Mathematical Modeling of Toxoplasmosis Considering a Time Delay in the Infectivity of Oocysts

Dynamics of toxoplasmosis in the cat's population with an exposed stage and a time delay

We propose a new mathematical model to investigate the effect of the introduction of an exposed stage for the cats who become infected with the T. gondii parasite, but that are not still able to produce oocysts in the environment. The model considers a time delay in order to represent the duration of the exposed stage. Besides the cat population the model also includes the oocysts related to the T. gondii in the environment. The model includes the cats since they are the only definitive host and the oocysts, since they are relevant to the dynamics of toxoplasmosis. The model considers lifelong immunity for the recovered cats and vaccinated cats. In addition, the model considers that cats can get infected through an effective contact with the oocysts in the environment. We find conditions such that the toxoplasmosis disease becomes extinct. We analyze the consequences of considering the exposed stage and the time delay on the stability of the equilibrium points. We numerically solve the constructed model and corroborated the theoretical results.

Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems.