a huge literature on spatial epiemic moels can be foun in [15]. The numerical simulation of spatial systems gives insight into ifferent scenarios of epiemic sprea an can be use to help plan actual interventions. However, the inclusion of large numbers (potentially many thousans) of interacting noes can be costly in terms of integrating the SIR processes, an simplifie moels (e.g., with the epiemic processes greatly simplifie within the noes) are wiely use. For example, uring the recent foot an mouth isease outbreak in the UK [9] real time moelling of the spatial sprea of the epiemic was performe, on the network of almost all sheep/cattle/pig livestock premises in the country. Following this experience, recommenations were mae for the construction, in avance, of well efine an unerstoo quantitative moels as part of attempts to prepare for future outbreaks of emerging iseases. A well-evelope simulation technique is base on passing to the continuous meia [18, 8, 14, 6, 3]. Usually this limit leas to PDE of the reaction-iffusion type. A well-known feature is the existence of the so-calle travelling wave solutions, that preserve spee an shape. Analogous patterns are systematically observe in epiemic ata sets [6]. However, situations are possible when the continuous limit is not applicable, although similar patters can occur. Analysis of the iscrete nature of spatial interactions is one our work goals. Many recent publications focus on the spatial aspect, an the propagation of isease (or computer viruses or rumours) in networks with ifferent topology attracts a lot of attention (see review [10] an papers cite therein, in particular, the Small Worl network [17]). Usually, these moels (i.e. the contact processes) apply a simplifie escription of the epiemic within a single noe (own to the simplest binary moel: infecte or not). As a step towars incluing more etaile epiemic ynamics into the noes of these lattice-type spatial moels, it is important to consier a system of locally interacting SIR moels an to stuy their global behaviour. Here, we consier a network with the simplest topology a 1D lattice with a large number of interacting noes. The ynamics within each noe is escribe by a stanar SIR moel, with aitional terms accounting for the interaction between noes. More precisely, we use an approximation of weak coupling an fast migration to obtain a relatively simple an explicit escription of epiemic propagation. Using an approximation of weak coupling, we assume that the interaction is ue to migration of a relatively small proportion of the population (incluing those infective) between noes, an the interaction is proportional to the small share of time this population spens in the neighbouring noes. The situation is therefore most applicable when populations are concentrate mainly in their own communities, an the sprea of an epiemic has a istinctive time elay cause by a lack of strong mixing in the overall population. We establish the existence of travelling wave type solutions in such systems, uner eterministic an ranom coupling an etermine its spee of propagation. It is shown that in an interesting parameter range it is impossible to approximate the finite ifferences by erivatives, i.e. to pass to reaction-iffusion equations, an a iscrete spatial approach is essential for the analysis. The paper is organize as follows. In Section 2, the limiting solutions of a single SIR moel 29

3 I. Sazonov et al. The spee of epiemic waves are introuce. They provie a goo approximation for an outbreak triggere by a small number of initial infectives, an provie the base for escribing travelling waves in the lattice of SIR noes in Section 4. In Section 3 the migration moel for sprea between noes is iscusse, an equations of interacting SIR moels are escribe. In Section 4, travelling waves in a 1D lattice of interacting ientical SIR moels are investigate. In Section 5, the stuy is extene for a lattice with ranom coupling. In Section 6, we stuy the limits of applicability of the approximation use in analysis, estimate the possible errors an consier the continuous case. Finally, in Section 7, we raw our conclusions, an iscuss the possible extensions of the moel. 2. Limiting solution of the SIR moel In the classical SIR moel, the numbers of susceptibles S, infectives I an remove (immune/ea ue to isease/vaccinate) R evolve in accorance with the equations t S = βsi, I = βsi αi, t R = αi (2.1) t where β is the infection rate, α the recovery rate (e.g. [18, 5]). Aing these equations an integrating the sum, we obtain S + I + R = const = N where N is the population number. The typical evolutionary problem is to integrate (2.1) if the initial number of infectives I 0 is set at t = t 0. We write equations (2.1) in the imensionless form s τ = ρsi (2.2) i τ = (ρs 1)i (2.3) where s = S/N, i = I/N, are, respectively, the shares of susceptibles an infectives, an τ = αt is the imensionless time. Here ρ = (β/α) N is calle the basic reprouction number usually enote by R 0 [18, 1, 7]. The ifferential equation for r can be omitte as r = 1 s i. The initial conitions in these variables can be written as i(τ 0 ) = i 0, s 0 := s(τ 0 ) = 1 i 0. (2.4) The initial number of infectives grows if ρs 0 > 1 [5] an reaches its maximal value [5] i max = 1 ρ 1 + ρ 1 ln(ρ 1 s 1 0 ) (2.5) at an instant which we enote as τ max. Then the outbreak time can be efine as [5] τ outb := τ max τ 0. In Figure 1a, the solutions i(τ) are plotte for ρ = 4 an various values of i 0. Initial instants τ 0 are chosen to achieve the maximum i max at the same instant. Observe that as i 0 ecreases, the 30

