Dynamics of an epidemic model with general incidence rate dependent on a class of disease-related contact functions

Starting from the idea of constructing the standard incidence rate, we take the effective contact times of individuals in the population per unit time as a contact function, $ T(\cdot) $, which depends on the population size. Considering the influence of disease on the contact function, the influence intensity factor of the disease affected by the infected person is integrated into the nonlinear incidence rate. We propose an epidemic model with a class of disease-related contact functions. Then, we analyze the well-posedness of the solutions of the model. By using the next generation matrix method, we get the basic reproduction number $ \mathcal{R}_0 $. We find that the existence and stability of the equilibria are not only related to $ \mathcal{R}_0 $, but also to the intensity of the disease affected for the infected person, $ \eta $, and the contact function, $ T(\cdot) $. We obtain some stability results under different assumptions about the contact function. Finally, we use MATLAB to simulate the system for several different contact functions. The numerical simulation results agree with our qualitative study. At the same time, we also prove that the system may have a Hopf bifurcation when the contact function $ T(\cdot) $ satisfies some corresponding conditions.