Travelling Waves in Wolbachia Spread Dynamics

TL;DR

The paper develops a Wolbachia-infected spread model using a bistable growth function coupled with spatial dispersal via four kernels in an integro-difference framework. It establishes existence and uniqueness (up to translation) of monotone traveling waves and derives a wave-speed representation linking growth, dispersal, and the wave profile. The analysis reveals that fat-tailed dispersal (Cauchy) yields faster waves and higher establishment thresholds, while thinner-tailed kernels slow spread and reduce thresholds; wave speed scales with dispersal MAD. Numerical experiments in 1D and 2D confirm theoretical predictions and quantify how landscape-mediated movement shapes invasion potential, informing targeted vector-control release strategies. Overall, the framework provides a rigorous link between dispersal mechanics and Wolbachia invasion dynamics across heterogeneous landscapes.

Abstract

Wolbachia, a maternally transmitted endosymbiont, offers a powerful biological control strategy for mosquito-borne diseases such as dengue, Zika, and malaria. We develop an integro-difference equation (IDE) model that integrates Wolbachia's nonlinear growth with spatially explicit mosquito dispersal kernels to study invasion dynamics in heterogeneous landscapes. Analytical results establish the existence and uniqueness of monotone traveling waves and provide explicit estimates of invasion speeds as functions of dispersal and growth parameters. Four kernels: Gaussian, Laplace, exponential square-root, and Cauchy, represent a continuum from short- to long-range movement. Fat-tailed kernels generate faster, broader wavefronts, while compact ones limit spread. We also identify a critical bubble, the minimal localized profile required for sustained invasion. Numerical simulations in one- and two-dimensional domains confirm theoretical predictions and reveal parameter regimes governing invasion success. This framework quantifies how dispersal mechanisms shape Wolbachia's spread, thus informing targeted and efficient vector-control strategies.

Figures (9)

• Figure 1: Threshold behavior for Wolbachia establishment in 1-D. Left: Infection decays to extinction when the release is below the threshold. Middle: Infection establishes locally and generates an outward-propagating traveling wave when the initial release exceeds the threshold. Right: Time series of the infection fraction at the release center, $x=0$, for the left and middle panels (near the threshold), illustrating the bistable dynamics. An approximation of the critical bubble is achieved around $t\approx 1000$.
• Figure 2: Dispersal kernels used in the simulations with controlled median absolute deviation $d = 200 m$. Kernel definitions and parameters are given in \ref{['tab:kernels', 'tab:kernels_PDFs']}.
• Figure 3: Space–time dynamics of solutions using, from left to right, Gaussian, Laplace, Exponential square-root, and Cauchy kernels. Kernel parameters are given in \ref{['tab:kernels']}. Color indicates infection fraction from 0 (blue) to 1 (yellow). Traveling-wave behavior is observed for all kernels; propagation speeds at final time rank Gaussian $<$ Laplace $<$ Exponential square-root $<$ Cauchy under chosen parameter settings.
• Figure 4: Solution profiles at $t=25,\,50,\,100$ for all the kernels. Same numerical configurations as \ref{['fig:comp_solu_3D']}. The Exponential square-root wavefront is caught up by the Cauchy wavefront around $t\approx 50$.
• Figure 5: Left: Time series of numerical wave velocities for Gaussian, Laplace, Exponential square-root, and Cauchy kernels at baseline. Right: Traveling wave velocity for varying MAD ($d$) for the four kernels. The vertical red dashed line indicates the baseline scenario seen in the left panel. The wave speed increases linearly in MAD.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Lemma 3.1: Monotonicity and Order Preservation
• Lemma 3.2: Tail limits
• Lemma 3.3: Contact
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Remark 3.6: Direction criterion