4 I. Sazonov et al. The spee of epiemic waves i(t) i t - tmax a s max b s Figure 1: Solutions i(τ) (a) an i(s) (b) for i 0 = 1/4, 1/8, 1/16, 1/32 an ρ = 4 (ashe curves: the smaller i 0 the thinner is the curve). The thin soli curve inicates the limiting solution. solutions ten to an unique limiting solution i lim (τ) starting from the point i = 0, s = 1 at the instant τ =. On the phase iagram {s, i}, the limiting solution i lim (s) is the lower bounary of all realistic trajectories (Figure 1b). The limiting solutions i lim (τ) are shift invariant with respect to τ, an from now on we set τ max = 0. They then form a one parameter family plotte in Figure i lim t Figure 2: Limiting solutions i(τ) for ρ = 2, 3,..., 10 in the logarithmic scale. Values of ρ are inicate in circles. The limiting solutions have simple asymptotic behaviors i lim = A 0 e λ 0τ if τ (2.6) i lim = A e λ τ if τ + (2.7) 31

5 I. Sazonov et al. The spee of epiemic waves where A 0 an A are some constants; λ 0 = ρ 1 > 0, λ = 1+ρs lim rates. Here s lim < 0 are the growth/ecay := lim s lim (τ) = 1 τ ρ W ( ) 0 ρe ρ (2.8) where W k (z) is the kth branch of the Lambert function [4] (see Appenix for etails). Any solution starte at τ 0 with small enough i 0 can be approximate by the limiting curve i(τ; ρ, i 0, τ 0 ) i lim (τ τ 0 τ outb ; ρ). (2.9) The maximal iscrepancy between usual an limiting solutions occurs in the vicinity of maximum of i an approximately equals max i i lim i max i lim max = ln(s 0) 1 ρ i 0 ρ. To apply approximation (2.9) it is necessary to calculate τ outb for given ρ an i 0. An integral formula for the outbreak time is s0 s τ outb = (2.10) ρ(s s 2 ) + s ln(s/s 0 ) where s max s max := s(τ max ) = ρ 1. (2.11) This value can be easily obtaine by setting i/τ = 0 in (2.3); substituting (2.11) into (A1) we obtain (2.5). When i 0 0, non-ecaying asymptotic terms of (2.10) are τ outb = 1 λ 0 ln 1 i 0 + C + O(i 0 ) (2.12) where C = C(ρ) is inepenent of i 0. The proof, an the integral formula for C(ρ) are given in the Appenix. The constant C is relate to the constant A 0 in (2.6) 3. Interacting SIR processes C = λ 1 0 ln A 0. (2.13) Consier the migration process between two interacting ientical centres. Then the rate of change of infectives/susceptibles will be etermine by migration as well, an aitional terms appear in the equations τ s 1 = ρs 1 i 1 + τ s 2 1 τ s 1 2 (3.1) τ i 1 = ρs 1 i 1 i 1 + τ i 2 1 τ i 1 2 (3.2) τ s 2 = ρs 2 i 2 + τ s 1 2 τ s 2 1 (3.3) τ i 2 = ρs 2 i 2 i 2 + τ i 1 2 τ i 2 1 (3.4) 32

6 I. Sazonov et al. The spee of epiemic waves where i m n an s m n are, respectively, the shares of infectives/susceptibles migrate from noe m to noe n. Let the population from noe 1 migrate between the two noes, spening a fraction of time ε in the secon noe. Suppose that at τ = 0, the share i 1 of infectives is concentrate in noe 1. Uner the assumption that the migration is escribe by linear equations, the share of infectives at noe 2 can be escribe by the convolution with a response function g(τ) of a linear system. For the problem in han, it is convenient to write this convolution in the form i 1 2 (τ) = [ i1 (τ) g(τ) ] := τ τ + i 1 (τ ) g(τ τ ) τ (3.5) where g(τ) = 0 if τ < 0 ue to the casuality principle. Generally, i 1 2 (τ) converges exponentially to the equilibrium value εi 1 with the characteristic time τ, for example, from the analogy with a iffusion process it can be g(τ) = ε ( 1 e τ/ τ) θ(τ) (3.6) where θ(τ) is the Heavisie unit-step function. The approximation of weak coupling consiere here means that ε 1. Then i m n 1 an s m n 1. If those first infecte appear in noe 1, then the outbreak evelops first in this noe. Its infectious iniviuals migrating to noe 2 trigger the outbreak there with a certain elay. A noe is sensitive to a small amount of infectives only before the outbreak, becoming almost insensitive uring an after it. Therefore, we can neglect transport terms in (3.1) (3.2) as they o not change the essential epiemic ynamics in noe 1. For the approximation of weak coupling, we can also neglect transport terms i 2 1, s 2 1 in (3.3) (3.4). The small term s 1 2 in (3.3) also oes not impact on the solution of the equation an can be roppe. Thus, in the approximation of weak coupling (ε 1) an when the outbreak is first initiate in noe 1, the only essential transport term is i 1 2 in (3.4) which accounts for infectives migrating from noe 1 to noe 2 an triggering the outbreak there. Here, we also restrict our consieration to an approximation of fast migration, when the characteristic time of the migration process τ is much smaller than 1/λ 0, i.e. the characteristic time of the epiemic growth on its linear stage In this approximation we can write τ 1/λ 0. (3.7) g(τ) ε θ(τ) an i 1 2 (τ) εi 1 (τ), (3.8) an equations (3.1) (3.4) coincie with (2.2) (2.3) whereas equations (3.3) (3.4) take the form τ s 2 = ρs 2 i 2 (3.9) τ i 2 = (ρs 2 1)i 2 + ε τ i 1. (3.10) 33

