Analysis of an S-I-R Model of Epidemics with Directed Spatial Diffusion

Analysis of an S-I-R Model of Epidemics with Directed Spatial Diffusion F.A.Milner R.Zhao Department of Mathematics, Purdue University, West Lafayette, IN 797-7 Abstract An S-I-R epidemic model is described in which susceptible individuals move away from foci of infection, and all individuals move away from overcrowded regions. It consists of hyperbolic partial differential equations, while their sum is parabolic. Some qualitative properties of solutions are discussed and illustrated through numerical examples. A new numerical approach to test the finite time blow-up of solutions is proposed. Introduction The Kermack-McKendrick model is the first one to provide a mathematical description of the kinetic transmission of an epidemic in an unstructured population [3]. In this model the total population is assumed to be constant and divided into three classes: susceptible, infected (and infective), and removed (recovered with permanent immunity). The propagation of an infection governed by this simple model that does not incorporate any structure due to age, sex, variability of infectivity, or spatial position is well-known and opened the door to the involvement of many scientists in the development of a wide range of better models to describe the evolution of various types of epidemics. Several extensions of the model have been considered, focusing on better describing many specific features of the disease kinetics, though few have focused on spatial issues. Webb [8] proposed and analyzed a model structured by spatial position in a bounded one-dimensional environment, [, L], L >. The spatial mobility is assumed in that paper to be governed by random diffusion with coefficients k and k for the susceptible and infected classes, respectively. The infection is assumed to be transmitted from infected to susceptible individuals at a per capita rate α > by a mass action contact term, and the infected are assumed to recover at a per capita rate γ >. Demographic changes are neglected under the assumption that the duration of the epidemic is short in comparison with the average life span of an individual and that the disease does

not affect fertility or mortality. This situation is typical, for example, of childhood diseases. Let S = S(x, t), I = I(x, t), and R = R(x, t) denote, respectively, the density of susceptible, infected, and removed individuals (recovered with immunity) at location x at time t. Then the evolution of the epidemic is described by the following coupled system of parabolic partial differential equations: S t = k S xx αsi, () I t = k I xx + αsi γi. All individuals are prevented from escaping the kinetics, i.e. the model is completed with homogeneous Neumann boundary conditions representing a closed environment, S x (, t) = S x (L, t) = I x (, t) = I x (L, t) =. () It was shown in [8] that () (), complemented with 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, Webb has shown that the infected class dies out while the susceptibles tend to a spatially uniform steady-state. A significant limitation of this model is the fact that the spatial variable is onedimensional, making the model of limited applicability. Moreover, the fact that the steady state is disease-free and spatially uniform leads to the conclusion that the spatial structure adds very little to the dynamics of the model, and nothing at all asymptotically. A different extension of the Kermack-McKendrick model was given by Gurtin and MacCamy []. For their model the susceptibles are assumed to move away from concentrations of infected while the disease renders these stationary. This model leads to overcrowding and, possibly, to finite time blow-up. We shall consider a model that incorporates spatial mobility of susceptibles away from infected regions, and of all individuals away from overcrowding. We assume in this paper that there is no incubation or latency period for this disease, so that the terms infected and infective will be used as synonyms. For one spatial dimension the model was first considered in []. We shall describe the model in two space dimensions, analyze some properties of its solution, and carry out numerical simulations that bring to light important epidemiological issues. For example, the unqualified movement away from infection can lead to catastrophic consequences of overcrowding accompanied by the impossibility of providing enough nourishment, housing, health care, and fresh water as sadly seen in several massive population displacements in avoidance of war. Since demography is not included, the model can also be applied to very lethal diseases by interpreting the removed class as those who die from the disease. When movement in avoidance of overcrowding is also in place, many of the problems just described are no longer so important, but the decreased movement away from infection may lead to increased morbidity in the long run. The present paper is structured as follows: in the next section we present our model with avoidance of infection and overcrowding; in section 3 we consider the special case of susceptibles avoiding just infection and in section that of

