Spreading-vanishing dichotomy in a diffusive epidemic model with Stefan condition

Spreading-vanishing dichotomy in a diffusive epidemic model with Stefan condition Inkyung Ahn Department of Mathematics Korea University April 1, 2016 2016 CMAC Workshop (joint-work with Seunghyun Baek)

What is the free boundary? If the domain is a priori unknown and has to be determined together with the solution, then it is called a free boundary problem and the boundary is called the free boundary. Figure: The classic example is the Stefan problem (around 1890), which treats the formation of ice in the polar seas. The boundary between water and ice is moving over time.

There are many types of free boundary problems; for example, the model for a tumor growth (the surface of the tumor is described by an free boundary) X. F. Chen & A. Friedman, SIAM J. Math. Anal. (2003), The model for European/American options(the optimal exercise boundary of option between the stopping region and the continuity region is an free boundary.) G. Peskir, A. Shiryaev, Optimal stopping and free-boundary problems, 2006, Birkhauser. L. S. Jiang & M. Dai, 2004, Convergence of binomial tree methods for European American path-dependent options. SIAM J. Numer. Anal..

The Stefan problem describing a logistic model Y. Du & Z. Lin, 2010, SIAM JMA (1-d) u t du xx = u(a bu), u x (t, 0) = 0, u(t, h(t)) = 0, t > 0, h (t) = µu x (t, h(t)), t > 0, h(0) = h 0, u(0, x) = u 0 (x), 0 x h 0, t > 0, 0 < x < h(t), (1) u 0 C 2 ([0, h 0 ]), u 0(0) = u 0 (h 0 ) = 0, u 0 > 0 in [0, h 0 ). This system models the spreading of a new or invasive species with population density u(t, x) over a one dimensional habitat.

The spreading-vanishing dichotomy The result shows that a spreading-vanishing dichotomy occurs for the logistic model, namely, as time t, the population u(t, x) either successfully establishes itself in the new environment (henceforth called spreading), in the sense that h(t) and u(t, x) a/b, or the population fails to establish and vanishes eventually (called vanishing), namely h(t) h π 2 d a and u(t, x) 0.

Comparison with other theoretical results Much previous mathematical investigation on the spreading of population has been based on the diffusive logistic equation over the entire space (, + ): u t d u = u(a bu), t > 0, x (, + ), (2) in the pioneering works of R. A. Fisher, Ann. Eugenics 7 (1937). A.N. Kolmogorov, et al, Bull. Univ. Moscou Sér. Internat. A1 (1937)

Comparison with other theoretical results Traveling wave solutions have been found for (2): For any c c min := 2 ad, there exists a solution u(t, x) := W (x ct) with the property that W (y) < 0 for y R 1, W ( ) = a, W (+ ) = 0; b No such solution exists if c < c min. The number c min is called the minimal speed of the traveling waves.

Comparison with other theoretical results Let c be the spreading speed of a new population u(t, x), it was shown that c = c min = 2 ad and for such u(t, x), lim u(t, x) = a/b, lim t, x (c ɛ)t u(t, x) = 0 t, x (c +ɛ)t for any small ɛ > 0. (see D.G. Aronson and H.F. Weinberger,Partial Differential Equations and Related Topics, 446, 1975 and J.G. Skellam, Biometrika, 38, (1951). ) For Stephan model, the asymptotic spreading speed k 0 (0, c ).

Free Boundary Condition Assume that as the expanding front propagates, the population suffers a loss of k units per unit volume at the front. By Fick s first law, the number of individuals of the population that enters the region (through diffusion, or random walk) bounded by the old front x = h(t) and new front x = h(t + t) is approximated by d x u t (surface area of { x = h(t)}) = d x u t Nω N h(t) N 1 where ω N denotes the volume of the N-dimensional unit ball, and x u is calculated at x = h(t).

Free Boundary Condition The population loss in this region is approximated by k (volume of the region) = k [ω N h(t + t) N ω N h(t) N ]. The average density of the population in the region bounded by the two fronts is given by d x u Nω N h(t) N 1 t[ω N h(t + t) N ω N h(t) N ] 1 = k. The limit of this quantity as t 0 is the population density at the front, namely u(t, h(t)), which by assumption is 0. It is easily checked that this limit equals with x = h(t). d x u /h (t) = k

Free Boundary Condition Therefore d x u = kh (t) with x = h(t). So h (t) = µu r (t, r) at r = h(t) with µ = dk 1. It can be shown that the Cauchy problem corresponds to the limiting case that µ =, that is, the free boundary problem reduces to the Cauchy problem if the population loss at the front is 0.