7 I. Sazonov et al. The spee of epiemic waves Note that the moel (3.9) (3.10) is ifferent from the well-known moel of simple epiemics in interacting groups (see [5], 2.2) where interaction terms appear without the time erivative. In the fast migration approximation, if the epiemic ynamics are temporarily neglecte (sprea between iniviuals an removal are switche off), equations (3.9) (3.10) imply that i 2 i 1 2 = εi 1, i.e. the migration provies the only source of infectives in noe 2. Analogous equations can be erive for many interacting noes with the external forces, (/τ)i m n, n, m = 1, 2,..., m n, proportional to the rate of change of the number of infectives arriving from other noes couple with the nth noe. They can also escribe networks of interacting noes of ifferent topology (e.g. [10, 17], etc); although in this work we consier the simplest network topology only. 4. One-imensional lattice of SIR moels Consier a lattice of interacting ientical SIR noes. Let the ynamics in the first noe (n = 1) be escribe by (2.2) (2.3) with initial conitions (2.4), an the ynamics in all subsequent noes (n = 2, 3,...) by (3.9) (3.10) which, assuming fast migration, take the form τ s n = ρs n i n (4.1) τ i n = (ρs n 1)i n + τ i n 1 n (4.2) with i n 1 n = εi n 1. Numerical integration of the ODEs (4.1) (4.2) with ifferent ρ, ε an i 0 (see Figure 3) shows in t Figure 3: Numerical simulation of lattice with couple SIR noes (first 10 noes) for ρ = 3 an for ε = that the time ifference T n between outbreaks in the (n 1)th an nth noes tens to a constant T (ρ, ε) (see Figure 4). The perio of initial growth of T n is more clearly visible in the logarithmical scale (see Figure 4, curve 2). The solution i n (τ), s n (τ) for n 1 almost replicates the solution 34

8 I. Sazonov et al. The spee of epiemic waves n Tn T max n Figure 4: Time interval T n between outbreaks in two neighbouring noes versus n for ρ = 3 an ε = 10 3 for linear (curve 1) an logarithmic (curve 2) n-scale. The approximation (4.11) is shown by the ashe line. i n 1 (τ), s n 1 (τ) in shape, with a elay T i n (τ) = i n 1 (τ + T ), s n (τ) = s n 1 (τ + T ). (4.3) Thus the numerical integration shows that for n the solution tens to the universal one elaye in time by nt : i n (τ) i tr (τ nt ). This inicates that a type of travelling non-linear wave propagates with the velocity c = T 1 in the iscretize lattice of iniviual SIR noes with unit istances between noes, whereas the variable n plays the role of a spatial coorinate. Substituting the solution of the form s n (τ) = s tr (τ nt ), i n (τ) = i tr (τ nt ) into (4.1) (4.2), we obtain a system of two equations Its solution must satisfy the initial conitions τ str = ρs tr i tr (4.4) τ itr = (ρs tr 1)i tr + ε τ itr (τ + T ). (4.5) s tr ( ) = 1, i tr ( ) = 0. (4.6) Here T = T (ρ, ε) is an eigenvalue of the problem (4.4) (4.6). As the equations are invariant with respect to a shift in time, we fix τ = 0 as an instant when i tr is maximal. As the solution i tr ecays for τ > 0, the secon terms on the r.h.s. of (4.5) are essential only for τ < T. Hence, for τ > T the solution is close to the solution for the iniviual noe consiere above. In the limit ε 0 it tens to the limiting solution i lim in (2.9). Numerical simulation confirms that the travelling wave an limiting solution iffer only for τ < T (see 35

