István Faragó, Róbert Horváth. Numerical Analysis of Evolution Equations, Innsbruck, Austria October, 2014.

Transcription

1 s István, Róbert 1 Eötvös Loránd University, MTA-ELTE NumNet Research Group 2 University of Technology Budapest, MTA-ELTE NumNet Research Group Analysis of Evolution Equations, Innsbruck, Austria October, 2014.

2 Outline 1 s s

3 s s

4 Mamatical s of diseases s Basic assumptions. population is presumed to be constant in size; y are divided into three classes: 1 I(t): infected individuals who can pass on disease to ors; 2 S(t): susceptibles who have yet to contract disease and become infectious, 3 R(t): members who have been infected but cannot transmit disease for some reason, e.g., y have been isolated from rest of population.

5 Mamatical s of diseases(cont d) s Kermack and McKendrick, 1927 S = asi, I = asi bi, R = bi, I = I(t): number of infective, S = S(t): number of susceptible and R = R(t): number of recovered (removed) members. a > 0: contact rate; b > 0: recovery coefficient This has been improved several times taking into account also births, deaths, latent periods, reinfections, incubations etc.

6 Inclusion of spatial dependence s How spatial dependence can be introduced into? Consider subpopulations that are connected into a network. Allow motion of individuals in population. We will consider anor approach: speed of motion of individuals can be neglected (compared to speed of disease) a member of population can infect only members in its well defined spatial neighbourhood.

7 Inclusion of spatial dependence s Mamatical with spatial dependence ( ) S t(x, t) = W ( x x )I(x, t) dx S(x, t), N(x) ( ) I t(x, t) = W ( x x )I(x, t) dx S(x, t) bi(x, t), N(x) R t(x, t) = bi(x, t), S = S(x, t), I = I(x, t) and R = R(x, t) are now spatial dependent densities. The nonnegative weighting function W depends only on distance of points x and x, and N(x) is a prescribed neighbourhood of point x.

8 Inclusion of spatial dependence s Simplifications [Jones, Sleeman, 2011]: We consider 1D problems. N(x) = [x δ, x + δ]. I is approximated with its second order spatial Taylor series. Simplified with spatial dependence where θ = δ S t = S ( θi + φi xx), I t = S ( θi + φi xx) bi, R t = bi, δ W ( u ) du, φ = 1 2 δ δ u 2 W ( u ) du are positive constants that can be computed from.

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10 Some qualitative properties of s Simplified with spatial dependence Our requirements are: P1 P2 P3 S t = S ( θi + φi xx), I t = S ( θi + φi xx) bi, R t = bi. Additivity property S + I + R is constant at a fixed spatial position. Monotonicity property S monotone decreases in time R monotone increases in time Nonnegativity property S > 0, I 0, R 0 at t = 0 S, I, R 0.

11 Some qualitative properties of s Lemma. If condition θi + φi xx 0 (1) is satisfied n properties [P2] and [P3] are true. Property [P1] is true without any restrictions. Remark. The condition (1) is also necessary. Remark. The above condition is hardly checkable: it depends on values of solution I in whole solution domain.

12 Travelling wave solutions s In order to epidemic waves we are looking for travelling wave solutions. Let us set S(x, t) = S(x ct), I(x, t) = Ĩ(x ct), R(x, t) = R(x ct), where c is constant wave speed, Ĩ and S have properties lim Ĩ(ξ) = 0, ξ ± lim Ĩ (ξ) = 0, lim S(ξ) = S > 0. ξ ± ξ

13 Travelling wave solutions s After some manipulations we get: Form of system after inserting wave form solutions S = c φ log( S/ S ) θc bφ (Ĩ + S S ), Ĩ = b cĩ c φ log( S/ S ) + θc bφ (Ĩ + S S ), R = b cĩ.