susceptibles avoiding only overcrowding. In section 5 we present the results of some numerical simulations, and in section we make some final remarks. A Model with Avoidance of Infection and Overcrowding In [] Meade and Milner described an epidemic model with spatial diffusion designed to overcome the effects of overcrowding that may lead to finite time blow-up in the Gurtin-MacCamy model. A second diffusion process is introduced that causes each of the three epidemic classes to disperse avoiding large concentrations of the total population N = N(x, t) = S + I + R: S t = k (SN x ) x + k (SI x ) x αsi, I t = k (IN x ) x + αsi γi, (3) = k (RN x ) x + γi. R t The differential system is complemented with non-negative initial distributions S (x), I (x), R (x), and no-flux boundary conditions (k N + k I) x (, t) = (k N + k I) x (L, t) = k N x (, t) = k N x (L, t) =, () to represent a closed environment. When k > and k > the boundary conditions clearly imply that I x = N x = on the boundary. However, in order to show that S x = and R x = on the boundary requires the use of the differential equations and the relation N = S + I + R. Compatibility of the initial and boundary data is also assumed. Some qualitative properties of the solution were given in [] []. Expanding N as S + I + R in (3) we obtain the equivalent system: S t = k S(S xx + I xx + R xx ) + k S x (S x + I x + R x ) + k SI xx + k S x I x αsi, I t = k I(S xx + I xx + R xx ) + k I x (S x + I x + R x ) + αsi γi, R t = k R(S xx + I xx + R xx ) + k R x (S x + I x + R x ) + γi. In order to understand the behavior of solutions of this system, let us rewrite it in abstract general form in terms of the vector unknown E = [S, I, R] T : E t = A E xx + B Ex + C E, (5) where A = k S (k + k )S k S k I k I k I k R k R k R, 3

B = k (S x + I x + R x ) + k I x k (S x + I x + R x ) k (S x + I x + R x ) C = αi αs γ γ Let us compute the eigenvalues of the coefficient matrix A that is, solve for λ the equation det (λi 3 A) =, where I 3 is the 3 3 identity matrix. We have. (k S λ)(k I λ)(k R λ) + k 3 SIR + k (k + k )SIR k SR(k I λ) k IR(k S λ) k (k + k )SI(k R λ) =, and, combining similar terms, λ 3 k (S + I + R)λ k k SIλ =. Note that λ = is always a root of this equation, and when k = (movement only away from infection but not away from overcrowding), it is the only root with algebraic multiplicity 3 and geometric multiplicity. In this case the behavior of the system is essentially that of a first-order hyperbolic one that, in particular, may exhibit finite time blow-up. For k >=, the other two eigenvalues of A are the roots of the quadratic λ k (S + I + R)λ k k SI =, which for k, S, I, and R positive has one positive and one negative root. The three distinct eigenvalues allow for a linear change of unknowns that diagonalizes A leading to three equations that are semi-linearly coupled in their diffusion terms, and each will drive a different behavior: one similar to that of the heat operator, one similar to a hyperbolic operator, and another one similar to the backward heat operator. This coupling of so different types of equations leads in some cases to numerical instabilities. When k = there is a double eigenvalue and a positive one, leading to a well-behaved system analyzed in [5]. In the present work we extend this model to the much more realistic twodimensional case. Let Ω = [, L ] [, L ](L >, L > ) be the habitat for the population under consideration. For (x, y) Ω and t >, S(x, y, t), I(x, y, t), R(x, y, t) satisfy S t = k (S N) + k (S I) αsi, I t = k (I N) + αsi γi, () R t = k (R N) + γi.,

