Impact of Travel Between Patches for Spatial Spread of Disease

Transcription

1 Impact of Travel Between Patches for Spatial Spread of Disease Ying-Hen Hsieh Department of Applied Mathematics National Chung Hsing University Taichung, Taiwan P. van den Driessche Department of Mathematics and Statistics University of Victoria Victoria, BC, Canada Lin Wang Department of Mathematics and Statistics University of Victoria Victoria, BC, Canada April 21,

2 Corresponding author: P. van den Driessche Department of Mathematics and Statistics University of Victoria Victoria, BC, Canada Fax:

3 Abstract A multi-patch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation, R (i) 0, is obtained along with the basic reproduction number of the system of patches, R 0. Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of R 0 on the travel rates between the two patches are obtained. These results show that, while banning travel from the low prevalence patch to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high prevalence patch to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel from the high prevalence region to the low prevalence region could result in the low prevalence patch becoming disease-free, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in patch 1 alone exceeding the combined number of infectives in both regions if border control had not been in place. Our results demonstrate that border control, if properly implemented, could be useful to stop the spatial spread of disease. Keywords: multi-patch model, spatial spread, basic reproduction number, travel rate, border control, influenza. 1 Introduction Circulating in Asia since late 2003 and causing more than one hundred human deaths around the world by April 2006 (WHO, 2006a), the H5N1 avian influenza virus jumped to Africa and the Middle East, spread through Eu- 3

4 rope, affecting wild bird populations in 13 additional countries within a single month in early 2006 (WOAH, 2006). Seasonal migration alone probably cannot explain the westward spread in Europe, so imports of poultry and pet birds must also be considered as factors that might lead to the spread of H5N1 through countries and continents (Butler, 2006). Furthermore, its initial appearance on the African continent marks a huge leap in its geographical range, and opens up a whole new front where the vast bird reservoir could potentially spark a pandemic of human-to-human infections. The 2003 severe acute respiratory syndrome (SARS) global epidemic demonstrated the ability of infectious disease spreading due to modern globalization to countries in several continents within a matter of days. In its aftermath and with the potential threat of a flu pandemic, several models describing spatial spread of infectious diseases have been proposed. These include the use of multi-patch compartment models to explore the dynamics of spatial spread of disease(see, for example, Arino and van den Driessche, 2003, 2006; Arino et al. 2005; Hyman and LaForce, 2003; Ruan et. al., 2006; Salmani and van den Driessche, 2006; Sattenspiel and Herring, 2003; Wang and Mulone, 2003; Wang and Zhao, 2004). Moreover, previous modeling work has shown that compartmental models incorporating multiple patches (cities, countries, etc.) describe epidemic spread more accurately than non-spatial models, even at early epidemic phases. Examples of such include foot-and-mouth disease (Chowell et. al., 2006), tuberculosis in possums (Fulford et. al., 2002), in addition to models of infectious diseases of humans. In this work we propose a multi-patch model to study the spread of influenza among the patches. We add a subpopulation of partially immune individuals to account for this important feature of influenza. Although the model is intended to be used for theoretical studies of the spread of human influenza, it can also be used as a model for studying the spread of enzootic diseases such as avian flu (H5N1) among birds. Partial immunity might 4

5 not be as important for studies of infectious disease involving poultry bird populations where the birds are slaughtered for food after a fixed time, it nonetheless could be an important consideration for studies involving wild bird populations. The paper is organized as follows. We formulate the model in Section 2, and in Section 3, we give results regarding the basic reproduction number of the model. In Section 4, some global analysis is obtained for the model with two patches. Numerical simulations, with parameter values relevant for influenza, to complement our analytical results are given in Section 5. Finally in the last section, from our results, we discuss the impact of travel on the spatial spread of disease. 2 Model Formulation In this section we formulate a model describing the spread of a disease in a population with n patches taking into consideration travel among patches (Arino and van den Driessche, 2006) and also including partially immune individuals (Hyman and LaForce, 2003). The population in patch i is divided into compartments of susceptible, incubating (infected but not yet showing symptoms), infective (infected with symptoms), recovered and partially immune individuals. Let S i,e i,i i,r i, P i denote, respectively, the associated population size. Then the total population size in patch i is N i = S i + E i + I i + R i + P i for i = 1, 2,...,n. A partially immune compartment between the recovered and the susceptible compartments is introduced in an influenza model (Hyman and LaForce, 2003) to account for people who have partial immunity to the current strain of influenza from a previous infection by an earlier strain. We include this partially immune compartment, and assume that individuals in this compartment may become infected again (but with a reduced rate). Also the partial immunity wanes, with these 5

