Rolling carpet strategy to reduce mosquito populations in two-dimensional space

TL;DR

This work analyzes a two-dimensional reaction-diffusion mosquito model incorporating aquatic E, fertilized F, wild M, and sterile M_s stages to implement a rolling-carpet SIT strategy. By leveraging a monotone framework and constructing radial sub- and super-solutions, the authors prove the existence of forced traveling waves that eradicate the population behind a moving front and preserve the positive equilibrium ahead, under bistable dynamics with Γ(M)=1−e^{−γM} and γ>γ_c (with a quantitative criterion defining γ_0). When releases occur in an annulus and the strategy meets threshold conditions on the release amplitude Λ and width L, extinction propagates at speed c>0; the approach extends to heterogeneous environments K(x) via a comparison principle. Numerically, the rolling carpet reduces the required sterile-male releases from O(T^3) to O(T^2) over a time horizon T, illustrating practical gains for large-area control. The results provide a rigorous, radially symmetric 2D justification for the rolling-carpet SIT and offer guidance for field deployment under spatial heterogeneity.

Abstract

Mosquitoes are vectors of numerous diseases; a strategy to fight the spread of these diseases is to control the vector population. In this article, we focus on the use of the sterile insect technique. Starting from a reaction-diffusion system, we show the existence of 'forced' traveling waves obtained by translating the intervention zone at constant speed. This result is proved in a two-dimensional space by using the radial symmetry.

Figures (5)

• Figure 1: Time and spatial dynamics of the density of female mosquitoes $F$ solution of \ref{['syst0']} with the numerical parameters in Table \ref{['tab:parametre']} in the situation of Proposition \ref{['prop:TWMs0']} where $\gamma_c<\gamma_0<\gamma$. We observe invasion of the species into the domain.
• Figure 2: Time and spatial dynamics of the density of female mosquitoes $F$, solution of \ref{['syst0']}, in the situation where $\gamma_c<\gamma<\gamma_0$ for $\gamma=0.01$ (left) and $\gamma=2.355\, 10^{-3}$ (right). We observe that we may have invasion (left) or natural extinction (right) of the mosquito population.
• Figure 3: Spatial distribution represented at different times T of female mosquitoes F, solution of \ref{['syst0']} in a 2D homogeneous space without any releases of sterile males. Without control, we observe that mosquitoes are invading the central area.
• Figure 4: Spatial distribution represented at different time T of female mosquitoes F (first line) and sterile males (second line) solutions of \ref{['syst0']} in a 2D homogeneous space. With control, we observe extinction of the species in an expanding region.
• Figure 5: Schematic representations of the function $\Gamma(\phi_0)$ and its intersections with the affine function (dotted line) defined by the right hand side of \ref{['rel_stat3']} for small $\gamma$ (left) and for larger $\gamma$ (right). In the latter case there are two intersections $F_1^*<F^*$; moreover, since $\Gamma$ is increasing with respect to $\gamma$, we see clearly that the larger $\gamma$ is, the smaller $F_1^*$ and the larger $F^*$ are.

Theorems & Definitions (42)

• Lemma 1
• Proposition 1
• Lemma 2
• Theorem 1
• Corollary 1
• Lemma 3
• Definition 1
• Lemma 4: Comparison principle
• Lemma 5
• proof