System () is complemented by non-negative initial distributions S (x, y), I (x, y), R (x, y) and no-flux boundary conditions: for (x, y) Ω, t >, ( ) N k n + k I N (x, y, t) = k (x, y, t) =, (7) n n where n is the outward normal unit vector on Ω. When we substitute S + I + R for N, () becomes S t = k S(S xx + I xx + R xx ) + k S x (S x + I x + R x ) + k SI xx + k S x I x +k S(S yy + I yy + R yy ) + k S y (S y + I y + R y ) + k SI yy + k S y I y αsi, I t = k I(S xx + I xx + R xx ) + k I x (S x + I x + R x ) +k I(S yy + I yy + R yy ) + k I y (S y + I y + R y ) + αsi γi, R t = k R(S xx + I xx + R xx ) + k R x (S x + I x + R x ) +k R(S yy + I yy + R yy ) + k R y (S y + I y + R y ) + γi. Using the same notation as for the one-dimensional problem, E = [S, I, R] T, we can rewrite the above system in terms of E as E t = A E + B Ex + B Ey + C E (8) where A, B, C are the same as before, while B is similar to B, with derivatives with respect to x replaced by derivatives with respect to y. We shall now prove that solutions of ()-(7) are physically meaningful. Theorem. Assume S, I, R >. Then, any solution (S, I, R) of ()-(7) is positive as long as it exists. Proof. Assume (S, I, R) is a smooth solution. Generalizing ideas presented in [5] for the porous medium equation, we first compute the solution S along characteristics defined by the following family of initial value problems: X t (p, t) = (k N x (X(p, t), Y (q, t), t) + k I x (X(p, t), Y (q, t), t)), Y t (q, t) = (k N y (X(p, t), Y (q, t), t) + k I y (X(p, t), Y (q, t), t)), X(p, ) = p, Y (q, ) = q. Differentiating X t (p, t), Y t (q, t) with respect to p and q respectively, X pt (p, t) = [k N xx (X(p, t), Y (q, t), t) + k I xx (X(p, t), Y (q, t), t)] X p (p, t), Y qt (q, t) = [k N yy (X(p, t), Y (q, t), t) + k I yy (X(p, t), Y (q, t), t)] Y q (q, t), X p (p, ) =, Y q (q, ) =. 5

It is thus apparent that, for each (p, q) Ω, X p (p, ), Y q (q, ) is a solution of a system of homogeneous first-order linear ordinary 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 x (X(p, t), Y (q, t), t)x t (p, t) +S y (X(p, t), Y (q, t), t)y t (q, t) + S t (X(p, t), Y (q, t), 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) Note that S(p, q, ) = S(X(p, ), Y (q, ), ) = S(p, q, ) = S (p, q) and, consequently, S(p, q, t) = ( S (p, q) X p (p, t)y q (q, t) exp α t ) I(X(p, τ), Y (q, τ), τ)dτ. Since X p, Y q and S are positive, the positivity of S follows immediately. The positivity of I and R follows from an analogous argument with the characteristics defined by X t (p, t) = k N x (X(p, t), Y (q, t), t), Y t (q, t) = k N y (X(p, t), Y (q, t), t), X(p, ) = p, Y (q, ) = q. 3 Special Case I: Avoidance of infection only The model in which susceptibles move in avoidance of infection only is similar to the Gurtin-MacCamy model, which is based on the assumption that the susceptible class moves to avoid concentrations of infectives with velocity v S = k I, where k is a positive constant. The infectives have zero velocity v I =. The resulting system of PDE s is nonlinear and it has degenerate diffusion. In the one-dimensional case, the system reduces to: S t = k (SI x ) x αsi, I t = αsi γi. with nonnegative initial distribution S (x), I (x) and no-flux boundary conditions S x (, t) = S x (L, t) = I x (, t) = I x (L, t) =.

Remark 3. It was shown in [] that infectives cannot spread out of their original support. The distribution of the other epidemic classes is stationary where infectives are absent. Moreover, if the solution exists for all time, the infected asymptotically die out. Also, the susceptibles may accumulate on the boundary of the compact support of the infected, and the solution may blow up in finite time. We present now a numerical way to estimate the rate of blow-up in the case it occurs in a finite time. Example : Figure below reproduces the results in [] and [] for the one-dimensional model that exhibits finite time blow-up at a time t estimated by determining when characteristic curves first intersect. Thus we took for this simulation the same parameters and initial data as in those papers, k =, k =., α =, γ = 5, S (x) =, R (x) = and 5x, x., 5 I (x) = 7 (.8 x),. < x.8,.8 < x. 35 3 5 Susceptible population evolution t= t=. t=. t=. t=.7 3.5 Infective population evolution t= t=. t=. t=. t=.7.5 5 5.5....8....8 Susceptible population at t=.7 N= N= N=8.5.5. Infective population at t=.7 N= N= N=8.35 8.3.5..5..5....8....8 Figure : Finite time blow-up of susceptibles in one dimension According to the numerical simulation based on the method of characteristics in [], we estimate that the blow-up occurs at t =.7. We compute the 7

