Math 19. Lecture 26Traveling WavesT. JudsonFall 20051 A Model for the HantavirusSuppose that mice are infected with the hantavirus in California. We wishto model how fast the infected mice will disperse across the country. Hereare our assumptions.• Mice are infected with a virus that are harmless to them but virulentin people.• Once a mouse is infected, it stays infected.• Each mouse wanders over some small territor y at random and meetsuninfected mice. This is how the infection is spread.• Infected mice pass the virus to some percentage of the uninfected micethat they encounter.• At time t = 0, all of the mice in western California are infected, butmice in eastern California and the rest of the country are not infected.• We will simplify o ur model of the United States by viewing the U.S.as an infinitely long strip whose topography and width are irrelevant.The x coordinate is very negative in San Francisco a nd very positivein Boston.12 The Reaction-Diffusion Equation.Let u(t, x) denote the proportion of mice that are infected at time t andposition x. We will make the assumption that u(t, x) is controlled by anequation of the form∂u∂t=∂2u∂x2+ ru(1 − u), (1)where r > 0 is a number whose value can be determined by studying the rateat which laboratory mice are infected by the virus. Equation (1) is calledFisher’s equation.3 The Traveling Wave AssumptionWe might look for a solution to (1) that has the form of a traveling wave,u(t, x) = f(x − ct),where c > 0 . Such a function maintains the same shape over time but is awave that travels eastward at a constant speed c as t → ∞. Thus,∂u∂t=∂∂tf(x − ct) = −cdfds∂u∂x=∂∂xf(x − ct) =dfds∂2u∂x2=∂2∂x2f(x − ct) =d2fds2,where s = x − ct. Substituting into (1), we get−cdfds=d2fds2+ rf(1 − f)Our first goal is to find a solution f to this equation that satisfies the followingconditions:1. 0 ≤ f ≤ 1,2. f(s) → 1 as s → −∞,3. f(s) → 0 as s → ∞.Ideally, we wish to find a solution at t = 0. In this case, u(0, x) = f(x).24 A Standard Tr i ckWe can turn any second-order differentia l equation such as−cdfds=d2fds2+ rf(1 − f). (2)into a first-order system by introducing a new variable p such that p = f′.That is,dfds= p (3)dpds= −cp − rf(1 − f). (4)The f null cline is p = 0, while the p null cline isp = −rcf(1 − f ) =rcf2−rcfTherefore, we have two equilibrium points: f = 0, p = 0 and f = 1, p = 0.The stability matrices for the first and second equilibrium points areA =0 1−r −cand B =0 1r −c.Since the trace of A is negative and the determinant is positive, the firstequilibrium point, (0, 0), is stable. Since det(B) < 0, the second equilibriumpoint, (1, 0), is unstable.5 The Phase Plane SolutionWe are interested in a solution to (4) that tends to the origin as s → ∞ andgoes to unstable equilibrium point, (1, 0), as s → −∞. We will show whenc2> 4r, there is such a traj ectory in the lower half of the phase plane.• There is a trajectory in the p < 0 part of the phase plane that tendstowards (1, 0) as s → −∞ and that for moderately negative values ofs, has p < 0 and f < 1. Since f must be a decreasing function, werequire f′= p < 0.3• When c2≥ 4 r, then the triangle pictured below is a region (called abasin of attraction or a trapping region) for (3) and (4). We claim thatany trajectory in this region must tend towards (0, 0) as s → ∞. Oncea trajectory enters this region, it can never escape. You can think of abasin of attraction as a black hole.6 Some Observations1. On the line segment where p = 0 and 0 < f < 1, we know thatdp/ds < 0, so every trajectory that crosses this line moves from aboveto below.2. On the half- line where p < 0 and f = 1, we know that df/ds < 0, soevery trajectory that crosses this half-line moves f rom right to left.7 Trajecto ries th at Leave (1, 0)8 The Triangle as a Basis o f Att ract i onConsider the tria ngle where the hypotenuse is formed by the line p+cf/2 = 0.We wish to know how we can choose c so that our solution f will satisfy allof the conditions t hat we have specified.Readings and Referenc es• C. Taubes. Modeling Differential Equations in Biolog y. Prentice Hall,Upper Saddle River, NJ, 2001 . Chapter 21.• “Hantavirus Outbreak Yields to PCR,” pp. 358–363• “US Braces for Hantavirus Outbreak,” pp.
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