Literature (Scalar Eq) Kaneko and Y. Yamada, Adv. Math. Sci. Appl. (2011) : the boundary condition u x = 0 at x = 0 in (1) is replaced by u = 0

Literature (Scalar Eq) Kaneko and Y. Yamada, Adv. Math. Sci. Appl. (2011) : the boundary condition u x = 0 at x = 0 in (1) is replaced by u = 0 Y. Du and Z. Guo, (i) JDE (2011) : In a ball in higher dimension, the case when the domain and the solutions are radial symmetric (ii) JDE (2012) : In a general domain, the existence of weak solutions

Peng and Zhao, DCDS-A (2013) : the time periodicity accounts for the effect of the bad and good seasons

Peng and Zhao, DCDS-A (2013) : the time periodicity accounts for the effect of the bad and good seasons Y. Du, H. Matano and K. Wang, Arch. Rational Mech. (2014) : regularity and asymptotic behavior of nonlinear Stefan problem in high dimension Y. Du and B. Lou, JEMS(2014) : spreading and vanishing for nonlinear problem Y. Du, H. Matsuzawa, M. Zhou, (i) SIAM JMA (2014) : spreading speed for nonlinear Stefan problem in 1-dimension (ii) J de Math Pures et Appliq (2015) : spreading speed in high dimension

Literature (System) M. Mimura, Y. Yamada and S. Yotsutani (1985), Japan J. Appl. Math. : a two phase Stefan problem for a competition system, where the free boundary separates the two competitors from each other in a bounded interval Z. G. Lin (2007), Nonlinearity : a predator-prey system over a bounded interval, showing the free boundary reaches the fixed boundary in finite time J. S. Guo and Wu, J. Dyn DE (2012) : The weak competition case in 1-dim, both species share the same free boundary Du and Lin, DCDS-B (2014) ; Wang and Zhao, J. Dyn DE (2014) : competition models

A epidemic model in a spatially heterogeneous environment Capasso and Maddalena, 1981, JMB Oro-faecal transmitted disease in the European Mediterranean regions: The model is described by the following coupled parabolic system: u(x,t) t = d u(x, t) a 11 u(x, t) + a 12 v(x, t), v(x,t) t = a 22 v(x, t) + G(u(x, t)), (x, t) Ω (0, + ), u (3) η + αu = 0, (x, t) Ω (0, + ), u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x Ω, where u(x, t) and v(x, t) represent the spatial densities of bacteria and the infective human population in an urban community at a point x in the habitat Ω R n and time t 0, respectively, / η denotes the outward normal derivative.

The positive constant d denotes the diffusion constant of bacteria a 11 u describes the natural decay rate of the bacterial population a 12 v is the contribution of the infective humans to the growth rate of bacteria a 22 v describes the natural damping of the infective population due to the finite mean duration of the infectiousness of humans G(u) is the infection rate of humans under the assumption that total susceptible human population is constant during the evolution of the epidemic.

Assume that (A1) G C 1 ([0, )), G(0) = 0, G (z) > 0, z 0; (A2) G(z) z is decreasing and lim z + G(z) z An example is G(z) = a21z 1+z with a 21 < a11a22 a 12. < a11a22 a 12.

Assume that (A1) G C 1 ([0, )), G(0) = 0, G (z) > 0, z 0; (A2) G(z) z is decreasing and lim z + G(z) z An example is G(z) = a21z 1+z with a 21 < a11a22 a 12. < a11a22 a 12. For the corresponding O.D.E. system of (3), a threshold parameter exists such that R 0 := G (0)a 12 a 11 a 22 if 0 < R 0 < 1, then the epidemic always tends to extinction, while if R 0 > 1, a nontrivial endemic level appears which is globally asymptotically stable in the positive quadrant.

Capasso and Maddalena, 1981, JMB Define a threshold parameter such that R0 d G (0)a 12 := (a 11 + dλ 1 )a 22

Capasso and Maddalena, 1981, JMB Define a threshold parameter such that R0 d G (0)a 12 := (a 11 + dλ 1 )a 22 for 0 < R d 0 < 1 the epidemic eventually tends to extinction, while R d 0 > 1 a globally asymptotically stable spatially inhomogeneous stationary endemic state appears, where λ 1 is the first eigenvalue of the boundary value problem φ = λφ in Ω with φ + αφ = 0 on Ω. η

Literature (S. Wu, 2012, NARWA) There exists entire solutions of a bistable reaction-diffusion system (solutions behave like two monotone increasing traveling wave solutions propagating from both sides of the x-axis). (Y. Wang & Z. Wang, 2013, AMC) There exists entire solutions for time-delayed model.

