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

Abstract

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.

1. Introduction

Syphilis is a blood-borne disease caused by Treponema pallidum. World Health Organization, 2024 estimated that about 8 million adults aged 15 to 49 worldwide were infected with syphilis in 2022. Chinese Center for Disease Control and Prevention (2024) reported that there were 530,116 syphilis cases in China in 2023, with an incidence rate of 0.0376 %.

The progression of untreated syphilis unfolds through five distinct stages, each marked by varying symptoms and potential health impacts. Without syphilis treatment, the infected individuals usually experience five stages: the exposed stage, primary stage, secondary stage, latent stage and tertiary stage. The individuals in the exposed stage do not show any symptoms, although the Treponema pallidum enters the lymphatic system and spreads rapidly in the hosts. When the 3–4 week incubation period is reached, the individuals enter the primary stage (Garnett et al., 1997). The individuals mainly present with painless ulcerative chancre and lymph node enlargement in the primary stage lasting for 46 days (Garnett et al., 1997), of which the chancre may not be easily noticed and being healed within 3–10 days (World Health Organization, 2024). The individuals in the primary stage enter the secondary stage, exhibiting systemic symptoms such as condyloma lata and red rashes on the palms and soles for an average 3.6 months (Garnett et al., 1997), and then they arrive the latent stage. While, the individuals in the latent stage have no symptoms, and the latent stage is separated into the early latent stage and late latent stage according to their duration. Especially, some individuals may relapse to the primary stage and secondary stage in the early latent stage that lasts less than 1 year (Birnbaum et al., 1999), and the majority of these individuals return to the secondary stage (Iboi & Okuonghae, 2016). Afterwards, the individuals in the tertiary stage may develop cardiovascular or neurosyphilis, which can cause irreversible damage to the human body. It is noticed that the latent stage and tertiary stage usually last for 5–10 years or even longer (Wu et al., 2024).

The latent stage has been widely studied in the mathematical models for the transmission dynamics of syphilis. Okuonghae et al. (2019) designed a deterministic model with the early latent stage and evaluated the impact of treatment rates, condom use and compliance on community transmission. Momoh et al. (2021) considered a gender-structured model with three infection stages and concluded that the combination of condoms usage and treatment is the optimal control measure for infected males and females in the primary stage. Jing et al. (2021) proposed an SILR model and concluded that public awareness, condom usage rate and treatment rate have a positive impact on reducing the spread of syphilis. Wu et al. (2024) developed a novel reaction-diffusion syphilis model, and governed the surveillance data of syphilis from 30 provinces and cities in China to assess the effects of spatial heterogeneity and intervention measures. Wu & Ruan, 2025 further proposed an age-structured syphilis model, and concluded that more effective control measures should be taken for sexually active populations, and that prompt and standard treatment for the individuals in the secondary and latent stages of syphilis significantly reduced the number of infected individuals in the tertiary stage of syphilis. Chai et al. (2025) constructed a generalized SEIR model with the latent stage for evaluating the impact of the progress proportion from the exposed stage to the primary stage, and for evaluating the treatment rates of syphilis transmission in Yunnan Province.

The saturated incidence rate also plays an important role in the mathematical models, and takes the form of (βSI)/(1 + aI). When a = 0, the saturated incidence becomes the bilinear incidence rate. The current studies basically use the bilinear incidence rate (Chai et al., 2025; Chen et al., 2024; Wang et al., 2024, Wang et al., 2025, Wang et al., 2025) and the standard incidence rate (Lan et al., 2024; Momoh et al., 2021; Okuonghae et al., 2019; Wei et al., 2023) to better study the spread of infectious diseases such as syphilis and COVID-19. For a large number of infected individuals, the contacts between infected individuals and susceptible individuals can decrease due to the crowding effect of infected individuals and the protective measures taken by susceptible individuals and the contacts eventually tend to saturation level. And the saturated incidence rate is more in line with the psychological effect in actual epidemic situations. Capasso and Serio (1978) put forward the saturated incidence rate when they studied the cholera epidemic spread in Bari in 1973. Lahrouz et al. (2011) developed deterministic and stochastic SIRS models with saturated incidence rate and disease-inflicted mortality, demonstrating that measures such as isolation and media campaigns can enhance the saturation effect. Han and Wang (2015) extended the work of Lahrouz et al. (2011) by developing deterministic and stochastic SIRS models for multiple populations. Wei and Chen (2016) considered a stochastic SIRS epidemic model with the saturated incidence rate, and analyzed the unique global positive solution as well as the asymptotic behavior of the stochastic model around the disease-free equilibrium and the endemic equilibrium. Baba et al. (2019) developed a TB SEI model with saturated incidence rate, showing that as the psychological effect of general public for infective increases the population of infected individuals decreases. Avila-Vales et al. (2021) established a diffusive SIR epidemic model with a saturated incidence rate and Holling type II treatment. Shao and Shateyi (2021) established an SEIRS model with saturated incidence rate, and analyzed the stabilities of equilibrium points using Lyapunov functions and Volterra-Lyapunov matrices. Liu and Wei (2022) established an epidemic model with the saturated incidence rate and the environmental fluctuations, and studied the extinction and the persistence in the mean, stationary distribution of the solution as well. The saturated incidence rate is less discussed in the syphilis models.

Through the above analyses, we establish an SEILRS model with the saturated incidence rate and early latent stage, and the scenario investigation and tendency prediction of syphilis are paid more attention in the recent studies (Chen et al., 2024; Lan et al., 2024; Wang et al., 2024, Wang et al., 2025b; Wei et al., 2023). Then, basic reproduction number of the model is derived, and the global stability analysis of two types of the equilibrium points are extensively investigated in Section 3. Meanwhile, the 2005–2022 data fittings of syphilis in Fujian Province are performed, the sensitivity analysis of main parameters is conducted, and the 2023–2030 tendency predictions for syphilis in Fujian Province are also simulated in Section 4. Further, the main results of this study and some new suggestions for the controls of infection scales of syphilis are provided.

2. Model formulation

In this study, the total population N(t) is separated into eight compartments: the individuals who are in the susceptible stage S(t), in the exposed stage E(t), infected individuals who are in the primary stage I₁(t), secondary stage I₂(t), early latent stage L₁(t), late latent stage L₂(t) and tertiary stage T(t), and the individuals who are in the recovered stage R(t). We further assume that the individuals of the total population are homogenously mixed. The individuals who are in the exposed stage cannot be detected because they have not produced sufficient treponema pallidum antibodies or pathogens. We then propose a Susceptible-Exposed-Infected-Latent-Recovered-Susceptible (short for SEILRS) syphilis model with the saturated incidence rate and early latent stage as follows:

$$\left\{\begin{array}{ll}\dot{S}(t)& =\mathrm{\Lambda }-\left(\frac{{\beta }_1{I}_{1}S}{1+a_1{I}_1}+\frac{{\beta }_2{I}_{2}S}{1+a_2{I}_2}+\frac{{\beta }_3{L}_{1}S}{1+a_3{L}_1}\right)-\mu S+{\gamma }_{6}R,\\ \dot{E}(t)& =\frac{{\beta }_1{I}_{1}S}{1+a_1{I}_1}+\frac{{\beta }_2{I}_{2}S}{1+a_2{I}_2}+\frac{{\beta }_3{L}_{1}S}{1+a_3{L}_1}-({\gamma }_1+\mu )E,\\ {\dot{I}}_1(t)& =\eta {\gamma }_{1}E-({\delta }_1+{\gamma }_2+\mu )I_1+\rho \theta L_1,\\ {\dot{I}}_2(t)& ={\gamma }_2{I}_1-({\delta }_2+{\gamma }_3+\mu )I_2+(1-\rho )\theta L_1,\\ {\dot{L}}_1(t)& =(1-\eta ){\gamma }_{1}E+{\gamma }_3{I}_2-({\delta }_3+{\gamma }_4+\mu +\theta )L_1,\\ {\dot{L}}_2(t)& ={\gamma }_4{L}_1-({\delta }_4+{\gamma }_5+\mu )L_2,\\ \dot{T}(t)& ={\gamma }_5{L}_2-({\delta }_5+\alpha +\mu )T,\\ \dot{R}(t)& ={\delta }_1{I}_1+{\delta }_2{I}_2+{\delta }_3{L}_1+{\delta }_4{L}_2+{\delta }_{5}T-({\gamma }_6+\mu )R,\end{array}\right)$$ (1)

with the notation N(t) = S(t) + E(t) + I₁(t) + I₂(t) + L₁(t) + L₂(t) + T(t) + R(t). Here, Λ is the constant recruitment rate into the susceptible population, μ is the natural death rate and α is the disease mortality rate. The constant transmission rates of infected individuals in the primary, secondary and early latent stages are respectively denoted by β₁, β₂ and β₃. The psychological effect coefficients of infected individuals in the primary, secondary and early latent stages are respectively denoted by a₁, a₂ and a₃. The progression rates are denoted by γ_i (i = 1, 2, 3, 4, 5, 6). In other words, 1/γ₁ is the time that the individuals experienced from the exposed stage to the primary stage, 1/γ₂ is the time that the individuals experienced from the primary stage to the secondary stage, 1/γ₃, from the secondary stage to the early latent stage, 1/γ₄, from the early latent stage to the late latent stage, 1/γ₅, from the late latent stage to the tertiary stage, and 1/γ₆, from the recovered stage to the susceptible stage. η and 1 − η are the proportions of the individuals moving from the exposed stage to the primary stage and the early latent stage. δ_i (i = 1, 2, 3, 4, 5) are the treatment rates from the primary, secondary, early latent, late latent, and tertiary stages to the recovered stage. θ is the recurrent rate from the early latent stage to the primary stage and secondary stage. ρ and 1 − ρ are the proportions of the individuals who are in the early latent stage revert to the primary stage and the secondary stage. The flow diagram of model (1) is presented in Fig. 1.