6 individuals returning to the susceptible compartment. For patch i, let A i be the recruitment, α i,γ i, δ i and η i be the progression rates of the incubating, infective, recovered and partially immune individuals, respectively, d i > 0 be the natural death rate and ǫ i be the disease-related death rate. We assume that individuals do not change their disease state during travel, and m K ij for K = S,E,I,R,P are the constant travel rates from patch j to patch i for i j of susceptible, incubating, infective, recovered and partially immune individuals, respectively, with m K ii = 0. All parameters are assumed to be positive except that ǫ i,δ i,η i can be zero, and the travel rate matrices M K = [m K ij] for K = S,E,I,R,P are irreducible. For more detailed discussion on travel rates, see, e.g., Arino and van den Driessche (2006). The number of new individuals infected by infectives per unit time in patch i is given by β i (N i )S i I i. The term β i (N i )S i is the product of β i (N i )N i, the average number of contacts made by each individual in patch i per unit time, and S i N i, the proportion of susceptibles. It is assumed that infectivity β i (N i ) is a continuously differentiable, non-increasing function of N i with β i (0) <. Note that the above assumptions encompass the widely used mass action and standard incidence disease transmission terms defined for N i > 0 as well as many other forms of saturating incidence. Reduced infectivity of incubating and partially immune individuals is given by σ i β i (N i ) and ν i β i (N i ), respectively, where σ i,ν i [0, 1) and it is assumed that infective individuals can reinfect those who are partially immune. The model flow chart for patch i omitting travel and natural death is given in Figure 1. The above assumptions lead to the following SEIRP model for i = 1, 2,...,n: 6

7 ds i = A i β i (N i )S i (I i + σ i E i ) d i S i + η i P i + m S ijs j de i = β i (N i )[S i (I i + σ i E i ) + ν i P i I i ] (d i + α i )E i + di i = α i E i (γ i + ε i + d i )I i + dr i = γ i I i (d i + δ i )R i + m I iji j m R ijr j dp i = δ i R i (d i + η i )P i ν i β i (N i )P i I i + m R jir i m I jii i m E ije j m P ijp j m P jip i m S jis i m E jie i (2.1) Figure 1: Flow diagram of the model. Initially each variable is assumed to be non-negative with S i (0) > 0 and n i=1 E i(0) + I i (0) > 0. It follows that for a given set of non-negative initial conditions, there is a unique solution to system (2.1). The total population size in all n patches is N(t) = n i=1 N i(t). Let d = min{d 1,d 2,...,d n } and A = n i=1 A i. Then the following result, which can be proved in a similar way to that of Theorem 1.1 in Salmani and van den Driessche (2006), indicates that the model is well posed and all variables remain non-negative and bounded. 7

8 Theorem 2.1. Consider system (2.1) with non-negative initial conditions. Then for each i = 1, 2,...,n, E i (t),i i (t),r i (t),p i (t) remain non-negative, S i (t) and N i (t) remain positive and the total population N(t) is in the interval (0, max{a/ d,n(0)}]. Notice that our patch model is an extension of those in Arino and van den Driessche (2006), Salmani and van den Driessche (2006). It includes a new compartment, namely, partially immune individuals in each patch, and also the probability of disease transmission by incubating individuals. These additional features may be important for modeling diseases such as influenza and SARS. In Hyman and LaForce (2003), which does not include an incubating compartment in a model for the spread of influenza, travel is assumed to be independent of disease status and symmetric. Moreover, the disease does not cause death, and the population of each patch remains constant. Notice also that our patch model keeps track of individuals present in patch i at time t, but does not keep track of where an individual resides. Models that include this are developed in Arino and van den Driessche (2003, 2006) and Sattenspiel and Herring (2003), and Ruan et. al. (2006) use such a model to study the effect of travel on the spread of SARS. 3 The basic reproduction number In this section, we derive a formula to compute the basic reproduction number R 0 for the general model (2.1) and then give a lower bound for R 0. Both lower and upper bounds of R 0 are given for a special case. A disease free equilibrium (DFE) is a steady state solution of system (2.1) with S i = N i > 0 and all other variables E i,i i,r i,p i equal to 0 for i = 1, 2,...,n. Let S = (N 1,N 2,...,N n) T. Then from the first equation in (2.1), there is a DFE if and only if S satisfies CS = A with A = (A 1,A 2,...,A n ) T and C = D M S, where D = diag( n ms ji + d i ). Note 8

