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

Junchen Zou,Zizhen Zhang,Ranjit Kumar Upadhyay,A Pratap

doi:10.1186/s13662-020-02814-7

Abstract

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.

Highlights

Porcine reproductive respiratory syndrome, known as blue ear disease and epidemic abortion respiratory syndrome, is caused by Lelystad virus
(2020) 2020:351 use mathematical modeling describing the propagation of porcine reproductive respiratory syndrome virus among a swine population
Mathematical modeling has been extensively used to study and predict the propagation of infectious diseases in populations, and in view of this point, we propose a delayed SIR model for the transmission of porcine reproductive respiratory syndrome virus (PRRSV) among a swine population by incorporating two delays into the model formulated in [26]

Summary

Introduction

Known as blue ear disease and epidemic abortion respiratory syndrome, is caused by Lelystad virus. Since porcine reproductive respiratory syndrome is a devastating infectious disease among a swine population, it is reasonable to. Zou et al Advances in Difference Equations (2020) 2020:351 use mathematical modeling describing the propagation of porcine reproductive respiratory syndrome virus among a swine population. Arruda et al [20,21,22,23] proposed different forms of a stochastic model to investigate transmission dynamics of porcine reproductive respiratory syndrome virus. Where τ1 is latent period delay of the porcine reproductive respiratory syndrome virus; τ2 is time delay due to the period the infectious swines need to recover. Further one can obtain the equation with respect to ω as follows: ω12 + L35ω10 + L34ω8 + L33ω6 + L32ω4 + L31ω2 + L30 = 0.

Direction and stability of Hopf bifurcation

Conclusions