S-I-R Model with Directed Spatial Diffusion

Transcription

1 S-I-R Model with Directed Spatial Diffusion Fabio A. Milner Ruijun Zhao Department of Mathematics, Purdue Universit, West Lafaette, IN Abstract A S-I-R epidemic model is described in which susceptible individuals move awa from foci of infection, and all individuals move awa from overcrowded regions. It consists of hperbolic partial differential equations, the sum of these equations being parabolic. Positivit and regularit of solutions are discussed and finite time blow-up of some solutions is illustrated through numerical simulations. A numerical test of the finite time blow-up of solutions is proposed. Introduction Kermack and McKendrick (927) provided a mathematical description of the kinetic transmission of an epidemic in an unstructured population. The total population is assumed to be constant and divided into three classes: susceptible, infected (and infective), and removed (recovered with permanent immunit). The propagation of an infection governed b this simple model which incorporates neither age or se structure nor variabilit of infectivit or spatial position gave rise to man models of various tpes of epidemics. Few etensions have dealt with spatial issues. Webb (98) proposed and analzed a model structured b spatial position in a bounded onedimensional environment, [,L], L >. Spatial mobilit is assumed in Webb (98) to be governed b random diffusion with coefficients k and k 2 for the susceptible and infected classes. The infection is assumed to be transmitted from infected to susceptible at a rate per head α>b a mass action contact term, and the infected are assumed to recover at a rate per head γ>. Demographic changes are neglected under the assumption that the duration of the epidemic is short in comparison with the average life span

2 of an individual and that the disease affects neither fertilit nor mortalit. This situation is tpical, for eample, of childhood diseases. Let S = S(, t), I = I(, t), and R = R(, t) denote, respectivel, the densit of susceptible, infected, and removed individuals (recovered with immunit) at location at time t. The epidemic is described b the coupled sstem of parabolic partial differential equations: S t = k S αsi, () I t = k 2 I + αsi γi. The model is completed b homogeneous Neumann boundar conditions representing a closed environment, S (,t)=s (L, t) =I (,t)=i (L, t) =. (2) Webb (98) showed that Eq. ()-(2), completed b appropriate initial conditions, is well-posed and has positive solutions for all time if the initial data are positive. Moreover, in the case k = k 2, Webb (98) showed that the infected class disappears while the susceptible tend to a spatiall uniform stead state. A significant limitation is the fact that the spatial variable is one-dimensional, making the model of limited applicabilit: the spatial displacement of a ground-bound population is two-dimensional, while that of a water- or air-bound population is three-dimensional. Because the stead state is disease-free and spatiall uniform, the spatial structure adds ver little to the dnamics of the model, and nothing at all asmptoticall. Gurtin and MacCam (977) assumed that the susceptible move awa from concentrations of the infected while the disease renders these individuals stationar. This model leads to overcrowding and, possibl, to a finite time blow-up, which biologicall means overcrowding after short time. We shall incorporate spatial mobilit of the susceptible awa from infected regions, and of all individuals awa from overcrowding. We assume no incubation or latenc period for this disease, so that the terms infected and infective are snonms. Whenever such a model ehibits finite time blow-up, its interest and applicabilit is just for short time projection. Otherwise, it ma be useful for long term projections as well. For one spatial dimension the model was first considered b Meade and Milner (99). We shall describe the model in two space dimensions, analze properties of its solution and, in case theoretical results are not available, carr out numerical simulations to show that, for eample, the unrestricted movement awa from infection can lead to catastrophic consequences from overcrowding that 2