Fig. 1.

The flow diagram of model (1).

3. Mathematical analysis

Since the sum of eight equations of model (1) gives the expression

$$\dot{N}\le \mathrm{\Lambda }-\mu N,$$

the positively invariant set for model (1) is derived as follows:

$$\mathrm{\Omega }=\left\{(S,E,I_1,I_2,L_1,L_2,T,R)\in {\mathbb{R}}_{+}^8,S+E+I_1+I_2+L_1+L_2+T+R\le \frac{\mathrm{\Lambda }}{\mu }\right\}.$$

Let the right-hand sides of eight equations of model (1) be zero, and E = 0, I₁ = 0, I₂ = 0, L₁ = 0, L₂ = 0, T = 0. It is easy to find that the syphilis-free equilibrium point is derived by P⁰ = (Λ/μ, 0, 0, 0, 0, 0, 0, 0). Let $\dot{X}={(\dot{E},{\dot{I}}_1,{\dot{I}}_2,{\dot{L}}_1,{\dot{L}}_2,\dot{T})}^{\text{T}}=\mathcal{F}-\mathcal{V}$, by the next generation matrix (van den Driessche & Watmough, 2002), we obtain that

$$\mathcal{F}=\left(\begin{array}{l}\frac{{\beta }_1{I}_{1}S}{1+a_1{I}_1}+\frac{{\beta }_2{I}_{2}S}{1+a_2{I}_2}+\frac{{\beta }_3{L}_{1}S}{1+a_3{L}_1}\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right),\mathcal{V}=\left(\begin{array}{l}({\gamma }_1+\mu )E\\ -\eta {\gamma }_{1}E+({\delta }_1+{\gamma }_2+\mu )I_1-\rho \theta L_1\\ -{\gamma }_2{I}_1+({\delta }_2+{\gamma }_3+\mu )I_2-(1-\rho )\theta L_1\\ -(1-\eta ){\gamma }_{1}E-{\gamma }_3{I}_2+({\delta }_3+{\gamma }_4+\mu +\theta )L_1\\ -{\gamma }_4{L}_1+({\delta }_4+{\gamma }_5+\mu )L_2\\ -{\gamma }_5{L}_2+({\delta }_5+\alpha +\mu )T\end{array}\right).$$

The corresponding Jacobian matrices at the syphilis-free equilibrium point P⁰ are followed:

$$\begin{array}{l}F=\left(\begin{array}{llllll}0& {\beta }_1\frac{\mathrm{\Lambda }}{\mu }& {\beta }_2\frac{\mathrm{\Lambda }}{\mu }& {\beta }_3\frac{\mathrm{\Lambda }}{\mu }& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right),\\ V=\left(\begin{array}{llllll}{\gamma }_1+\mu & 0& 0& 0& 0& 0\\ -\eta {\gamma }_1& M_1& 0& -\rho \theta & 0& 0\\ 0& -{\gamma }_2& M_2& -(1-\rho )\theta & 0& 0\\ -(1-\eta ){\gamma }_1& 0& -{\gamma }_3& M_3+\theta & 0& 0\\ 0& 0& 0& -{\gamma }_4& M_4& 0\\ 0& 0& 0& 0& -{\gamma }_5& M_5\end{array}\right),\end{array}$$ (2)

where

$$\begin{array}{ll}& M_1={\delta }_1+{\gamma }_2+\mu ,M_2={\delta }_2+{\gamma }_3+\mu ,M_3={\delta }_3+{\gamma }_4+\mu ,\\ & M_4={\delta }_4+{\gamma }_5+\mu ,M_5={\delta }_5+\alpha +\mu .\end{array}$$ (3)

So, we derive

$$\begin{array}{l}F{V}^{-1}=\left(\begin{array}{llllll}\frac{\mathrm{\Lambda }B}{\mu C}& \ast & \ast & \ast & \ast & \ast \\ 0& 0& \ast & \ast & \ast & \ast \\ 0& 0& 0& \ast & \ast & \ast \\ 0& 0& 0& 0& \ast & \ast \\ 0& 0& 0& 0& 0& \ast \\ 0& 0& 0& 0& 0& 0\end{array}\right),\end{array}$$

where

$$B={\beta }_1{\gamma }_1[\eta M_2(M_3+\theta )-\eta {\gamma }_3\theta +(1-\eta )M_2\rho \theta +\eta {\gamma }_3\rho \theta ]+{\beta }_2{\gamma }_1[\eta {\gamma }_2(M_3+\theta )+(1-\eta )M_1(1-\rho )\theta +\eta {\gamma }_2\theta +(1-\eta ){\gamma }_2\rho \theta ]+{\beta }_3{\gamma }_1[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3],C=({\gamma }_1+\mu )[M_1{M}_2(M_3+\theta )-{\gamma }_3{M}_1\theta +{\gamma }_3{M}_1\rho \theta -{\gamma }_2{\gamma }_3\rho \theta ].$$ (4)

Since

$$\begin{array}{ll}& \eta M_2(M_3+\theta )-\eta {\gamma }_3\theta =\eta ({\delta }_2+{\gamma }_3+\mu )(M_3+\theta )-\eta {\gamma }_3\theta >\eta ({\delta }_2+\mu )\theta >0,\\ & {\gamma }_3{M}_1\rho \theta -{\gamma }_2{\gamma }_3\rho \theta ={\gamma }_3({\delta }_1+{\gamma }_2+\mu )\rho \theta -{\gamma }_2{\gamma }_3\rho \theta ={\gamma }_3({\delta }_1+\mu )\rho \theta >0,\\ & M_1{M}_2(M_3+\theta )-{\gamma }_3{M}_1\theta =M_1({\delta }_2+{\gamma }_3+\mu )(M_3+\theta )-{\gamma }_3{M}_1\theta >M_1({\delta }_2+\mu )\theta >0,\end{array}$$

it follows that B > 0 and C > 0. Thus, the basic reproduction number is given by

$${\mathcal{R}}_0=\frac{\mathrm{\Lambda }B}{\mu C}=f(\mathrm{\Lambda },{\beta }_1,{\beta }_2,{\beta }_3,{\delta }_1,{\delta }_2,{\delta }_3,{\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4,\eta ,\rho ,\theta ,\mu ,\alpha ),$$ (5)

where f(⋅) is a function that represents the relationship between the sixteen parameters. Further, we analyze the relationship between the basic reproduction number and the treatment rates, as well as the relationship between the basic reproduction number and the transmission rates. If the treatment rates (i.e., δ₁, δ₂, δ₃) tend towards infinity, then we obtain

$$\begin{array}{ll}& {\lim }_{{\delta }_1\to \infty }{\mathcal{R}}_0=\frac{\mathrm{\Lambda }[{\beta }_2(1-\eta ){\gamma }_1(1-\rho )\theta +{\beta }_3(1-\eta ){\gamma }_1{M}_2]}{({\gamma }_1+\mu )[M_2(M_3+\theta )-{\gamma }_3\theta +{\gamma }_3\rho \theta ]}>0,\\ & {\lim }_{{\delta }_2\to \infty }{\mathcal{R}}_0=\frac{\mathrm{\Lambda }[{\beta }_1{\gamma }_1\left(\eta \right)(M_3+\theta )+(1-\eta )\rho \theta +{\beta }_3{M}_1(1-\eta ){\gamma }_1]}{\mu ({\gamma }_1+\mu )M_1(M_3+\theta )}>0,\\ & {\lim }_{{\delta }_3\to \infty }{\mathcal{R}}_0=\frac{\mathrm{\Lambda }({\beta }_1\eta {\gamma }_1{M}_2+{\beta }_2\eta {\gamma }_1{\gamma }_2)}{\mu ({\gamma }_1+\mu )M_1{M}_2}>0.\end{array}$$ (6)

From (6), we obtain that the syphilis still persists though the treatment rates are very large. The partial derivative of the basic reproduce number $({\mathcal{R}}_0)$ with respect to δ₁, δ₂ and δ₃ are

$$\begin{array}{ll}& \frac{\partial {\mathcal{R}}_0}{\partial {\delta }_1}=-\frac{\mathrm{\Lambda }({\gamma }_1+\mu )A_1[M_2{M}_3+({\delta }_2+\mu )\theta +{\gamma }_3\rho \theta ]}{\mu C^2}<0,\\ & \frac{\partial {\mathcal{R}}_0}{\partial {\delta }_2}=-\frac{\mathrm{\Lambda }({\gamma }_1+\mu )A_2{M}_1(M_3+\theta )}{\mu C^2}<0,\\ & \frac{\partial {\mathcal{R}}_0}{\partial {\delta }_3}=-\frac{\mathrm{\Lambda }({\gamma }_1+\mu )A_3{M}_1{M}_2}{\mu C^2}<0,\end{array}$$ (7)

