Malaria Transmission Model with Transmission-Blocking Drugs and a Time Delay

A class of transmission-blocking drugs (TBDs) that block the transmission of parasites between humans and mosquitoes has recently been shown to be effective in controlling malaria transmission. In this paper, we develop a time-delay differential equation model for malaria using TBDs intervention, in which the human population consists of a treated class and a successfully treated class. In classifying the positive equilibria, the control reproduction number was obtained and the forward and backward branching cases were explored. Then, by constructing a Lyapunov function, the disease-free equilibrium is globally asymptotically stable under certain conditions. In addition, when , the model exhibits Hopf bifurcation, the positive equilibrium becomes unstable from stable, and the model exhibits a periodic solution due to the change of time delay. On the other hand, it is concluded that the use of TBDs has a positive effect on disease control when the treatment rate and the efficacy of TBDs meet certain conditions. Finally, numerical simulation was used to observe the effect of treatment rate and the efficacy of TBDs on , and it was found that the increase in the efficacy of TBDs had a more pronounced effect on disease control compared to treatment rate.