3 leads to the impossibilit of providing enough nourishment, housing, health care, and fresh water. Because demograph is ignored, the model can also be applied to ver lethal diseases b interpreting the removed class as those who die from the disease. The model could be applied, for eample, to the bubonic plague that killed almost half of the population of Europe in XIVth centur, with R representing the isolated infected who do not move nor transmit disease because of their isolation. The displacement of susceptible awa from infected areas created regions with a high densit of susceptible who were prone to cause large secondar outbreaks when an infected appeared among them. Refugee displacements in areas of armed conflict constitute a more modern eample of applicabilit of the model, with S representing the nonbattling and I the battling individuals, and R those wounded or otherwise immobilized and non-battling. The displacements of non-battling individuals awa from the battle zones create large accumulations of non-battling individuals along the border of the embattled countr, making them malnourished and prone to illness. 2 A Model with Avoidance of Infection and Overcrowding Meade and Milner (99) described an epidemic model with spatial diffusion designed to overcome the effects of overcrowding and which ma lead to finite time blow-up in the Gurtin-MacCam (977) model. A second diffusion process parameterized b k 2 causes each of the three epidemic classes to disperse avoiding large concentrations of the total population N = N(, t) = S + I + R: S t = k (SN ) + k 2 (SI ) αsi, I t = k (IN ) + αsi γi, (3) R t = k (RN ) + γi. The differential sstem is completed b non-negative initial distributions S (),I (), R (), and no-flu boundar conditions (k N + k 2 I) (,t)=(k N + k 2 I) (L, t) =k N (,t)=k 2 N (L, t) =, (4) to represent a closed environment. 3

4 When k > andk 2 >, Eq. (4) implies that I = N =onthe boundar. However, to show that S =andr = on the boundar requires differential equations and the relation N = S + I + R. The compatibilit of the initial with boundar data is assumed. Some qualitative properties of the solution were given b Meade and Milner (99, 992). Epanding N as S + I + R in Eq. (3) we obtain the equivalent sstem: S t = k S(S + I + R )+k S (S + I + R )+k 2 SI +k 2 S I αsi, I t = k I(S + I + R )+k I (S + I + R )+αsi γi, R t = k R(S + I + R )+k R (S + I + R )+γi, (5) which is, in terms of the unknown vector E =[S, I, R] T : E t = A E + B E + C E, (6) where A = k S (k + k 2 )S k S k I k I k I k R k R k R, B = k (S + I + R )+k 2 I k (S + I + R ) k (S + I + R ) C = αi αs γ γ. The eigenvalues λ of the coefficient matri A satisf where I 3 is the 3 3identitmatri. Simplifing Eq. (7),, det (λi 3 A) =, (7) λ 3 k (S + I + R)λ 2 k k 2 SIλ =. (8) λ = is alwas a root of Eq. (8), and when k = (movement awa from infection but not awa from overcrowding), it is the onl root with 4

5 algebraic multiplicit 3 and geometric multiplicit 2. In this case Sstem (3) is first-order hperbolic and it ma ehibit blow-up in a finite time. For k >, the other two eigenvalues of A are the roots of the quadratic λ 2 k (S + I + R)λ k k 2 SI =, (9) which for k 2, S, I, andr positive has one positive and one negative root. The three distinct eigenvalues allow for a linear change of unknowns. The diagonalization A leads to three equations which are semi-linearl coupled in their diffusion terms. Each corresponds a different behavior: one similar to the heat operator, one similar to the hperbolic operator, and one similar to the backward heat operator. This coupling of different tpes of equations ma lead to numerical instabilities. When k 2 = there is a double eigenvalue and a positive one, leading to a well-behaved sstem analzed b Meade and Milner (992). We etend this model to the more realistic two-dimensional case. Let Ω=[,L ] [,L 2 ](L >,L 2 > ) be the habitat for the population under consideration. For (, ) Ωandt>, S(,, t), I(,, t), R(,, t) satisf S t = k (S N)+k 2 (S I) αsi, I t = k (I N)+αSI γi, () R t = k (R N)+γI, where =(, )T denotes the gradient vector and the scalar product of vectors, so that is the divergence operator. Sstem () is completed b non-negative initial distributions S (, ), I (, ), R (, ) and no-flu boundar conditions: for (, ) Ω, t>, ( ) N k n + k I N 2 (,, t) =k (,, t) =, () n n where n is the outward normal unit vector on Ω. 5

