Dynamic analysis of syphilis model with the saturated incidence and early latent stage.

Syphilis is a blood-borne disease with multiple hidden-transmission stages caused by Treponema pallidum, and most infected individuals are asymptomatic as reported by the World Health Organization. This study establishes a Susceptible-Exposed-Infected-Latent-Recovered-Susceptible syphilis model with the nonlinear incidence and early latent stage incorporating the psychological effect. Firstly, the basic reproduction number of the SEILRS syphilis model is derived using the next generation matrix method. Then, the local asymptotic stability of the syphilis-free equilibrium point is proved. The global asymptotic stabilities of the syphilis-free equilibrium point and an endemic equilibrium point are shown by LaSalle's invariance principle. Further, the key parameters of the SEILRS syphilis model are estimated by the least squares method against the surveillance data of Fujian Province, China. The numerical simulation demonstrates that the changes of transmission rate in the early latent stage, treatment rate in the secondary stage and psychological effect in the early latent stage present the significant influences on the infection scale of syphilis. The 2023-2030 tendency predictions of the infection scale with scenarios indicate that the transmission rates are most critical for the prevalence of syphilis. As a consequence, to more effectively reduce the transmission rates of syphilis, it is recommended to enhance testing and screening for high-risk groups, to ensure the effective and complete treatment for infected individuals in the secondary stage, and to vigorously publicize the asymptomatic but infectious characteristics of infected individuals in the early latent stage.