9 that C is irreducible, has positive column sums d 1,d 2,...,d n and negative off-diagonal entries. Thus C is a non-singular M-matrix (page 141, Berman and Plemmons, 1979), and therefore C 1 > 0. Hence S = C 1 A > 0 is the unique solution of CS = A. This shows that the DFE always exists and is unique. In the absence of disease system (2.1) reduces to just the first equation and S is stable. Next we consider the local stability of the DFE for system (2.1). To this end, we order the infected variables by E 1,E 2,...,E n,i 1,I 2,...,I n and make use of the result in van den Driessche and Watmough (2002) to obtain [ ] [ ] F11 F 12 diag(σi β i (Ni )Ni ) diag(β i (Ni )Ni ) F = = with where [ ] V11 0 V =, V 21 V 22 V 11 = diag(d i + α i + m E ji) M E, V 22 = diag(γ i + ǫ i + d i + m I ji) M I, V 21 = diag(α i ), i = 1, 2,...,n. Note that V 11 and V 22 are both irreducible non-singular M matrices with positive column sums and hence V 1 11 > 0, V 1 22 > 0. (3.1) The basic reproduction number for the system, denoted by R 0 is then the spectral radius of FV 1, i.e., R 0 = ρ{fv 1 } where [ ][ FV 1 F11 F 12 = V V22 1 V 21 V11 1 V22 1 ].

10 Therefore R 0 = ρ(f 11 V F 12 V 1 22 V 21 V 1 11 ). (3.2) The first term accounts for infection from incubating individuals, while the second term accounts for infection from infective individuals who survive the incubating compartment. Travel rates influence the average time spent in the incubating and infective compartments. Note that R 0 does not depend on ν i,δ i,η i for i = 1, 2,...,n. The basic reproduction number gives an important threshold for the disease, as shown in the following result. Theorem 3.1. Consider model (2.1). If R 0 < 1, then the DFE is locally asymptotically stable and if R 0 > 1, the DFE is unstable. Moreover, if the disease transmission is standard incidence, then the DFE is globally asymptotically stable provided that R 0 < 1. Proof. It follows from Theorem 2 of van den Driessche and Watmough (2002) that the DFE is locally asymptotically stable if R 0 < 1 and is unstable if R 0 > 1. If the disease transmission is standard incidence, then β i (N i )N i = β i for i = 1, 2,...,n. Note that S i N i and S i + ν i P i N i. This gives the inequality de i β ii i + σ i β i E i (d i + α i )E i + Consider the linear system m E ije j m E jie i. de i = β i I i + σ i β i E i (d i + α i )E i + di i = α i E i (γ i + ε i + d i )I i + m E ije j m I iji j m I jii i. m E jie i (3.3) The right hand side of the above system has F V as its coefficient matrix. Again, by proof of Theorem 2 of van den Driessche and Watmough (2002), 10

11 each eigenvalue of F V has negative real part if R 0 < 1. Thus any solution of (3.3) satisfies lim E i = 0 and lim I i = 0 for i = 1, 2,...,n. Using a t t comparison theorem (Thorem B.1, Smith and Waltman, 1995), each solution of system (2.1) satisfies lim E i = 0 and lim I i = 0 for i = 1, 2,...,n. Using t t a similar argument as in the proof of Theorem 2.2, Salmani and van den Driessche (2006), lim R i (t) = 0, and similarly lim P i (t) = 0. Thus the DFE t t is globally asymptotically stable provided that R 0 < 1. Let a i = γ i + ǫ i + d i, b i = β i (N i )N i, c i = d i + α i for i = 1, 2,...,n. In the case that there is no travel between patch i and all other patches, the basic reproduction number in patch i in isolation is given by R (i) 0 = σ ib i + b iα i. (3.4) c i a i c i Define a modified reproduction number (modified by travel) in patch i, R (i) 0 = σ i b i c i + n + me ji b i α i (a i + n mi ji )(c i + n me ji ). (3.5) The following result gives bounds for the basic reproduction number for system (2.1) in terms of the numbers defined in (3.4) and (3.5) for each patch. Theorem 3.2. For system (2.1), R 0 max 1 i n R (i) 0. (3.6) Furthermore, if a i = a,α i = α, σ i = σ and d i = d for i = 1, 2,...,n, then ( ) max max R (i) 0, min 1 i n 1 i n R(i) 0 R 0 max 1 i n R(i) 0. (3.7) Proof For j = 1, 2, let V jj [1 ] denote the matrix V jj with row and column 1 deleted, Y = [y ij ] and Z = [z ij ] denote V 1 11 and V 1 22, respectively. Let W = [w ij ] = G + H, where G = [g ij ] = F 11 Y, H = [h ij ] = F 12 ZV 21 Y. It follows from (3.1) that y ij > 0,z ij > 0 for i,j = 1, 2,...,n. Then, by Corollary , Horn and Johnson (1985), R 0 = ρ(w) w ii = g ii + h ii for i = 1, 2,...,n. 11