6 Substituting S + I + R for N, Eq. () becomes S t = k S(S + I + R )+k S (S + I + R ) +k S(S + I + R )+k S (S + I + R ) +k 2 SI + k 2 S I + k 2 SI + k 2 S I αsi, I t = k I(S + I + R )+k I (S + I + R ) +k I(S + I + R )+k I (S + I + R )+αsi γi, R t = k R(S + I + R )+k R (S + I + R ) +k R(S + I + R )+k R (S + I + R )+γi. (2) Using the same notation as for the one-dimensional problem, E =[S, I, R] T, we rewrite Sstem (2) in terms of E as E t = AΔ E + B E + B 2 E + C E (3) where A, B,C are the same as before, B 2 is similar to B, with derivatives with respect to replaced b derivatives with respect to. We now prove that solutions of Eq. ()-() are meaningful. Theorem 2. If S,I,R >, ansolution(s, I, R) of Eq. ()-() is positive as long as it eists. Proof. Assume (S, I, R) is a smooth solution. Etending on the technique in Meade and Milner (992) for the porous medium equation, we first compute the solution S along characteristics defined b the following famil of initial value problems: X t (p, t) = (k N (X(p, t),y(q,t),t)+k 2 I (X(p, t),y(q,t),t)), Y t (q,t) = (k N (X(p, t),y(q,t),t)+k 2 I (X(p, t),y(q,t),t)), (4) X(p, ) = p, Y (q,) = q. 6

7 Differentiating X t (p, t),y t (q,t) with respect to p and q respectivel, X pt (p, t) = [k N (X(p, t),y(q,t),t) +k 2 I (X(p, t),y(q,t),t)] X p (p, t), Y qt (q,t) = [k N (X(p, t),y(q,t),t) +k 2 I (X(p, t),y(q,t),t)] Y q (q,t), X p (p, ) =, Y q (q,) =. (5) For each (p, q) Ω, X p (p, ),Y q (q, ) is a solution of a sstem of homogeneous first-order linear ordinar differential equations with positive initial data. Hence X p and Y q are positive. Let S(p, q, t) :=S(X(p, t),y(q,t),t). Then S t (p, q, t) =S t (X(p, t),y(q,t),t)+s (X(p, t),y(q,t),t)x t (p, t) +S (X(p, t),y(q,t),t)y t (q,t) ( Xpt (p, t) = X p (p, t) + Y ) qt(q,t) + αi(x(p, t),y(q,t),t) S(p, q, t), Y q (q,t) (6) and S(p, q, t) = S(p, q, ) = S(X(p, ),Y(q,), ) = S(p, q, ) = S (p, q), (7) ( S (p, q) X p (p, t)y q (q,t) ep α t ) I(X(p, τ),y(q,τ),τ)dτ. (8) As X p,y q and S are positive, the positivit of S follows. The positivit of I and R follows from an analogous argument with the characteristics defined b X t (p, t) = k N (X(p, t),y(q,t),t), Y t (q,t) = k N (X(p, t),y(q,t),t), (9) X(p, ) = p, Y (q,) = q. 7

8 3 Special Case I: Avoidance of infection The model in which the susceptible move in avoidance of infection is similar to the Gurtin-MacCam s (977) model, which is based on the assumption that the susceptible class moves to avoid concentrations of the infected with velocit v S = k 2 I, wherek 2 is a positive constant. The infective have zero velocit v I =. The resulting sstem of PDE is nonlinear and diffusion is degenerate. In the one-dimensional case, Sstem (3) reduces to: S t = k 2 (SI ) αsi, (2) I t = αsi γi, with nonnegative initial distribution S (),I () and no-flu boundar conditions S (,t)=s (L, t) =I (,t)=i (L, t) =. (2) Remark 3. Meade (992) showed that the infected cannot spread out of their original support. The distribution of the other epidemic classes is stationar where the infected are absent. Moreover, if the solution eists at all time, the infected disappears asmptoticall. Also, the susceptible ma accumulate on the boundar of the compact support of the infected, and the solution ma blow up in finite time. We now estimate the rate of blow-up when it occurs in finite time through an eample. Eample : We took the same parameters and initial data for Eq. (2)-(2) as Meade (992) k =,k 2 =., α =,γ =5,S () =6,R () =, and ,., I () = 25 7 (.8 )2,. <.8,.8 <. Meade (992) conjectured that the solution of Eq. (2) ma break down with the formation of a shock b rewriting Eq. (2) as a hperbolic sstem. B tracking the characteristic line of the hperbolic sstem in Eample, Meade (992) estimated that the solution breaks down (characteristic lines cross each other) at =.8 andt =.7. We introduce a numerical test of 8

