Dynamics of a delayed SIR model for the transmission of PRRSV among a swine population

The objective of this paper is to propose a delayed susceptible-infectious-recovered (SIR) model for the transmission of porcine reproductive respiratory syndrome virus (PRRSV) among a swine population, including the latent period delay of the virus and the time delay due to the period the infectious swines need to recover. By taking different combinations of the two delays as the bifurcation parameter, local stability of the disease-present equilibrium and the existence of Hopf bifurcation are analyzed. Sufficient conditions for global stability of the disease-present equilibrium are derived by constructing a suitable Lyapunov function. Directly afterwards, properties of the Hopf bifurcation such as direction and stability are studied with the aid of the normal form theory and center manifold theorem. Finally, numerical simulations are presented to justify the validity of the derived theoretical results.