Dynamical analysis of the \textit{SEIB} model for brucellosis transmission to the dairy cows with immunological threshold (Q2424578)

Summary: As we all know, bacteria is different from virus which with certain types can be killed by the immune cells in the body. The brucellosis, a bacterial disease, can invade the body by indirect transmission from environment, which has not been researched by combining with immune cells. Considering the effects of immune cells, we put a minimum infection dose of brucellosis invading into the dairy cows as an immunological threshold and get a switch model. In this paper, we accomplish a thorough dynamics analysis of a \textit{SEIB} switch model. On the one hand, we can get a disease-free and bacteria-free steady state and up to three endemic steady states which may be thoroughly analyzed in different cases of a minimum infection dose in a switch model. On the other hand, we calculate the basic reproduction number \(R_0\) and know that the disease-free and bacteria-free steady state is a global stability when \(R_0 < 1\), and the one of the endemic steady state is a conditionally global stability when \(R_0 > 1\). We find that different amounts of \(R_0\) may lead to different steady states of brucellosis, and considering the effects of immunology is more serious in mathematics and biology.