9 Table : Numerical solution of Eq. (2) using Scheme (38), with point values S at t =.7 andat =.8 /M for different M. Partition number M Numerical solution S Ratio r finite time blow-up of solutions b looking at the numerical solution derived from the numerical scheme provided in Section 5. Numerical solutions of the susceptible at time t =.7 and =.8 /M are listed in Table, where M is the total number of spatial mesh points, S the numerical solution at =.8 /M, r the ratio between two consecutive values for S, corresponding to M and 2M spatial mesh points. From Table, we conjecture that the ratios continue to increase when M increases. For a function of the form /(.8 ) α for some α>, the ratio computed at =.8 /M is r = 2 α, a constant. The solution of susceptible blows up at =.8, indeed faster than /(.8 ) α. This happens because the susceptible cease to move as soon as the escape from the infection, when the reach the first infection-free point ( =.8, the boundar of the compact support of the infective). Remark 3.2 B comparing the ratios, we test whether a solution ehibits blow-up or not at a specific spatial location and a specific instant. Numerical simulations suggest that the blow-up happens where and when a pseudooscillation first occurs, that is at the location and instant for which the numerical solution oscillates no matter how finel time and spatial intervals are divided. In the two-dimensional case with avoidance of infection, Eq. () becomes S t = k 2 (S I) αsi, (22) I t = αsi γi, with nonnegative initial distributions and no-flu boundar conditions, S(,, ) = S (, ), I(,, ) = I (, ), (23) S n (,, t) =, I (,, t) = on Ω. (24) n 9

10 Concerning the eistence of solutions of Eq. (22), we know that global solutions do not eist in general, because in Eample 2 Sstem (22) ehibits finite time blow-up. As for local eistence, we conjecture that the following result holds, though we do not have a proof for it et. Theorem 3. Suppose S > and I > on Ω. Then, there is a T > such that Eq. (22) (24) has a unique solution on Ω [,T). Remark 3.3 If we let ζ(s, t) =e t [αs(,,τ) γ]dτ, it follows from Eq. (22.ii) that I(,,t)=I (, ) ζ(s, t), (25) I = ζ(s, ) I + I ζ(s, ), for an function S(,, t). We rewrite Eq. (22.i) as S t k 2 [ (I ζ(s, )+ζ(s, ) I ) S +2S I ζ(s, ) + I S ζ ζ 2 + ( ΔI α ] ) I Sζ(S, ) = k 2 αi Sζ k 2 This equation is of the form t ΔS(τ) dτ. (26) S t + F( S)+G(S) =H(S) t ΔS(τ) dτ. (27) Even though some eistence results are available for the semi linear case with constant H and no integral in the gradient or Laplacian terms, such as in Smoller (994), nothing is known about the quasi linear case with strong gradient nonlinearities in general. Theorem 3.2 If S = S (, ) and I = I (, ) are a stead state solution to Eq. (22)-(24), then I. Proof. From Eq. (22.ii), I =ors = γ α. If S = γ α, then the reduced boundar-value problem Eq. (22.ii)-(24), ΔI α k 2 I =, in Ω, I n =, on Ω, where Δ = ( ), has the unique solution I. (28)

11 We now show that a finite time blow-up solution ma eist in the twodimensional case through an eample. Eample 2 For Eq. (22)-(24), we take k =,k 2 =., S (, ) =6, R (, ) =, and I (, ) equal to the tensor product of the initial distribution from Eample in and in. Our numerical scheme is capable of capturing shocks because shocks confront to a peak of the solution. Figure shows the spatial distribution and contour plots of the densit of the susceptible at t =,.5,, 4. The susceptible accumulate near the compact support of the initial distribution of infected (.8,) (,.8),,.8. We conjecture that a finite time blow-up solution eists, just as in the one-dimensional case. However, unlike the one-dimensional case, Eq. (22)- (24) cannot be written as a hperbolic sstem. The characteristic method in Meade (992) cannot be applied and we have to run man simulations to locate the pseudo-oscillation to find the blow-up time and position. Remark 3.4 Eample 2 can be viewed as representative of an outbreak of a severe infectious disease appearing in a limited environment such as an island. The susceptible individuals move awa from the region of infection and the infected sta wherever the are located because the are too sick to move. If the infected do not occup the entire habitat at the onset of the epidemic, the susceptible accumulate on the boundar of the support of the initial distribution of infection, because the infected can never spread out of their initial location and eventuall disappear, while the susceptible from the infected region move towards the boundar of the infected region in their attempt to escape infection, but the stop moving as soon as the reach the boundar because there are no infected individuals there from whom to escape. This ma lead to unhealth overcrowding along the boundar of the infected region but it also suggests that this model cannot accuratel represent the dnamics of the epidemic for a long time. 4 Special Case II: Avoidance of overcrowding If k >,k 2 =, diffusion is directed awa from overcrowding but not awa from infection, we then write N = S + I + R, sum all three equations, and obtain the porous media equation in one dimension: N t =( 2 N 2 ) (29)

