Global stabilization and emergence tracking via aquatic control in an age-structured mosquito model

TL;DR

This work develops an age-structured, non-autonomous logistic model for malaria-vector mosquitoes with a targeted aquatic-stage control input $P(t)$. A Lyapunov-based feedback design stabilizes the aquatic population, and a combined feedforward–feedback controller enables tracking of a time-varying reference $y_d(t)$ for the emergent adult density $y(t)=\int_0^A w(a) I(a,t)\,da$, proven via semigroup well-posedness and nonlinear Lyapunov arguments. The main contributions are a global asymptotic stabilization result under aquatic control, an explicit dynamic tracking controller with proven exponential convergence, and numerical demonstrations of robustness to seasonal forcing. The findings suggest that manipulating the aquatic population can effectively regulate the whole vector population, offering a complementary avenue to genetic control methods for malaria management and enabling future extensions to controllability, turnpike properties, and spatial models.

Abstract

This paper presents an age-structured, non-autonomous logistic model describing the aquatic and adult stages of the dynamics of malaria-vector mosquitoes. We propose a biological control strategy targeting the aquatic compartment and implement a tracking control for its emergence. A feedback control law guarantees stabilization of the emergent population density, specifically the global asymptotic stability of the logistic model. Additionally, a feedforward controller combined with feedback is introduced to steer the emergent density toward a time-varying reference trajectory. The analytical findings are corroborated and illustrated by numerical simulations.

Figures (7)

• Figure 1: With $\eta_{I0}=0.3$ as the initial condition, the environmental parameters are chosen to be periodic: $K(t)=K^*(1+0.2\sin(3\pi t/T)),\; \Gamma(t)=\Gamma^*(1+0.3\sin(4\pi t/T))\; \text{and}\; \gamma(t)=\gamma^*(1+0.2\cos(3\pi t/T)).$ By choosing $P(t)=P^*$, we observe that the temporal heterogeneity of these covariates undermines the effectiveness of a static control $P^*$ in ensuring the system’s stability.
• Figure 2: Autonomous case : $\eta_{I0}=0.3\,\text{the initial condition}$ and $P(t)=P^*,\;K(t)=K^*,\;\Gamma(t)=\Gamma^*,\;\gamma(t)=\gamma^*$(see Remark \ref{['re7.5']})
• Figure 3: The time-dependent (non-autonomous) case: the evolution of $\eta_I$ in \ref{['e3.55']} under the time-dependent control given in \ref{['e3.75']}, with initial value $\eta_{I0}=1.007.$
• Figure 4: $\eta_{I0}=0.03;\; K(t)=K^*+0.5K^*e^{-t/10},\; \Gamma(t)=\Gamma^ * (1.0 + 0.25 e^{-t / 40.0}\sin(2\pi t / 15.0)),\; \gamma(t)=\gamma^*(1+0.3e^{-t/8}\sin(2\pi t/20))$ with $P(t)=P^*.$
• Figure 5: $\eta_{I0}=0.03;\; K(t)=K^*+0.5K^*e^{-t/10},\; \Gamma(t)=\Gamma^ * (1.0 + 0.25 e^{-t / 40.0}\sin(2\pi t / 15.0)),\; \gamma(t)=\gamma^*(1+0.3e^{-t/8}\sin(2\pi t/20))$ with the control \ref{['e3.75']}.

Theorems & Definitions (17)

• Remark 3.1
• Lemma 3.2
• Remark 3.3
• Theorem 4.1
• Remark 4.2
• Theorem 4.3
• Theorem 4.4
• Remark 4.5
• Proposition 5.1
• Remark 5.2