Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 437–492 | Cite as

Mathematical Analysis of the Transmission Dynamics of HIV Syphilis Co-infection in the Presence of Treatment for Syphilis

  • A. Nwankwo
  • D. Okuonghae
Original Research


The re-emergence of syphilis has become a global public health issue, and more persons are getting infected, especially in developing countries. This has also led to an increase in the incidence of human immunodeficiency virus (HIV) infections as some studies have shown in the recent decade. This paper investigates the synergistic interaction between HIV and syphilis using a mathematical model that assesses the impact of syphilis treatment on the dynamics of syphilis and HIV co-infection in a human population where HIV treatment is not readily available or accessible to HIV-infected individuals. In the absence of HIV, the syphilis-only model undergoes the phenomenon of backward bifurcation when the associated reproduction number (\({\mathcal {R}}_{T}\)) is less than unity, due to susceptibility to syphilis reinfection after recovery from a previous infection. The complete syphilis–HIV co-infection model also undergoes the phenomenon of backward bifurcation when the associated effective reproduction number (\({\mathcal {R}}_{C}\)) is less than unity for the same reason as the syphilis-only model. When susceptibility to syphilis reinfection after treatment is insignificant, the disease-free equilibrium of the syphilis-only model is shown to be globally asymptotically stable whenever the associated reproduction number (\({\mathcal {R}}_{T}\)) is less than unity. Sensitivity and uncertainty analysis show that the top three parameters that drive the syphilis infection (with respect to the associated response function, \({\mathcal {R}}_{T}\)) are the contact rate (\(\beta _S\)), modification parameter that accounts for the increased infectiousness of syphilis-infected individuals in the secondary stage of the infection (\(\theta _1\)) and treatment rate for syphilis-only infected individuals in the primary stage of the infection (\(r_1\)). The co-infection model was numerically simulated to investigate the impact of various treatment strategies for primary and secondary syphilis, in both singly and dually infected individuals, on the dynamics of the co-infection of syphilis and HIV. It is observed that if concerted effort is exerted in the treatment of primary and secondary syphilis (in both singly and dually infected individuals), especially with high treatment rates for primary syphilis, this will result in a reduction in the incidence of HIV (and its co-infection with syphilis) in the population.


Co-infection Mathematical model Backward bifurcation Global statbility Sensitivity and uncertainty analysis 



The authors will like to thank the anonymous reviewers for their invaluable and constructive comments.


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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BeninBenin CityNigeria

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