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Analysis and numerical simulation of a spatio-temporal Ricker-type model for the control of Aedes aegypti mosquitoes with Sterile Insect Techniques

Oscar Eduardo Escobar-Lasso, Stefan Frei, Reinhard Racke, Olga Vasilieva

Abstract

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

Analysis and numerical simulation of a spatio-temporal Ricker-type model for the control of Aedes aegypti mosquitoes with Sterile Insect Techniques

Abstract

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

Paper Structure

This paper contains 15 sections, 4 theorems, 51 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A generalized SIT-based model
  3. Analysis of the model
  4. Discretization
  5. Temporal Discretization
  6. Spatial Discretization by Finite Elements
  7. Numerical results
  8. Homogeneous Releases of Sterile Insects
  9. Nonhomogeneous Releases and Numerical Convergence Study
  10. Time-stepping Schemes
  11. Influence of Initial Conditions
  12. Location of the Control
  13. Impulsive Control
  14. Conclusions
  15. Acknowledgement

Key Result

Proposition 1

For each $X=(M,F,M_S)$ and $\widehat{X}=(\widehat{M},\widehat{F},\hat{M}_S)$ with $M,F,M_S, \widehat{M},\widehat{F},\hat{M}_S \geq 0$ and $M+\gamma M_S >0$, we have where Moreover, we have for $f$

Figures (11)

  • Figure 1: Evolution of $\|M \|_\Omega$ and $\|F \|_\Omega$ over time for constant releases $\Lambda =0.9\Lambda^{crit}$ (top) resp. $\Lambda =1.1\Lambda^{crit}$ (bottom) and $M_0=F_0=80$ (left) resp. $M_0=F_0=85$ (right).
  • Figure 2: Left: Illustration of $M_S$ plotted over the domain $\Omega$ at times $t=2$ and $t=10$ (left to right), right: Illustration of $M$ over $\Omega$ at times $t=2$ and $t=10$.
  • Figure 3: Plot over the diagonal line $\Gamma_D$ of $M$ at times $t\in \{4,10,20,50\}$ for different time-stepping schemes ($\theta$) and time-step sizes $\delta t$.
  • Figure 4: Visualization of the three initial conditions: (a) uniform, (b) sinusoidal, (c) Gaussian.
  • Figure 5: Plot of $M$ over the diagonal line $\Gamma_D$ at times $t=0$, $t=5$ and $t=20$ (left to right) for different initial conditions $M_{0}$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 1
  • proof