Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental effects

In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasible equilibria with or without antibody response are established. The basic reproduction number $ R_{b0} $ is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if $ R_{b0} < 1 $. Two endemic equilibria exist if $ R_{b0} < 1 $, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number $ R_{0} $ is calculated, from which the disease-free equilibrium is globally asymptotically stable if $ R_0\leq1 $, and the unique endemic equilibrium is globally asymptotically stable if $ R_0 > 1 $. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.