Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

Abstract

A five-dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self-diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans-critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease-free and endemic equilibria are established for the reaction-diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease.

Mathematical modelling of Toxoplasma gondii transmission: A systematic review

“…Three papers focused on cat-human transmission ( Ferreira et al, 2017 ; González-Parra et al, 2009 ; Yongzhen et al, 2018 ). González-Parra et al (2009) developed a compartment model for exploring the dynamics of T. gondii transmission to humans (SIC-type: susceptible, infected, controlled (people in treatment)) with cat (SI-type: susceptible, infected) as a vector of transmission.…”

“… Ferreira et al (2017) and Yongzhen et al (2018) extended the model of González-Parra et al (2009) to describe the transmission of T. gondii between cat and human populations. Both models assumed that horizontal transmission of the parasite to humans and cats was only through contact with infected cats.…”

“…Both models assumed that horizontal transmission of the parasite to humans and cats was only through contact with infected cats. Ferreira et al (2017) extended the model with a spatial version and similar to González-Parra et al (2009) , they showed that the R 0 depends on the contact rate of susceptible cats with infected cats, the birth/immigration rate of cats and the probability of effective infectious contact among cats. In the study by Yongzhen et al (2018) , vertical transmission was considered (both cats and humans born from an infected mother had probabilities ( p 1 and p 2 ) of not being infected), and the human population size was a variable.…”

“…The limited number of T. gondii mathematical models may largely be due to the complicated life cycle of the parasite, i.e., multiple hosts and different transmission pathways, which makes it very difficult to model. Thus, models do not always represent reality, for example, the models by González-Parra et al (2009) , Ferreira et al (2017) and Yongzhen et al (2018) did not include transmission via intermediate hosts (i.e., infection of cats via infected prey, and infection of humans via undercooked or raw meat containing tissue cysts), and have the underlying assumption that vertically infected cats excrete oocysts and infected cats remain infectious for life (i.e., infectious for humans and cats via contact), and the model did not differentiate between the sex of the cats, thus all infected cats were assumed to produce infected offspring at a varying probability. It is questionable whether the conclusions from these models have value in spite of these important omissions compared to current knowledge on T. gondii transmission.…”

Dynamics of Toxoplasmosis Disease in Cats population with vaccination

“…Then they spread by all over epithelial cells and subsequently are expelled to the environment through the cat feces. Ferreira et al [11], studied the dynamical behaviours of both deterministic and spatial models of toxoplasmosis disease in cat and human populations. They showed that the deterministic model exhibits a trans-critical bifurcation and the system has no periodic orbits inside the positive octant.…”