Optimization approaches to Wolbachia-based biocontrol

TL;DR

It turns out that a direct discrete-time optimization (second method) renders better quantifiable results compared to transforming a continuous-time optimal release function into a sequence of suboptimal impulses (first method).

Abstract

This paper proposes two realistic and biologically viable methods for Wolbachia-based biocontrol of Aedes aegypti mosquitoes, assuming imperfect maternal transmission of the Wolbachia bacterium, incomplete cytoplasmic incompatibility, and direct loss of Wolbachia infection caused by thermal stress. Both methods are based on optimization approaches and allow for the stable persistence of Wolbachia-infected mosquitoes in the wild Ae. aegypti populations in a minimum time and using the smallest quantity of Wolbachia-carrying insects to release. The first method stems from the continuous-time optimal release strategy, which is further transformed into a sequence of suboptimal impulses mimicking instantaneous releases of Wolbachia-carrying insects. The second method constitutes a novel alternative to all existing techniques aimed at the design of release strategies. It is developed using metaheuristics ($ε$-constraint method combined with the genetic algorithm) and directly produces a discrete sequence of decisions, where each element represents the quantity of Wolbachia-carrying mosquitoes to be released instantaneously and only once per a specified time unit. It turns out that a direct discrete-time optimization (second method) renders better quantifiable results compared to transforming a continuous-time optimal release function into a sequence of suboptimal impulses (first method). As an illustration, we provide examples of daily, weekly, and fortnightly release strategies designed by both methods for two Wolbachia strains, wMel and wMelPop.

Figures (9)

• Figure 1: Phase portraits to the system \ref{['system']} with $u(t)=0$ under the conditions \ref{['WB-cond']}, \ref{['cond-viab']}, \ref{['coex-ex']} and parameter values presented in Table \ref{['tab:model parameters']}
• Figure 2: Illustration of the Crossover operation with two crossover points $r1=2$ and $r2=5$.
• Figure 3: Illustration of the Mutation operation on the gene's sequence from gene $r3=3$ to gene $r4=5$.
• Figure 4: Numerical solutions to the OCP \ref{['ocp']} for wMel strain: (a) optimal control $u^{*}(t), t \in [ 0, T^{*} ]$ with $T^{*}=13.72$ days; (b) optimal state trajectories $x^{*}(t)$ and $y^{*}(t), \ t \in [0, 400]$
• Figure 5: Numerical solutions to the OCP \ref{['ocp']} for wMelPop strain: (a) optimal control $u^{*}(t), t \in [ 0, T^{*} ]$ with $T^{*}=64.87$ days; (b) optimal state trajectories $x^{*}(t)$ and $y^{*}(t), \ t \in [0, 400]$