12 detv Note that g 11 = σ 1 b 1 y 11 = σ 1 b 11 [1 ] 1 detv 11. By virtue of Fischer s inequality (page 117, Horn and Johnson, 1991), it follows that ( ) detv 11 c 1 + detv 11 [1 ]. Therefore, Similarly, thus, g 11 h 11 = b 1 α 1 z 11 y 11 + m E j1 σ 1 b 1 c 1 + n. me j1 b 1 α k z 1k y k1, h 11 b 1 α 1 z 11 y 11 = b 1 α 1 detv 22 [1 ] detv 22 detv 11 [1 ] detv 11. Again, by Fischer s inequality, it follows that k=2 h 11 ( a 1 + n mi j1 b 1 α )( 1 c 1 + n me j1 ). Adding these shows that Similarly, it can be shown that R 0 R (1) 0. R 0 R (i) 0 for i = 2, 3,...,n and this gives (3.6). If a i = a,α i = α, σ i = σ and d i = d (thus c i = c) for i = 1, 2,...,n, then w ij = σb i y ij + αb i n k=1 z iky kj for i,j = 1, 2,...,n. Without loss of generality, assume that 0 < b 1 b 2 b n. From the fact that the matrix V 11 has each column sum equal to c > 0 and the matrix V 22 has each column sum equal to a > 0, it follows that n i=1 y ij = 1 c, n i=1 z ij = 12

13 1 a for j = 1, 2,...,n. Therefore, for the matrix W, the sum of column j is given by w ij = i=1 σb i y ij + i=1 αb i i=1 σb n y ij + αb n i=1 = σb n y ij + αb n i=1 = σb n + αb n c ac = R (n) 0. i=1 z ik y kj k=1 z ik y kj k=1 ( z ik )y kj Similarly, n i=1 w ij σb 1 c + αb 1 ac = R (1) 0. From Theorem , Horn and Johnson (1985), ρ(w) lies between the minimum and maximum column sums of W, thus k=1 i=1 min 1 i n R(i) 0 R 0 = ρ(w) max 1 i n R(i) 0. Combining with (3.6) immediately gives the desired (3.7) The analytical results give the basic reproduction numbers in isolation of the individual patches as upper and lower bounds for the basic reproduction number of the system. However, its usefulness is rather limited in practice since the range for the bounds could be too large. In the next section, we consider the simple case of only two patches, and obtain more explicit results giving insight about the impact of travel on the spatial spread of disease between patches. 13

14 4 The Model with 2 Patches From now on we consider the special case of (2.1) with only 2 patches, i.e., we consider the system ds 1 = A 1 β 1 (N 1 )S 1 (I 1 + σ 1 E 1 ) d 1 S 1 + η 1 P 1 + m S 12S 2 m S 21S 1 de 1 = β 1 (N 1 )[S 1 (I 1 + σ 1 E 1 ) + ν 1 P 1 I 1 ] (d 1 + α 1 )E 1 + m E 12E 2 m E 21E 1 di 1 = α 1 E 1 (γ 1 + ε 1 + d 1 )I 1 + m I 12I 2 m I 21I 1 dr 1 = γ 1 I 1 (d 1 + δ 1 )R 1 + m R 12R 2 m R 21R 1 dp 1 = δ 1 R 1 (d 1 + η 1 )P 1 ν 1 β 1 (N 1 )P 1 I 1 + m P 12P 2 m P 21P 1 ds 2 = A 2 β 2 (N 2 )S 2 (I 2 + σ 2 E 2 ) d 2 S 2 + η 2 P 2 + m S 21S 1 m S 12S 2 de 2 = β 2 (N 2 )[S 2 (I 2 + σ 2 E 2 ) + ν 2 P 2 I 2 ] (d 2 + α 2 )E 2 + m E 21E 1 m E 12E 2 di 2 = α 2 E 2 (γ 2 + ε 2 + d 2 )I 2 + m I 21I 1 m I 12I 2 dr 2 = γ 2 I 2 (d 2 + δ 2 )R 2 + m R 21R 1 m R 12R 2 dp 2 = δ 2 R 2 (d 2 + η 2 )P 2 ν 2 β 2 (N 2 )P 2 I 2 + m P 21P 1 m P 12P 2 (4.1) The DFE of (4.1) is given by S = Also [ ] [ N 1 d1 + m S 21 m S 12 = N2 m S 21 d 2 + m S 12 ] 1 [ A1 A 2 ]. F 11 = diag(σ 1 b 1,σ 2 b 2 ), F 12 = diag(b 1,b 2 ), V 21 = diag(α 1,α 2 ), and [ c1 + m E 21 m E 12 ] [ a1 + m I 21, V 22 = m I 12 m I 21 a 2 + m I 12 ] V 11 = m E 21 c 2 + m E 12. Hence R 0 = ρ(w), where W = F 11 V F 12 V 1 22 V 21 V 1 11 = 14 [ w11 w 12 w 21 w 22 ] (4.2)

