Mathematical modelling and optimal control of the transmission dynamics of enterovirus

We propose a mathematical model for the transmission dynamics of enterovirus, with novelty of incorporating indirect transmission from the environment, inclusion of control interventions (sanitation and hygiene) and optimal control study. We prove that if the basic reproduction number  0 ≤ 1 , a suitable Lyapunov function is used to establish the global stability of the disease free equilibrium, in which case the infection will die out over time. Our analysis further establish the global stability of the endemic equilibrium based on the approach of Volterra-Lyapunov matrices if  0 > 1 . Our findings show that when  0 > 1 , the endemic equilibrium is globally asymptotically stable. In this case, the enterovirus will invade the population. It is shown that by reducing direct transmission rate by 80%, the basic reproduction number can be reduced below one and thus controlling the infection. Using optimal control, it is shown that the disease can be controlled within a shorter period of time as compared to minimizing the direct contact rate by 80%. Numerical simulations are provided to illustrate the results.