CHIN.PHYS.LETT.
Vol. 25, No. 6 (2008) 2296
Emergence of Strange Spatial Pattern in a Spatial Epidemic Model
∗
SUN Gui-Quan(? ), JIN Zhen(@ )∗∗ , LIU Quan-Xing(4,), LI Li(os) Department of Mathematics, North University of China, Taiyuan 030051
(Received 25 February 2008) Pattern formation of a spatial epidemic model with nonlinear incidence rate kI 2 S/(1 + αI 2 ) is investigated. Our results show that strange spatial dynamics, i.e., filament-like pattern, can be obtained by both mathematical analysis and numerical simulation, which are different from the previous results in the spatial epidemic model such as stripe-like or spotted or coexistence of both pattern and so on. The obtained results well extend the finding of pattern formation in the epidemic model and may well explain the distribution of the infected of some epidemic.
PACS: 87. 23. Cc, 82. 40. Ck, 05. 45. Pq Since spatial models can be used to estimate the formation of spatial patterns on the large scale and the transmission velocity of diseases, and in turn guide policy decisions,[1,2] pattern formation of the spatial epidemic model is received more and more attention. Many important epidemiological and ecological phenomena are strongly influenced by spatial heterogeneities because of the localized nature of transmission or other forms of interaction.[3,4] Furthermore, both mathematicians and epidemiologists want to know that how populations diseases transmit in both space and time, which can enhance the understanding of the epidemiological features of diseases in the populations. Thus, spatial epidemic models are more suitable for describing the process of epidemiology. In general, the population, in which a pathogenic agent is active, comprises three subgroups: the healthy individuals who are susceptible (S) to infection, the already infected individuals (I) who can transmit the disease to the healthy ones and the removed (R) who cannot get the disease or transmit it: either they have a natural immunity, or they have recovered from the disease and are immune from getting it again, or they have been placed in isolation, or they have died. We firstly pay attention to the reduced SIRS model, which is as follows:
viduals who lose immunity and return to susceptible class. The incidence rate is that g(I) = kI l /(1 + αI h ) (here l = h = 2), which was proposed by Liu et al.[5] and used by a number of authors, where kI l measures the infection force of the disease and 1/(1 + αI h ) measures the inhibition effect from the behavioural change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. The detail about model (1) can be found in Ref. [6]. Following Ref. [6] with the scaling √ √ kI kR X= , Y = , τ = (d + ν)t, (2) d+ν d+ν
dI kI 2 (N0 − I − R) = − (d + γ)I, dt 1 + αI 2 dR = γI − (d + ν)R, dt
In the present study, we investigate the pattern formation of the spatial model. As a result, the model we employ is as follows:
(1a) (1b)
where N0 is the density of the total population, d is the death rate of the population, γ is the recovery rate of infective individuals, ν is the rate of removed indi-
and still using the variables I,R,t instead of X,Y and τ , we have dI I 2 (A − I − R) = − mI, dt 1 + pI 2 dR = qI − R, dt
(3a) (3b)
where α(d + ν) p= , A = N0 k d+γ γ m= , q= . d+ν d+ν
√
k , d+ν
I 2 (A − I − R) ∂I = − mI + DI ∇2 I, ∂t 1 + pI 2 ∂R = qI − R + DR ∇2 R, ∂t
(4a) (4b)
(5a) (5b)
∗ Supported by the National Natural Science Foundation of China under Grant No 60771026, the Programme for New Century Excellent Talents in University (NCET050271), and the Special Scientific Research Foundation for the Subjects of Doctors in University (20060110005). ∗∗ To whom correspondence should be addressed. Email:
[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd °
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SUN Gui-Quan et al. 2
2
∂ ∂ + 2 is the usual Laplacian operator 2 ∂x ∂y in two-dimensional space, DI and DR are the diffusion coefficients. From the biological point of view, we assume that all the parameters are positive constants. To study patterns of the system given by Eq. (3), we must consider a spatially homogeneous system. Thus we firstly find the positive steady state of the nospatial model. From Ref. [6], we know that (i) There is no positive equilibrium if A2 < 4m(mp + q + 1). (ii) There is one positive equilibrium if A2 = 4m(mp + q + 1). (iii) There are two positive equilibriums if A2 > 4m(mp + q + 1). We focus our attention on the case of A2 = 4m(mp + q + 1). Then we obtain a unique positive equilibrium E ∗ = (I ∗ , R∗ ), where where ∇2 =
A , I∗ = 2(mp + q + 1) R∗ = qI ∗ .
