Fractional-order delayed Ross-Macdonald model for malaria transmission.

This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.