12 with nonnegative initial distribution N () and no-flu boundar conditions In two dimensions, the resulting equation is N (,t)=n (L, t) =. (3) N t = (N N) (3) with nonnegative initial distribution N (, ) and no-flu boundar conditions N = on Ω. (32) n Meade and Milner (992) established the eistence and uniqueness of the solution to the problem in one dimension under the condition S + I >. Using an identical argument, we obtain the same result in the twodimensional case: Theorem 4. For k > and k 2 =, Eq. ()-() admit a unique solution in Ω [, ) for an initial data S (, ) and I (, ) such that S (, )+ I (, ) > in an point (, ) Ω. Remark 4. Unlike in the case of avoidance of infection, a global solution eists here as long as N >. Moreover, the stead state solution for the total population is spatiall uniform, the infection vanishes, and the susceptible and removed densities converge to their stead states. The net eample shows that the total population can reach its stead state much faster than the epidemic classes do. Eample 3 describes a population with an outbreak of a mild disease that is not smmetric across the line = in the spatial domain, and all individuals move awa from large concentrations of population but not from the infection, which is assumed to be mild. Eample 3 We take k =., k 2 =,S (, ) =6,R (, ) =, and I (, ) = ( 3( )ep ( 75 7 (.8 )2 ep ),., <.8, ),. <.8, <.8,,.8 <,.8. 2

13 The simulation is based on the numerical scheme described in Section 5. Figures 2 and 3 show the densit of the susceptible, the infected, and the total population from the initial distribution to the stead state. The total population N is close to its stead state at time t = 4. However, the subclasses S and I are close to their stead states onl at time t = 5. This is because the equation for N is independent of S and I. At the beginning, the movement force awa from overcrowding governs the spatial flow of all subclasses while, as soon as the total population is close to its stead state, the sstem behaves like the Kermack-McKendrick s (927) model. Remark 4.2 Meade and Milner (992) claimed that the densit at t =4 was near a stead state for Eq. (3)-(4) when k =., k 2 =,α =,γ =5, R =,S =6and I () = ( 2 ) 3ep 2.8 2, <.8,,.8. However, our simulations show that the stead state cannot be reached in such a short time. Figure 4a shows that the stead state of Eq. (3)-(4) in Meade and Milner (992) is reached onl at time t = 8. If the infected invade the entire habitat before the disappear, the asmptotic behavior is governed b the Kermack-McKendrick model, because the total population converges to its stead state in a ver short time. Figure 4b (S =3,other parameters unchanged) shows that the infected disappear before the invade the whole domain, so that onl the susceptible occup the habitat in the region that the infected never reached. 5 Numerical Schemes In Section 2, we noted that Sstems (3) and () behave as hperbolic, forward or backward parabolic, depending on the diffusion coefficients and the sizes of the epidemic classes. In order to appl an appropriate numerical method, we treat the hperbolic and the parabolic cases separatel. For highl virulent epidemic diseases, the dnamics of the spatial displacements is more likel to be dominated b infection. We introduce a numerical method combining a splitting method, Runge- Kutta Discontinuous Galerkin Method (RKDG) (Cockburn and Shu, 2), and the standard Finite Element Method (FEM). As in Section 4, N = S + I + R denotes the total population size. It satisfies the equation N t = k (N N)+k 2 (S I). (33) 3

