Disease extinction in finite time for a continuous-time SIRS model

This paper addresses the problem of controlling the spread of an epidemic through a multiplicatively controlled SIRS model, in which the control input modulates the infection rate in a bilinear manner. Classical approaches based on asymptotic stability only guarantee disease eradication as time tends to infinity, which may be inadequate for timely and effective intervention. To overcome this limitation, we investigate the finite-time behavior of the infected population, ensuring its convergence to zero either in finite time (FTS) or within a fixed time independent of the initial conditions (FxTS). Moreover, through an appropriate choice of the control gain, this convergence time can be arbitrarily prescribed, leading to prescribed-time stability (PrTS). We propose explicit feedback control laws under which the infected population vanishes in finite, fixed, or prescribed time, while the susceptible and recovered populations converge exponentially to their equilibrium values as time tends to infinity. Numerical simulations are provided to validate the theoretical results, illustrating rapid disease eradication and global system stability. These findings demonstrate the effectiveness of finite, fixed, and prescribed-time control strategies for controlled epidemic models and offer practical guidelines for the design of responsive and robust public health interventions.