Dynamical Analysis of the SEIB Model for Brucellosis Transmission to the Dairy Cows with Immunological Threshold

Abstract

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 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 R0 and know that the disease-free and bacteria-free steady state is a global stability when R0<1, and the one of the endemic steady state is a conditionally global stability when R0>1. We find that different amounts of R0 may lead to different steady states of brucellosis, and considering the effects of immunology is more serious in mathematics and biology.

Transmission Dynamics of Fractional Order Brucellosis Model Using Caputo–Fabrizio Operator

“…Researchers in mathematics have been engaged in the last decade in the formulation of mathematical models of Brucellosis to investigate the stability and dynamical behaviours of the disease [24,25]. e primary model of these kinds, which examine the indirect transmission mechanism of Brucellosis with the influence of immunology, was jointly investigated by Zhang et al [26]. In [1], Li et al proposed a model that quantified the spread of Brucellosis from sheep to man and among sheep.…”

“…ere are four compartments into which the Brucellosis model in [26] is separated, denoted as susceptible dairy cows S(t), the exposed dairy cow E(t), infectious dairy cows I(t), and the Brucella contaminated environment B(t). e sum of the population at any time t is represented by N � S + E + I + B.…”

“…e sum of the population at any time t is represented by N � S + E + I + B. In [26], Zhang et al proposed a Brucellosis model and analyzed the qualitative characteristics of the suggested model which is denoted by…”

“…Motivated by the previously mentioned literature, we have applied Caputo-Fabrizio fractional order ϕ, where ϕ ∈ (0, 1) to explore dynamics of the Brucellosis model discussed in Zhang et al [26]:…”

“…e typical model in [26] is derived for ϕ � 1. As the Caputo-Fabrizio fractional order quantifies the previously mentioned phenomenon accurately.…”