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Optimal strategies for Wolbachia mosquito replacement technique: influence of the carrying capacity on spatial releases

Luis Almeida, Jesús Bellver Arnau, Gwenaël Peltier, Nicolas Vauchelet

TL;DR

The paper addresses optimizing Wolbachia-based mosquito replacement in landscapes with spatially varying carrying capacity by deriving a reduced scalar model for the infection proportion under high fertility and negligible mobility. It formulates a global-release optimal control problem tied to an initial condition $p_0$ and proves structural results: for short horizons ($T\le T_0$) the optimal release is unique up to rearrangement and concentrates where $K(x)$ is large; for long horizons ($T> T_0$) the problem reduces to a one-dimensional optimization over a Lagrange multiplier, with a secondary shape problem in regions where the solution is not uniquely determined. Numerical implementations in 1D and 2D demonstrate how carrying capacity heterogeneity shapes release strategies and confirm the theoretical insights, offering practical guidelines for spatially targeted releases. The study advances the understanding of spatially heterogeneous optimization in population replacement and lays groundwork for extensions to include diffusion and more realistic mobility patterns."

Abstract

This work is devoted to the mathematical study of an optimization problem regarding control strategies of mosquito population in a heterogeneous environment. Mosquitoes are well-known vectors of diseases. For some diseases, such as dengue, it has been found that mosquitoes have a reduced vector capacity when carrying the endosymbiotic bacterium Wolbachia. We consider a mathematical model of a replacement technique consisting in rearing and releasing Wolbachia-infected mosquitoes to replace the wild population. Our goal is to optimize the release protocol to maximize replacement effectiveness in a spatially inhomogeneous environment. Using a scalar model with space-dependent carrying capacity, we explore the existence and properties of an optimal release profile maximizing the replacement across the domain. In particular, neglecting mosquito mobility and under some assumptions on the biological parameters, we characterize the optimal releasing strategy for a short time horizon, and we reduce the case of a long time horizon to a one-dimensional optimization problem. Our theoretical results are illustrated with several numerical simulations.

Optimal strategies for Wolbachia mosquito replacement technique: influence of the carrying capacity on spatial releases

TL;DR

The paper addresses optimizing Wolbachia-based mosquito replacement in landscapes with spatially varying carrying capacity by deriving a reduced scalar model for the infection proportion under high fertility and negligible mobility. It formulates a global-release optimal control problem tied to an initial condition and proves structural results: for short horizons () the optimal release is unique up to rearrangement and concentrates where is large; for long horizons () the problem reduces to a one-dimensional optimization over a Lagrange multiplier, with a secondary shape problem in regions where the solution is not uniquely determined. Numerical implementations in 1D and 2D demonstrate how carrying capacity heterogeneity shapes release strategies and confirm the theoretical insights, offering practical guidelines for spatially targeted releases. The study advances the understanding of spatially heterogeneous optimization in population replacement and lays groundwork for extensions to include diffusion and more realistic mobility patterns."

Abstract

This work is devoted to the mathematical study of an optimization problem regarding control strategies of mosquito population in a heterogeneous environment. Mosquitoes are well-known vectors of diseases. For some diseases, such as dengue, it has been found that mosquitoes have a reduced vector capacity when carrying the endosymbiotic bacterium Wolbachia. We consider a mathematical model of a replacement technique consisting in rearing and releasing Wolbachia-infected mosquitoes to replace the wild population. Our goal is to optimize the release protocol to maximize replacement effectiveness in a spatially inhomogeneous environment. Using a scalar model with space-dependent carrying capacity, we explore the existence and properties of an optimal release profile maximizing the replacement across the domain. In particular, neglecting mosquito mobility and under some assumptions on the biological parameters, we characterize the optimal releasing strategy for a short time horizon, and we reduce the case of a long time horizon to a one-dimensional optimization problem. Our theoretical results are illustrated with several numerical simulations.
Paper Structure (22 sections, 8 theorems, 100 equations, 9 figures, 1 table)

This paper contains 22 sections, 8 theorems, 100 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of a mathematical model
  3. The starting point: a two populations model with diffusion
  4. Model simplification
  5. High fertility.
  6. Negligible mobility and initial release
  7. Optimal control problem studied
  8. Analysis of the optimal problem
  9. Preliminaries
  10. Optimality conditions
  11. Study of the switch function
  12. Sign of $\frac{\partial w}{\partial p_0}$ at $p_0=0$.
  13. Conclusion.
  14. The case $T\leq T_{0}$
  15. The case $T > T_{0}$
  16. ...and 7 more sections

Key Result

Lemma 3.1

If $u_0^*=K(x)G(p_0^*)$ is an optimal solution of the optimal control Problem prob:u0 and $M|\Omega|>C$, then $\int_\Omega u_0^*(x)\,dx = C$, or, equivalently, $\int_\Omega K(x) G(p_0^*(x))\,dx = C$. Also, if $M|\Omega| \leq C$, the optimal solution is given by $u_0^* = M$ or equivalently $p_0^* = p

Figures (9)

  • Figure 1: Typical shape of $p_0\mapsto w_{T}(p_0)$, in the case $T\leq T_0$.
  • Figure 2: Schematic representation of $w$, as a function of $p_0$ in case $T > T_0$. As $p_M$ increases (from top to bottom) the three diagrams, that we call A, B and C, show the three possible relative positions of $w(0)$, $w(p_M)$ and $\min_{p_0} w$.
  • Figure 3: Solutions to problems \ref{['prob:u0']}, $u_0^*(x)$, and \ref{['prob:p0']}, $p_0^*(x)$, with a continuous carrying capacity $K(x):=K_S(x)$ as defined in \ref{['K_simulations']}, in $\Omega=[0,1]$, for different time horizons and amounts of available mosquitoes. $p^*(T,x)$ stands for the solution of equation \ref{['eq:psimpl']} with initial data $p_0^*(x)$. The values of $K(x)$ and $u_0^*(x)$ are read on the left axis, the values of $p_0^*(x)$ and $p^*(T,x)$ are read on the right axis. In green, the threshold $p=\theta$.
  • Figure 4: Solutions to problems \ref{['prob:u0']}, $u_0^*(x)$, and \ref{['prob:p0']}, $p_0^*(x)$, with a piece-wise constant carrying capacity $K(x):=K_P(x)$ as defined in \ref{['K_simulations']}, in $\Omega=[0,1]$, for different time horizons and amounts of available mosquitoes. $p^*(T,x)$ stands for the solution of equation \ref{['eq:psimpl']} with initial data $p_0^*(x)$. The values of $K(x)$ and $u_0^*(x)$ are read on the left axis, the values of $p_0^*(x)$ and $p^*(T,x)$ are read on the right axis. In green, the threshold $\theta$.
  • Figure 5: Alternative arrangement of solutions to problems \ref{['prob:u0']}, $u_0^*(x)$, and \ref{['prob:p0']}, $p_0^*(x)$, with a piece-wise constant carrying capacity $K(x):=K_P(x)$ as defined in \ref{['K_simulations']}, in $\Omega=[0,1]$, with $C=30$ and $T=25$. $p^*(T,x)$ stands for the solution of equation \ref{['eq:psimpl']} with initial data $p_0^*(x)$. The values of $K(x)$ and $u_0^*(x)$ are read on the left axis, the values of $p_0^*(x)$ and $p^*(T,x)$ are read on the right axis. In green, the threshold $\theta$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Remark 3.7
  • Theorem 3.8
  • Remark 3.9
  • Corollary 3.10
  • ...and 2 more