9 8 I. Sazonov et al. The spee of epiemic waves max 0 i a t -T 0 b - T 0 Figure 5: Traveling wave i tr (τ; ρ, ε) (soli) an limiting solution i lim (τ; ρ) (ashe) versus time for ρ = 3 an ε = 10 3 (a). Plot of the growth rate over time, λ(τ) = (/τ) ln(i) (b). Figure 5). For τ < T the value s(τ) iffers negligibly from 1, an the equation (4.5) can be well approximate by a linear one τ itr = λ 0 i tr + ε τ itr (τ + T ). (4.7) Here the solution of (4.7) has an exponential form i tr = exp{λτ} where the parameter λ is relate to the eigenvalue T of the non-linear problem λ = λ 0 + ελe λt. (4.8) A general approach escribe [22, 16] can be applie to fin T, an provies the results well supporte by numerical evience. More precisely, in some non-linear systems, the velocity of a travelling wave correspons to the lowest value taken over a range of wavenumbers for the linearize equation. This shoul be true for a system in which non-linearity results in slowing own the wave. In fact, the velocity of travelling wave is proportional to the growth rate λ 0 of infinitesimal solution (see below). The higher the wave amplitue i, the smaller is the share of susceptibles s, hence the smaller is the growth rate which is λ = ρs 1 < λ 0. Therefore we fin the smallest spee of propagation (greatest T ) for the linear solution an compare it with the results of numerical simulation. Substituting the plane wave solution of the form i tr (τ, n) = exp (λτ nt ) (4.9) to (4.1) (4.2) an neglecting non-linear terms there, we obtain the characteristic equation (cf. (4.8)) L(λ, T ) := λ λ 0 ελe λt = 0. (4.10) 36

10 I. Sazonov et al. The spee of epiemic waves Solving the system of equations L(λ, T ) = 0, L λ (λ, T ) = 0 where L λ = L/ λ, we obtain explicit formulas for the maximal value of T an express the corresponent value of λ in terms of the Lambert function T max = 1 λ 0 (W 0 1) 2 W 0, λ max = λ 0 W 0 W 0 1. (4.11) Here W 0 = W 0 (e/ε) is the main branch of the Lambert function, e is Euler s constant. Depenence of λ 0 T on ε is plotte in Figure 6. l T e Figure 6: Delay time λ 0 T versus ε for ρ = 2, 3,..., 10. Circles inicate numerical results. In this scale, all circles for ifferent ρ an the same ε are merge. Dashe line is rawn by use of (4.11). The solution has the following expansion for small ε T max 1 λ 0 [ ln 1 ε ln ln 1 ε 1] if ε 1. (4.12) This converges slowly as an expansion with respect to (ln ln(ε 1 )) k whereas the Lambert function can be compute easily by fast converging iterations like W (k+1) (x) = ln x ln W (k) (x). Numerical integration shows that the compute outbreak time ifference T n is very close to T max for sufficiently large n but noticeably excees it when n = O(10) (see Figure 4). Figure 5b presents a plot of the current growth/ecay rate of i λ(τ) = ln(i). (4.13) τ The limiting solution λ(τ) passes from the value λ 0 to λ, an the travelling wave λ(τ) is close to λ th if τ is small enough. For ρ = 3 an ε = 10 3 we obtain λ th 2.39, λ 0 = 2, λ These asymptotes are shown by horizontal otte lines. 37

11 I. Sazonov et al. The spee of epiemic waves 5. Lattice with ranomize interaction coefficients Finally, we consier a lattice of SIR noes with the same reprouction numbers ρ for every noe but ranomize interaction coefficients ε n, i.e the lattice is escribe by equations τ s n = ρs n i n (5.1) τ i n = (ρs n 1)i n + ε n τ i n 1. (5.2) Let all ε n be statistically inepenent, positive, with the mean ε an variance σ 2 ε. We consier the simplest moel, with coefficients ε n having the log-normal istribution, i.e. ln ε n have a normal istribution with mean µ an variance σ 2. Then T n are also normally istribute in the main term of its asymptotic, see (4.12). Values ε an σ ε can be etermine from the following relations (e.g. [21]) ε = e µ+σ2 /2, σ 2 ε = ε 2( e σ2 1 ). (5.3) Resolving (5.3), we fin µ an σ µ = ln ε 1 2 ln(1 + v), σ2 = ln(1 + v) (5.4) where v = σ 2 ε/ ε 2 is the variance normalize by the square of the mean value. An example of numerical simulation of this moel is shown in Figure in t Figure 7: Numerical simulation of lattice with couple SIR noes (first 10 noes) for ρ = 3 an for ranomize ε n with ε = 10 3 an (σ ε / ε) 2 = 2. Assume that in the case ε 1 an σ ε ε all time intervals T n between outbreaks in the (n 1)th an nth noes are escribe by (4.11), i.e. λ 0 T n ( W0 (e/ε n ) 1 ) 2. (5.5) W 0 (e/ε n ) 38

