Asymptotic Analysis for a nonlinear Reaction-Diffusion System Modeling an Infectious Disease

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as cholera. One of the features in this model is that a certain portion of the recovered human hosts lost a lifetime immunity and could be infected again. Another feature in the model is that the mobility for each species is assumed to be dependent upon location and time. We also assume that the whole group is susceptible with the bacteria. This leads to a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on known parameters and functions. Moreover, the long-time behavior and stability analysis for the solution are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability for all steady-state solutions. These results obtained in this paper answered several open questions raised in the previous literature

Kinetics of the Catalytic Combustion of Ethanol and Ethyl Acetate with Estimation of Activation Energy and Rate Constants: An Analytical Study

: A mathematical model for the combustion of ethanol and ethyl acetate mixtures using Mn9Cu1 (mixture of manganese and copper with a weight ratio of 9:1) catalyst is discussed. The model’s kinetic mechanism is expressed in terms of nonlinear reaction-diffusion equations with common initial and boundary conditions in a finite planar, cylindrical, and spherical geometry. A Taylor series approach is used to derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various values of rate constants, diffusion, and kinetic parameters. The effect of shape factor for the planar, cylindrical, and spherical geometry of dispersed particles was examined for the first time. Activation energy and rate constant at the reference temperature of ethanol, acetaldehyde, and ethyl acetate are also obtained from the rate equations. A direct comparison with numerical simulations confirms the accuracy of the derived analytical results. Background: A mathematical model for the combustion of ethanol and ethyl acetate mixtures using Mn9Cu1 (mixture of manganese and copper with a weight ratio of 9:1) catalyst is discussed. The model’s kinetic mechanism is expressed in terms of nonlinear reaction-diffusion equations with common initial and boundary conditions in a finite planar, cylindrical, and spherical geometry. Objective: Derive general approximate analytical expressions of ethanol, acetaldehyde, and ethyl acetate molar concentrations inside the particle and reactor phase for various parameter values. Method: We employ the simple and reliable Taylor series method. Results: semi-analytic expressions of the concentration and bulk concentration of ethanol, ethyl acetate, and acetaldehyde. Conclusion: Approximate analytical expressions of the concentrations of ethanol, acetaldehyde and ethyl acetate were derived for arbitrary catalyst particle (planar, cylindrical and spherical) by using a simple, reliable, and robust method. In addition, the concentration of the species in reactor phase was also reported. The effects of the kinetic parameters, which are influenced by adsorption equilibrium constant, effective diffusivity, activation energy, on concentration, were discussed.