Global stabilizing feedback law for a problem of biological control of mosquito-borne diseases

Abstract

Abstract-The control of the spread of dengue fever by introduction of the intracellular parasitic bacterium Wolbachia in populations of the vector Aedes aegypti, is presently one of the most promising tools for eliminating dengue, in the absence of an efficient vaccine. The success of this operation requires locally careful planning to determine the adequate number of mosquitoes carrying the Wolbachia parasite that need to be introduced into the natural population. The latter are expected to eventually replace the Wolbachia-free population and guarantee permanent protection against the transmission of dengue to human.In this paper, we propose and analyze a model describing the fundamental aspects of the competition between mosquitoes carrying Wolbachia and mosquitoes free of the parasite. We then introduce a simple feedback control law to synthesize an introduction protocol, and prove that the population is guaranteed to converge to a stable equilibrium where the totality of mosquitoes carry Wolbachia. The techniques are based on the theory of monotone control systems, as developed after Angeli and Sontag. Due to bistability, the considered inputoutput system has multivalued static characteristics, but the existing results are unable to prove almost-global stabilization, and ad hoc analysis has to be conducted.

Establishing wMelPop Wolbachia Infection among Wild Aedes aegypti Females by Optimal Control Approach

“…have fewer possibilities to produce viable offspring than their Wolbachia-infected coevals. This fact comes from scientific evidence and has been reflected in other models describing Wolbachia invasion [3,5,6,35,39]. Furthermore, our model allows to estimate the threshold in the population size of wild female mosquitoes below which an average uninfected female produces less viable offspring than an average Wolbachia-infected female.…”

“…The above arguments are reflected in different mathematical models describing Wolbachia invasion in wild mosquito populations (see, e.g., [3,5,6,35,39] among others) and all these models have the common feature that consists in the bistable nature of Wolbachia dynamics and the existence of certain threshold in the infection frequency 3 above which the invasion and stabilization of Wolbachia can be achieved. The latter has a simple explanation since wild males and females a scarce at high frequencies of Wolbachia infection, while Wolbachia-carriers (both males and females) are abundant.…”

“…In this case, the model can be still considered realistic only if we set K 0 = 0. However, our interest consist in modeling the interaction (competition) of two mosquito species; therefore, the presence of Wolbachia-carrying female in the dynamical system (3) is compulsory and 3 By infection frequency we understand the share of infected individuals with respect to total mosquito population, that is, W /(F +W ). cannot be ignored.…”

Galton-Watson Process and Bayesian Inference: A Turnkey Method for the Viability Study of Small Populations

“…(II) How sharply are numbers expected to fall in the short term? (III) How can we proceed to avoid extinction [14,15], to maintain a population size [43] or wipe out some population [6,33,23]. For instance, if we are able to add some individuals in a population, how many re-introductions are needed to reach a threshold of variability for the population with high probability?…”

A mathematical model for mosquito infestation

“…The introduction of bacteria in mosquitoes is called bacterial infestation, many researchers propose mathematical predictions for biological control of different host-vector diseases. These kind of differential equation models are described in ( [1], [2], [4]) . We refer to the model of infested population only models, type S-I epidemic models.…”