where

$$A_1=({\gamma }_1+\mu )[M_2(M_3+\theta )-{\gamma }_3\theta +{\gamma }_3\rho \theta ]B-[{\beta }_2(1-\eta ){\gamma }_1(1-\rho )\theta +{\beta }_3(1-\eta ){\gamma }_1{M}_2]C>({\gamma }_1+\mu )[M_2(M_3+\theta )-{\gamma }_3\theta +{\gamma }_3\rho \theta ][{\beta }_2(1-\eta ){\gamma }_1{M}_1(1-\rho )\theta +{\beta }_3(1-\eta ){\gamma }_1{M}_1{M}_2]-[{\beta }_2(1-\eta ){\gamma }_1(1-\rho )\theta +{\beta }_3(1-\eta ){\gamma }_1{M}_2]({\gamma }_1+\mu )[M_1{M}_2(M_3+\theta )-{\gamma }_3{M}_1\theta +{\gamma }_3{M}_1\rho \theta ]=0,A_2=({\gamma }_1+\mu )M_1(M_3+\theta )B-[{\beta }_1{\gamma }_1(\eta (M_3+\theta )+(1-\eta )\rho \theta )+{\beta }_3(1-\eta ){\gamma }_1{M}_1]C>({\gamma }_1+\mu )M_1(M_3+\theta )[{\beta }_1{\gamma }_1(\eta M_2(M_3+\theta )+(1-\eta )M_2\rho \theta )+{\beta }_3(1-\eta ){\gamma }_1{M}_1{M}_2]-[{\beta }_1{\gamma }_1(\eta (M_3+\theta )+(1-\eta )\rho \theta )+{\beta }_3(1-\eta ){\gamma }_1{M}_1]({\gamma }_1+\mu )\left[M_1{M}_2(M_3+\theta )\right]=0,A_3=({\gamma }_1+\mu )M_1{M}_{2}B-[{\beta }_1\eta {\gamma }_1{M}_2+{\beta }_2\eta {\gamma }_1{\gamma }_2]C>({\gamma }_1+\mu )M_1{M}_2[{\beta }_1\eta {\gamma }_1{M}_2(M_3+\theta )+{\beta }_2\eta {\gamma }_1{\gamma }_2(M_3+\theta )]-[{\beta }_1\eta {\gamma }_1{M}_2+{\beta }_2\eta {\gamma }_1{\gamma }_2]({\gamma }_1+\mu )\left[M_1{M}_2(M_3+\theta )\right]=0.$$

From (7), we obtain that the basic reproduction number decreases as the treatment rates (i.e., δ₁, δ₂, δ₃) increase. If the transmission rates (i.e., β₁, β₂, β₃) tend towards infinity, then we get

$$\begin{array}{ll}& {\lim }_{{\beta }_1\to \infty }{\mathcal{R}}_0=\infty ,\\ & {\lim }_{{\beta }_2\to \infty }{\mathcal{R}}_0=\infty ,\\ & {\lim }_{{\beta }_3\to \infty }{\mathcal{R}}_0=\infty .\end{array}$$ (8)

The partial derivative of the basic reproduce number $({\mathcal{R}}_0)$ with respect to β₁, β₂ and β₃ are

$$\begin{array}{ll}& \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_1}=\frac{\mathrm{\Lambda }{\gamma }_1[\eta M_2(M_3+\theta )-\eta {\gamma }_3\theta +(1-\eta )M_2\rho \theta +\eta {\gamma }_3\rho \theta ]}{\mu C}>0,\\ & \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_2}=\frac{\mathrm{\Lambda }{\gamma }_1[\eta {\gamma }_2(M_3+\theta )+(1-\eta )M_1(1-\rho )\theta +\eta {\gamma }_2\theta +(1-\eta ){\gamma }_2\rho \theta ]}{\mu C}>0,\\ & \frac{\partial {\mathcal{R}}_0}{\partial {\beta }_3}=\frac{\mathrm{\Lambda }{\gamma }_1[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3]}{\mu C}>0.\end{array}$$ (9)

From (8), (9), the high transmission rates (i.e., β₁, β₂, β₃) increase the transmission risk of syphilis.

Theorem 3.1

If ${\mathcal{R}}_0$<1, then the syphilis-free equilibrium point P⁰ is locally asymptotically stable.

Proof The Jacobian matrix $J_{P^0}$ of model (1) at P⁰ is given by

$$\begin{array}{l}{J}_{P^0}=\left(\begin{array}{lll}-\mu & G_1& {\gamma }_6\\ 0& F-V& 0\\ 0& G_2& -({\gamma }_6+\mu )\end{array}\right),\end{array}$$

where F and V are given in (2), and $G_1=\left(0,-{\beta }_1\frac{\mathrm{\Lambda }}{\mu },-{\beta }_2\frac{\mathrm{\Lambda }}{\mu },-{\beta }_3\frac{\mathrm{\Lambda }}{\mu },0,0\right)$, G₂ = (0, δ₁, δ₂, δ₃, δ₄, δ₅).

After a similar transformation, we derive

$$\begin{array}{l}G=\left(\begin{array}{lll}F-V& 0& 0\\ G_2& -({\gamma }_6+\mu )& 0\\ G_1& {\gamma }_6& -\mu \end{array}\right).\end{array}$$

Clearly, the eigenvalues of G include μ, γ₆ + μ, and the eigenvalues of F − V. Moreover, F − V is irreducible and has non-negative off-diagonal elements. It is known (van den Driessche & Watmough, 2002) that we obtain that ${\mathcal{R}}_0<1$ is equal to s(F − V) < 0 and ${\mathcal{R}}_0>1$ is equal to s(F − V) > 0, where s(F − V) = max{Re λ: λ is an eigenvalue of F − V}.

Hence, if ${\mathcal{R}}_0<1$, then the eigenvalues of $J_{P^0}$ have negative real parts. The syphilis-free equilibrium point P⁰ is locally asymptotically stable if only and if ${\mathcal{R}}_0<1$. The proof is complete.

Theorem 3.2

If ${\mathcal{R}}_0$<1, then the syphilis-free equilibrium point P⁰ is globally asymptotically stable.

Proof Consider the Lyapunov function

$$W_1(S,E,I_1,I_2,L_1,L_2,T,R)=E_1+m_1{I}_1+m_2{I}_2+m_3{L}_1,$$

m₁, m₂ and m₃ are constants that will be determined.

The derivative of W₁ with respect to t along the solution of model (1) gives that

$${\dot{W}}_1=\dot{E}\left(t\right)+m_1{\dot{I}}_1\left(t\right)+m_2{\dot{I}}_2\left(t\right)+m_3{\dot{L}}_1\left(t\right)=\left[\left(\frac{{\beta }_1{I}_{1}S}{1+a_1{I}_1}+\frac{{\beta }_2{I}_{2}S}{1+a_2{I}_2}+\frac{{\beta }_3{L}_{1}S}{1+a_3{L}_1}\right)-({\gamma }_1+\mu )E\right]+m_1(\eta {\gamma }_{1}E-M_1{I}_1+\rho \theta L_1)+m_2[{\gamma }_2{I}_1-M_2{I}_2+(1-\rho )\theta L_1]+m_3[(1-\eta ){\gamma }_{1}E+{\gamma }_3{I}_2-(M_3+\theta )L_1]\le [{\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S-({\gamma }_1+\mu )E]+m_1(\eta {\gamma }_{1}E-M_1{I}_1+\rho \theta L_1)+m_2[{\gamma }_2{I}_1-M_2{I}_2+(1-\rho )\theta L_1]+m_3[(1-\eta ){\gamma }_{1}E+{\gamma }_3{I}_2-(M_3+\theta )L_1]\le [m_1\eta {\gamma }_1+m_3(1-\eta ){\gamma }_1-({\gamma }_1+\mu )]E+\left(\frac{\mathrm{\Lambda }}{\mu }{\beta }_1-m_1{M}_1+m_2{\gamma }_2\right)I_1+\left(\frac{\mathrm{\Lambda }}{\mu }{\beta }_2-m_2{M}_2+m_3{\gamma }_3\right)I_2+\left[\frac{\mathrm{\Lambda }}{\mu }{\beta }_3+m_1\rho \theta +m_2(1-\rho )\theta -m_3(M_3+\theta )\right]L_1=\frac{({\mathcal{R}}_0-1)L_1}{{\gamma }_{1}C[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3]},$$

here,

$$\begin{array}{ll}{m}_1& =\frac{\mu ({\gamma }_1+\mu ){\gamma }_2{\gamma }_3+\mathrm{\Lambda }{\beta }_1(1-\eta ){\gamma }_1{M}_2+\mathrm{\Lambda }{\beta }_2(1-\eta ){\gamma }_1{\gamma }_2}{\mu {\gamma }_1[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3]},\\ m_2& =\frac{\mu M_1({\gamma }_1+\mu ){\gamma }_3-\mathrm{\Lambda }{\beta }_1\eta {\gamma }_1{\gamma }_3+\mathrm{\Lambda }{\beta }_2(1-\eta ){\gamma }_1{M}_1}{\mu {\gamma }_1[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3]},\\ m_3& =\frac{\mu ({\gamma }_1+\mu )M_1{M}_2-\mathrm{\Lambda }{\beta }_1\eta {\gamma }_1{M}_2+\mathrm{\Lambda }{\beta }_2\eta {\gamma }_1{\gamma }_2}{\mu {\gamma }_1[(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3]}.\end{array}$$

If ${\mathcal{R}}_0=(\mathrm{\Lambda }B)/(\mu C)<1$, then

$$\Lambda {\beta }_1\eta {\gamma }_1[M_2(M_3+\theta )+{\gamma }_3\rho \theta -{\gamma }_3\theta ]=\Lambda {\beta }_1{\gamma }_1[\eta M_2(M_3+\theta )+\eta {\gamma }_3\rho \theta -\eta {\gamma }_3\theta ]<\Lambda \left[{\beta }_1{\gamma }_1(\eta M_2(M_3+\theta )+(1-\eta )M_2\rho \theta +\eta {\gamma }_3\rho \theta -\eta {\gamma }_3\theta )\right)+{\beta }_2{\gamma }_1(\eta {\gamma }_2(M_3+\theta )+(1-\eta )M_1(1-\rho )\theta +\eta {\gamma }_2\theta +(1-\eta ){\gamma }_2\rho \theta )\left(+{\beta }_3{\gamma }_1((1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3)\right]=\Lambda B<\mu C=\mu ({\gamma }_1+\mu )[M_1{M}_2(M_3+\theta )+{\gamma }_3{M}_1\rho \theta -{\gamma }_2{\gamma }_3\rho \theta -{\gamma }_3{M}_1\theta ]<\mu ({\gamma }_1+\mu )[M_1{M}_2(M_3+\theta )+{\gamma }_3{M}_1\rho \theta -{\gamma }_3{M}_1\theta ]=\mu ({\gamma }_1+\mu )M_1[M_2(M_3+\theta )+{\gamma }_3\rho \theta -{\gamma }_3\theta ].$$

Thus, we deduce Λβ₁ηγ₁γ₃ < μ(γ₁ + μ)γ₃M₁ and Λβ₁ηγ₁M₂ < μ(γ₁ + μ)M₁M₂, and further deduce m₁ > 0, m₂ > 0, m₃ > 0.