12 I. Sazonov et al. The spee of epiemic waves Assumption (5.5) agrees well with numerical simulation an implies that the mean time T interval can be calculate via the integral ( W0 (e/ε) 1 ) 2 λ 0 T = f(ε) ε, W 0 (e/ε) 0 λ 0 T = 1 2π + (ln ε µ) 2 f(ε) = e 2 σ 2. (5.6) 2πσε Here f(ε) is the probability ensity of the log-normal istribution. The substitution ε = e µ+σx = εe σx σ2 /2 in (5.6) gives ( [W e 1 σx+σ 2 /2 ) ] 2 0 ε 1 W 0 ( e 1 σx+σ 2 /2 ε ) e x2 2 x. (5.7) Now we expan the integran into series with respect to σ an integrate it term-by-term. Terms with o powers of σ vanish after integration. The first two terms approximate (5.7) very well even for σ exceeing unity λ 0 T = (W 0 1) 2 W 0 + σ 2 W 0 (W ) 8 (W 0 + 1) 5 + O(σ4 ). (5.8) Here W 0 = W 0 (e/ ε). (See Figure 8: close circles are fitte well by the otte line). l T l T a b Figure 8: Depenence of λ 0 T on v = (σε / ε) 2 for ρ = 3, ε = 10 3 (a) an ρ = 5, ε = 10 2 (b). Numerical results are inicate by open squares. Dotte line inicates the approximation (5.8), close circles inicate the irect numerical integration of (5.7). In numerical simulations for a lattice of 401 noes, the last 200 outbreak intervals T n were taken for averaging. We performe 50 realizations for every v, thus every mean value T n was average by 10 4 inepenent values. The results are plotte in Figure 8. Note that the above assumption fits surprisingly well for rather large values of the variance: σ ε > ε. 39

13 I. Sazonov et al. The spee of epiemic waves 6. Limits of applicability for the approximations Here we apply the general approach evelope in ([22, 16]) to ifferent moifications of our basic moel an compare how these moelling assumptions are reflecte in changes to the preicte epiemic spee. First we consier all the interaction terms between neighbouring lattice noes ( / τ) s n = ρi n s n + ( / τ) [s n 1 n + s n+1 n s n n 1 s n n+1 ] (6.1) ( / τ) i n = (ρs n 1)i n + ( / τ) [i n 1 n + i n+1 n i n n 1 i n n+1 ]. (6.2) In the linear approximation the equation for i n is inepenent of s n. The linearize equation (6.2) in the fast migration approximation reas as ( / τ) i n = λ 0 i n + ε ( / τ) [i n 1 + i n+1 2i n ]. (6.3) Substituting the solution in the form i n (τ) = e λ(τ nt ), we obtain the following characteristic function L(λ, T ) = λ λ 0 ελ [ e λt + e λt 2 ]. (6.4) We solve equations L(λ, T ) = 0, L λ (λ, T ) = 0 numerically (as they have no solution in a close form) an compare with the expressions (4.11) above. The results for comparison are given in Table 1 for ε = 10 1, 10 2, Note that an error of neglecting the transport terms is O(ε). This error is mainly relate to neglecting (outgoing) transport terms i n n±1 which ecreases the epiemic growth. Thus, the reuce form of equations (4.1) (4.2) is justifie in the approximation of weak coupling. Table 1: Comparison of T max for ifferent approximations ε λ 0 T max, (6.4) λ 0 T max, (4.11) error, % λ 0 T max, (6.8) error, % Next we compare the results for the epiemic spee in iscrete an continuous spatial moels. For this, we introuce the spatial coorinate x instea of n i n (τ) = i(x, τ), i n+1 (τ) = i(x + x, τ) an, similarly, for s n. Here x is the istance between noes (later on we set it to unity). Then we can write i n±1 (τ) = i(x ± x, τ) = i(x, τ) ± x ( x)2 2 i(x, τ) + x 2 x i(x, τ) + 2 O(( x)3 ). (6.5) 40