(6a) (6b)
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where trk = a11 + a22 − k 2 (DI + DR ), ∆k = a11 a22 − a12 a21 − k 2 (a11 DR + a22 DI ) + k 4 DI DR .
(8)
The onset of Hopf instability corresponds to the case, when a pair of imaginary eigenvalues cross the real axis from the negative to the positive side. This situation occurs only when the diffusion vanishes. Mathematically speaking, the Hopf bifurcation occurs when Im(λ(k)) 6= 0,
Re(λ(k)) = 0,
at k = 0.
(9)
Im(λ(k)) = 0, Re(λ(k)) = 0, at k = kT 6= 0,
(10)
The Turing bifurcation occurs when
and the wavenumber kT satisfies √ ∆0 . kT2 = DI DR
(11)
We are interested in studying the stability behaviour of the equilibrium point E ∗ following the theory of Turing.[7−11] The Jacobian matrix corresponding to this equilibrium point is that ( J=
a11 a21
a12 a22
) ,
where a11 = (−16m + 4A2 + 24A2 m2 p2 q + 24A2 mpq 2 + 40A2 mpq + 8A2 q 3 + 20A2 q 2 + 16A2 q− A4 p + 8A2 m3 p3 + 20A2 m2 p2 + 16A2 mp
Fig. 1. (Colour online) Dispersion relation for different DI : (a) DI = 0.09, (b) DI = 0.1, (c) DI = 0.15.
− A4 mp2 − 192m3 p2 q − 192m2 pq 2 − 64m2 p − 64mq − 96m3 p2 − 96mq 2 − 64m4 p3 − 64mq 3 − 16mq − 16m p − 192m pq − 64m p q 4
5 4
2
4 3
− 96m3 p2 q 2 − 64m2 pq 3 )/(4m2 p2 + 8mpq a12
+ 8mp + 4q 2 + 8q + 4 + pA2 )2 , ( = −A2 4m2 p2 + 8mpq + 8mp + 4q 2 + 8q )−1 , + 4 + pA2 a21 = q,
a11 + a22 < 0,
(DR a11 + DI a22 )2 > 4DI DR (a11 a22 − a12 a21 ).
a22 = −1.
1( trk ± 2
√
) tr2k − 4∆k ,
a11 a22 − a12 a21 > 0,
DI a22 + DR a11 > 0,
Imposing the zero-flux condition on the boundary, which means that the pattern does not flow out of the region considered, results in solutions with the form (5) only being admissible for a set of discrete values of k, which are called the admissible modes. The temporal growth rate λ can easily be found as a function of the possible modes indicated by k, which is as follows: λ1,2 (k) =
A general linear analysis[12−14] shows that the necessary conditions for yielding Turing patterns are given by
(7)
To well see the effects of the diffusion coefficient, we keep that m = 0.115, p = 0.06, q = 10, A = √ 5.063174, DR = 1.2 and plot in Fig. 1 the dispersion relation corresponding to several values of DI . We assume that the diffusion coefficient of the recovered is larger than that of the infected, which is common in nature. The dynamics behaviour of the spatial epidemic model can not be studied by using analytical methods or normal forms. Thus we must perform simulations by computer. To solve differential equations by computers, one must discretize the space and time of the
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SUN Gui-Quan et al.