approximation of the solution for the susceptibles at x =.8 M, where.8 is assumed to be the blow-up position (i.e. the internal boundary of the support of infection). In the following table we let M be the number of spatial mesh points, S be the solution at x =.8 M, r be the ratio between two consecutive values for S, corresponding to M and M spatial mesh points. M 8 3 8 S 7..9 3.3897 9.8799 73.35 9.559.533 r.59.3.5.7.89.559 In view of this table, it is reasonable to conjecture that the ratios increase when we increase M. As we can immediately see, the ratio for a function of the form (.8 x) for some α > computed in the same way as above keeps α r = α constant. Therefore, the solution of susceptibles blows up at x =.8, indeed faster than (.8 x). This happens because the susceptibles cease to α move as soon as they escape from the infection, i.e. as soon as they reach the first infection-free point (x =.8, the boundary of the compact support of infectives). Remark 3. By comparing the ratios, we are effectively introducing an efficient way to test whether a solution exhibits 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 pseudo-oscillation first occurs, that is at the location and instant for which the numerical solution develops oscillations no matter how finely the time and spatial intervals are divided. In the two-dimensional case with avoidance of infection only, system () becomes S t = k (S I) αsi, (9) I t = αsi γi, with nonnegative initial distributions S(x, y, ) = S (x, y), and no-flux boundary conditions, I(x, y, ) = I (x, y), () S n (x, y, t) =, I (x, y, t) = on Ω. () n Concerning the existence of solutions of (9), we know that global solutions do not exist in general, since we can see from numerical examples that the system may exhibit finite time blow-up. As for local existence, we conjecture the following result holds, though we do not have a proof for it at present. 8

Theorem 3. Suppose S > and I > on Ω. Then, there is a T > such that the problem (9) () has a unique solution on Ω [, T ). Remark 3.3 If we let ζ(s, t) = e R t [αs(,,τ) γ]dτ. It follows from (9.ii) that I(,, t) = I (, ) ζ(s, t), I = ζ(s, ) I + I ζ(s, ), for any function S(x, y, t). We can now rewrite (9.i) as S t k [ (I ζ(s, ) + ζ(s, ) I ) S + S I ζ(s, ) + I S ζ ζ + ( I α k I ) S ζ(s, ) ] = k α I S ζ t S(τ) dτ. () This equation is of the form S t + F( S) + G(S) = H(S) t S(τ) dτ. Even though some existence results are available for the semi linear case with constant H and no integrals in the gradient or Laplacian terms, e.g. [7], nothing is known about the quasi linear case with strong gradient nonlinearities in general. Theorem 3. If S = S (x, y) and I = I (x, y) are a steady-state solution to (9)-(), then I. Proof. From (9.ii), I = or S = γ α. If S = γ α, then the reduced boundaryvalue problem (9.ii)-(), I α k I =, in Ω, I n =, has the unique solution I. on Ω, Remark 3. Numerical simulations, see Example 5, show that the dynamics in the two-dimensional case is much richer than in the one-dimensional. Taking the initial data for the infected as the tensor product of that in the onedimensional case with itself and the other coefficients to be same, the susceptibles move towards the boundary of the compact support of the infectives and blow up starting from the outer boundary and moving towards the inside. 9