14 I. Sazonov et al. The spee of epiemic waves Substituting (6.5) into (6.3), neglecting terms O(( x) 3 ) an setting x = 1, we obtain a linear PDE of the form i τ = λ 0i + ε 3 i 2 x τ. (6.6) Substituting the solution of the form i = e λ(τ xt ) into (6.6), we obtain the function L(λ, T ) = λ λ 0 ελ 3 T 2. (6.7) Note that the same equation emerges by expaning e ±λt in series with respect to small λt in (6.4). Solving L(λ, T ) = 0, L λ (λ, T ) = 0 we obtain T max = 2 3λ 0 1 3ε, λ max = 3λ 0 2. (6.8) Note that the asymptotic behaviour of T max for small ε is ifferent: ε 1/2 in (6.8) in contrast with ln(ε 1 ) in (4.12). The results for comparison are also given in Table 1. We conclue that for ε 10 2 the weak coupling approximation gives goo accuracy, whereas the error for the continuous moel becomes large. If ε approaches 0.1, the continuous approximation gives more accurate result for epiemic spee than the small coupling approximation although the latter still has the reasonable accuracy O(ε). It worth to mention that if in iscrete moel we account for the outgoing migration terms (i n n±1 ) but neglect the incoming migration from a forwar noe (i n+1 n ) (the characteristic function in this case has the form L(λ, T ) = λ λ 0 ε(e λt 2)) then the system L = 0, L λ = 0 has an explicit solution expresse via the Lambert function of somewhat ifferent argument T = (W 0 1) [(1 + 2ε)W 0 1] W 0, λ = W 0 (W 0 1)(1 + 2ε), W 0 = W 0 [ (1 + 2ε)e Then the relative errors in T compare to the numerical solution of (6.4) are 10 5, , for ε = 10 3, 10 2, 10 1, respectively. This is the most accurate explicit formula for T. Also base on approach escribe in [22, 16], we can estimate the accuracy of the fast migration approximation. The linearize equation in this case takes the form τ i n(τ) = λ 0 i n (τ) + 2 τ 2 τ ε ]. i n 1 (τ )g(τ τ )τ. (6.9) Here we use the kernel g(τ) escribe by (3.6). Substituting i n = e λ(τ nt ), we obtain the characteristic function L(λ, T ) = λ λ 0 ε λ τ + 1. (6.10) The results of numerical solutions for L(λ, T ) = 0, L λ (λ, T ) = 0 are plotte in Figure 9. Observe that for small ε the increment T (ε, τ) T (ε, 0) epens slightly on ε (see Figure 9a). This leas to a smaller relative change of the epiemic spee for small ε (see Figure 9b). 41 λeλt

15 I. Sazonov et al. The spee of epiemic waves T T T T a b 3 Figure 9: Change of T cause by finite migration characteristic time τ: absolute (a) an relative (b) increment of T (τ) for ε = 10 1 (curve 1), 10 2 (curve 2) an 10 3 (curve 3). 7. Discussion A 1D lattice of interacting SIR epiemic moels is consiere in the approximation of weak interactions an fast migration. Numerical simulations confirm that over a wie parameter range (ε = , ρ = 2 10), the solution tens to an universal one which behaves as a travelling wave preserving its shape an spee. In the framework of this moel, a simple explicit formula is obtaine for the spee of the epiemic sprea in a lattice with either constant or ranom interaction coefficients. Compare with numerical simulations, the solutions for the ranomize case emonstrate surprisingly goo accuracy over a wie range of applicability, i.e. when the variance of the coefficient excees the square of its mean value. In the weak coupling approximation the eveloping outbreak in every noe of the lattice is well approximate by the limiting solution obtaine for a single SIR moel. A similar analysis is performe for the finite characteristic migration time τ, an responses of the form given in (3.5) (3.6). The qualitative behaviour of the travelling waves remains the same, with the spee reuce compare to the approximation of fast migration. The shape of the travelling wave aroun the outbreak will presumably remain the same an be well approximate by the limiting solution. The numerical simulation of a lattice with the finite τ an ranom coupling parameters will be a subject of future work. Note that the results of this paper can be straightforwarly extene to a response function of form g(τ) = εθ(τ δ). We simply a δ n to T n obtaine numerically (see Figure 4) or theoretically (see (4.11)), i.e. Tn δ = T n + δ n where δ n is the migration elay between the nth an (n+1)th noes. In the case of a Gaussian istribution of coupling coefficients, the istribution of Tn δ remains Gaussian in the main term of the approximation with the mean value of Tn δ being the sum of T (5.8) an the mean of δ n. In the present work, we concentrate on an asymptotic case when the number of introuce infectives is small compare to the number of cases generate uring the outbreak. In the lattice 42

