Stability and bifurcation analysis of a heroin model with diffusion, delay and nonlinear incidence rate

As we all know, the use of heroin and other drugs in Europe and more specifically in Ireland and the resulting prevalence are well documented. A huge population is still dying using heroin every day. This may happen due to, several reasons like, excessive use of painkiller, lack of awareness etc. It has also inspired mathematical modelers to develop dynamical systems predicting the use of heroin in long run. In this work, the effect of heroin in Europe has been discussed by constructing a suitable mathematical model. Our model describes the process of treatment for heroin users by consolidating a sensible utilitarian structure that speaking to the restricted accessibility of treatment. In the treatment time frame, because of the discretion of the medication clients, some kind of time delay called immunity delay might be found. The effect of immunity delay on the system's stability has been examined. The existence of positive solution and its boundedness has been established. Also, the local stability of the interior equilibrium point has been studied. Taking the immunity delay as the key parameter, the condition for Hopf-bifurcation has been studied. Using normal form theory and center manifold theorem, we have likewise talked about the direction and stability of delay induced Hopf-bifurcation. The corresponding reaction diffusion system with Dirichlet boundary condition has been considered and the Turing instability has been studied. Obtained solutions have also been plotted by choosing a suitable value of the parameters as the support of our obtained analytical results.