Modeling disease transmission in a mixed-species grazing environment

In this article, we propose and analyzed a model describes the dynamics of a disease affects two different herbivores populations co-grazing together in the same environment under vaccination strategies and cross-immunity between the two species. Results show that the disease free equilibrium point (DFE) is locally asymptotically stable when the basic reproduction number, R₀, is less than unity, and unstable when R₀ is greater than unity, and our model undergoes a backward bifurcation, where R₀ < 1 is not sufficient for the disease elimination, as R₀ passes throw unity. Numerical results show that cross-immunity plays an important role in the eradication of the disease from both populations, however it plays also a negative role for both populations in the presence of vaccination strategies.