Dengue disease, basic reproduction number and control

Abstract

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.Comment: This is a preprint of a paper whose final and definitive form has appeared in International Journal of Computer Mathematics (2011), DOI: 10.1080/00207160.2011.55454

Understanding Dengue Control for Short- and Long-Term Intervention with a Mathematical Model Approach

“…The intervention has not yet been used into the mathematical model [16]. In [19], the mathematical model with only insecticide campaign intervention is discussed. It has been shown that, with a steady insecticide campaign, it is possible to reduce the number of infected humans and mosquitoes and prevent an outbreak that could transform an epidemiological episode to an endemic disease [19].…”

“…In [19], the mathematical model with only insecticide campaign intervention is discussed. It has been shown that, with a steady insecticide campaign, it is possible to reduce the number of infected humans and mosquitoes and prevent an outbreak that could transform an epidemiological episode to an endemic disease [19]. A year after the research discussed in [19], it was updated [20], and the mathematical model for dengue was updated continuously with all controls included, that is, (1) proportion of larvicide, (2) proportion of adulticide, and (3) proportion of mechanical control.…”

“…(2) Humans and mosquitoes are assumed to be born susceptible, there is no natural protection, and dengue is not passed onto the next generation (no vertical transmission) [19]. (3) The transmission process in susceptible and vaccinated humans is simply by the bite of an infected mosquito.…”

Optimal control strategies for tuberculosis treatment: A case study in Angola

“…This is the expected number of secondary cases produced in a complete susceptible population, by a typical infected individual during his/her entire period of infectiousness [15]. In the absence of controls (u 1 = u 2 ≡ 0), it is known (see [5]) that the basic reproduction number for system (1) is proportional to the transmission coefficient β:…”

“…While the usefulness of optimal control theory in epidemiology is nowadays well recognized [2,9,10,11,14,15], results pertaining to tuberculosis are a rarity [7], and specific studies for the situation of Angola nonexistent. Our aim in this paper is thus to use real data from Angola to study optimal strategies for the minimization of the number of active TB infectious and persistent latent individuals, taking into account the cost of the measures for the treatment of these individuals.…”