Special Case II: Avoidance of overcrowding only If k >, k =, there is diffusion directed away from overcrowding but not away from infection, we then write N = S + I + R, sum all three equations and obtain with the following porous media equation in one dimension: N t = ( N ) xx (3) with nonnegative initial distribution N (x) and no-flux boundary conditions In two dimensions, the resulting equation is N x (, t) = N x (L, t) =. () N t = (N N) (5) with nonnegative initial distribution N (x, y) and no-flux boundary conditions N n = on Ω. () In [5] the existence and uniqueness of the solution to the problem in one dimension was established under the condition S + I >. Using an identical argument, we can obtain the same result for the two-dimensional case: Theorem. Let k > and k = in ()-(7). Then, the system admits a unique solution in Ω [, ) for any initial data such that S (x, y)+i (x, y) >. Remark. Unlike the case of avoidance of infection only, a global solution exists here as long as N >. Moreover, the steady-state solution for the total population is a constant (spatially uniform), the infection dies out, and the susceptible and removed densities approach their steady-states. Remark. In numerical simulations it transpires that the total population reaches its steady-state much faster than the subclasses. In the initial evolution the movement of the total population governs the flow of all subclasses, while after the total population is close to its steady-state, the system behaves like the Kermack-McKendrick model [3]. Remark.3 In [] and [] it was claimed that the density at t= was near steady-state. However, in our simulations we found that the steady-state can t be reached in such short time and the susceptibles and the removed approach a nearly constant asymptotic distribution in the region outside of the original compact support of the infectives who asymptotically die out. Moreover, the steady-state constants depend on the relation between infectivity rate, recovery rate and the local density of the initial susceptible population.

9 Population density distribution at t=8 S I R N 7 Population density distribution at t=8 S I R N 8 5 7 5 3 3....8....8 Figure : Spatial density of N, S, I, and R at steady state. Remark. Numerical simulations show that the infected can spread out of their original support, which is impossible in the case of avoidance of infection only. Figure a (left) shows that the infected spread out to the whole habitat before they die out and the susceptible and removed coexist everywhere asymptotically. Figure b (right) shows that the infected die out before they reach the whole domain so that only the susceptible fill in the habitat in some region. 5 Numerical Examples In Section, we noted that the systems (3) and () can behave as hyperbolic, and forward or backward parabolic, depending on the relative sizes of the diffusion coefficients and the sizes of the epidemic classes. In order to apply an appropriate numerical method, we treat the hyperbolic and the parabolic cases separately. For highly virulent epidemic diseases, the dynamics of the spatial displacements is more likely to be infection dominated. We introduce a numerical method that combines a splitting method, Runge- Kutta Discontinuous Galerkin Method (RKDG) [], and the standard Finite Element Method (FEM). Just as in Section, we let N = S + I + R denote the total population size. It satisfies the equation N t = k (N N) + k (S I) where the gradient vector = x is just first order differentiation in one dimension, and = ( x, y )T is the vector of first order partial derivatives in two dimensions. In all simulations, the per capita infectivity rate is α = and the per capita recovery rate is γ = 5. The habitat is taken as the unit interval in the onedimensional case and as the unit square in the two-dimensional case. To be more specific about the nature of the numerical method we propose and utilize, let S, S, I, I, R, R represent the intermediate steps in the approximations to S, I, R. We solve systems (3) and () by approximating N, I, S, R sequentially applying the standard finite element method to the equation for N, and

splitting the equations for S, I, R into conservation laws and residues. While the conservation laws are solved by the Runge-Kutta Discontinuous Galerkin Method, the residues can be first solved for analytically and then the necessary integrals approximated using the trapezoidal rule. For any time t n, we solve (3) (or ()) for t n t t n+ using the following scheme: N t = k (N N) + k (S I), t n t t n+, 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 (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+, R n+ = R n+. Since in the case of avoidance of infection only, k =, k >, the system is closed in S and I, we only solve the first two equations and ignore the third one (corresponding to R). The dynamics of infection in this case does not depend on the size or spatial distribution of the immune (removed) class, since the variable R does not appear in the equations for S or I. Thus, the procedure for this case is as follows: I t = αsi γi, t n t t n+ ; S t = k (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+.