If ${\mathcal{R}}_0$<1, then it follows that ${\dot{W}}_1(t)\le 0$. Indeed, ${\dot{W}}_1(t)=0$ if and only if S = Λ/μ, E = 0, I₁ = 0, I₂ = 0, L₁ = 0, L₂ = 0, T = 0, R = 0. Further, it is proved that {P⁰} is the largest compact invariant subset of ${\mathrm{\Omega }}_1=\{(S,E,I_1,I_2,L_1,L_2,T,R)\in {\mathbb{R}}_{+}^8:{\dot{W}}_1(t)=0\}$. By LaSalle's invariance principle, the syphilis-free equilibrium point P⁰ of model (1) is globally asymptotically stable. The proof is complete.

Garnett et al. (1997) pointed that the duration of protective immunity was significantly longer than the durations for the primary and secondary stages of syphilis. If the duration of protective immunity of syphilis is the life-long period, then the reinfection risk is declined to zero. Hence, let γ₆ = 0 be reasonable. If psychological effects are not taken into account in the population (i.e., a₁ = a₂ = a₃ = 0) and ${\mathcal{R}}_0$>1, then the endemic equilibrium point of model (1) is given by $P^{\ast }=(S^{\ast },E^{\ast },I_1^{\ast },I_2^{\ast },L_1^{\ast },L_2^{\ast },T^{\ast },R^{\ast })$, where

$$\begin{array}{ll}{S}^{\ast }& =\frac{\mathrm{\Lambda }}{\mu {\mathcal{R}}_0},E^{\ast }=\frac{\mu C({\mathcal{R}}_0-1)}{({\gamma }_1+\mu )B},\\ I_1^{\ast }& =\frac{\mu {\gamma }_1(\eta M_2(M_3+\theta )+(1-\eta )M_2\rho \theta +\eta {\gamma }_3\rho \theta -\eta {\gamma }_3\theta )({\mathcal{R}}_0-1)}{B},\\ I_2^{\ast }& =\frac{\mu {\gamma }_1(\eta {\gamma }_2{M}_3+(1-\eta )M_1(1-\rho )\theta +\eta {\gamma }_2\theta +(1-\eta ){\gamma }_2\rho \theta )({\mathcal{R}}_0-1)}{B},\\ L_1^{\ast }& =\frac{(1-\eta ){\gamma }_1{E}^{\ast }+{\gamma }_3{I}_2^{\ast }}{M_3+\theta },L_2^{\ast }=\frac{{\gamma }_4{L}_1^{\ast }}{M_5},T^{\ast }=\frac{{\gamma }_4{\gamma }_5{L}_1^{\ast }}{M_5{M}_6},\\ R^{\ast }& ={\delta }_1{I}_1^{\ast }+{\delta }_2{I}_2^{\ast }+{\delta }_3{L}_1^{\ast }+{\delta }_4{L}_2^{\ast }+{\delta }_5{T}^{\ast }.\end{array}$$ (10)

Theorem 3.3

Let γ₆ = 0, a₁ = a₂ = a₃ = 0. If ${\mathcal{R}}_0$>1, then the endemic equilibrium point P∗ in (10) of model (1) is globally asymptotically stable.

Proof Let γ₆ = 0, a₁ = a₂ = a₃ = 0. It is shown from model (1) at P∗ that

$$\begin{array}{ll}& \mathrm{\Lambda }={\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast }+\mu S^{\ast },\\ & ({\gamma }_1+\mu )E^{\ast }={\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast },\\ & M_1{I}_1^{\ast }=\eta {\gamma }_1{E}^{\ast }+\rho \theta L_1^{\ast },\\ & M_2{I}_2^{\ast }={\gamma }_2{I}_1^{\ast }+(1-\rho )\theta L_1^{\ast },\\ & (M_3+\theta )L_1^{\ast }=(1-\eta ){\gamma }_1{E}^{\ast }+{\gamma }_3{I}_2^{\ast },\end{array}$$ (11)

where M₁, M₂, and M₃ are given in (3). Define a Lyapunov function

$$W_2=\left(S-S^{\ast }-S^{\ast }\ln \frac{S}{S^{\ast }}\right)+\left(E-E^{\ast }-E^{\ast }\ln \frac{E}{E^{\ast }}\right)+k_1\left(I_1-I_1^{\ast }-I_1^{\ast }\ln \frac{I_1}{I_1^{\ast }}\right)+k_2\left(I_2-I_2^{\ast }-I_2^{\ast }\ln \frac{I_2}{I_2^{\ast }}\right)+k_3\left(L_1-L_1^{\ast }-L_1^{\ast }\ln \frac{L_1}{L_1^{\ast }}\right),$$

k₁, k₂ and k₃ are constants that will be checked. The derivative of W₂ with respect to t along the solution of model (1) gives that

$$\begin{array}{ll}{\dot{W}}_2=& \left(1-\frac{S^{\ast }}{S}\right)\dot{S}+\left(1-\frac{E^{\ast }}{E}\right)\dot{E}+k_1\left(1-\frac{I_1^{\ast }}{I_1}\right){\dot{I}}_1+k_2\left(1-\frac{I_2^{\ast }}{I_2}\right){\dot{I}}_2+k_3\left(1-\frac{L_1^{\ast }}{L_1}\right){\dot{L}}_1,\end{array}$$

in which

$$\left(1-\frac{S^{\ast }}{S}\right)\dot{S}=\left(1-\frac{S^{\ast }}{S}\right)[\mathrm{\Lambda }-({\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S)-\mu S]=\left(1-\frac{S^{\ast }}{S}\right)\left[\left({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast }+\mu S^{\ast }-({\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S)-\mu S\right)\right]-\mu \frac{{(S-S^{\ast })}^2}{S}+{\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast }-{\beta }_1{I}_{1}S-{\beta }_2{I}_{2}S-{\beta }_3{L}_{1}S-{\beta }_1{I}_1^{\ast }{S}^{\ast }\frac{S}{S^{\ast }}-{\beta }_2{I}_2^{\ast }{S}^{\ast }\frac{S}{S^{\ast }}-{\beta }_3{L}_1^{\ast }{S}^{\ast }\frac{S}{S^{\ast }}+{\beta }_1{I}_1^{\ast }{S}^{\ast }\frac{I_1}{I_1^{\ast }}+{\beta }_2{I}_2^{\ast }{S}^{\ast }\frac{I_2}{I_2^{\ast }}+{\beta }_3{L}_1^{\ast }{S}^{\ast }\frac{L_1}{L_1^{\ast }},\left(1-\frac{E^{\ast }}{E}\right)\dot{E}=\left(1-\frac{E^{\ast }}{E}\right)[({\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S)-({\gamma }_1+\mu )E]=\left(1-\frac{E^{\ast }}{E}\right)\left[({\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S)-({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast })\frac{E}{E^{\ast }}\right]={\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S-{\beta }_1{I}_1^{\ast }{S}^{\ast }\frac{E}{E^{\ast }}-{\beta }_2{I}_2^{\ast }{S}^{\ast }\frac{E}{E^{\ast }}-{\beta }_3{L}_1^{\ast }{S}^{\ast }\frac{E}{E^{\ast }}-{\beta }_1{I}_1^{\ast }{S}^{\ast }\frac{I_{1}S{E}^{\ast }}{I_1^{\ast }{S}^{\ast }E}-{\beta }_2{I}_2^{\ast }{S}^{\ast }\frac{I_{2}S{E}^{\ast }}{I_2^{\ast }{S}^{\ast }E}-{\beta }_3{L}_1^{\ast }{S}^{\ast }\frac{L_{1}S{E}^{\ast }}{L_1^{\ast }{S}^{\ast }E}+{\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast },k_1\left(1-\frac{I_1^{\ast }}{I_1}\right){\dot{I}}_1=k_1\left(1-\frac{I_1^{\ast }}{I_1}\right)(\eta {\gamma }_{1}E-M_1{I}_1+\rho \theta L_1)=k_1\left(1-\frac{I_1^{\ast }}{I_1}\right)\left(\eta {\gamma }_{1}E+\rho \theta L_1-\eta {\gamma }_1{E}^{\ast }\frac{I_1}{I_1^{\ast }}-\rho \theta L_1^{\ast }\frac{I_1}{I_1^{\ast }}\right)=k_1\eta {\gamma }_1{E}^{\ast }\frac{E}{E^{\ast }}+k_1\rho \theta L_1^{\ast }\frac{L_1}{L_1^{\ast }}-k_1\eta {\gamma }_1{E}^{\ast }\frac{I_1}{I_1^{\ast }}-k_1\rho \theta L_1^{\ast }\frac{I_1}{I_1^{\ast }}-k_1\eta {\gamma }_1{E}^{\ast }\frac{E{I}_1^{\ast }}{E^{\ast }{I}_1}-k_1\rho \theta L_1^{\ast }\frac{L_1{I}_1^{\ast }}{L_1^{\ast }{I}_1}+k_1\eta {\gamma }_1{E}^{\ast }+k_1\rho \theta L_1^{\ast },k_2\left(1-\frac{I_2^{\ast }}{I_2}\right){\dot{I}}_2=k_2\left(1-\frac{I_2^{\ast }}{I_2}\right)({\gamma }_2{I}_1-M_2{I}_1+(1-\rho )\theta L_1)=k_2\left(1-\frac{I_2^{\ast }}{I_2}\right)\left({\gamma }_2{I}_1+(1-\rho )\theta L_1-{\gamma }_2{I}_1^{\ast }\frac{I_2}{I_2^{\ast }}-(1-\rho )\theta L_1^{\ast }\frac{I_2}{I_2^{\ast }}\right)=k_2{\gamma }_2{I}_1^{\ast }\frac{I_1}{I_1^{\ast }}+k_2(1-\rho )\theta L_1^{\ast }\frac{L_1}{L_1^{\ast }}-k_2{\gamma }_2{I}_1^{\ast }\frac{I_2}{I_2^{\ast }}-k_2(1-\rho )\theta L_1^{\ast }\frac{I_2}{I_2^{\ast }}-k_2{\gamma }_2{I}_1^{\ast }\frac{I_1{I}_2^{\ast }}{I_1^{\ast }{I}_2}-k_2(1-\rho )\theta L_1^{\ast }\frac{L_1{I}_2^{\ast }}{L_1^{\ast }{I}_2}+k_2{\gamma }_2{I}_1^{\ast }+k_2(1-\rho )\theta L_1^{\ast },$$

