Population dynamics of a mathematical model for syphilis

• A new mathematical model for syphilis is proposed.
• Loss of immunity can induce the backward bifurcation phenomenon.
• Reversions of early latent cases to the infectious stages affect syphilis control.
• Treatment and progression rates in the primary and secondary stages are connected.

A new multistage deterministic model for the transmission dynamics of syphilis is designed and used to qualitatively assess the role of loss of transitory immunity in the transmission process. It is shown that loss of transitory (natural) immunity can induce the phenomenon of backward bifurcation when the associated reproduction number is less than unity. For the period when there is no loss of transitory immunity, after recovery from infection, and in populations where early latent cases of syphilis do not revert to the primary and secondary stages of infection, it is shown that the disease-free equilibrium of the model is globally asymptotically stable whenever the associated reproduction number is less than unity; it is also further shown that the unique endemic equilibrium of the model is globally asymptotically stable whenever the reproduction number is greater than unity, for the same situations described above. Analytical and numerical results show an interesting relationship between the rates of progression, from the primary and secondary stages of infection, the treatment rates, for individuals in the primary and secondary stages, and the reproduction number and incidence of syphilis in the population. Numerical simulations of the model suggest that high treatment rates for individuals in the primary and secondary stages of infection have a positive cascading effect on the number of infected individuals in the remaining stages of infection.