Stability and Hopf bifurcation analysis of cryptosporidiosis transmission dynamics with time delays

Abstract

Cryptosporidiosis, caused by a protozoan parasite known as cryptosporidium, is a zoonotic disease that affects animals and humans. In this paper we have developed a mathematical model with a non linear incidence called Beddington-DeAngelis function to describe the spread of Cryptosporidiosis from the animals to human and from the animals to animal. We also consider the incubation periods of cryptosporidium in this model with different time delay in the infective animal and human populations. by analyzing behavior of the model, we calculate the basic reproduction number (R_ha) and investigate the local and global stability of the disease-free equilibrium of the system. We also establish the sufficient conditions for the stability of the endemic equilibrium in presence of delays and we investigate the occurrence of Hopf bifurcation when certain conditions are satisfied. Finally, numerical simulations are performed and displayed graphically to support the analytical results.