Example This example shows that our numerical method is consistent in how it simulates the epidemic classes and the total population. We use the data from [5], k =., k =, R (x) =, and ( x ) 3 exp I (x) = x.8, x <.8,,.8 x. However, in order to investigate the steady state when the infected die out before they occupy the whole habitat we make a change to the initial distribution of susceptibles and take S (x) = 3 and keep all other parameters the same. Let M denote the number of mesh points in each space direction, and let L and L in the tables below denote the magnitude of the errors in total population in L and L, that is S + I + R N L, S + I + R N L. Table shows that the sum S + I + R converges to N; Table shows that the method is stable in the sense that the error stays bounded as time increases. M 8 3 L.973.599.83.5.7.79 L.3938.79.9557.9.95.5 Table : Errors in the one-dimensional case as a function of M at t=.5 We see in this table numerical convergence of first order in both L and L. Time.5.5.5 L..93.7.8.8.8 L.97.3.95.53.533.533 Table : Errors in the one-dimensional case as time increases, M = 3 In our model, avoidance of overcrowding is introduced so that the system does not blow up in finite time. To see that it accomplishes this goal we present the results of a simulation with the same initial distributions from Example, while the diffusion coefficients are taken as k =., k =.. Example 3 As in Remark 3., we find the approximate blow-up time t =. for the coefficients taken as k =, k =.. Figure 3 displays side-by-side the density of susceptibles at time t =. for k =., with k = and with k =.. We see that when susceptibles only move away from a local infection (i.e. I vanishes in some part of Ω with positive area), they accumulate at the boundary of the support of the infected region and produce a finite time blow-up (the maximum value of S(.8,.) grows unbounded as the space-time mesh is refined). In contrast, when there is even very minor movement away from overcrowding, the blow-up stops (S(.8,.) remains bounded as the mesh is refined). We also see that the numerical method develops oscillations that impede the prolongation of the simulation when the mesh is too fine. 3

Density of Susceptible Population at t=. 9 Density of Susceptible Population at t=. 8 7 8 5 3....8 k =, N=....8 k =., N= 3 Density of Susceptible Population at t=. Density of Susceptible Population at t=. 5 8 5 5....8....8 k =, N= k =., N= Density of Susceptible Population at t=. Density of Susceptible Population at t=. 5 3 8....8....8 k =, N=8 k =., N=8 9 Density of Susceptible Population at t=. 5 Density of Susceptible Population at t=. 8 7 5 3 5....8 k =, N=....8 k =., N= Figure 3: Comparison between the effects of avoidance of overcrowding only and avoidance of both infection and overcrowding

We show next the results of a simulation for a two-dimensional population with an outbreak of a mild disease that is not symmetric in x and y. All epidemic classes move away from large concentrations of population but not from the infection, since it is mild. Example We take k =., k =, S (x, y) =, R (x, y) =, and I (x, y) equal to the product of the initial distribution from Example in x and that from Example in y. Figures 8 below show the evolution of the spatial distributions of the total population, as well as those of susceptibles and infected individuals. Density of Total Population at t= Density of Total Population at t=.5 Density(N) 9 8 7 Density(N) 7.95.9.85.8.75.5....8.5....8 Density of Total Population at t= Density of Total Population at t=.89.89.89.89 Density(N).89.89 Density(N).89.89.89.89.89.89.5....8.5....8 Figure : Spatial density of N at selected times (k =., k = ) Density of Susceptible Population at t= Density of Infective Population at t= 7.5 5.5 5 3 5.8.5..5......8 Figure 5: Initial distribution of S and I (k =., k = ) 5

Density of Susceptible Population at t=.5 Density of Infective Population at t=.5 7 5 3.8....5....8.5....8 Density of Susceptible Population at t= Density of Infective Population at t= 8.5..3...5....8.5....8 Density of Susceptible Population at t= Density of Infective Population at t= 8..3...5....8.5....8 Density of Susceptible Population at t= Density of Infective Population at t= 8.3....5....8.5....8 Figure : Distribution of the susceptible and infected classes at selected times, from early on to nearly steady-state, when susceptible and recovered ultimately coexist in the whole habitat (k =., k = )

