[Paper Review] Control of dengue disease: a case study in Cape Verde

Helena Sofia Rodrigues, M. Teresa T. Monteiro+2 more|arXiv (Cornell University)|Jun 30, 2010

Mathematical and Theoretical Epidemiology and Ecology Models3 citations

TL;DR

This study develops a nonlinear mathematical model to analyze dengue transmission dynamics in Cape Verde, incorporating a control variable for adult mosquito spraying. It demonstrates that maintaining a constant insecticide spray level above 0.0837 can reduce the basic reproduction number ($\mathcal{R}_0$) below 1, thereby preventing endemic transmission and significantly lowering both human and mosquito infection levels.

ABSTRACT

A model for the transmission of dengue disease is presented. It consists of eight mutually-exclusive compartments representing the human and vector dynamics. It also includes a control parameter (adulticide spray) in order to combat the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) --- where humans, with or without mosquitoes, live without the disease --- and another endemic equilibrium (EE). In the literature it has been proved that a DFE is locally asymptotically stable, whenever a certain epidemiological threshold, known as the basic reproduction number, is less than one. We show that if a minimum level of insecticide is applied, then it is possible to maintain the basic reproduction number below unity. A case study, using data of the outbreak that occured in 2009 in Cape Verde, is presented.

Motivation & Objective

• To analyze the transmission dynamics of dengue fever in Cape Verde using a compartmental mathematical model.
• To evaluate the impact of adult mosquito spraying as a control intervention on disease transmission.
• To determine the threshold level of insecticide application required to reduce the basic reproduction number ($\mathcal{R}_0$) below unity.
• To assess the effectiveness of vector control in preventing endemicity in a dengue-naive population during a first outbreak.

Proposed method

• Formulates a system of seven nonlinear ordinary differential equations to model human and mosquito population dynamics across four epidemiological states each.
• Incorporates a time-invariant control parameter $c(t)$ representing the intensity of adult mosquito spraying.
• Derives the basic reproduction number $\mathcal{R}_0$ using the next-generation matrix method, accounting for transmission between humans and *Aedes aegypti* mosquitoes.
• Analyzes equilibrium points (disease-free and endemic) and their stability using Lyapunov functions and eigenvalue analysis.
• Performs numerical simulations using Scilab with parameter values calibrated from Cape Verde and Brazil, assuming initial outbreak conditions.
• Evaluates the effect of varying $c$ on $\mathcal{R}_0$ and infection dynamics, identifying a critical threshold for control efficacy.

Experimental results

Research questions

• RQ1 What is the threshold level of insecticide spraying ($c$) required to reduce $\mathcal{R}_0$ below 1 in Cape Verde?
• RQ2 How does the inclusion of a constant control parameter $c$ affect the stability of disease-free and endemic equilibria?
• RQ3 To what extent can adult mosquito spraying reduce the number of infected humans and mosquitoes during a dengue outbreak?
• RQ4 How do the dynamics of dengue transmission differ between controlled and uncontrolled scenarios in a dengue-naive setting like Cape Verde?

Key findings

• The basic reproduction number $\mathcal{R}_0$ drops below 1 when the insecticide control level exceeds $c = 0.0837$, ensuring the disease-free equilibrium is locally asymptotically stable.
• With $c = 0.084$, simulations show a dramatic reduction in both infected humans and mosquitoes compared to the uncontrolled scenario, with infected mosquito populations driven nearly to zero.
• The model predicts that even a small, sustained spraying campaign can prevent the transition from an epidemic to an endemic state in Cape Verde.
• The Biologically Realistic Disease-Free Equilibrium (BRDFE) exists only when $\mathcal{M} > 0$, which depends on mosquito population parameters and is satisfied under the given Cape Verde conditions.
• Numerical results confirm that $\mathcal{R}_0 < 1$ is achievable with a constant $c > 0.0837$, making long-term disease control feasible through targeted spraying.
• The model highlights the importance of active monitoring and surveillance to validate and sustain the effectiveness of control programs in real-world settings.