Abstract
In this study, we develop a nonlinear ordinary differential equation to study the dynamics of syphilis transmission incorporating controls, namely prevention and treatment of the infected males and females. We obtain syphilis-free equilibrium (SFE) and syphilis-present equilibrium (SPE). We obtain the basic reproduction number, which can be used to control the transmission of the disease, and thus establish the conditions for local and global stability of the syphilis-free equilibrium. The stability results show that the model is locally asymptotically stable if the Routh–Hurwitz criteria are satisfied and globally asymptotically stable. The bifurcation analysis result reveals that the model exhibits backward bifurcation. We adopted Pontryagin’s maximum principle to determine the optimality system for the syphilis model, which was solved numerically to show that syphilis transmission can be optimally best control using a combination of condoms usage and treatment in the primary stage of infection in both infected male and female populations.
Highlights
Syphilis is one of the infectious diseases, most commonly caused by sexual contact
The results reveal that health education leading to enhanced biological and behavioral protection against infection and the development of effective vaccine is the most effective way to control syphilis transmission in a highrisk population. [17] presented a new multistage deterministic model for the transmission dynamics of syphilis to qualitatively assess the role of loss of transitory immunity in the transmission process
5 Numerical simulations we investigate the impact of interventions on the transmission of syphilis in a population
Summary
Syphilis is one of the infectious diseases, most commonly caused by sexual contact. Spirochete Treponema pallidum is a spiral-shaped bacteria that causes syphilis [23]. The population of recovered males at time t, Rm(t), is increased due to progression of treated males from primary, secondary, and latent stages of syphilis (Imp, Ims, Lm), respectively, at rates σm , σm2 , and σm , which are the treatment rates of syphilis infection in male population This population is reduced due to natural death at rate μ and is further decreased due to loss of immunity acquired as a result of treatment and transfer of such individuals to susceptible male population at rate φm, so that dRm dt. The population of recovered females at time t, Rf (t), is increased due to progression of treated females from primary, secondary, and latent stages of syphilis infections (Ifp, Ifs, Lf ), respectively, at rates ρf , ρf2 , and ρf3 This population is reduced due to natural death at rate μ and is further decreased due to loss of immunity acquired as a result of treatment and transfer of such an individual to the susceptible male population at rate φf : dRf dt.
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