$$k_3\left(1-\frac{L_1^{\ast }}{L_1}\right){\dot{L}}_1=k_3\left(1-\frac{L_1^{\ast }}{I_1}\right)[(1-\eta ){\gamma }_{1}E+{\gamma }_3{I}_2-(M_3+\theta )L_1]=k_3\left(1-\frac{L_1^{\ast }}{L_1}\right)\left[(1-\eta ){\gamma }_{1}E+{\gamma }_3{I}_2-(1-\eta ){\gamma }_1{E}^{\ast }\frac{L_1}{L_1^{\ast }}-{\gamma }_3{I}_2^{\ast }\frac{L_1}{L_1^{\ast }}\right]=k_3(1-\eta ){\gamma }_1{E}^{\ast }\frac{E}{E^{\ast }}+k_3{\gamma }_3{I}_2^{\ast }\frac{I_2}{I_2^{\ast }}-k_3(1-\eta ){\gamma }_1{E}^{\ast }\frac{L_1}{L_1^{\ast }}-k_3{\gamma }_3{I}_2^{\ast }\frac{L_1}{L_1^{\ast }}-k_3(1-\eta ){\gamma }_1{E}^{\ast }\frac{E{L}_1^{\ast }}{E^{\ast }{L}_1}-k_3{\gamma }_3{I}_2^{\ast }\frac{I_2{L}_1^{\ast }}{I_2^{\ast }{L}_1}+(1-\eta ){\gamma }_1{E}^{\ast }+k_3{\gamma }_3{I}_2^{\ast }.$$

Further, we obtain

$$\begin{array}{ll}{\dot{W}}_2=& -\mu \frac{{(S-S^{\ast })}^2}{S}+{\beta }_1{I}_1^{\ast }{S}^{\ast }\left(2-\frac{E}{E^{\ast }}-\frac{S^{\ast }}{S}+\frac{I_1}{I_1^{\ast }}-\frac{E^{\ast }{I}_{1}S}{E{I}_1^{\ast }{S}^{\ast }}\right)\\ & +{\beta }_2{I}_2^{\ast }{S}^{\ast }\left(2-\frac{E}{E^{\ast }}-\frac{S^{\ast }}{S}+\frac{I_2}{I_2^{\ast }}-\frac{E^{\ast }{I}_{2}S}{E{I}_2^{\ast }{S}^{\ast }}\right)\\ & +{\beta }_3{L}_1^{\ast }{S}^{\ast }\left(2-\frac{E}{E^{\ast }}-\frac{S^{\ast }}{S}+\frac{L_1}{L_1^{\ast }}-\frac{E^{\ast }{L}_{1}S}{E{L}_1^{\ast }{S}^{\ast }}\right)+k_1\eta {\gamma }_1{E}^{\ast }\left(\frac{E}{E^{\ast }}-\frac{I_1}{I_1^{\ast }}-\frac{E{I}_1^{\ast }}{E^{\ast }{I}_1}+1\right)\\ & +k_3(1-\eta ){\gamma }_1{E}^{\ast }\left(\frac{E}{E^{\ast }}-\frac{L_1}{L_1^{\ast }}-\frac{E{L}_1^{\ast }}{E^{\ast }{L}_1}+1\right)+k_2{\gamma }_2{I}_1^{\ast }\left(\frac{I_1}{I_1^{\ast }}-\frac{I_2}{I_2^{\ast }}-\frac{I_1{I}_2^{\ast }}{I_1^{\ast }{I}_2}+1\right)\\ & +k_3{\gamma }_3{I}_2^{\ast }\left(\frac{I_2}{I_2^{\ast }}-\frac{L_1}{L_1^{\ast }}-\frac{I_2{L}_1^{\ast }}{I_2^{\ast }{L}_1}+1\right)+k_1\rho \theta L_1^{\ast }\left(\frac{L_1}{L_1^{\ast }}-\frac{I_1}{I_1^{\ast }}-\frac{I_1^{\ast }{L}_1}{I_1{L}_1^{\ast }}+1\right)\\ & +k_2(1-\rho )\theta L_1^{\ast }\left(\frac{L_1}{L_1^{\ast }}-\frac{I_2}{I_2^{\ast }}-\frac{I_2^{\ast }{L}_1}{I_2{L}_1^{\ast }}+1\right).\end{array}$$

According to the inequality of 1 − (1/x) ≤ ln x for x > 0, we derive

$$1-\frac{E^{\ast }{I}_{1}S}{E{I}_1^{\ast }{S}^{\ast }}\le \ln \frac{E{I}_1^{\ast }{S}^{\ast }}{E^{\ast }{I}_{1}S}=\ln \frac{E}{E^{\ast }}-\ln \frac{I_{1}S}{I_1^{\ast }{S}^{\ast }}=\ln \frac{E}{E^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}-\ln \frac{S}{S^{\ast }},1-\frac{S^{\ast }}{S}\le \ln \frac{S}{S^{\ast }}.$$ (12)

Hence,

$$\begin{array}{ll}2-\frac{E}{E^{\ast }}-\frac{S^{\ast }}{S}+\frac{I_1}{I_1^{\ast }}-\frac{E^{\ast }{I}_{1}S}{E{I}_1^{\ast }{S}^{\ast }}& \le \frac{I_1}{I_1^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}-\frac{E}{E^{\ast }}+\ln \frac{E}{E^{\ast }}+1-\frac{S^{\ast }}{S}-\ln \frac{S}{S^{\ast }}\\ & \le \frac{I_1}{I_1^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}-\frac{E}{E^{\ast }}+\ln \frac{E}{E^{\ast }}.\end{array}$$

By the similar discussion, we obtain

$$\begin{array}{ll}{\dot{W}}_2\le & -\mu \frac{{(S-S^{\ast })}^2}{S}+{\beta }_1{I}_1^{\ast }{S}^{\ast }\left(\frac{I_1}{I_1^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}-\frac{E}{E^{\ast }}+\ln \frac{E}{E^{\ast }}\right)\\ & +{\beta }_2{I}_2^{\ast }{S}^{\ast }\left(\frac{I_2}{I_2^{\ast }}-\ln \frac{I_2}{I_2^{\ast }}-\frac{E}{E^{\ast }}+\ln \frac{E}{E^{\ast }}\right)+{\beta }_3{L}_1^{\ast }{S}^{\ast }\left(\frac{L_1}{L_1^{\ast }}-\ln \frac{L_1}{L_1^{\ast }}-\frac{E}{E^{\ast }}+\ln \frac{E}{E^{\ast }}\right)\\ & +k_1\eta {\gamma }_1{E}^{\ast }\left(\frac{E}{E^{\ast }}-\ln \frac{E}{E^{\ast }}-\frac{I_1}{I_1^{\ast }}+\ln \frac{I_1}{I_1^{\ast }}\right)+k_3(1-\eta ){\gamma }_1{E}^{\ast }\left(\frac{E}{E^{\ast }}-\ln \frac{E}{E^{\ast }}-\frac{L_1}{L_1^{\ast }}+\ln \frac{L_1}{L_1^{\ast }}\right)\\ & +k_2{\gamma }_2{I}_1^{\ast }\left(\frac{I_1}{I_1^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}-\frac{I_2}{I_2^{\ast }}+\ln \frac{I_2}{I_2^{\ast }}\right)+k_3{\gamma }_3{I}_2^{\ast }\left(\frac{I_2}{I_2^{\ast }}-\ln \frac{I_2}{I_2^{\ast }}-\frac{L_1}{L_1^{\ast }}+\ln \frac{L_1}{L_1^{\ast }}\right)\\ & +k_1\rho \theta L_1^{\ast }\left(\frac{L_1}{L_1^{\ast }}-\ln \frac{L_1}{L_1^{\ast }}-\frac{I_1}{I_1^{\ast }}+\ln \frac{I_1}{I_1^{\ast }}\right)+k_2(1-\rho )\theta L_1^{\ast }\left(\frac{L_1}{L_1^{\ast }}-\ln \frac{L_1}{L_1^{\ast }}-\frac{I_2}{I_2^{\ast }}+\ln \frac{I_2}{I_2^{\ast }}\right).\\ =& -\mu \frac{{(S-S^{\ast })}^2}{S}+[k_1\eta {\gamma }_1{E}^{\ast }+k_3(1-\eta ){\gamma }_1{E}^{\ast }-{\beta }_1{I}_1^{\ast }{S}^{\ast }-{\beta }_2{I}_2^{\ast }{S}^{\ast }-{\beta }_3{L}_1^{\ast }{S}^{\ast }]\left(\frac{E}{E^{\ast }}-\ln \frac{E}{E^{\ast }}\right)\\ & +[{\beta }_1{I}_1^{\ast }{S}^{\ast }-k_1\eta {\gamma }_1{E}^{\ast }+k_2{\gamma }_2{I}_1^{\ast }-k_1\rho \theta L_1^{\ast }]\left(\frac{I_1}{I_1^{\ast }}-\ln \frac{I_1}{I_1^{\ast }}\right)\\ & +[{\beta }_2{I}_2^{\ast }{S}^{\ast }-k_2{\gamma }_2{I}_1^{\ast }+k_3{\gamma }_3{I}_2^{\ast }-k_2(1-\rho )\theta L_1^{\ast }]\left(\frac{I_2}{I_2^{\ast }}-\ln \frac{I_2}{I_2^{\ast }}\right)\\ & +[{\beta }_3{L}_1^{\ast }{S}^{\ast }-k_3(1-\eta ){\gamma }_1{E}^{\ast }-k_3{\gamma }_3{I}_2^{\ast }+k_1\rho \theta L_1^{\ast }+k_2(1-\rho )\theta L_1^{\ast }]\left(\frac{L_1}{L_1^{\ast }}-\ln \frac{L_1}{L_1^{\ast }}\right),\end{array}$$

