Assessing Syphilis transmission among MSM population incorporating low and high-risk infection: a modeling study

Abstract

Globally, there is a high incidence of Syphilis among men who have sex with men (MSM). This is due to risk factors, such as having multiple partners, condomless sex, or substance abuse. In this paper, we present a mathematical model for Syphilis transmission dynamics among the MSM popion that incorporates high/low-risk transmission classes. The model equilibrium points and basic reproduction number ($\mathcal {R}_{0}$) are computed, and a bifurcation analysis is performed. Analytical results show that in the absence of infection-acquired immunity, the disease-free equilibrium is globally asymptotically stable when $\mathcal {R}_{0}<1.$ Global sensitivity analysis was performed using the Latin-hypercube sampling technique to determine sensitive parameters. The results indicate that the most sensitive parameters are the high-risk transmission rate, progression rate from primary infection to secondary infection, human recruitment, and mortality rates. Results from numerical simulations suggest that increasing the Syphilis treatment rates and reducing high/low-risk infection rates is essential for controlling Syphilis spread in the MSM population. We envisage that our findings can serve as a tool for Syphilis risk assessment to help alleviate public health concerns associated with Syphilis transmission within the MSM population.

Appendices

Appendix A

The coefficients $a_0,~ a_1,$ and $a_2$ for the polynomial (13) are given below

$$\begin{aligned} a_0= & {} (1-\mathcal {R}_0) K {\xi }^{2} \left( \mu +\kappa \right) \left( \mu +\tau _{{3}} \right) \mu \, \left( \alpha +\mu \right) \left( \mu +\sigma +\tau _{{2}} \right) \left( \mu +\xi +\tau _{{1}} \right) \\{} & {} \left( \mu +\omega _{{1}}+\omega _{{2 }} \right) \left( \gamma +\mu \right) , \\ a_1= & {} -\xi ^{2} \left( \xi q_{1}+\mu +\sigma +\tau _{2}\right) \alpha \pi \beta _{l} \left( \mu +\kappa \right) \left( \mu +\tau _{3}\right) \left( \mu +\gamma \right) q_{2}-\xi \\{} & {} \left( \xi q_{1}+\mu +\sigma +\tau _{2}\right) \dots \\{} & {} \left[ \alpha \pi \left( \mu +\kappa \right) \left( \mu +\tau _{3}\right) \left( \tau _{2}+\mu +\sigma \right) \left( \mu +\gamma \right) \beta _{l} \dots \right. \\{} & {} -\mu \left( \left( \mu +\kappa \right) \left( \mu +\tau _{3}\right) \left( \tau _{2}+\mu +\sigma \right) \left( \tau _{1}+\mu +\xi \right) \left( \omega _{1}+\mu \right) \left( \mu +\gamma \right) \dots \right. \\{} & {} + \gamma \kappa \,\mu ^{2} \left( \omega _{1}+\tau _{1}+\tau _{2}+\tau _{3}+\mu +\sigma +\xi \right) + \gamma \kappa \mu \\{} & {} \left( \sigma \xi +\sigma \omega _{1}+\sigma \tau _{1}+\sigma \tau _{3}+\xi \omega _{1}+\xi \tau _{2}\right) \dots \\{} & {} + \gamma \kappa \mu \left( \xi \tau _{3}+\omega _{1} \tau _{2}+\omega _{1} \tau _{3}+\tau _{1} \tau _{2}+\tau _{1} \tau _{3}+\tau _{2} \tau _{3}\right) \dots \\{} & {} + \kappa \left( \left( \left( \xi +\sigma +\tau _{2}\right) \omega _{1}+\left( \sigma +\tau _{2}\right) \left( \xi +\tau _{1}\right) \right) \tau _{3}+\sigma \xi \omega _{1}\right) \gamma \dots \\{} & {} + \left. \left( \tau _{3}+\mu \right) \left( \tau _{2}+\mu +\sigma \right) \left( \tau _{1}+\mu +\xi \right) \left( \omega _{1}+\mu \right) \left( \gamma +\kappa +\mu \right) \right] .\\ a_2= & {} \beta _h(\mathcal {R}_0) \beta _l \mu (\xi q_1 + \mu + \sigma + \tau _2) (\xi q_2 + \mu + \sigma + \tau _2) \\{} & {} \left[ (\tau _3+\mu )(\tau _2+\mu +\sigma )((\alpha +\mu )(\alpha +\gamma +\mu )+\gamma \kappa )\tau _1 + (\mu +\xi )(\alpha +\mu )(\tau _3+\mu )\right. \\{} & {} \left. (\tau _2+\mu +\sigma )(\gamma +\mu ) \right. \\{} & {} + \kappa (\alpha +\mu )(\tau _3+\mu )(\tau _2+\mu +\sigma )(\gamma +\mu ) \\{} & {} +\left. \left. \xi (\tau _3+\mu )(\tau _2+\mu +\sigma )(\gamma +\mu ) + \xi \alpha ((\tau _3+\mu ) (\tau _2+\gamma +\mu +\sigma ) + \gamma \sigma )\right) \right] , \end{aligned}$$

with $K = (\mu +\alpha ) (\mu +\xi +\tau _1)(\mu +\psi +\tau _2)$.

Appendix B

Table 3 Pairwise PRCC comparisons (unadjusted p values)

Table 4 Pairwise PRCC comparisons (FDR-adjusted p values)