16 I. Sazonov et al. The spee of epiemic waves moel, this is equivalent to small interaction coefficients (weak coupling approximation). In this approximation, the outbreak time (in a single SIR) an the time between outbreaks in neighbouring noes are logarithmically large: see (2.12) for a single SIR moel an (4.12) for the lattice. The numerical solutions in Section 6 (see Table 1) emonstrate that an approximation of finite ifferences by erivatives, i.e. the moels of reaction-iffusion PDEs [18, 8, 14, 6, 3] lea to significant errors in the value of propagation spee, an thus a iscrete spatial moel is essential for the analysis (cf. [19, 12]). Clearly, the problem of etermining whether the iscrete or continuous nature of the phenomena shoul be consiere is rather serious (see iscussion in the paper The importance of being iscrete an spatial [19]). In contrast to this paper we o not consier a population of iscrete iniviuals, however we show that the step from continuous meia to iscreteness in space can itself lea to ifferent preictions as emonstrate here. Our formulae for the spee an shape of a travelling wave in a lattice of SIR moels potentially provies the means to estimate the interaction coefficients between urban populations in relevant ecological atabases. One of the key examples from epiemiology, that links an extensive atabase to an unerlying travelling wave moel is base on the recors of measles outbreaks in an aroun Lonon since the 1940s [6]. In the case of the measles ata however, it was conclue that the outbreaks in large cities were in reality synchronize by strong linkage, enemic status an a share seasonal forcing term in the contact rates (ue to the school year). In contrast, in a lattice of towns an small population centres, travelling waves coul persist if there was local extinction of isease in small centres. The outbreaks in these centres will therefore ten to lag behin the epiemic generate in a larger community until the weak coupling between the towns sparks a new outbreak. We note that our erivation of time to the peak of the outbreak can also provie a simple approximation for the fae-out time of the outbreak (e.g., by efining the extinction time when the number of infecte iniviuals is reuce below 1). If travelling waves in epiemiology are cause by an interaction of weak coupling, an local centre extinction, our moel provies a useful analytical framework for investigating the unerlying mechanisms of observe travelling waves using long term historical recors such as certain measles scenarios, an other irectly transmitte infections such as the influenza viruses. The problems aresse here are characterize by ifferent time scales, T, τ an 1/λ 0. A more elicate use of separation of scales phenomena an backwar influence will be exploite in forthcoming papers. We also inten to stuy other types of eterministic or ranom coupling an other istributions of ranom coupling constants. Although a network of the simplest topology is consiere here (intuitively it correspons to a sequence of towns locate along a railway or highway), a more general interpretation of the results is require. A 2D system woul clearly fit more closely with the geographical sprea of real iseases. From a physical viewpoint, the 1D lattice correspons to a plane wave approximation, which is stanar for a cylinrical wave far from the source. We believe that away from a localize source, a cylinrical epiemic follows more an more the pattern escribe by the 1D moel consiere in our work. A more specific escription of this phenomena will be a subject of future work. Although very well stuie, the SIR moel continues to provie analytical insight into the basic mechanisms of epiemic processes. This basic moel cannot represent all the features of real-life 43

17 I. Sazonov et al. The spee of epiemic waves phenomena, it is often foun that the more complex simulation-type moels are use in the case of specific outbreaks an to plan actual interventions. However, there can be concerns over assessing the valiity of complex moels, that inclue many noes, an how the large number of assumptions an parameter values etermine the moel output [13]. There is therefore a nee for continuous evelopment of analytical moels of the spatial evolution of epiemics (e.g. [2, 11]) to provie a theoretical basis for those use in planning scenarios. Acknowlegements The work was supporte by EPSRC Project EP/F014015/1. We woul like to thank anonymous referees for helpful comments. References [1] R.M. Anerson, R.M. May. Infectious Disease of Humans: Dynamics an Control. Oxfor Univ. Press, Oxfor, [2] D. Brockmann, L. Hufnagel, T. Geisel. Dynamics of moern epiemics. In SARS: A Case Stuy in Emerging Infections. Eite by A.R. McLean, R.M. May, J. Pattison, an R.A. Weiss. Oxfor Univ. Press, Oxfor, [3] R.S. Cantrell, C. Cosner. Spatial Ecology via Reaction-Diffusion Moels. Mathematical Surveys an Monographs. Wiley, Chichester, [4] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, an D.E. Knuth. On the Lambert W function. Avances in Computational Math., 5 (1996), [5] D.J. Daley, J. Gani. Epiemic Moelling. Cambrige Univ. Press, Cambrige, [6] B.T. Grenfell, O.N. Bjørnsta, J. Kappey. Traveling waves an spatial hierarchies in measles epiemics Nature, 414 (2001), No. 13, [7] J.A.P. Heesterbeek. A brief history of R 0 an a recipe for its calculation. Acta Biotheor., 50 (2002) No. 5, [8] Y. Hosono, B. Ilyas. Traveling waves for a simple iffusive epiemic moel. Math. Moels & Methos in Appl. Sciences., 5 (1995), No. 7, [9] R.R. Kao. The role of mathematical moelling in the control of the 2001 FMD epiemic in the UK. Trens in Microbiology, 10 (2002), [10] M.J. Keeling, K.T.D. Eames. Networks an epiemic moels. J. R. Soc. Interface, 2 (2005),