here,

$$\begin{array}{ll}{k}_1& =\frac{H_1}{H_4}>0,k_2=\frac{H_2}{H_4}>0,k_3=\frac{H_3}{H_4}>0,\\ H_1& =({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast })I_1^{\ast }{I}_2^{\ast }{\gamma }_2{\gamma }_3+{\beta }_1{E}^{\ast }{I}_1^{\ast }{L}_1^{\ast }{S}^{\ast }(1-\eta ){\gamma }_1(1-\eta )\theta \\ & +({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast })E^{\ast }{I}_1(1-\eta ){\gamma }_1{\gamma }_2>0,\\ H_2& ={\beta }_2{(E^{\ast })}^2{I}_2^{\ast }{S}^{\ast }\eta (1-\eta ){\gamma }_1^2+({\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast })E^{\ast }{I}_1^{\ast }\eta {\gamma }_1{\gamma }_3\\ & +{\beta }_2{E}^{\ast }{I}_2^{\ast }{L}_1^{\ast }{S}^{\ast }(1-\eta ){\gamma }_1\rho \theta +({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast })I_2^{\ast }{L}_1^{\ast }{\gamma }_3\rho \theta >0,\\ H_3& =({\beta }_1{I}_1^{\ast }{S}^{\ast }+{\beta }_2{I}_2^{\ast }{S}^{\ast }+{\beta }_3{L}_1^{\ast }{S}^{\ast })[L_1^{\ast }\rho (1-\rho )\theta +I_1^{\ast }{L}_1^{\ast }{\gamma }_2\rho \theta ]\\ & +{\beta }_3{E}^{\ast }{I}_1^{\ast }{L}_1^{\ast }{S}^{\ast }\eta {\gamma }_1{\gamma }_2+{\beta }_2{E}^{\ast }{I}_2^{\ast }{L}_1^{\ast }{S}^{\ast }\eta {\gamma }_1(1-\rho )\theta >0,\\ H_4& =E^{\ast }{\gamma }_1\left[{(L_1^{\ast })}^2(1-\eta )\rho (1-\rho )\theta +E^{\ast }{I}_1^{\ast }\eta (1-\eta ){\gamma }_1{\gamma }_2+I_1^{\ast }{I}_2^{\ast }\eta {\gamma }_2{\gamma }_3\right)\\ & \left(+E^{\ast }{L}_1^{\ast }\eta (1-\eta ){\gamma }_1(1-\rho )\theta +I_1^{\ast }{L}_1^{\ast }(1-\eta ){\gamma }_2\rho \theta \right]>0.\end{array}$$

Thus, for ${\mathcal{R}}_0$>1, we obtain

$${\dot{W}}_2(t)=-\frac{\mu {(S-S^{\ast })}^2}{S}<0.$$

Indeed, ${\dot{W}}_2(t)=0$ if and only if S = S∗ and $\dot{S}=0$.

Let γ₆ = 0 and a₁ = a₂ = a₃ = 0. The first and second equations of model (1) become

$$\begin{array}{ll}\dot{S}(t)& =\mathrm{\Lambda }-({\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S)-\mu S,\\ \dot{E}(t)& ={\beta }_1{I}_{1}S+{\beta }_2{I}_{2}S+{\beta }_3{L}_{1}S-({\gamma }_1+\mu )E.\end{array}$$

Then

$$\dot{E}=\mathrm{\Lambda }-\mu S^{\ast }-({\gamma }_1+\mu )E,E=\frac{\mathrm{\Lambda }-\mu S^{\ast }}{{\gamma }_1+\mu }.$$

From the first and second equations of (11), we obtain

$$E^{\ast }=\frac{\mathrm{\Lambda }-\mu S^{\ast }}{{\gamma }_1+\mu }.$$

So, we get E = E∗. From (12), we derive $E/E^{\ast }=I_1/I_1^{\ast }$, and hence $I_1=I_1^{\ast }$. Similarly, we obtain $I_2=I_2^{\ast }$ and $L_1=L_1^{\ast }$. Further, we obtain $L_2=L_2^{\ast }$, T = T∗ and R = R∗. It is clear that {P∗} is the largest compact invariant subset of ${\mathrm{\Omega }}_2=\{(S,E,I_1,I_2,L_1,L_2,T,R)\in {\mathbb{R}}_{+}^8:{\dot{W}}_2(t)=0\}$. By LaSalle's invariance principle, the endemic equilibrium point P∗ of model (1) is globally asymptotically stable. The proof is complete.

Corollary 3.1

Let γ₆ = 0 if the psychological effects in the population (i.e., a₁ > 0, a₂ > 0, a₃ > 0) are considered and ${\mathcal{R}}_0$>1, then the existence and uniqueness of the endemic equilibrium point P∗∗=(S∗∗, E∗∗, $I_1^{\ast \ast }$, $I_2^{\ast \ast }$, $L_1^{\ast \ast }$, $L_2^{\ast \ast }$, T∗∗, R∗∗) of model (1) holds.

Proof The endemic equilibrium point P∗∗=(S∗∗, E∗∗, $I_1^{\ast \ast }$, $I_2^{\ast \ast }$, $L_1^{\ast \ast }$, $L_2^{\ast \ast }$, T∗∗, R∗∗) satisfies

$$\left\{\begin{array}{ll}0& =\mathrm{\Lambda }-\left(\frac{{\beta }_1{I}_1^{\ast \ast }{S}^{\ast \ast }}{1+a_1{I}_1^{\ast \ast }}+\frac{{\beta }_2{I}_2^{\ast \ast }{S}^{\ast \ast }}{1+a_2{I}_2^{\ast \ast }}+\frac{{\beta }_3{L}_1^{\ast \ast }{S}^{\ast \ast }}{1+a_3{L}_1^{\ast \ast }}\right)-\mu S^{\ast \ast },\\ 0& =\left(\frac{{\beta }_1{I}_1^{\ast \ast }{S}^{\ast \ast }}{1+a_1{I}_1^{\ast \ast }}+\frac{{\beta }_2{I}_2^{\ast \ast }{S}^{\ast \ast }}{1+a_2{I}_2^{\ast \ast }}+\frac{{\beta }_3{L}_1^{\ast \ast }{S}^{\ast \ast }}{1+a_3{L}_1^{\ast \ast }}\right)-({\gamma }_1+\mu )E^{\ast \ast },\\ 0& =\eta {\gamma }_1{E}^{\ast \ast }-({\delta }_1+{\gamma }_2+\mu )I_1^{\ast \ast }+\rho \theta L_1^{\ast \ast },\\ 0& ={\gamma }_2{I}_1^{\ast \ast }-({\delta }_2+{\gamma }_3+\mu )I_2^{\ast \ast }+(1-\rho )\theta L_1^{\ast \ast },\\ 0& =(1-\eta ){\gamma }_1{E}^{\ast \ast }+{\gamma }_3{I}_2^{\ast \ast }-({\delta }_3+{\gamma }_4+\mu +\theta )L_1^{\ast \ast },\\ 0& ={\gamma }_4{L}_1^{\ast \ast }-({\delta }_4+{\gamma }_5+\mu )L_2^{\ast \ast },\\ 0& ={\gamma }_5{L}_2^{\ast \ast }-({\delta }_5+\alpha +\mu )T^{\ast \ast },\\ 0& ={\delta }_1{I}_1^{\ast \ast }+{\delta }_2{I}_2^{\ast \ast }+{\delta }_3{L}_1^{\ast \ast }+{\delta }_4{L}_2^{\ast \ast }+{\delta }_5{T}^{\ast \ast }-\mu R^{\ast \ast }.\end{array}\right)$$ (13)