15 with w 11 = σ 1 b 1 (c 2 + m E 12) c 1 c 2 + c 1 m E 12 + c 2 m E 21 + α 1b 1 (c 2 + m E 12)(a 2 + m I 12) + α 2 b 1 m I 12m E 21 (c 1 c 2 + c 1 m E 12 + c 2 m E 21)(a 1 a 2 + a 1 m I 12 + a 2 m I 21) w 12 = w 21 = w 22 = σ 1 b 1 m E 12 c 1 c 2 + c 1 m E 12 + c 2 m E 21 σ 2 b 2 m E 21 c 1 c 2 + c 1 m E 12 + c 2 m E 21 σ 2 b 2 (c 1 + m E 21) c 1 c 2 + c 1 m E 12 + c 2 m E 21 It follows from (3.2) that R 0 = α 1b 1 m E 12(a 2 + m I 12) + α 2 b 1 m I 12(c 1 + m E 21) (c 1 c 2 + c 1 m E 12 + c 2 m E 21)(a 1 a 2 + a 1 m I 12 + a 2 m I 21) + α 1b 2 m I 21(c 2 + m E 12) + α 2 b 2 m E 21(a 1 + m I 21) (c 1 c 2 + c 1 m E 12 + c 2 m E 21)(a 1 a 2 + a 1 m I 12 + a 2 m I 21) + α 1b 2 m E 12m I 21 + α 2 b 2 (c 1 + m E 21)(a 1 + m I 21) (c 1 c 2 + c 1 m E 12 + c 2 m E 21)(a 1 a 2 + a 1 m I 12 + a 2 m I 21). (w 11 + w 22 + (w 11 w 22 ) 2 + 4w 12 w 21 ). (4.3) From Theorem 3.1, we have the following result. Theorem 4.1. For the two-patch model, assume that the disease transmission is standard incidence. Then the DFE is globally asymptotically stable if R 0 < 1 and is unstable if R 0 > 1, where R 0 is given by (4.3). As expected, an increase in σ i or b i increases R 0. To investigate how R 0 changes with the other parameters, we first assume that all travel rates for incubating and infective individuals are equal, namely, m E ij = m I ij = m for i,j = 1, 2. The proof of the following result is given in the Appendix. Theorem 4.2. If m E ij = m I ij = m for i,j = 1, 2 and σ 1 = σ 2 = σ,α 1 = α 2 = α,a 1 = a 2 = a,c 1 = c 2 = c,b 1 b 2, then an increase in a or m decreases R 0. Remark 4.1. With the conditions of the above theorem, as the travel rate becomes large, the basic reproduction number R 0 in (4.3) approaches the mean value of R (1) 0 and R (2) 0, i.e. R (R(1) 0 + R (2) 0 ) as m. This can be seen from the form of W in the proof of Theorem 4.2. In this limit case, the two patches merge into one. 15

16 Using a similar technique as in proof of Theorem 4.2, we can prove the following result in the case that infectives of both patches are too sick to travel. Theorem 4.3. If m E ij = m,m I ij = 0 for i,j = 1, 2 and σ 1 = σ 2,α 1 = α 2,a 1 = a 2,c 1 = c 2,b 1 b 2, then R 0 m < 0, i.e., the basic reproduction number R 0 decreases as m increases. 5 Numerical Simulations Assuming standard incidence disease transmission, we present some numerical simulations for 2 patches to illustrate how R 0 changes with travel rates, with the choice of parameters relevant for human influenza (Ferguson et. al., 2005). In the next section, we present simulations for restricted travel. The model parameters with time unit as one day are taken as: α 1 = α 2 = , γ = γ 2 = (the average incubating time is 1.48 days 2.6 and the average infective time is 2.6 days), ǫ 1 = ǫ 2 = , d 1 = d 2 = , b 1 = 0.6, b 2 = 0.1,σ 1 = σ 2 = These parameters yields the respective basic reproduction numbers in isolation of R (1) > 1 and R (2) < 1. Hence we consider the hypothetical scenario of disease spread between a high-prevalence endemic region (patch 1) and a low-prevalence region where a minor outbreak could be eradicated (patch 2). We first keep m E 21 = m I 21 fixed with values at 0.6, 0.8 and 1. We let m E 12 = m I 12 vary from 0 to 1. The curves of R 0 are given in Figure 2. A 3-d plot of R 0 versus m E 12 = m I 12 and m E 21 = m I 21 is given in Figure 3. Figure 2 shows that by keeping the same travel rate from patch 1 to patch 2, an increase in the travel rate from patch 2 (the patch with lower disease transmission rate) to patch 1 (the patch with higher disease transmission rate) results in an increase in R 0. Consequently, we can conclude that travel between the two patches may cause the disease to become endemic in both 16