14 Let S,S,I,I,R, R represent the intermediate steps in the approimations to S, I, R. We solve Eq. (3) and () b approimating N,I,S,R sequentiall appling the standard finite element method to Eq. (33) for N, and splitting the equations for S, I, R into conservation laws and residues. The conservation laws are solved b the Runge-Kutta Discontinuous Galerkin Method, the residues are solved analticall and the integrals approimated b trapezoids. For an time t n, we solve Eq. (3) or () for t n t t n+ using the scheme: N t = k (N N)+k 2 (S n I), t n t t n+, (34) I t = k (I N), t n t t n+, I n = I n, I t = αsi γi, t n t t n+, I n = I n+, I n+ = I n+ ; S t = k (S N)+k 2 (S I), t n t t n+, S n = S n, S t = αsi, t n t t n+, S n = S n+, S n+ = S n+ ; R t = k (R N), t n t t n+, R n = R n, R t = γi, t n t t n+, R n = R n+, (35) (36) (37) R n+ = R n+. Because in the case of avoidance of infection, k =,k 2 >, the sstem is closed in S and I, we solve the first two equations and ignore the third one corresponding to R. The dnamics of infection does not depend on the size or spatial distribution of the immune (removed) class, because the variable 4

15 R appears neither in Eq. (3.a)-(3.b) nor in Eq. (6.a)-(6.b) for S or I. The procedure for this case is: I t = αsi γi, t n t t n+, S t = k 2 (S I), t n t t n+, S n = S n, S t = αsi, t n t t n+, S n = S n+ (38) S n+ = S n+. Remark 5. In the case of avoidance of infection, computing I b differentiating the analtical solution of the ODE and then using the trapezoidal rule for spatial integration leads to numerical oscillations in the vicinit of the region where the solution blows up. In this case, we update I using centered differences of the values of I. We show that our numerical method is consistent to simulate epidemic classes and the total population through Eample 3 (two-dimensional) and the eample in Remark 4.2 (one-dimensional). The discrete L and L 2 norms S + I + R N L, S + I + R N L 2 corresponding to different partitions and different time t are listed in Table 2, 3, 4, 5. InTable2,3,4,5,letM denote the total number of mesh points in each space direction, and let L and L 2 denote S + I + R N L and S + I + R N L 2, respectivel. Tables 2 and 3 show that the method is stable in the sense that the error stas bounded for chosen partition as time increases; Tables 4 and 5 show that the sum of S + I + R converges to N for chosen time as partition refines. Moreover, from Tables 4 and 5, the numerical scheme converges at first order in both L and L 2. Table 2: Numerical errors S + I + R N L, S + I + R N L 2 for the eample in Remark 4.2 (one-dimensional case), with partition M = 32, at t =.25,.25,.5,, 2, 4. Time L L

16 Table 3: Numerical errors S +I +R N L, S +I +R N L 2 for Eample 3 (two-dimensional case), with partition M = 8, at t =.25,.25,.5,, 2, 4. Time L L Table 4: Numerical errors S + I + R N L, S + I + R N L 2 for the eample in Remark 4.2 (one-dimensional case), with partition M = 2, 4, 8, 6, 32, 64, at t =.5. M L L Table 5: Numerical errors S +I +R N L, S +I +R N L 2 for Eample 3 (two-dimensional case), with partition M = 2, 4, 8, at t =.25. M L L Remark 5.2 Figure 5 shows numerical results of Eample generated from the numerical scheme provided in this section. It has a great coincidence with the numerical simulations in Meade and Milner (99), which shows that our scheme is just as reliable. Avoidance of overcrowding is introduced in Curtin-MacCam s (977) model so that the sstem does not blow up in finite time. Eample 4 We compare the numerical solutions of Eq. (3)-(4), with parameter values k =,k 2 =., with k =.,k 2 =. andtherestare thesameasinremark4.2. Figure 6 displas side-b-side the densit of the susceptible at time t =.22 for k 2 =., with k =andwithk =.. When the susceptible move awa from a local infection (I vanishes in part of Ω), the accumulate at the interior boundar of the infected region and produce a finite time blow-up (the maimum value of S(.8,.22) grows unbounded as the spacetime mesh is refined). In contrast, even with a little movement awa from overcrowding, the blow-up stops (S(.8,.22) remains bounded as the mesh 6

