Stability and Global Hopf Bifurcation Analysis of a Schistosomiasis Transmission Model with Multi-Delays

In this paper, a Schistosomiasis model with multi-delays is proposed to study the impacts of delays on disease transmission. First, the exact expression of the basic reproduction number [Formula: see text] of the model is obtained, which is proved to be the threshold parameter. More precisely, when [Formula: see text], the disease is extinct, and when [Formula: see text], the disease is persistent. Next, the existence of the Hopf bifurcation of the model is discussed with delays as parameters, and the direction and stability of the Hopf bifurcation induced by delays are investigated according to the paradigm theory and the central prevalence theory. In addition, the global extensionality of the local Hopf bifurcation is discussed, and the length of delays is estimated using the Nyquist criterion to determine the stability of the endemic equilibrium. Finally, numerical simulations are presented to demonstrate the main theoretical results and the effect of delays on the disease.