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.

Thermal transport in dimerized harmonic lattices: Exact solution, crossover behavior, and extended reservoirs.

We present a method for calculating analytically the thermal conductance of a classical harmonic lattice with both alternating masses and nearest-neighbor couplings when placed between individual Langevin reservoirs at different temperatures. The method utilizes recent advances in analytic diagonalization techniques for certain classes of tridiagonal matrices. It recovers the results from a previous method that was applicable for alternating on-site parameters only, and extends the applicability to realistic systems in which masses and couplings alternate simultaneously. With this analytic result in hand, we show that the thermal conductance is highly sensitive to the modulation of the couplings. This is due to the existence of topologically induced edge modes at the lattice-reservoir interface and is also a reflection of the symmetries of the lattice. We make a connection to a recent work that demonstrates thermal transport is analogous to chemical reaction rates in solution given by Kramers' theory [Velizhanin et al., Sci. Rep. 5, 17506 (2015)]2045-232210.1038/srep17506. In particular, we show that the turnover behavior in the presence of edge modes prevents calculations based on single-site reservoirs from coming close to the natural-or intrinsic-conductance of the lattice. Obtaining the correct value of the intrinsic conductance through simulation of even a small lattice where ballistic effects are important requires quite large extended reservoir regions. Our results thus offer a route for both the design and proper simulation of thermal conductance of nanoscale devices.