Burcu Gürbüz

As offered by the World Health Organisation (WHO), close to half of the population in the world's resides in dengue-risk zones. Dengue viruses are transmitted to individuals by Aedes mosquito species infected bite (Ae. Albopictus of Ae. aegypti). These mosquitoes can transmit other viruses, including Zika and Chikungunya. In this research, a mathematical model is formulated to reflect different time delays considered in both extrinsic and intrinsic incubation periods, as well as in the recovery periods of infectious individuals. Preliminary results for the non-delayed model including positivity and boundedness of solutions, non-dimensionalization and equalibria analysis are presented. The threshold parameter (reproduction number) of the model is obtained via next generation matrix schemes. The stability analysis of the model revealed that various dynamical behaviour can be observed depending on delay parameters, where in particular the effect of delay in the recovery time of infectious individuals may lead to substantial changes in the dynamics. The ideas presented in this paper can be applied to other infectious diseases, providing qualitative evaluations for understanding time delays influencing the transmission dynamics.

Being an example for a relaxation oscillator, the FitzHugh-Nagumo model has been widely employed for describing the generation of action potentials. In this paper, we begin with a biological interpretation of what the subsequent mathematical and numerical analyses of the model entail. The interaction between action potential variable and recovery variable is then revisited through linear stability analysis around the equilibrium and local stability conditions are determined. Analytical results are compared with numerical simulations. The study aims to show an alternative approach regarding Taylor polynomials and constructed difference scheme which play a key role in the numerical approach for the problem. The robustness of the schemes is investigated in terms of convergency and stability of the techniques. This systematic approach by the combination of numerical techniques provides beneficial results which are uniquely designed for the FitzHugh-Nagumo model. We describe the matrix representations with the collocation points. Then the method is applied in order to acquire a system of nonlinear algebraic equations. On the other hand, we apply finite difference scheme and its stability is also performed. Moreover, the numerical simulations are shown. Consequently, a comprehensive investigation of the related model is examined.

In this investigation, we study for a dynamical system aimed at elucidating a disease model under the influence of environmental stress from a broad perspective. The model is articulated through both standard differential equations and their Caputo fractional form. Our methodology involves a numerical approach using the Adams-Bashforth-Moulton technique to solve the system of differential equations, including the initial conditions. The existence, uniqueness and convergence of the technique are also briefly discussed. This study aims not only to improve the current technique, but also to introduce a novel design for obtaining numerical solutions to issues discussed in the existing literature, thus paving the way for further research. We also perform a stability analysis focusing on the coexistence equilibrium. In addition, we present visualisations of the results to elucidate the behaviour of the system, time evolution and phase plane plots with respect to specific parameters.

The theory and applications of differential equations have played an essential role both in the development of mathematics and in exploring new horizons in the applied sciences