Viable control of an epidemiological model

In mathematical epidemiology, epidemic control often aims at driving the number of infected humans to zero, asymptotically. However, during the transitory phase, the number of infected individuals can peak at high values. Can we limit the number of infected humans at the peak? This is the question we address. More precisely, we consider a controlled version of the Ross-Macdonald epidemiological dynamical model: proportions of infected individuals and proportions of infected mosquitoes (vector) are state variables, and vector mortality is the control variable. We say that a state is viable if there exists at least one admissible control trajectory - time-dependent mosquito mortality rates bounded by control capacity - such that, starting from this state, the resulting proportion of infected individuals remains below a given infection cap for all times. The so-called viability kernel is the set of viable states. We obtain three different expressions of the viability kernel, depending on the couple control capacity-infection cap. In the comfortable case, the infection cap is high, the viability kernel is maximal and all admissible control trajectories are viable. In the desperate case, both control capacity and infection cap are too low and the viability kernel is the zero equilibrium without infection. In the remaining viable case, the viability kernel is neither zero nor maximal and not all admissible control trajectories are viable. We provide a numerical application in the case of the dengue outbreak in 2013 in Cali, Colombia.