Spread gradually The solution of system subject to the Neumann or Dirichlet boundary conditions is always positive for any t > 0 no matter what the nonnegative nontrivial initial date is.

Spread gradually The solution of system subject to the Neumann or Dirichlet boundary conditions is always positive for any t > 0 no matter what the nonnegative nontrivial initial date is. It means that the disease spreads to the whole area immediately even infectious is limited a small part at the beginning. It doesn t match the observed fact that disease always spreads gradually.

An epidemic model with a free boundary How are bacteria spreading spatially over further to large area? Focus on the changing of the infected domain and consider an epidemic model with the free boundary, which describes the spreading frontier of bacteria.

Assumptions The human population in the whole habitat (, ) is constant Environment in g(t) < x < h(t) is infected by bacteria

Assumptions The human population in the whole habitat (, ) is constant Environment in g(t) < x < h(t) is infected by bacteria No bacteria or infective humans on the rest part The right spreading frontier of the infected environment is represented by the moving boundary x = h(t) h(t) grows at a rate that is proportional to bacteria population gradient at the frontier, so the condition on the right frontier is u(h(t), t) = 0, µ u x (h(t), t) = h (t). Similarly, the conditions on the left frontier (free boundary) is u(g(t), t) = 0, µ u x (g(t), t) = g (t).

Model The problem for u(x, t) and v(x, t) with free boundaries x = g(t) and x = h(t) such that u(x,t) t = d 2 u(x,t) x a 2 11 u(x, t) + a 12 v(x, t), g(t) < x < h(t), t > 0, v(x,t) t = a 22 v(x, t) + G(u(x, t)), g(t) < x < h(t), t > 0, u(x, t) = v(x, t) = 0, x = g(t) or x = h(t), g(0) = h 0, g (t) = µ u x (g(t), t), t > 0, h(0) = h 0, h (t) = µ u x (h(t), t), t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), h 0 x h 0, (4) where x = g(t) and x = h(t) are the moving left and right boundaries to be determined, h 0 and µ are positive constants, and the initial functions u 0 and v 0 are nonnegative and satisfy

Local Existence Theorem For any given (u 0, v 0 ) satisfying (5), and any α (0, 1), there is a T > 0 such that problem (4) admits a unique solution moreover, (u, v; g, h) [C 1+α,(1+α)/2 (D T )] 2 [C 1+α/2 ([0, T ])] 2 ; u C 1+α,(1+α)/2 (D T ) + v C 1+α,(1+α)/2 (D T ) + g C 1+α/2 ([0,T ]) + h C 1+α/2 ([0,T ]) C, where D T = {(x, t) R 2 : x [g(t), h(t)], t [0, T ]}, C and T depend only on h 0, α, u 0 C 2 ([ h 0,h 0]) and v 0 C 2 ([ h 0,h 0]). Lemma Let (u, v; g, h) be a solution to problem (4) defined for t (0, T 0 ] for some T 0 (0, + ). Then there exists a constant C 3 independent of T 0 such that 0 < g (t), h (t) C 3 for t (0, T 0 ].

Global Existence Since u, v and g (t), h (t) are bounded in (g(t), h(t)) (0, T 0 ] by constants independent of T 0, the global solution is guaranteed. Theorem The solution of problem (4) exists and is unique for all t (0, ).

x = h(t) is monotonic increasing, x = g(t) is monotonic decreasing and therefore there exist h, g (0, + ] such that lim t + h(t) = h and lim t + g(t) = g. The next lemma shows that if h <, then g <, and vice versa. That is, the double free boundary fronts x = g(t) and x = h(t) are both finite or infinite simultaneously. Lemma Let (u, v; g, h) be a solution to problem (4) defined for t [0, + ) and x [g(t), h(t)]. Then we have 2h 0 < g(t) + h(t) < 2h 0 for t [0, + ).

Definition Definition The disease vanishes if h g < and while the disease spreads if lim ( u(, t) C(g(t),h(t)])+ v(, t) C([g(t),h(t)]) ) = 0, t + h g = and lim inf t + ( u(, t) C([g(t),h(t)])+ v(, t) C([g(t),h(t)]) ) > 0.