17 E m = E m 21 = E m 21 =1.0 R E I m =m12 12 Figure 2: R 0 vs m E 12 = m I 12 for fixed m E 21 = m I 21. The three curves from top to bottom correspond to : m E 21 = m I 21 = 0.6, 0.8, R E I m =m E I m =m21 21 Figure 3: R 0 vs m E 12 = m I 12 and m E 21 = m I

18 patches if the travel rate from patch 2 to patch 1 is large enough. The numerical simulation in Figure 3 shows that, for the chosen parameters and keeping m E 12 = m I 12 fixed, an increase in m E 21 = m I 21 leads to a decrease in R 0. In particular, R 0 could conceivably decrease to a value less than one if the travel rate from the high-prevalence patch to the low-prevalence patch, namely m E 21 = m I 21, is high enough. If all travel rates from one patch to the other are the same, i.e. m E 12 = m I 12 = m E 21 = m I 21 = m, then an increase in m leads to a decrease in R 0 (Figure 4), as predicted by Theorem 4.2. Note that, for these parameter values, R 0 < 1 if m > R m Figure 4: R 0 vs m, where m = m E 12 = m I 12 = m E 21 = m I 21. Taking m = 0.5 and subsequently R 0 < 1, the disease will die out eventually. To investigate how the infective population sizes in the two patches change over time, we numerically simulate the system (4.1) with i = 1, 2, and plot I 1 and I 2 against time in Figure 5. In addition to the parameter values given at the beginning of this section, we take A 1 = A 2 = 100,η 1 = η 2 = δ 1 = δ 2 = 0.2,ν 1 = ν 2 = 0.1. The infective population sizes in both 18

19 patches increase initially then slowly decrease to zero since the DFE is globally asymptotically stable if R 0 < I 1 and I I 1 20 I t Figure 5: Numerical solution of system (4.1) with parameters as in the text and m = 0.5 showing I 1 and I 2 vs time t. Initial conditions are: S 1 (0) = 400,E 1 (0) = 50,I 1 (0) = 20,R 1 (0) = P 1 (0) = 0;S 2 (0) = 800,E 2 (0) = 30,I 2 (0) = 10,R 2 (0) = P 2 (0) = 0. Now we use the same parameter values except that m = 0.2, and subsequently R 0 > 1. Numerical simulation of this case given in Figure 6 shows that in both patches there is an initial increase in the number of infectives before the infective populations decrease to their endemic levels. However, the existence and stability of such an endemic equilibrium remain unproved analytically. Some interesting observations can be made from our results regarding the role that travel plays in the spatial spread of a disease. Figures 2 and 3 demonstrate the possibility that, for a lower-prevalence patch with a minor disease outbreak (basic reproduction numbers in isolation R (2) 0 less than one) open travel with a high-prevalence patch could lead to the disease becoming 19

20 45 40 I 1 and I I 2 I t Figure 6: Numerical solution of system (4.1) showing I 1 and I 2 vs time t. Parameter values and initial conditions are the same as in Figure 5 except that m = 0.2. endemic. On the other hand, for a high-prevalence patch with endemic disease in isolation (basic reproduction numbers in isolation R (1) 0 greater than one), open travel with a high-prevalence patch could eradicate the disease. Essentially, under appropriate parameter ranges, travel between patches dilutes the overall prevalence to the point that it could either lessen the severity of an endemic patch or worsen a minor outbreak region. Further evidence can be found in Figures 4 and 5 in which assuming all travel rates are equal, the disease can be eventually eradicated if the travel rates are sufficiently large. 6 Discussion of Travel Restrictions To consider hypothetical intervention scenarios, we let m I 21 = 0 and keep all other parameters the same as in Figure 2. This models a situation in which the authority bans all travel of symptomatic travelers from patch 1, the high 20

