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Bistable waves in an epidemic model. (English) Zbl 1073.34027

A model of the cholera epidemic which spreads in the European Mediterranean regions in 1973 was proposed by V. Capasso and L. Maddalena [SIAM J. Appl. Math. 43, 417–427 (1983; Zbl 0531.92026)] \[ \partial U_1/\partial t=d\partial ^2U_1/\partial x^2-a_{11}U_1+a_{12}U_2, \]
\[ \partial U_2/\partial t=-a_{22}U_2+g(U_1),\quad U_i=U_i(x,t), \] where \(d\), \(a_{ij}\) are positive constants, \(U_1\) and \(U_2\) denotes the spatial densities of infectious agent and the infective human population at a point \(x\) in the habitat at time \(t\geq 0\), \(1/a_{11}\) is the mean lifetime of the agent in the environment, \(1/a_{22}\) is the infectious period of the human infections, \(a_{12}\) is the multiplicative factor of the mean infectious agent due to the human populations, and \(g(x)\) is the infection rate of human under the assumption that total susceptible human population is constant during the evolution of the epidemic.
The paper deals with existence, uniqueness and global exponential stability of travelling waves of the system in the case of a bistable nonlinearity. The first section contains an elaborated introduction into the topic as well as a useful survey of the literature. In the second section, the existence of travelling waves is established by imposing that \(g\in C^2(\mathbb{R}_{+})\), \(g(0)=0\), \(g^{\prime }(0)\geq 0\), \(g^{\prime }(z)>0\) for \(z>0\), lim\(_{z\rightarrow +\infty }g(z)=1\) and there exists a \(\xi >0\) such that \(g^{\prime \prime }(z)>0\) in \((0,\xi )\) and \(g^{\prime \prime }(z)<0\) for \(z>\xi \). The system is rescaled to \[ \partial U_1/\partial t=d\partial ^2U_1/\partial x^2-U_1+\alpha U_2, \]
\[ \partial U_2/\partial t=-\beta U_2+g(U_1), \] where \(\alpha =a_{12}/a_{11}^2\), \(\beta =a_{22}/a_{11}\). Introducing the travelling wave \(U_i(x,t)=u_i(x+ct)\), Theorem 2.1 reads as:
Assume that \(g^{\prime }(0)<\gamma <\gamma _{crit}\) (this value regards the question of equilibria to the associated cooperative system \[ dU_1/dt=-U_1+\alpha U_2,\quad dU_2/dt=-\beta U_2+g(U_1), \] namely \(E^{-}=(0,0)\) - stable node, \(E^0=(a,a/\alpha )\)-saddle point, and \(E^{+}=(b,b/\alpha )\) - stable node, with \(0<a<b\)). Then there exists a wave speed \(c^{*}\) such that the system has a nontrivial strictly monotone wave solution connecting \(E^{-}\) and \(E^{+}\), and \(c^{*}\) has the same sign as \(\int_0^b((\alpha /\beta )g(z)-z)dz\). Moreover, \(c^{*}=0\) if and only if the integral vanishes. In the third section, a more general quasi-monotone system is investigated. Precisely, the term \(\alpha U_2\) in the rescaled system is replaced with \(F^1(U_1,U_2)\), where \(F^1\in C^2((-l,+\infty )^2,\mathbb{R})\) for a certain \(l>0\), \(\partial F^1/\partial u_1<0\), \(\partial F^1/\partial u_2>0\), \(F^1(0,0)=0\) and for any \(l_2\geq 1/\beta \) there exists \(l_1>0\) such that \(F^1(l_1,l_2)<0\). Introducing \(X\) as the Banach space of all bounded and uniformly continuous functions from \(\mathbb{R}\) to \(\mathbb{R}^2\) with the typical sup-norm and the cone \(X_{+}=\{(\psi _1,\psi _2),\psi _i\geq 0\}\), Theorem 3.1 reads as:
Let \(\varphi (x-ct)\) be a monotone travelling wave solution of the more general system introduced above and \(U(x,t,\psi )\) be a solution of this system with \(U(.,0,\psi )=\psi (.)\in X_{+}\). Then, for any \(\psi \in X_{+}\) with limsup\(_{\xi \rightarrow -\infty }\psi (\xi )<<E^0<<\)liminf\(_{\xi \rightarrow \infty }\psi (\xi )\) (it is assumed that the new cooperative system has two nonegative equilibria, the first being the saddle point \(E^0\); here, “\(<<\)” refers to inequality for components), there exists \(s(\psi )\in \mathbb{R}\) such that lim\(_{t\rightarrow +\infty }\left\| U(x,t,\psi )-\varphi (x-ct+s(\psi ))\right\| =0\) uniformly for \(x\in \mathbb{R}\). Moreover, any travelling wave solution of the system connecting \(E^{-}\) and \(E^{+}\) is a translate of \(\varphi \). The fourth section gravitates around Theorem 4.1:
Let \(\varphi (x-ct)\) be a monotone travelling wave solution with \(c\neq 0\). Then, there exists a positive constant \(\mu \) such that for every \(\psi \in X_{+}\) satisfying the condition in Theorem 3.1, the solution \(U(x,t,\psi )\) satisfies \(\left\| U(x,t,\psi )-\varphi (x-ct+s(\psi ))\right\| \leq C(\psi )\)exp\((-\mu t)\) for all \(x\), \(t\), for some constant \(C(\psi )>0\) and \(s(\psi )\).
The paper closes with numerical simulations. The only minus of this excellently written paper is the large amount of misprints.

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
92D30 Epidemiology

Citations:

Zbl 0531.92026
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