Reproduction number Introduce the basic reproduction number R F 0 (t) for the free boundary problem (4) by G (0) a12 R0 F (t) := R0 D a ((g(t), h(t))) = 22 ( a 11 + d π h(t) g(t) where we use R D 0 (Ω) to denote the basic reproduction number for the corresponding problem in Ω with null Dirichlet boundary condition on Ω. Lemma R F 0 (t) is strictly monotone increasing function of t, that is if t 1 < t 2, then R F 0 (t 1 ) < R F 0 (t 2 ). Moreover, if h(t) as t, then R F 0 (t) R 0 as t. ) 2,

Bacteria Vanishing Theorem If R 0 1, then h g < and lim t + ( u(, t) C([g(t),h(t)]) + v(, t) C([g(t),h(t)]) ) = 0. Idea of proof. Recall that R 0 := G (0)a 12 a 11 a 22. Use the fact that If h <, then lim t + ( u(, t) C([g(t),h(t)]) + v(, t) C([g(t),h(t)]) ) = 0. d dt h(t) g(t) = d µ (h (t) g (t)) + [u(x, t) + a 12 a 22 v(x, t)]dx h(t) g(t) a 11 u(x, t) + a 12 a 22 G(u(x, t))dx.

(continued Proof) Integrating from 0 to t (> 0) gives h(t) g(t) Since G(z) z [u + a 12 a 22 v](x, t)dx = h(0) g(0) + d µ (h(0) g(0)) d (h(t) g(t)) µ + t h(s) 0 g(s) [u + a 12 a 22 v](x, 0)dx (6) a 11 u(x, t) + a 12 a 22 G(u(x, t))dxds, t 0.(7) G (0) by the assumption (A2), it follows from R 0 1 that a 11 u(x, t) + a12 a 22 G(u(x, t)) 0 for x [g(t), h(t)] and t 0, d (h(t) g(t)) µ h(0) g(0) [u + a 12 v](x, 0)dx + d (h(0) g(0)) a 22 µ for t 0, which in turn gives that h g <.

Vanishing (R 0 = 0.5 < 1 for all t 0) d = 0.01; a 11 = 0.1; a 12 = 0.1; a 22 = 0.1; a 21 = 0.05; mu = 0.01; x = [ 10, 10]; h 0 = 1

Theorem If R F 0 (0) < 1 and u 0 (x) C([ h0,h 0]), v 0 (x) C([ h0,h 0]) are sufficiently small. Then h g < and lim t + ( u(, t) C([g(t),h(t)]) + v(, t) C([g(t),h(t)]) ) = 0. Idea of proof. Construct a suitable upper solution to problem (4). Since R0 F (0) < 1, it follows from Lemma 3.3 that there is a λ 0 > 0 and 0 < ψ(x) 1 in ( h 0, h 0 ) such that { dψxx = a 11 ψ + G (0) a12 a 22 ψ + λ 0 ψ, h 0 < x < h 0, (8) ψ(x) = 0, x = ±h 0. Therefore, there exists a small δ > 0 such that 1 δ + ( (1 + δ) 2 1) a 11 + G (0) a 12 1 + [ a 22 (1 + δ) 2 1 4 ]λ 0 0.

Set σ(t) = h 0 (1 + δ δ 2 e δt ), t 0, and u(x, t) = εe δt ψ(xh 0 /σ(t)), σ(t) x σ(t), t 0. v(x, t) = ( G (0) a 22 + λ 0 4a 12 )u(x, t), σ(t) x σ(t), t 0. h 0 h 0 1+δ/2 )( G (0) a 22 If u 0 L εψ( 1+δ/2 ) and v 0 L εψ( h then u 0 (x) εψ( 0 x 1+δ/2 ) u(x, 0) = εψ( 1+δ/2 ) and h v 0 (x) εψ( 0 1+δ/2 )( G (0) a 22 + λ0 4a 12 ) v(x, 0) for x [ h 0, h 0 ]. + λ0 4a 12 ), Conclude that g(t) σ(t) and h(t) σ(t) for t > 0. It follows that h g lim t 2σ(t) = 2h 0 (1 + δ) <.

Spreading Theorem If R F 0 (0) 1, then h = g = and lim inf t + ( u(, t) C([g(t),h(t)]) ) > 0, that is, spreading occurs.

Spreading Theorem If R F 0 (0) 1, then h = g = and lim inf t + ( u(, t) C([g(t),h(t)]) ) > 0, that is, spreading occurs. Remark It follows from the above proof that spreading occurs, if there exists t 0 0 such that R F 0 (t 0 ) 1. Idea of proof. Construct a suitable lower solution to (4), and we define u(x, t) = δψ(x), v = ( G (0) a 22 + λ 0 4a 12 )δψ(x) for h 0 x h 0, t 0, where δ is sufficiently small. u(x, t) u(x, t) and v(x, t) v(x, t) in [ h 0, h 0 ] [0, ). It follows that lim inf t + u(, t) C([g(t),h(t)]) δψ(0) > 0 and therefore h g = +.

