Traveling Waves of Diffusive Predator-Prey Systems: Disease Outbreak Propagation

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1 Traveling Waves of Diffusive Predator-Prey Systems: Disease Outbreak Propagation Xiang-Sheng Wang Teaching Postdoctoral Fellow Memorial University of Newfoundland St. John s, Newfoundland, Canada joint work with Professor Haiyan Wang (Arizona State University) Professor Jianhong Wu (York University) Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 1

2 Outline A 2-dim diffusive disease model Main theorem Proof of main theorem 1. constructing convex invariant cone 2. existence of traveling waves (Schauder fixed point theorem) 3. non-existence of traveling waves (two-side Laplace transform) Extension to a 3-dim diffusive disease model Discussion Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 2

3 Reaction-Diffusion system We study the following reaction-diffusion system in one-dimensional space. t S = d 1 xx S βsi/(s + I); t I = d 2 xx I + βsi/(s + I) γi The basic reproduction number of the corresponding ODE system (Diekmann et al., 1990; van den Driessche & Watmough, 2002) R 0 = β/γ. Minimal traveling speed c = 2 d 2 (β γ). Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 3

4 Traveling wave solution pair Non-trivial traveling wave solutions (S(x + ct), I(x + ct)) cs = d 1 S βsi/(s + I); ci = d 2 I + βsi/(s + I) γi. Boundary conditions S( ) = S, S( ) = S, I(± ) = 0. S is non-increasing, I is non-negative and 0 S < S. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 4

5 Main theorem conditions R 0 > 1 and c > c R 0 1 or c < c non-trivial traveling wave existence non-existence Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 5

6 Fixed point Traveling wave solution (α 1 and α 2 are sufficiently large) 1 S := d 1 S + cs + α 1 S = α 1 S βsi/(s + I); 2 I := d 2 I + ci + α 2 I = α 2 I + βsi/(s + I) γi. Define F = (F 1, F 2 ): F 1 (S, I) := 1 1 [α 1S βsi/(s + I)]; F 2 (S, I) := 1 2 [α 2I + βsi/(s + I) γi]. We shall prove that F has a fixed point (S, I) in a suitable convex invariant cone. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 6

7 Convex invariant cone Define Γ be the set of all (S, I) such that S S S + and I I I +. S + (x) = S ; S (x) = max{s (1 M 1 e ε1x ), 0}; I + (x) = e λx ; I (x) = max{e λx (1 M 2 e ε2x ), 0}, where λ is the smaller root of the characteristic equation f(λ) := d 2 λ 2 + cλ (β γ) = 0. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 7

8 Schauder fixed point theorem F maps Γ into Γ. F is continuous and compact on Γ with respect to the norm where µ > 2λ is given. (φ 1, φ 2 ) µ := max{sup φ 1 (x) e µ x, sup φ 2 (x) e µ x }, x R x R F has a fixed point (S, I) in Γ. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 8

9 Boundary conditions Squeeze lemma (sandwich rule) gives S( ) = S and I( ) = 0. Integral representations of 1 1 and 1 2, together with L hôpital s rule yield S ( ) = 0 and I ( ) = 0. From differential equations we obtain S ( ) = 0 and I ( ) = 0. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 9

10 Boundary conditions Squeeze lemma (sandwich rule) gives S( ) = S and I( ) = 0. Integral representations of 1 1 and 1 2, together with L hôpital s rule yield S ( ) = 0 and I ( ) = 0. From differential equations we obtain S ( ) = 0 and I ( ) = 0. It can be proved that S is non-increasing with S( ) < S( ), and I( ) = 0. Furthermore, we have S ( ) = I ( ) = S ( ) = I ( ) = 0. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 10

11 Non-existence of traveling waves If R 0 = β/γ > 1 and c < c := 2 d 2 (β γ), then the characteristic equation f(µ) := d 2 µ 2 + cµ + β γ is always negative on the real line. Taking two-sided Laplace transformation on d 2 I + ci (β γ)i = βi 2 /(S + I) yields f(µ) e µx I(x)dx = e µx βi(x)2 S(x) + I(x) dx. For large µ we have f(µ) + βi(x) S(x)+I(x) f(µ) + β < 0 and e µx I(x)[f(µ) + βi(x) ]dx < 0. S(x) + I(x) Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 11

12 A further extension We study the following 3-dim diffusive disease model t S = βsi d 1 xx S S + I + R ; t I = βsi d 2 xx I + γi δi; S + I + R t R = d 3 xx R + γi. We look for traveling wave solutions of the form (S(x + ct), I(x + ct), R(x + ct)). cs = d 1 S ci = d 2 I + cr = d 3 R + γi. βsi S + I + R ; βsi (γ + δ)i; S + I + R Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 12

13 Main result For any S > 0, if R 0 := β/(γ + δ) > 1, c > c := 2 d 2 (β γ δ) and d 3 < 2d (c /c) 2, then there exist S < S and a traveling wave solution (S, I, R) such that S( ) = S, S( ) = S, I(± ) = 0, R( ) = γ(s S )/(γ + δ) and R( ) = 0. Furthermore, S(x) is decreasing, 0 I(x) S S, R(x) is increasing, and (γ + δ)i(x)dx = βs(x)i(x) S(x) + I(x) + R(x) dx = c(s S ). On the other hand, if c < c or R 0 1, then there does not exist a non-trivial and non-negative traveling wave solution (S, I, R) such that S( ) = S, S( ) < S, I(± ) = 0 and R( ) = 0. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 13

14 Convex invariant cone Construct the super- and sub-solutions: S + (x) := S ; S (x) := max{s (1 M 1 e ε1x ), 0}; I + (x) := e λ0x ; I (x) := max{e λ0x (1 M 2 e ε2x ), 0}; R + (x) := γe λ 0x γeλ 0x cλ 0 d 3 λ 2 ; R (x) := max{ 0 cλ 0 d 3 λ 2 (1 M 3 e ε3x ), 0}, 0 where λ 0 is the smaller root of the characteristic equation f(λ) := d 2 λ 2 + cλ (β γ δ) = 0. Remark: we require an additional condition d 3 < c λ 0 = 2d (c /c) 2. Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 14

15 Future work What is S( ) = S? Is the traveling wave solution unique? What happens for the limit case R 0 > 1 and c = c? The standard argument of taking limit c c fails because I is not monotone. What if the additional condition d 3 < c λ 0 = 2d (c /c) 2 is violated? What if the model parameters are spatially periodic? Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 15

16 Thank you! Traveling Waves of Diffusive Predator-Prey Systems First Previous Next Last 16