Density of Susceptible Population at t=5 Density of Infective Population at t=5 x 3.5 3.5.5.5.5.5.5.5....8.5....8 Figure 7: Distribution of the susceptibles and infected at steady-state, with susceptibles and recovered coexisting in the whole habitat (k =., k = ) Contour Plof of Density of Susceptible Population at t=5 Contour Plof of Density of Infective Population at t=.9.8.7.9.8.7..5...5..3...3.....3..5..7.8.9...3..5..7.8.9 Contour Plof of Density of Infective Population at t= Contour Plof of Density of Infective Population at t=.9.8.7.9.8.7..5...5..3...3.....3..5..7.8.9...3..5..7.8.9 Figure 8: Contour plots of susceptible and infected classes at selected times, when infection spreads to the whole habitat before dying out (k =., k = )) 7

Just as in Example, let M denote the number of mesh points in each space direction, and let L and L in the tables below denote the magnitude of the errors in total population in L and L, that is S + I + R N L, S + I + R N L. Table 3 shows that the sum S + I + R converges to N; Table shows that the numerical method is stable in the sense that the error stays bounded as time increases. M 8 L.958.999.53 L..37.7 Table 3: Errors in the two-dimensional case as a function of M at t=.5 Time.5.5.5 L.87.53.575.57.587.599 L.7.7.888.8.887.857 Table : Errors in the two-dimensional case as time increases, M = 8 Example 5 Here we show the results of a simulation that is symmetric in x and y for an outbreak of a severe disease appearing in a two-dimensional environment. Susceptible individuals move away from the region of infection. We take k =, k =., S (x, y) =, R (x, y) =, and I (x, y) equal to the tensor product of the initial distribution from Example in x and in y. Figures 9 and at the end of the article show the evolution of the spatial distributions of susceptible and infected individuals. It is interesting to see that the accumulation and eventual finite time blow-up of susceptibles at the interior boundary of the support of the infected class, (.8, y) (x,.8), x, y.8, starts where the interior boundary meets the outer boundary Ω, that is at (.8, ) and (,.8), and then it moves inward along the interior boundary towards the interior vertex of the support of the infected, (.8,.8). Remark 5. In the case of avoidance of infection only, computing I by differentiating the analytical solution of the ODE and then using the trapezoidal rule for integration leads to numerical oscillations in the vicinity of the region where the solution blows up. In this case, we need to update I by central differences of the values of I. Conclusions We have described and partially analyzed an epidemic model of S-I-R type in which susceptible individuals move away from infection and from overcrowded areas. This model generalizes to two spatial dimensions the ones proposed for one spatial dimension in [, 5, ]. 8

