Lourdes Esteva

Analysis of a dengue disease transmission model

A model for the transmission of dengue fever in a constant human population and variable vector population is discussed. A complete global analysis is given, which uses the results of the theory of competitive systems and stability of periodic orbits, to establish the global stability of the endemic equilibrium. The control measures of the vector population are discussed in terms of the threshold condition, which governs the existence and stability of the endemic equilibrium.

Influence of vertical and mechanical transmission on the dynamics of dengue disease

We formulate a non-linear system of differential equations that models the dynamics of transmission of dengue fever. We consider vertical and mechanical transmission in the vector population, and study the effects that they have on the dynamics of the disease. A qualitative analysis as well as some numerical examples are given for the model.

Optimal control of Aedes aegypti mosquitoes by the sterile insect technique and insecticide

We present a mathematical model to describe the dynamics of mosquito population when sterile male mosquitoes (produced by irradiation) are introduced as a biological control, besides the application of insecticide. In order to analyze the minimal effort to reduce the fertile female mosquitoes, we search for the optimal control considering the cost of insecticide application, the cost of the production of irradiated mosquitoes and their delivery as well as the social cost (proportional to the number of fertilized females mosquitoes). The optimal control is obtained by applying the Pontryagins Maximum Principle.

Modelling the dynamics of dengue real epidemics

In this work, we use a mathematical model for dengue transmission with the aim of analysing and comparing two dengue epidemics that occurred in Salvador, Brazil, in 1995–1996 and 2002. Using real data, we obtain the force of infection, Λ, and the basic reproductive number, R0, for both epidemics. We also obtain the time evolution of the effective reproduction number, R(t), which results in a very suitable measure to compare the patterns of both epidemics. Based on the analysis of the behaviour of R0 and R(t) in relation to the adult mosquito control parameter of the model, we show that the control applied only to the adult stage of the mosquito population is not sufficient to stop dengue transmission, emphasizing the importance of applying the control to the aquatic phase of the mosquito.

Qualitative study of transmission dynamics of drug-resistant malaria

This paper presents a deterministic model for monitoring the impact of drug resistance on the transmission dynamics of malaria in a human population. The model has a disease-free equilibrium, which is shown to be globally-asymptotically stable whenever a certain threshold quantity, known as the effective reproduction number, is less than unity. For the case when treatment does not lead to resistance development, the model has a wild strain-only equilibrium whenever the reproduction number of the wild strain exceeds unity. It is shown, using linear and nonlinear Lyapunov functions, coupled with the LaSalle Invariance Principle, that this equilibrium is globally-asymptotically stable for a special case. The model has a resistant strain-only equilibrium, which is globally-asymptotically stable whenever its reproduction number is greater than unity and exceeds that of the wild strain. In this case, the two strains undergo competitive exclusion, where the strain with the higher reproduction number displaces the other. Further, for the case when treatment does not lead to resistance development, the model can have no coexistence equilibrium or a continuum of coexistence equilibria. When treatment leads to resistance development, the model can have a unique coexistence equilibrium or a resistant-only equilibrium. This coexistence equilibrium is shown to be locally-asymptotically stable, using a technique based on Krasnoselskii sub-linearity argument. Numerical simulations of the model show that for high treatment rates, the resistant strain can dominate, and drive out, the wild strain. Finally, when the two strains co-exist, the proportion of individuals with the resistant strain at steady-state decreases with increasing rate of resistance development.

A MODEL FOR VECTOR TRANSMITTED DISEASES WITH SATURATION INCIDENCE

A model for a disease that is transmitted by vectors is formulated. All newborns are assumed susceptible, and human and vector populations are assumed to be constant. The model assumes a saturation effect in the incidences due to the response of the vector to change in the susceptible and infected host densities. Stability of the disease free equilibrium and existence, uniqueness and stability of the endemic equilibrium is investigated. The stability results are given in terms of the basic reproductive number R0.

Control measures for Chagas disease

Chagas disease, also known as American trypanosomiasis, is a potentially life-threatening illness caused by the protozoan parasite, Trypanosoma cruzi. The main mode of transmission of this disease in endemic areas is through an insect vector called triatomine bug. Triatomines become infected with T. cruzi by feeding blood of an infected person or animal. Chagas disease is considered the most important vector borne infection in Latin America. It is estimated that between 16 and 18 millions of persons are infected with T. cruzi, and at least 20,000 deaths each year. In this work we formulate a model for the transmission of this infection among humans, vectors and domestic mammals. Our main objective is to assess the effectiveness of Chagas disease control measures. For this, we do sensitivity analysis of the basic reproductive number R₀ and the endemic proportions with respect to epidemiological and demographic parameters.

Seasonality and Outbreaks in West Nile Virus Infection

In this paper we analyze the impact of seasonal variations on the dynamics of West Nile Virus infection. We are interested in the generation of new epidemic peaks starting from an endemic state. In many cases, the oscillations generated by seasonality in the dynamics of the infection are too small to be observable. The interplay of this seasonality with the epidemic oscillations can generate new outbreaks starting from the endemic state through a mechanism of parametric resonance. Using experimental data we present specific cases where this phenomenon is numerically observed.

ASSESSING THE SUITABILITY OF STERILE INSECT TECHNIQUE APPLIED TO AEDES AEGYPTI

The efficacy of biological control of Aedes aegypti mosquitoes using Sterile Insect Technique (SIT) is analyzed. This approach has shown to be very efficient on agricultural plagues and has become an alternative control strategy to the usual technique of insecticide application, which promotes resistance against chemical controls and is harmful to other species that live in the same mosquito habitat. By using a discrete cellular automata approach we have shown that in the case of Aedes aegypti, the spatially heterogeneous distribution of oviposition containers and the mosquito behavior, especially with respect to mating, make the application of STI difficult or impracticable.

Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations

We formulate a mathematical model that describes the population dynamics of bacteria exposed to multiple antibiotics simultaneously, assuming that acquisition of resistance is through mutations due to antibiotic exposure. Qualitative analysis reveals the existence of a free-bacteria equilibrium, resistant-bacteria equilibrium and an endemic equilibrium where both bacteria coexist.