Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays

Infectious diseases have been ranked in the top ten causes of death by WHO in 2016 and despite of availability of various types of vaccines and antibiotics, a huge population is still dying by infectious diseases every day. This may happen due to, several reasons like, resistance of pathogens to antibiotics, improper hygiene and various types of difficulties in vaccination. It has also inspired mathematical modelers to develop dynamical systems predicting the infections in long run. During their spread in a particular population, infectious diseases show various kind of delays which essentially affects the dynamics. In this paper, a susceptible - vaccinated- exposed - infectious - removed (SVEIR) epidemic model is developed with vaccination and two discrete time delays. The first time delay has been incorporated for the time period used to cure the infectious population and another time delay denote the temporary immunity period. The existence of solution and its boundedness have been established. The local stability of disease free equilibrium in respect of both the delays have been discussed explicitly and we have found threshold values of both delay parameters for the local stability of disease free equilibrium. We have also established the local stability of interior equilibrium following the existence of Hopf-bifurcation. Theoretical result shows that considered model system undergoes a Hopf-bifurcation around the interior equilibrium when the time delay due to time period used to cure the infectious population crosses a threshold value. We have also discussed the direction and stability of delay induced Hopf bifurcation using normal form theory and centre manifold theorem. In presence of delay, by constructing a Lyapunov function, local asymptotic stability of the positive equilibrium point is discussed. The length of delay has been estimated to preserve the stability using Nyquist criterion. With the suitable choices of the parameters, some numerical simulations have been presented in the support of our analytical results. (C) 2019 Elsevier Ltd. All rights reserved.