14 Travelling wave solutions s Let us introduce notations S = lim S(ξ), S = lim S(ξ). ξ ξ Lemma. The necessary condition of travelling wave solution is S > b/θ. Moreover S < b/θ, that is epidemic wave does not leave enough susceptible members back to be able to sustain a new wave. If necessary condition is satisfied n c 2 S φ( S θ b) is a lower bound for wave speed.

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16 Finite difference scheme s We define a uniform spatial grid ω h = {x k [0, L] x k = kh, k = 0,..., N, h = L/N} and a positive time step τ > 0. We apply notations s n k S(kh, nτ), etc. We define difference scheme as follows. s n+1 ( k θi n k + φin k 1 2in k + in k+1 i n+1 k r n+1 k s n k = s n k τ i n k = s n k τ rk n = bi n k τ, h 2 ( θi n k + φin k 1 2in k + in k+1 h 2 k = 0,..., N, s n 1 = sn 1, sn N+1 = sn N 1, etc. ), ) bi n k,

17 numerical solution s Lemma. The finite difference scheme satisfies discrete version of [P1], and if relation 0 θi n k + φin k 1 2in k + in k+1 h 2 is true for all possible indices k and n, and time step satisfies condition { } 1 τ min M(θ + 4φ/h 2 ), 1/b, where M = max x [0,L] {S(x, 0) + I(x, 0) + R(x, 0)}, n discrete versions of properties [P2] and [P3] are also satisfied. Under above conditions scheme is stable in maximum norm.

18 numerical solution (cont d) s The condition 0 θi n k + φin k 1 2in k + in k+1 h 2 is an a posteriori condition. How to guaranty it a priori? We introduce notation: j n k := θin k + φin k 1 2in k + in k+1 h 2. Suppose that re are positive numbers P, Q, p, q > 0 such that Q u δ q W (u) P u δ p, if u [0, δ]. (2)

19 numerical solution (cont d) s Lemma. Assume that we have { } 1 τ min M(θ + 4φ/h 2 ), 1/b, or h 2P/Q δp/2 q/2+1 q + 1 (p + 1)(p + 2)(p + 3) and time-step satisfies { τ min 1 M(θ + 4φ/h 2 ), 1/ ( b + 2φM )} h 2. Then nonnegativity of i n k, sn k, rn k and jn k imply nonnegativity of next approximations i n+1 k, s n+1 k, r n+1 k and j n+1 k.

20 numerical solution (cont d) Lemma. Assume that we have h 2P/Q δp/2 q/2+1 q + 1 (p + 1)(p + 2)(p + 3) s and or time-step satisfies { τ min { } 1 τ min M(θ + 4φ/h 2 ), 1/b, 1 M(θ + 4φ/h 2 ), 1/ ( b + 2φM )} h 2. Then nonnegativity of i n k, sn k, rn k and jn k imply nonnegativity of next approximations i n+1 k, s n+1 k, r n+1 k and j n+1 k.

21 numerical solution (cont d) s Hence we have: Theorem. Under conditions of previous lemma, nonnegativity of initial values i 0 k, s0 k, r0 k and j0 k imply discrete versions of qualitative properties [P2] and [P3], and scheme is stable in maximum norm, too. Remark. For asymptotic case (h 0) condition has form: τ h 2 1 4φM.

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23 Parameter setting to numerical s simulation We solve problem on interval [0,10] with standard finite difference schemes. In epidemic case parameters are as follows: parameters: b = 0.5, φ =2.86e-005, θ = 0.7. Mesh parameters: x = 1/199, t = 1 ( t max = 1.34) Condition of wave propagation: b/θ > 0.71

24 Summary s We have formulated basic qualitative properties of continuous and discrete epidemic s. We gave necessary and sufficient condition for continuous. We gave a priori checkable sufficient condition for discrete. We analyzed travelling wave solutions in continuous s and in numerical examples.

25 Future work s More complex discrete s (not only diffusion). 1D 2D, 3D. Or qualitative properties? Or semidiscretization?

26 s Thank you for your attention