Spreading ( R F 0 (0) = 2 > 1) d = 0.01; a 11 = 0.1; a 12 = 0.1; a 22 = 0.1; a 21 = 0.2; mu = 0.01; x = [ 10, 10]; h 0 = 1

The above theorem shows if R F 0 (0) < 1, vanishing occurs for small initial size of infected bacteria, and the previous theorem implies that if R 0 1, vanishing always occurs for any initial values. The next result shows that spreading occurs for large values. Theorem Suppose that R F 0 (0) < 1 < R 0. Then h g = and lim inf t + ( u(, t) C([g(t),h(t)]) ) > 0 if u 0 (x) C([ h0,h 0]) and v 0 (x) C([ h0,h 0]) are sufficiently large. Idea of proof. Construct a vector (u, v, h) such that u u, v v in [ h(t), h(t)] [0, T 0 ], and also g(t) h(t), h(t) h(t) in [0, T 0 ]. If we can choose T 0 such that R D 0 (( h( T 0 ), h( T 0 )) > 1, then h g =.

R F 0 (0) 1 R 0 = 1.1 ; d = 0.01; a 11 = 0.1; a 12 = 0.1; a 22 = 0.1; a 21 = 0.11; mu = 0.01; x = [ 10, 10] (i) h 0 = 1 (ii) h 0 = 4

Sharp threshold Theorem Suppose that R 0 > 1. Fixed µ, h 0. Let (u, v; g, h) be a solution of (4) with (u 0, v 0 ) = (σφ(x), σψ(x)) for some σ > 0. Then there exists σ = σ (φ, ψ) [0, ) such that spreading happens when σ > σ, and vanishing happens when 0 < σ σ.

Sharp threshold If we consider µ instead of u 0 as a varying parameter, the following result holds; Theorem (Sharp threshold) Suppose that R 0 > 1, with fixed h 0, u 0 and v 0. Then there exists µ [0, ) such that spreading occurs when µ > µ, and vanishing occurs when 0 < µ µ.

Convergence Next, we consider the asymptotic behavior of the solution to (4) when the spreading occurs. Theorem Suppose that R 0 > 1. If spreading occurs, then the solution of free boundary problem (4) satisfies lim t + (u(x, t), v(x, t)) = (u, v ) uniformly in any bounded subset of (, ), where (u, v ) is the unique positive equilibrium of (4).

Spreading-vanishing Dichotomy Combining the above results, we immediately obtain the following spreading-vanishing dichotomy: Theorem Suppose that R 0 > 1. Let (u(x, t), v(x, t); g(t), h(t)) be the solution of free boundary problem (4). Then, the following alternatives hold: Either (i) Spreading: h g = + and lim t + (u(x, t), v(x, t)) = (u, v ) uniformly in any bounded subset of (, ); or (ii) Vanishing: h g h with G (0) a 12 a 22 a 11+d( π h )2 = 1 and lim t + ( u(, t) C([g(t),h(t)]) + v(, t) C([g(t),h(t)]) ) = 0.

Summary There is a unique solution (u(x, t), v(x, t), h(t), g(t)) for all t > 0, u, v > 0, h (t) > 0, g (t) < 0.

Summary There is a unique solution (u(x, t), v(x, t), h(t), g(t)) for all t > 0, u, v > 0, h (t) > 0, g (t) < 0. R F 0 (t) R 0 and R F 0 (t) R 0 if (g(t), h(t)) (, + ) as t.

Summary There is a unique solution (u(x, t), v(x, t), h(t), g(t)) for all t > 0, u, v > 0, h (t) > 0, g (t) < 0. R F 0 (t) R 0 and R F 0 (t) R 0 if (g(t), h(t)) (, + ) as t. If R 0 1, the bacteria are always vanishing. The result is the same as that for the corresponding ODE system. However, if R F 0 (t 0 ) 1 for some t 0 0, the bacteria are spreading. For the case R F 0 (0) < 1 < R 0, the spreading or vanishing of the bacteria depends on the initial size of bacteria, or the ratio (µ) of the expansion speed of the free boundary and the population gradient at the expanding fronts.

Ecologically, our main results reveal that if the multiplicative factor of the infectious bacteria is small, the bacteria will die out eventually and the humans are safe.

Ecologically, our main results reveal that if the multiplicative factor of the infectious bacteria is small, the bacteria will die out eventually and the humans are safe. Otherwise, the spreading or vanishing of the bacteria depends on the initial infected habitat, the diffusion rate, and other factors. In particular, the initial number of bacteria plays a key role. A large initial number can induce the spreading of bacteria easily.

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