21 prevalence region, to patch 2, the low prevalence region. In comparison with Figure 2, Figure 7 shows this gives an overall increase in the value of the basic reproduction number R 0. More significantly, there is a range of parameter values for m I 12 = m E 12 in which stopping all travel of symptomatic travelers from patch 1 to patch 2 could adversely impact the epidemic by driving the basic reproduction number above one, thus prolonging the epidemic that otherwise would be eradicated E m = R E m = E m = E I m =m12 12 Figure 7: R 0 vs m E 12 = m I 12 for fixed m I 21 = 0. The three curves from top to bottom correspond to : m E 21 = 0.6, 0.8, 1.0 Conversely, Figure 8 shows that stopping all travel of symptomatic travelers from patch 2 to patch 1 (letting m I 12 = 0) would alleviate the epidemic, lowering the basic reproduction number to below one, thus successfully preventing further spread of the outbreak. Therefore, the policy of border control to ban travel of symptomatic travelers only from the high prevalence patch to the low prevalence patch could affect the containment of the outbreak adversely (Figures 2 and 7). However, banning travel of symptomatic travelers only from the low prevalence 21

22 R E m 12. Figure 8: R 0 vs m E 12 with m I 12 = 0 for fixed m E 21 = m I 21 = 0.6 patch to the high prevalence patch always has a positive impact (Figures 2 and 8). We note here that by banning all travels (both exposed but asymptomatic and symptomatic individuals) would result in an outbreak in the high prevalence patch. We now consider the case in which the disease is endemic, as in Figure 6. We again let m I 21 = 0 and keep all other parameters the same as in Figure 6. Figure 9 shows that the disease is still endemic in both patches, but with significantly larger numbers of infectives in both patches. Thus once again, banning all travel of symptomatic travelers from the high prevalence region to the low prevalence region is detrimental to the intervention and control of the outbreak. If we let m I 21 = m E 21 = 0, i.e., banning all travel from the high prevalence region to the low prevalence region (Figure 10), patch 2 becomes disease-free, while patch 1 is even more prevalent with the number of infectives in patch 1 alone exceeding the total number of infectives in the two regions if border control had not been in place. Note that in this case the infected equations for patch 2 uncouple. Matrix W given by (4.2) has 22

23 I 1 I 1 and I I t Figure 9: Numerical solution of system (4.1) with two patches showing I 1 and I 2 vs time t. Parameters as in Figure 6 except that m I 21 = 0. w 21 = 0 and so is reducible. Thus the reproduction number in patch 1 is R (1) (2) 0 > 1 and the reproduction number in patch 2 is R 0 < 1. Now we let m I 12 = 0 and keep all other parameters the same as in Figure 6, i.e., banning travel of symptomatic travelers from the low prevalence patch to the high prevalence patch only, the disease is eradicated in both regions (Figure 11). This indicates the importance of border control out of a low prevalence patch and into a high prevalence patch. We could also consider a slightly different scenario in which both patches have the same infectivity, but initially one patch is disease free while the other patch has infectives, i.e., E 2 (0) = I 2 (0) = 0 and E 1 (0) + I 1 (0) > 0. However, since the analytical results do not depend on the initial conditions, the resulting simulations are asymptotically similar to Figures During the 2003 SARS outbreak, travel warnings to all affected areas were issued by WHO to prevent travelers from entering and becoming infected. Dozens of countries also issued border control either banning travelers from 23

24 I 1 I 1 and I I t Figure 10: Numerical solution of system (4.1) with two patches showing I 1 and I 2 vs time t. Parameters are the same as in Figure 6 except that m I 21 = m E 21 = I 1 I 1 and I I t Figure 11: Numerical solution of system (4.1) with two patches showing I 1 and I 2 vs time t. Parameters as in Figure 6 except that m I 12 = 0. 24

25 entering or placing them under quarantine. From a simple public health point of view, it is imperative for the low-prevalence region to stop travel of sick (both exposed and asymptomatic) individuals from the high-prevalence region for intervention purposes. See (Ruan et. al., 2006) for a study of a patch model for the spread of SARS. Our results show that border control does not necessarily always have a positive impact on the overall spread of disease and it is more important to ban travel from a low prevalence patch to a high prevalence patch. Moreover, an increase of travel rates in the opposite direction (from a high prevalence patch to a low prevalence patch), while theoretically alleviating the spatial spread of the disease, is not likely an implementable policy. Furthermore, letting m I 12 = m E 12 = 0, i.e., banning all travel from the low prevalence patch to the high prevalence patch, numerical simulations in Figure 12 show R 0 decreasing drastically as m I 21 = m E 21 increase from 0 to 1. Note that in this case the infected equations in patch 1 uncouple, and the matrix W in (4.2) is reducible, thus (1) R 0 = max{ R 0, R (2) 0 }. On the other hand, banning all travel from the high prevalence patch to the low prevalence patch (m I 21 = m E 21 = 0) results in R 0 = R (1) > 1 and the outbreak becomes endemic. On the basis of our model assumptions and the parameters considered, we could therefore conclude that, retrospectively, WHO acted properly with travel warnings for travelers to avoid all but essential travel to affected areas (WHO, 2006b). Moreover, screening at border for travelers in and out is important mainly for the purpose of quick identification of sick individuals. As a final remark, we note again that, while the model is proposed with the spread of human influenza in mind, it also can be used, with some appropriate modifications and parameter changes, as the basis of a theoretical model to study the spread of enzootic diseases such as avian flu (H5N1) among birds, both that of importation of poultry and pet birds, as well as wild migratory bird populations. 25

