An age-structured syphilis model, I: well-posedness and stability

Syphilis is a highly contagious bacterial disease with different stages that results in substantial morbidity and mortality worldwide. In particular, it has been resurging among all age groups in both developed and developing countries in the last few years after the COVID-19 pandemic. Since sexual contact is the main transmission mode, the age structure of the host is crucial for the spread of syphilis. In this series of two papers, we develop an age-structured epidemic model with four infection stages (primary, secondary, latent and tertiary) to study the transmission dynamics of syphilis. In part I, first we establish the well-posedness of the initial-boundary value problem. Next, we derive an explicit expression of the basic reproduction number. Finally, we study the global stability of the disease-free steady state and the uniqueness and local stability of the endemic steady state. In part II, we will investigate the optimal control and numerical simulation of the age-structured syphilis model and use the model to calibrate the syphilis data from China.