  In mathematical epidemiology, epidemic control often aims at driving the
number of infected individuals to zero, asymptotically. However , during the
transitory phase, the number of infected can peak at high values. In this
paper, we consider mosquito vector control in the Ross-Macdonald
epidemiological model, with the goal of capping the proportion of infected by
dengue at the peak. We formulate this problem as one of control of a dynamical
system under state constraint. We allow for time-dependent fumigation rates to
reduce the population of mosquito vector, in order to maintain the proportion
of infected individuals by dengue below a threshold for all times. The
so-called viability kernel is the set of initial states (mosquitoes and
infected individuals) for which such a fumigation control trajectory exists.
Depending on whether the cap on the proportion of infected is low, high or
medium, we provide different expressions of the viability kernel. We also
characterize so-called viable policies that produce, at each time, a fumigation
rate as a function of current proportions of infected humans and mosquitoes,
such that the proportion of infected humans remains below a threshold for all
times. We provide a numerical application in the case of control of a dengue
outbreak in 2013 in Cali, Colombia.
