Front propagation in hybrid reaction-diffusion epidemic models with spatial heterogeneity. Part I: Spreading speed and asymptotic behavior

TL;DR

This work analyzes front propagation in a spatially heterogeneous, two-species hybrid reaction-diffusion epidemic model, establishing spreading speeds from the linearized leading edge via the λ-periodic principal eigenvalue k(λ). It proves a hair-trigger-type spreading under suitable spectral conditions and demonstrates global convergence to a positive equilibrium in the homogeneous case, with analogous homogenization results for rapidly oscillating coefficients. The speeds and equilibria of the oscillatory system converge to those of the homogenized system as the period tends to zero, with explicit expressions c_R^* = inf_{λ>0} k(λ)/λ and c_L^* = inf_{λ<0} k(λ)/(-λ) and large-time behavior governed by λ_A and σ. These results provide a rigorous framework for understanding epidemic front propagation in periodically structured environments and the effective behavior under homogenization, with implications for predicting invasion speeds and the long-time distribution of two competing/mutating pathogen strains.

Abstract

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold for the whole system. We study spreading properties of solution fronts when the infection is localized initially. We show that there is a well-defined spreading speed both in the right and left directions and that it can be computed from the linearized equation at the leading edge of the propagation front. Next we study the case where the coefficients are spatially homogeneous and show that, when spreading occurs, every solution to the Cauchy problem converges to the unique positive stationary solution as $t\to\infty$. Finally we consider the case of rapidly oscillating coefficients, that is, when the spatial period of the coefficients, denoted by $\varepsilon$, is very small. We show that there exists a unique positive stationary solution, and that every positive solution to the Cauchy problem converges to this stationary solution as $t\to\infty$. We then discuss the homogenization limit as $\varepsilon\to 0$.

Figures (2)

• Figure 1: Profiles of propagating fronts of \ref{['eq:main-sys']} for different parameter values. (a) Spatially homogeneous coefficients with large mutation rates $\mu_u$, $\mu_v$. In this case, the cooperative zone of system \ref{['eq:main-sys']} is rather large, and the propagating front lies entirely in this zone. As a result, both $u$ and $v$ have monotone profiles, just as in the case of scalar equations. (b) Spatially homogeneous coefficients with small mutation rates $\mu_u$, $\mu_v$. In this case, a large part of the profile of the propagating front lies outside the cooperative zone, and a hump appears on $v$. (c) Spatially periodic case. The coefficients are the same as in (b), except $r_u(x)$ and $r_v(x)$, which have a cosine-like periodic perturbation.
• Figure 2: Illustration of the homogenization process of propagating fronts. In each figure, the reproduction rates $r_u(x), r_v(x)$ are periodic functions of the form $r_i(x) = r_i+A_i\cos((x+\varphi_i)\times(2\pi)/L)$ for $i= u, v$, where $L$ is a varying parameter, and all other coefficients including $A_i$ are spatially homogeneous and fixed. $L$ is relatively large in (a), smaller in (b) and very small in (c). As $L$ decreases, the amplitude of oscillation of the front profiles becomes smaller and the shape ultimately converges to that of a traveling wave for the homogenized problem.

Theorems & Definitions (56)

• Definition 1: Periodic principal eigenpair
• Definition 2: $\lambda$-periodic principal eigenpair
• Definition 3: Dirichlet principal eigenpair
• Proposition 2.1: Properties of $k(\lambda)$
• Proposition 2.2: On the Dirichlet principal eigenvalue for cooperative systems
• Theorem 2.3: Comparison between Dirichlet and $\lambda$-periodic principal eigenvalues
• Proposition 2.4
• Proposition 2.5: Basic boundedness estimate
• Definition 4
• Theorem 2.6: Spreading speeds for front-like initial data