18 I. Sazonov et al. The spee of epiemic waves [11] E. Kenah, J.M. Robins. Secon look at the sprea of epiemics on networks. Phys. Rev. E, 76 (2007) No. 3, Pt. 2, [12] S.A. Levin, R. Durrett. From Iniviuals to epiemics Philos. Trans.: Biological Sciences, 351 (1996), [13] R.M. May. Uses an abuses of mathematics in biology. Science, 303 (2004), [14] V. Menez. Epiemic moels with an infecte-infectious perio. Phys. Rev. E., 57 (1998) No. 3, [15] D. Mollison, e. Epiemic Moels: Their Structure an Relation to Data, Cambrige Univ. Press, Cambrige, [16] D. Mollison. Depenence of Epiemic an Population Velocities on Basic parameters. Math. Biosciences, 107 (1991), [17] C. Moore, M. Newman. Epiemics an percolation in small-worl networks. Phys. Rev. E, 61 (2000), [18] J.D. Murray. Mathematical Biology. Springer-Verlag. Lonon, [19] R. Durrett, S.A. Levin. The importance of being iscrete (an spatial), Theor. Pop. Biol. 46 (1994), [20] L. Rass, J. Racliffe. Spatial eterministic epiemics. Vol. 102 of Mathematical Surveys an Monographs. Amer. Math. Soc. Provience RI, USA, [21] Y. Suhov, M. Kelbert. Probability an Statistics by Example. Vol. I. Cambrige Univ. Press, Cambrige, [22] F. van en Borsch, J.A.J. Metz, O. Diekmann. The velocity of spatial population expansion, J. Math. Biol., 28 (1990), A Approximations for the outbreak time Diviing (2.3) by (2.2) an integrating the obtaine equation, taking into account (2.4), we obtain the epenence of i versus s (cf. [18]) i = 1 s + 1 ρ ln s s 0 (A1) The inverse epenence can be written via the Lambert W k (x) function, i.e. the kth branch of solution of W exp(w ) = x [4] (alternatively calle ProuctLog in the algebraic package MATHEMATICA) s = 1 ρ W k ( ρs0 e ρ(i 1)) (A2) 45

19 I. Sazonov et al. The spee of epiemic waves where k = 1 for the growing part of the solution an k = 0 for the ecaying part. Setting i = 0 we obtain an explicit formula for s s = 1 ρ W ( 0 ρs0 e ρ) which gives (2.8) by setting s 0 = 1. Substituting (A1) into (2.2) an integrating τ(s) in the limits [s max, s 0 ], we obtain formula (2.10). To stuy the asymptotic behaviour of (2.10) for i 0 0 we make the substitution s = 1 ξ τ outb = 1 ρ 1 ( where f(ξ) = (1 ξ) ρξ + ln ( (1 ξ)/(1 i 0 ) )). i 0 ξ f(ξ) Denote the root of f(ξ) with the minimal absolute value by ξ 0 = i 0 (ρ 1) 1 + O(i 2 0). Consier the function F (ξ; i 0 ) = 1 f(ξ) 1 f (ξ 0 )(ξ ξ 0 ). (A4) It is regular in the vicinity of ξ = 0, an it is easy to show that F (ξ; i 0 ) = F (ξ; 0) + O(i 0 ), ξ [0, 1]. Hence, 1 ρ 1 i 0 F (ξ; i 0 )ξ = 1 ρ 1 0 F (ξ; 0)ξ + O(i 0 ) Our goal is to represent τ outb as a term inepenent of i 0 plus a correction [ ] 1 ρ 1 1 ρ 1 τ outb ξ = F (ξ; 0)ξ + O(i 0 ) + f (ξ 0 )(ξ ξ 0 ). 0 To this aim, we estimate the correction represente by the secon integral 1 ρ 1 i 0 ξ f (ξ 0 )(ξ ξ 0 ) = 1 λ 0 i 0 [ ln 1 ] +2 ln(1 ρ 1 ) + O(i 0 ). i 0 This argument also yiels an expression for constant C(ρ) in (2.12) C(ρ) = 1 ρ 1 0 (A3) F (ξ; 0)ξ + 2 ln(1 ρ 1 ) λ 0. (A5) The function f(ξ) vanishes at ξ = 0 an ξ = 1. As the integran in (A3) grows towars the limits of the integration omain, its bounaries are the main contributors, especially the integration near its lower limit (ξ = i 0 ). Approximating f(ξ) by the secon orer polynomial P 2 (ξ) = λ 0 ξ (1 ξ) 46

20 I. Sazonov et al. The spee of epiemic waves C r Figure 10: Outbreak time coefficient C 0 (a) vs ρ: exact soli, approximation ashe line. which vanishes at ξ = 0, 1 an matches the first erivative f (0), an neglecting non-essential terms at i 0 0, we obtain C ln(λ 0) (A6) λ 0 that gives A 0 λ 0 in accorance with (2.13). Comparison between C(ρ) compute numerically an approximate by (A6) is shown in Figure