problem. That is to say, transform it from an infinitedimensional (continuous) to a finite-dimensional (discrete) form. In practice the continuous problem defined by the reaction-diffusion system in 2D is solved in a discrete domain with M × N lattice sites. The spacing between the lattice points is defined by the lattice constant ∆h. In the discrete system the Laplacian describing diffusion is calculated using finite differences, i.e., the derivatives are approximated by differences over ∆h. For ∆h → 0 the differences approach the derivatives. The time evolution is also discrete, i.e., the time goes in steps of ∆t. The time evolution can be solved by using the Euler method, which means approximating the value of the concentration at the next time step based on the change rate of the concentration at the previous time step.[15] In the present study, we set ∆h = 1.25, ∆t = 0.05 and M = N = 200. All our numerical simulations employ the periodic Neumann (zero-flux) boundary conditions.
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In order to see the spread of the epidemic, we restrict our analysis of pattern formation of the infected. In Fig. 2, we show the evolution of the spatial pattern of infected population at 50000, 60000, 70000, 80000, 87000, 96000, 200000 and 300000 iterations, with small random perturbation of the stationary solution I ∗ and R∗ of the spatially homogeneous systems. As is seen, the random initial distribution leads to the formation of a strongly irregular transient pattern in the domain. After the irregular pattern forms, filament-like pattern emerge [see Fig. 2(f)]. Finally, stationary filament-like patterns [see Figs. 2(g)–2(h)] prevail in the whole domain and do not undergo any change. The obtained patterns are different from the patterns in Ref. [1,2], such as stripe-like or spotted or coexistence of both pattern and so on. Biologically speaking, the epidemic only exists in the local region. Our finding in this study enriches the results in the pattern formation of spatial epidemic model.
Fig. 2. Snapshots of contour pictures of the time evolution of the infected at different instants with the values √ of the parameters m = 0.115, p = 0.06, q = 10, A = 5.063174, DR = 1.2 and DR = 0.1: (a) 50000 iterations, (b) 60000 iterations, (c) 70000 iterations, (d) 80000 iterations, (e) 87000 iterations, (f) 96000 iterations, (g) 200000 iterations, (h) 300000 iterations.
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In summary, we have investigated the pattern formation of the spatial epidemic model and obtained a strange pattern which is not found in the previous works. The application of spatial and temporal analytical techniques to the epidemic model are of immense help in unravelling the pressing scientific and managerial questions faced by those involved in the spread of diseases in the spatial world. However, noise may play an important role on the formation of patterns of the epidemic model,[16] which needs further investigation.
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[4] Hastings A 1990 Ecology 71 426 [5] Liu W M, Hethcote H W and Levin S A 1987 J. Math. Biol. 25 359 [6] Ruan S and Wang W 2003 J. Diff. Equations 188 135 [7] Turing A M 1952 Philos. Trans. R. Soc. London B 237 7 [8] Baurmann M, Gross T and Feudel U 2007 J. Theor. Biol. 245 200 [9] Segel L A and Jackson J L 1972 J. Theor. Biol. 37 545 [10] Pascual M 1993 Proc. R. Soc. London B 251 1 [11] Medvinsky A B, Petrovskii S V, Tikhonova I A, Malchow H and Li B L 2002 SIAM Rev. 44 311 [12] Murray J D 1993 Mathematical Biology 2nd edn (Berlin: Springer) [13] Arag´ on J L, Torres M, Gil D, Barrio R A and Maini P K 2002 Phys. Rev. E 65 051913 [14] Cross M C and Hohenberg P C 1993 Rev. Mod. Phys. 65 851 [15] Leppnen T 2004 PhD Thesis (Helsinki University of Technology, Finland) [16] Dushoff J, Plotkin J B, Levin S A and Earn D J D 2004 Proc. Natl. Acad. Sci. U.S.A. 101 16915
