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In this article, a tuberculosis (TB) epidemic model with Beddington-DeAngelis incidence and distributed delay is proposed to characterize the interaction between latent period, endogenous reactivation, treatment of latent TB infection (LTBI), as well as relapse. The basic reproduction number R0 is defined, and the globally asymptotic stability of disease-free equilibrium is shown when R0 <1, while if R0 >1, the globally asymptotic stability of endemic equilibrium is also acquired. Theoretical results are validated through performing numerical simulations, wherein we detect that TB dynamic behavior between models with discrete and distributed delays could be same and opposite. The effect on the protection levels of susceptible and infectious individuals is discussed as well.

A tuberculosis (TB) epidemic model with Beddington–DeAngelis incidence and distributed delay is proposed to characterize the interaction between latent period, endogenous reactivation, treatment of latent TB infection, as well as relapse. The basic reproduction number is defined, and the globally asymptotic stability of disease-free equilibrium is shown when , while if the globally asymptotic stability of endemic equilibrium is also acquired. Theoretical results are validated through performing numerical simulations, wherein we detect that TB dynamic behavior between models with discrete and distributed delays could be same and opposite, and TB is more persistent in the model with distributed delay. Besides, increasing the protection level of susceptible and infectious individuals is crucial for the control of TB.

A hepatitis B or C virus (HBV or HCV) epidemic model with intra- and extra-hepatic coinfection, immune delay and saturation incidence, as well as antiviral therapy is proposed in this paper. The existence of equilibria (infection-free, immune-free and immune-activated), the basic reproduction numbers, i.e., R0, R1, are given respectively, by which the criteria on (local and global) stability of above equilibria are established. Furthermore, if the immune delay τ>τ0, the existence of Hopf bifurcation is obtained at the immune-activated equilibrium E*. Finally, the theoretical results are demonstrated by numerical simulations. We derive that the immune delay and intra- and extra-hepatic coinfection have significant influence on the transmission of HBV/HCV, could cause more complicated dynamics at E* from stability to unstablity untill bifurcation, which greatly increases the difficulty of disease control. While effective antiviral therapy could evidently decrease the spread of HBV/HCV.

In this paper, a coupling transmission epidemic model with mutualistic two-strain of virus in body and vitro of host is proposed, in which humoral immune response only works for strain 1 due to immunity evasion of mutation. For the within-host subsystem, the global stability of all feasible equilibria with and without environmental influence are discussed. For the between-host subsystem, the basic reproduction number R 0 is obtained. When R 0 < 1 , the disease-free equilibrium is local stable, while the endemic equilibrium is local stable and the disease is uniformly persistent if R 0 > 1 . Meanwhile, backward bifurcation would occur when there exists immune response within host. Finally, numerical examples are provided to illustrate obtained conclusions, by which we find that the mutualism between two strains during co-infection leads to a more persistent disease than single strain, even the basic reproduction number is small than 1 in each single strain.