17 is refined). The numerical method develops oscillations which prevent the simulation from proceeding further when the mesh is too fine. 6 Conclusion We have partiall analzed an epidemic model of S-I-R tpe in which susceptible individuals move awa from infection and from crowded areas. This model etends the spatiall one-dimensional model (Meade and Milner, 99, 992) to two spatial dimensions. The theoretical results and numerical simulations for the model with movement awa from infection highlight that it ma lead to overcrowding. Limiting such displacements b avoiding overcrowded regions increases the number of infected individuals through the course of the epidemic. When the susceptible move awa from infection but not awa from overcrowded areas, numerical simulations show that the solution can blow up in finite time, both in one and two space dimensions. We provide a qualitative description of the tpe of blow up, as faster than /(.8 ) α for the special case presented. Unlike the case of avoidance of infection, a global solution eists when there is movement awa from overcrowding, as long as N >. Moreover, the stead state solution for the total population is spatiall uniform, the infection disappears, and the susceptible and removed densities converge to their stead states, which are not uniform in space. These results were alread established in the case of one space dimension (Meade and Milner, 992). Numerical simulations for the case of avoidance of overcrowding, k 2 =, indicate that the total population ma reach its stead state much faster than the subclasses. Initiall the movement force awa from overcrowding governs the spatial flow of all subclasses but soon, when the total population is close to its stead state, the sstem behaves according to Kermack- McKendrick because the gradient terms have vanished. A local infection can move out from its original location and become global. The infected ma spread out to the whole habitat before disappearing. The susceptible and recovered individuals coeist everwhere asmptoticall. Depending on the initial ratio of susceptible to infected, the infected ma also disappear before the spread out to the whole domain so that onl susceptible fill the habitat in some region once the infected disappear. Numerical simulations for this case also indicate that susceptible and recovered converge to a given distribution in the region outside of the orig- 7

18 inal compact support of the infective, who disappear asmptoticall. These stead state values depend on the infectivit and recover rates, as well as on the initial distribution of susceptible. The numerical method we described and used for the case of avoidance of overcrowding is consistent in the sense that the numerical approimations of N and of S +I +R are close to one another. The sum of the approimated values for the three subclasses, S + I + R, converges to the approimated value for the total population, N, for all time. We do not have a proof for this fact but it seems to hold for simulations over a wide parameter range. The problem of eistence of solutions for the model is a ver difficult one and remains open. References Cockburn, B. and Shu, C.-W. (2). Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 6: Gurtin, M. E. and MacCam, R. C. (977). On the diffusion of biological populations. Math.Biosc.33: Kermack, O. and McKendrick, A. G. (927). A contribution to the mathematical theor of epidemics. Proc. Roal. Soc. London A 5: Meade, D. B. (992). Qualitative analsis for an epidemic model with diffusion to avoid infection. IMA Preprint Series 96. Meade, D. B. and Milner, F. A. (992). S-I-R epidemic models with directed diffusion. Mathematical Aspects of Human Diseases, Giuseppe Da Prato (Ed.), Applied Mathematics Monographs 3. Pisa: Giardini Editori. Meade, D. B. and Milner, F. A. (99). An S-I-R model for epidemics with diffusion to avoid infection and overcrowding. Proceedings of the 3th IMACS World Congress on Computation and Applied Mathematics v.3: Smoller, J. (994). Shock waves and reaction-diffusion equations (second edition). New York: Springer Verlag. 8

19 Webb, G. F. (98). A reaction-diffusion model for a deterministic diffusion epidemic. J. Math. Anal. Appl. 84:

20 a. t= b. t= c. t= d. t= a. t= b. t= c. t= d. t= Figure : plots (top four) and contour plots (bottom four) of susceptible for Eample 2. 2

21 a. t= b. t= c. t=4 d. t= a. t= b. t= c. t=4 d. t= Figure 2: plots of susceptible (top four) and infected (bottom four) in Eample 3. 2

22 a. t= b. t= c. t= d. t= Figure 3: plot of total population in Eample 3. 9 a. S =6 S I R N 7 6 b. S =3 S I R N Figure 4: Distribution of total population and subclasses at stead states corresponding to different initial data. 22

23 4 Susceptible population 35 3 t= t=.2 t=.4 t=.6 t=.7 25 (S) Position() Infective population t= t=.2 t=.4 t=.6 t=.7 (I) Position() Figure 5: Distribution of susceptible (top) and infected (bottom) for infection with compact initial support and diffusion to avoid infection (k =,k 2 =.), as Eample. 23

24 2 a. k =, N=2 a2. k =., N= b. k =, N=4 3.5 b2. k =., N= c. k =, N=8 5 c2. k =., N= d. k =, N=6.5 d2. k =., N= Figure 6: Comparison of the effect of avoidance of infection and avoidance of both infection and overcrowding. Left column is the densit of susceptible for k = and right column for k =. at t =