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# Function pt↦-f(pt)g(pt)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_t\mapsto -\frac{f(p_t)}{g(p_t)}$$\end{document} represented between pt=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_t=0$$\end{document} and pt=0.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_t=0.3$$\end{document} for the parameters at Table 1

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Vector-borne diseases, in particular arboviruses, represent a major threat to human health. In the fight against these viruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. In this work, we investigate optima...

## Citations

... First, the optimisation of the temporal distribution of the releases, neglecting the spatial dependency, has been studied in e.g. [11,6,4,2]. In particular, using a reduced model, the authors in [6] show that the best strategy is close to a bang-bang strategy where the maximum number of mosquitoes available is released at one time at the beginning or end (depending of the number of mosquitoes available) of the release period. ...

This work is devoted to the mathematical study of an optimization problem regarding control strategies of mosquito population in a heterogeneous environment. Mosquitoes are well known to be vectors of diseases, but, in some cases, they have a reduced vector capacity when carrying the endosymbiotic bacterium Wolbachia. We consider a mathematical model of a replacement strategy, consisting in rearing and releasing Wolbachia-infected mosquitoes to replace the wild population. We investigate the question of optimizing the release protocol to have the most effective replacement when the environment is heterogeneous. In other words we focus on the question: where to release, given an inhomogeneous environment, in order to maximize the replacement across the domain. To do so, we consider a simple scalar model in which we assume that the carrying capacity is space dependent. Then, we investigate the existence of an optimal release profile and prove some interesting properties. In particular, neglecting the mobility of mosquitoes and under some assumptions on the biological parameters, we characterize the optimal releasing strategy for a short time horizon, and provide a way to reduce to a one-dimensional optimization problem the case of a long time horizon. Our theoretical results are illustrated with several numerical simulations.

... We also study the robustness of optimal strategies with respect to the convexity of the function chosen to model the cost of the mosquito releases. This chapter is taken from [4]. -Chapter 2: Optimal initial time strategies for mosquito population replacement: influence of the carrying capacity on spatial releases In Chapter 2 we study the optimal spatial distribution of a single initial release in an inhomogeneous environment, assuming that mosquitoes do not diffuse in the domain. ...

... It appears in the definition of α 0 and hereafter due to the fact that g(0) = 1/K. 4. In order to prove this we recall that x → x 2+x is an increasing function of x, we have α 0 ⩽ α max if and only if ...

... Therefore, considering a time window of size T , we want to find u minimizing T 0 I H (t) dt. Other works have studied related problems in the case of Wolbachia [158], or problems involving only the mosquito population [4,6,3,7] considering controls in L ∞ (0, T ). ...

With vector-borne diseases rising globally and mosquitoes expanding their habitats due to climate change,mosquito control is undoubtedly one of the main challenges for human health in the years to come. This thesis isdevoted to the modeling, analysis and simulation of mosquito and mosquito-borne diseases optimal control strategies using modified vector releases. We first investigate optimal population replacement strategies. These consist in replacing optimally the wild population by a population carrying the endosymbiotic bacterium Wolbachia, since it has been shown that mosquitoes carrying this bacterium are less likely to transmit some arboviruses. By considering a high fecundity limit we reduce the study of the mosquito population to a single equation on the proportion of Wolbachia-infected mosquitoes. First, we study strategies optimizing a convex combination of both the cost of the releases and the performance of the technique. We fully analyse this problem, proving a time monotonicity property on the proportion of Wolbachia-infected mosquitoes and using a reformulation of the problem based on a suitable change of variable. Next, we consider the spatial optimization of the releases, optimizing a single instantaneous release at the initial time maximising the final proportion of Wolbachia-infected mosquitoes throughout the domain at a given time horizon. We fully characterize the solutions under some hypothesis in the non-diffusive case. Moreover, simulations are carried for the case with diffusion. Finally, we extend the focus of the study to humans. We consider an epidemiological model in which both populations are taken into account as well as the dynamics of a vector-borne disease with exclusively human-mosquito and mosquito-human transmission like dengue. In this setting, we minimise the amount of human infections during an outbreak using instantaneous releases of modified vectors, represented by linear combinations of Dirac measures with positive coefficients determining their intensity. Optimal strategies for both population replacement and the sterile insect technique are studied numerically using ad-hoc algorithms, based on writing first-order optimality conditions characterizing the best combination of Dirac measures.