From (13), let $S^{\ast \ast },E^{\ast \ast },I_1^{\ast \ast },I_2^{\ast \ast },L_1^{\ast \ast },L_2^{\ast \ast },R^{\ast \ast }$ be uniquely determined by T∗∗. Here,

$$\begin{array}{ll}{S}^{\ast \ast }& =\frac{\mathrm{\Lambda }}{\mu }-\frac{({\gamma }_1+\mu )D_1{G}_3}{\mu D_2{\gamma }_1{\gamma }_4{\gamma }_5}{T}^{\ast \ast },E^{\ast \ast }=\frac{D_1{G}_3}{D_2{\gamma }_1{\gamma }_4{\gamma }_5}{T}^{\ast \ast },I_1^{\ast \ast }=\frac{G_1}{D_2{\gamma }_4{\gamma }_5}{T}^{\ast \ast },\\ I_2^{\ast \ast }& =\frac{G_2}{D_2{\gamma }_4{\gamma }_5}{T}^{\ast \ast },L_1^{\ast \ast }=\frac{G_3}{{\gamma }_4{\gamma }_5}{T}^{\ast \ast },L_2^{\ast \ast }=\frac{M_5}{{\gamma }_5}{T}^{\ast \ast },\\ R^{\ast \ast }& =\frac{1}{\mu }({\delta }_1{I}_1^{\ast \ast }+{\delta }_2{I}_2^{\ast \ast }+{\delta }_3{L}_1^{\ast \ast }+{\delta }_4{L}_2^{\ast \ast }+{\delta }_5{T}^{\ast \ast }).\end{array}$$ (14)

where

$$D_1=M_1{M}_2(M_3+\theta )-{\gamma }_3{M}_1(1-\rho )\theta -{\gamma }_2{\gamma }_3\rho \theta =M_1{M}_2{M}_3+M_1{M}_2\theta -{\gamma }_3{M}_1\theta +{\gamma }_3{M}_1\rho \theta -{\gamma }_2{\gamma }_3\rho \theta =M_1{M}_2{M}_3+M_1\theta (M_2-{\gamma }_3)+{\gamma }_3\rho \theta (M_1-{\gamma }_2)\ge M_1{M}_2{M}_3>0,D_2=(1-\eta )M_1{M}_2+\eta {\gamma }_2{\gamma }_3>0,G_1=[\eta M_2(M_3+\theta )+(1-\eta )M_2\rho \theta -\eta {\gamma }_3(1-\rho )\theta ]G_3>0,G_2=[\eta {\gamma }_2(M_3+\theta )+(1-\eta )M_1(1-\rho )\theta +(1-\eta ){\gamma }_2\rho \theta ]G_3>0,G_3=M_3{M}_4>0.$$

Substituting (14) into (13), we obtain

$$f_1(T^{\ast \ast })T^{\ast \ast }=f_2(T^{\ast \ast })T^{\ast \ast },$$ (15)

where

$$\begin{array}{ll}{f}_1(T)& =\frac{\mathrm{\Lambda }}{\mu }-\frac{({\gamma }_1+\mu )D_1{G}_3}{\mu D_2{\gamma }_1{\gamma }_4{\gamma }_5}T,\\ f_2(T)& =\frac{({\gamma }_1+\mu )D_1{G}_3}{\mu D_2{\gamma }_1{\gamma }_4{\gamma }_{5}G(T)},\\ G(T)& =\frac{{\beta }_1{G}_1}{a_1{G}_{1}T+D_2{\gamma }_4{\gamma }_5}+\frac{{\beta }_2{G}_2}{a_2{G}_{2}T+D_2{\gamma }_4{\gamma }_5}+\frac{{\beta }_3{G}_3}{a_3{G}_{3}T+{\gamma }_4{\gamma }_5}.\end{array}$$ (16)

From (15), we obtain T∗∗ = 0 or f₁(T∗∗) = f₂(T∗∗). Due to T∗∗ is positive solution, T∗∗ = 0 is dropped. Next, we only consider f₁(T∗∗) = f₂(T∗∗). From (16), it is obvious that f₁(T) is a strictly decreasing function for T > 0 and f₂(T) is a strictly increasing function for T ≥ 0 because G(T) is a strictly decreasing function for T > 0. Meanwhile, f₁(0) = Λ/μ and f₂(0) = C/B < Λ/μ = f₁(0) when ${\mathcal{R}}_0=(\mathrm{\Lambda }B)/(\mu C)>1$ holds. Hence, f₁(T) and f₂(T) have a unique solution T∗∗ to (15). The existence and uniqueness of P∗∗=(S∗∗, E∗∗, $I_1^{\ast \ast }$, $I_2^{\ast \ast }$, $L_1^{\ast \ast }$, $L_2^{\ast \ast }$, T∗∗, R∗∗) holds. The proof is complete.

4. Numerical simulations

4.1. Data fitted with parameter estimation

The Chinese government implemented five control measures for curbing the spread of syphilis in 2006 (National Health Commission of the People’s Republic of China, 2006), 2010 (National Health Commission of the People’s Republic of China, 2010), 2013 (National Health Commission of the People’s Republic of China, 2013), 2015 (National Health Commission of the People’s Republic of China, 2015), and 2020 (National Health Commission of the People’s Republic of China, 2020). The detection methods of syphilis were updated to increase the coverage of treatments in 2016 (National Health Commission of the People’s Republic of China, 2016). The monthly surveillance data of syphilis of Fujian Province, spanning from January 2005 to December 2022, are obtained from Fujian Province Center for Disease Control and Prevention (i.e., Fujian CDC). The corresponding population size of Fujian Province is collected from the Fujian Statistical Yearbook in 2004 (Fujian Provincial Bureau of Statistics, 2023). The 2005–2022 monthly proportion tendency of latent syphilis is increasing as of December 2022 (Fig. 2). The awareness delay of an infectious disease is referred as the time period between the date of the first infection of an individual and the date of the first confirmation for the same individual, which is independent of the models and the surveillance data, but relating to the infection scale and the basic reproduction number when the assessments are needed in (Chen et al., 2024; Huang et al., 2020; Lan et al., 2024; Wang et al., 2025b; Wei et al., 2023). So, the awareness delay of syphilis in Fujian Province is chosen as one-month since the incubation period ranges from three-week to four-week. The step size is set as 0.01 for the performance of the numerical simulations.

Fig. 2.

Monthly proportion of syphilis cases at different infection stages from 2005 to 2022.

The parameter estimations of the models are crucial for describing the epidemiological indices such as the basic reproduction number, infection scale and transmission risk. The initial values of syphilis in Fujian Province are set as N(0) = 35, 290, 000, E(0) = 387, I₁(0) = 242, I₂(0) = 194, L₁(0) = 145, L₂(0) = 92, T(0) = R(0) = 0. The curve for cumulative infection scale of syphilis is conducted against the curve for 2005–2022 cumulative surveillance data from Fujian CDC by the least squares method (Chen et al., 2024; Wang et al., 2025b). The corresponding parameter values are taken from Table 1. The basic reproduction number of syphilis showed a declining tendency by phases due to the implementations of five national policies from 2006 to 2020, of which the lowest value 1.40 was reached in 2020, the highest value 1.69 was reached in 2013 (Fig. 3). If the awareness delay of syphilis is taken as three-month or six-month (Fig. 4), then the corresponding infection scales are approximately increased by 9327 cases or 16091 cases, respectively. The numerical simulation results point out that the infection scales and the transmission risk depend on value of the awareness delay.

Table 1.

Main parameters and their values of syphilis in Fujian Province from 2005 to 2022.

Parameter	Value	Period (Start - End)a	Source	Parameter	Value	Period (Start - End)a	Source
Λ	39,334	2004-12 - 2022-12	FBSb	δ5	0.7292	2004-12 – 2016-11	Fitted
β1	1.3898 × 10−8	2004-12 - 2006-02	Fitted	δ5	0.8492	2016-12 – 2022-12	Fitted
β1	1.2878 × 10−8	2006-03 - 2010-05	Fitted	a1	0.00020	2004-12 - 2006-02	Fitted
β1	1.1599 × 10−8	2010-06 - 2012-12	Fitted	a1	0.00023	2006-03 - 2010-05	Fitted
β1	1.3523 × 10−8	2013-01 - 2015-05	Fitted	a1	0.00026	2010-06 - 2012-12	Fitted
β1	1.2999 × 10−8	2015-06 - 2020-11	Fitted	a1	0.00023	2013-01 - 2015-05	Fitted
β1	1.2092 × 10−8	2020-12 - 2022-12	Fitted	a1	0.00026	2015-06 - 2020-11	Fitted
β2	1.2056 × 10−8	2004-12 - 2006-02	Fitted	a1	0.00029	2020-12 - 2022-12	Fitted
β2	1.0791 × 10−8	2006-03 - 2010-05	Fitted	a2	0.00023	2004-12 - 2006-02	Fitted
β2	0.9656 × 10−8	2010-06 - 2012-12	Fitted	a2	0.00026	2006-03 - 2010-05	Fitted
β2	1.1992 × 10−8	2013-01 - 2015-05	Fitted	a2	0.00029	2010-06 - 2012-12	Fitted
β2	1.1589 × 10−8	2015-06 - 2020-11	Fitted	a2	0.00026	2013-01 – 2015 -05	Fitted
β2	1.0891 × 10−8	2020-12 - 2022-12	Fitted	a2	0.00029	2015-06 - 2020-11	Fitted
β3	1.7594 × 10−8	2004-12 - 2006-02	Fitted	a2	0.00032	2020-12 - 2022-12	Fitted
β3	1.6007 × 10−8	2006-03 - 2010-05	Fitted	a3	0.00026	2004-12 - 2006-02	Fitted
β3	1.5590 × 10−8	2010-06 - 2012-12	Fitted	a3	0.00029	2006-03 - 2010-05	Fitted
β3	1.7899 × 10−8	2013-01 - 2015-05	Fitted	a3	0.00032	2010-06 - 2012-12	Fitted
β3	1.7519 × 10−8	2015-06 - 2020-11	Fitted	a3	0.00029	2013-01 - 2015-05	Fitted
β3	1.5899 × 10−8	2020-12 - 2022-12	Fitted	a3	0.00032	2015-06 - 2020-11	Fitted
α	2.3048 × 10−8	2004-12 - 2022-12	Fitted	a3	0.00035	2020-12 - 2022-12	Fitted
μ	4.9264 × 10−4	2004-12 - 2022-12	FBSb	γ1	1.0714	2004-12 - 2022-12	Garnett et al. (1997)
δ1	0.3760	2004-12 - 2016-11	Fitted	γ2	0.6522	2004-12 - 2022-12	Garnett et al. (1997)
δ1	0.4102	2016-12 - 2022-12	Fitted	γ3	0.2778	2004-12 - 2022-12	Garnett et al. (1997)
δ2	0.4495	2004-12 - 2016-11	Fitted	γ4	0.0833	2004-12 - 2022-12	Garnett et al. (1997)
δ2	0.4612	2016-12 - 2022-12	Fitted	γ5	0.0417	2004-12 - 2022-12	Wu et al. (2024)
δ3	0.1025	2004-12 - 2016-11	Fitted	γ6	0.0167	2004-12 - 2022-12	Wu et al. (2024)
δ3	0.1256	2016-12 - 2022-12	Fitted	ρ	0.4069	2004-12 - 2022-12	Fitted
δ4	0.1018	2004-12 - 2016-11	Fitted	θ	0.2500	2004-12 - 2022-12	Fitted
δ4	0.1135	2016-12 - 2022-12	Fitted	η	0.6480	2004-12 - 2022-12	Fitted

^a2004–12 is for YYYY - MM with Y (year) and M (month).

