Dengue disease, basic reproduction number and control

Dengue is one of the major international public health concerns. Although progress is underway, developing a vaccine against the disease is challenging. Thus, the main approach to fight the disease is vector control. 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 (insecticide) in order to fight the mosquito. The model presents three possible equilibria: two disease-free equilibria (DFE) and another endemic equilibrium. 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 we apply a minimum level of insecticide, it is possible to maintain the basic reproduction number below unity. A case study, using data of the outbreak that occurred in 2009 in Cape Verde, is presented.

💡 Research Summary

The paper presents a deterministic compartmental model for the transmission dynamics of Dengue fever, incorporating both human and vector (Aedes aegypti mosquito) populations. The human population is divided into four epidemiological classes—susceptible (S_h), exposed (E_h), infectious (I_h), and recovered (R_h)—while the mosquito population is split into an aquatic stage (A_m) and three adult stages—susceptible (S_m), exposed (E_m), and infectious (I_m). The total human population N_h is assumed constant, and the model neglects immigration of infected individuals. A novel feature of the work is the inclusion of a control variable c, representing the intensity of adulticide (insecticide) application, which directly reduces the adult mosquito compartments (S_m, E_m, I_m) but does not affect the aquatic stage.

The system of eight ordinary differential equations is derived from standard mass-action transmission assumptions. Key parameters include the daily biting rate B, transmission probabilities β_mh (mosquito‑to‑human) and β_hm (human‑to‑mosquito), human and mosquito mortality rates (μ_h, μ_m), intrinsic and extrinsic incubation periods (ν_h, η_m), and several demographic rates governing mosquito development (μ_b, μ_A, η_A). The control parameter c enters linearly as a mortality term for adult mosquitoes, reflecting the effect of spraying campaigns.

Mathematical analysis focuses on the existence and stability of equilibria. The authors define a composite parameter M = –