The theoretical results and numerical examples provided bring to light some important consequences of indiscriminate escape from infected regions: it may lead to severe overcrowding with possible catastrophic consequences. On the other hand, limiting such displacements by avoiding overcrowded regions increases the number of infected individuals through the course of the epidemic. If we were to assign a cost to the ills of overcrowding and another to those of disease, then an epidemiologically useful mathematical optimization problem would be that of finding the optimal ratio between the diffusion coefficients that minimizes the sum of both costs. This problem will be addressed elsewhere. The terms used in the model to avoid overcrowding may actually not very good when using Dirichlet boundary conditions, as they would force everyone to leave the domain by moving away from any concentration of individuals, however small its density. A better way of modeling movement away from large concentrations is by putting a lower bound on the population density form which individuals will move away. For example, one could use in the equation for the susceptibles [S (N k) + ], for some fixed k > representing the minimum density individuals will move away from, with similar terms in the equations for the remaining epidemic classes. When susceptibles move away from infection only but not away from overcrowded areas, numerical simulations show that the solution may blow up in finite time, both in one and two space dimensions. We provide a qualitative description of the type of blow up for some special cases. Unlike the case of avoidance of infection only, a global solution exists when there is movement away from overcrowding only, as long as N >. Moreover, the steady-state solution for the total population is constant (spatially uniform), the infection dies out, and the susceptible and removed densities approach their steady-states, which are spatially non-constant. These results had been established in the case of one space dimension in [5]. Through numerical simulations for the case of avoidance of overcrowding only (k = ), we can see that the total population may reach its steady-state much faster than the subclasses. In consequence, initially the movement of the total population governs the spatial flow of all subclasses but soon after, when the total population is close to its steady-state, the system behaves like Kermack-McKendrick s [3] since the gradient terms have essentially vanished. Then a local infection can move out of its original support and even become global. The infected may spread out to the whole habitat before dying out and then susceptible and immune individuals asymptotically coexist everywhere. Depending on the initial conditions, the infected may also die out before they spread out to the whole domain so that only susceptibles fill the habitat in some region. Numerical simulations for this case also indicate that susceptibles and recovered approach a nearly constant asymptotic distribution in the region outside of the original compact support of the infectives who asymptotically die out. These steady-state values depend on the infectivity and recovery rates, as well as on the initial distribution of susceptibles. The numerical method we described and used for the case of avoidance of 9

overcrowding only is consistent, in the sense that the numerical approximations of N and of S + I + R are close. More specifically, the sum of the approximated values for the three subclasses, S + I + R, converges to the approximated value for the total population, N, for all time. The problem of existence of solutions for the proposed model is a very difficult one and remains open. References [] B. Cockburn and C.-W. Shu, Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems, J. Sci. Comp. (). [] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosc. 33 (977), 35-9. [3] O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Royal Soc. London A, 5 (97), 7-7. [] D. B. Meade, Qualitative Analysis for an Epidemic Model with Diffusion to Avoid Infection, IMA Preprint Series 9 99. [5] D. B. Meade and F. A. Milner, S-I-R epidemic models with directed diffusion, in Mathematical Aspects of Human Diseases, Giuseppe Da Prato (Ed.), Applied Mathematics Monographs 3, Giardini Editori, Pisa, 99. [] D. B. Meade and F. A. Milner, An S-I-R model for epidemics with diffusion to avoid infection and overcrowding, in Proceedings of the 3th IMACS World Congress on Computation and Applied Mathematics (v.3), R. Vichnevetsky and J. J. H. Miller (Eds.), IMACS, Dublin, Ireland, 99. [7] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second Edition, Springer Verlag, New York, 99. [8] G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl. 8 (98), 5-.

Density of Infective Population at t= 7.5 3 Density of Susceptible Population at t= 5.5 5.8.5.8.5...... Density of Infective Population at t=.5.8 3. Density of Susceptible Population at t=.5...8.5.8.5..... Density of Infective Population at t= Density of Susceptible Population at t=.5. 8..5..5.8.5.8.5..... Density of Susceptible Population at t=. Density of Infective Population at t= 3 x. 8.8....8.5.8.5...... Figure 9: Distribution of susceptible and infected classes for an infection with compact initial support and diffusion to avoid infection only (k =, k =.).

Contour Plot of Density of Susceptible Population at t= Contour Plot of Density of Infective Population at t=.9.8.7.9.8.7..5...5..3...3......8...3..5..7.8.9 Contour Plot of Density of Susceptible Population at t=.5 Contour Plot of Density of Infective Population at t=.5.9.8.7.9.8.7..5...5..3...3.....3..5..7.8.9...3..5..7.8.9 Contour Plot of Density of Susceptible Population at t= Contour Plot of Density of Infective Population at t=.9.8.7.9.8.7..5...5..3...3.....3..5..7.8.9...3..5..7.8.9 Contour Plot of Density of Susceptible Population at t= Contour Plot of Density of Infective Population at t=.9.8.7.9.8.7..5...5..3...3.....3..5..7.8.9...3..5..7.8.9 Figure : Contour plots of density of susceptible and infected populations for an infection with compact initial support and diffusion to avoid infection only (k =, k =.).