Study of immune response in a latent tuberculosis infection model

A simple mathematical model describing the immune response during the stage latent tuberculosis infection is established and analyzed. The main purpose of this study is to explore the sustained immune response of the immune system against invaded Mycobacterium tuberculosis in the stage of latent tuberculosis infection. First, the threshold R 0 is defined to determine the occurrence of sustained immune response. Then, stability conditions are derived to show that the sustained immune response may converge to a constant or to a stable periodical oscillation, implying that the Mycobacterium tuberculosis , the infected macrophages, the activated uninfected macrophages, and the immune cells coexist to form the tuberculous granuloma structure. This structure may appear calcified if the system solution converges to a constant, or maintains a dynamic balance if the system solution undergoes a periodical oscillation. These findings well demonstrate the process of sustained immune response in the latent tuberculosis infection as the Mycobacterium tuberculosis is changing. Numerical examples are presented to illustrate the theoretical predictions.