26 R I E m 21 =m 21 Figure 12: R 0 vs m I 21 = m E 21 for m E 12 = m I 12 = 0. Appendix: Proof of Theorem 4.2 Proof. It is clear that an increase in a decreases each entry in the matrix W, and so decreases the basic reproduction number R 0. Under the parameter assumptions where u = v = W = [ b1 u b 1 v b 2 v b 2 u σ(c + m) c(c + 2m) + α(2m2 + (a + c)m + ac) ac(c + 2m)(a + 2m) σm αm(a + c + 2m) + c(c + 2m) ac(c + 2m)(a + 2m) By (4.3), R 0 is the larger root of the quadratic equation ], λ 2 (b 1 + b 2 )uλ + b 1 b 2 (u 2 v 2 ) = 0. (6.1) It follows from the above expressions that u + v = α + σa, u v = ac 26 α + σ(a + 2m) (c + 2m)(a + 2m),

27 giving u 2 v 2 = (u + v)(v v) = α + σa ac It follows from u + v = α+σa ac respect to m for (6.1) gives α + σ(a + 2m) (c + 2m)(a + 2m). that u = v. Taking partial derivative with m m (2λ (b 1 + b 2 )u) λ m = λ(b 1 + b 2 ) u m b (u 2 v 2 ) 1b 2 m ( ) α + σa u = λ(b 1 + b 2 ) 2b 1 b 2 ac m From the definition of u, u m = α(4m + a + c) + σ(4m2 + 4am + a 2 ) < 0. [(c + 2m)(a + 2m)] 2 Note that 2λ λ=r0 > (b 1 + b 2 )u. Therefore, to show that R 0 m show λ m λ=r 0 < 0, it suffices to show that for λ = R 0, (b 1 + b 2 )λ 2b 1 b 2 α + σa ac We first claim that λ > 1 2 (b 1 + b 2 )(u + v). Since λ = 1 2 it is equivalent to show that > 0. [ (b 1 + b 2 )u + ] (b 1 + b 2 ) 2 u 2 4b 1 b 2 (u 2 v 2 ) (b 1 + b 2 ) 2 u 2 4b 1 b 2 (u 2 v 2 ) > (b 1 + b 2 ) 2 v 2, < 0, i.e., to, or (b 1 b 2 ) 2 (u 2 v 2 ) > 0. This is automatically true since u > v > 0. It follows from λ > 1 2 (b 1+b 2 )(u+ v) that (b 1 + b 2 )λ > 1 2 (b 1 + b 2 ) 2 (u + v) = α + σa (b 1 + b 2 ) 2. ac

28 Thus as required (b 1 + b 2 )λ 2b 1 b 2 α + σa ac 1 2 α + σa (b 1 b 2 ) 2 > 0. ac Acknowledgements. YHH is supported by grant (NSC M ) from the National Science Council of Taiwan. YHH is also grateful to the Canadian government for their generous financial support to fund YHH s visit to University of Victoria under a Faculty Research Award (623-2-FRP ). PvdD is partially supported by NSERC of Canada, MITACS, and LW is supported by PIMS and MITACS PDF fellowships. References Arino, J. and van den Driessche, P., Disease spread in metapopulations. Fields Institute Communications, in: Nonlinear Dynamics and Evolution Equations, 48. Arino, J. and van den Driessche, P., A multi-city epidemic model. Mathematical Population Studies, 10, Arino, J., Jordan,R. and van den Driessche, P., Quarantine in a multispecies epidemic model with spatial dynamics. Math Biosci. Dec 9, [Epub ahead of print]. Berman, A. and Plemmons, R. J., Non-negative Matrices in the Mathematical Sciences. Academic Press, New York. Butler, D., Doubts hang over source of bird flu spread, Nature, 439,

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