^bFBS is Fujian Provincial Bureau of Statistics in Fujian Provincial Bureau of Statistics (2023).

Fig. 3.

Optimal fitting with one-month awareness delay by least squares method (Red solid curve) against all cases (Blue dashed curve) and latent cases (Purple dashed curve) for syphilis in the 2005–2022 monthly surveillance data of Fujian Province. Basic reproduction number for each phase (Orange solid line). The rectangular blocks of different colors represent the time periods when the Chinese government implements new policies for syphilis.

Fig. 4.

Comparisons of the cumulative numbers of infection cases for syphilis with three-month awareness delay and six-month awareness delay in Fujian Province from 2005 to 2022.

4.2. Sensitivity analysis of ${\mathcal{R}}_0$

The sensitivity analyses of ${\mathcal{R}}_0$ against transmission rates and treatment rates are displayed in Fig. 5. It is found that ${\mathcal{R}}_0$ exhibits positive correlation with transmission rates. The hightest positive correlation of ${\mathcal{R}}_0$ is transmission rate in the early latent stage, then follows transmission rates in the secondary stage and primary stage. It is also revealed that ${\mathcal{R}}_0$ exhibits negative correlation with treatment rates. The hightest negative correlation of ${\mathcal{R}}_0$ is treatment rate in the secondary stage, then follows treatment rates in the early latent stage and primary stage. If transmission rate in the early latent stage of Table 1 is increased or decreased by 20 %, then the corresponding infection scales for syphilis are approximately increased by 101,364 cases or decreased by 103,811 cases in Fig. 6, respectively. If treatment rate in the secondary stage of Table 1 is increased or decreased by 20 %, then the corresponding infection scales for syphilis are approximately decreased by 44,867 cases or increased by 51,317 cases in Fig. 6, respectively.

Fig. 5.

The contour plots of sensitivities for ${\mathcal{R}}_0$ with respect to transmission rates (Left) and treatment rates (Right) in the primary, secondary and early latent stages.

Fig. 6.

Impacts of transmission rate in the early latent stage (Top) and treatment rate in the secondary stage (Bottom) on syphilis infection scale.

4.3. Psychological effect coefficients

The psychological effect coefficients are enhanced as the intensity of media campaigns is increased (Baba et al., 2019). In this study, the impacts of psychological effect coefficients in the primary, secondary, and early latent stages on the infection scale are explored. The scenarios (1, 0, 0), (0, 1, 0), and (0, 0, 1) represent single-measure scenarios. Scenarios (1, 1, 0), (1, 0, 1), and (0, 1, 1) represent double-measure scenarios. Scenario (1, 1, 1) is for a triple-measure scenario and scenario (0, 0, 0) is for none-measure scenario. Precisely, the psychological effect coefficients reveal that triple-measure scenario carries out the smallest infection scale, none-measure scenario leads to the largest infection scale in Fig. 7, Fig. 8. The increases of the psychological effect coefficients (i.e., a₁, a₂, a₃) reduce the contact number between susceptible individuals and infected individuals, which in turn decrease the transmission rates (i.e., β₁, β₂, β₃). Since the expression of the basic reproduction number linearly depends on the transmission rates, both the basic reproduction number and infection scale would decline. Meanwhile, the psychological effect coefficient in the early latent stage has the largest impact on the infection scale, compared with the ones in other two stages. If the psychological effect coefficient in the early latent stage of Table 1 is increased or decreased by 20 %, then the corresponding infection scales for syphilis are approximately decreased by 38,487 cases or increased by 51,422 cases in Fig. 9, respectively.

Fig. 7.

Impacts of psychological effect coefficients (none-measure and single-measure) on syphilis infection scale.

Fig. 8.

Impacts of psychological effect coefficients (double-measure and triple-measure) on syphilis infection scale.

Fig. 9.

Impact of psychological effect coefficient in the early latent stage on syphilis infection scale.

4.4. Tendency prediction

The control policies of syphilis are implemented by the Chinese government in December 2022 (National Health Commission of the People’s Republic of China, 2022) and September 2025 (The National People’s Congress of People’s Republic of China, 2025) respectively. The changes of the main parameters including transmission rates, treatment rates, and psychological effect coefficients are extensively discussed by scenarios. The 2023–2030 tendency predictions of syphilis infection scales of Fujian Province are provided by scenarios in Fig. 10.

Fig. 10.

The 2023–2030 tendency predictions of the cumulative numbers of infection cases for syphilis in Fujian Province.

The baseline scenario is referred as the situation that the parameter values of Table 1 from 2020 to 2022 are kept same to predict 2023–2030 infection scale. The β_H scenario is for the situation that the values of the transmission rates of Table 1 are raised up 10 % in December 2022, and an extra increase 10 % in September 2025. The δ_H scenario is for the situation that the values of the treatment rates of Table 1 are raised up 10 % in December 2022, and an extra increase 10 % in September 2025. The a_H scenario is for the situation that the values of the psychological effect coefficients of Table 1 are raised up 10 % in December 2022, and an extra increase 10 % in September 2025. The β_L scenario is for the situation that the values of the transmission rates of Table 1 are decreased 10 % in December 2022, and an extra decline 10 % in September 2025. The δ_L scenario is for the situation that the values of the treatment rates of Table 1 are decreased 10 % in December 2022, and an extra decline 10 % in September 2025. The a_L scenario is for the situation that the values of the psychological effect coefficients of Table 1 are decreased 10 % in December 2022, and an extra decline 10 % in September 2025.

The tendency predictions of syphilis infection scale of Fujian Province with scenarios are performed up to the end of 2030 in Table 2, of which the transmission rates have the largest impacts on the infection scale, then the treatment rates and psychological effect coefficients are followed. The numerical simulations demonstrate that the infection scale of the baseline scenario is 5.6884 × 10⁵ cases, the largest infection scale is 7.2933 × 10⁵ cases for the (β_H, δ_L, a_L) scenario, and the smallest infection scale is 4.6788 × 10⁵ cases for the (β_L, δ_H, a_H) scenario. Since the basic reproduction number is not related to psychological effect coefficients in (5), the basic reproduction number of the baseline scenario is the same as that of the a_H and a_L scenarios, the basic reproduction number of the (β_H, δ_L) scenario is the same as that of the (β_H, δ_L, a_L) scenario, and the basic reproduction number of the (β_L, δ_H) scenario is the same as that of the (β_L, δ_H, a_H) scenario. Meanwhile, the average reproduction number in Table 3 reaches the maximum value in the (β_H, δ_L) scenario, and the minimum value in the (β_L, δ_H) scenario.

Table 2.

The prediction of the syphilis infection scale with scenarios in Fujian Province by the end of 2030.

	Baseline	βH	δL	aL	(βH, δL)	(βH, aL)	(δL, aL)	(βH, δL, aL)
Infection scale a	5.6884	6.4208	6.0235	5.9831	6.7694	6.8703	6.3899	7.2933
Saved scale a,b	0	0.7324	0.3351	0.2947	1.0810	1.1819	0.7015	1.6049

	Baseline	βL	δH	aH	(βL, δH)	(βL, aH)	(δH, aH)	(βL, δH, aH)
Infection scale a	5.6884	5.0508	5.3425	5.4630	4.7580	4.9254	5.1726	4.6788
Saved scale a,b	0	−0.6376	−0.3459	−0.2254	−0.9304	−0.7630	−0.5158	−1.0096

^aUnits of the infection scale and saved scale are × 10⁵.

^bSaved scale is calculated by taking the difference between the infection scale with each scenario and the infection scale with baseline scenario.

Table 3.

The basic reproduction numbers of syphilis by scenarios in Fujian Province from 2023 to 2030.

Period (Start - End)a	${\mathcal{R}}_0$	${\mathcal{R}}_0$			${\mathcal{R}}_0$			${\mathcal{R}}_0$
	Baseline	βH	δL	aL	(βH, δL)	(βH, aL)	(δL, aL)	(βH, δL, aL)
2023-01 - 2025-09	1.40	1.54	1.53	1.40	1.69	1.54	1.53	1.69
2025-10 - 2030-12	1.40	1.70	1.67	1.40	2.02	1.70	1.67	2.02
Average b	1.40	1.65	1.62	1.40	1.91	1.65	1.62	1.91

Period (Start - End)a	${\mathcal{R}}_0$	${\mathcal{R}}_0$			${\mathcal{R}}_0$			${\mathcal{R}}_0$
	Baseline	βL	δH	aH	(βL, δH)	(βL, aH)	(δH, aH)	(βL, δH, aH)
2023-01 - 2025-09	1.40	1.29	1.30	1.40	1.17	1.29	1.30	1.17
2025-10 - 2030-12	1.40	1.16	1.20	1.40	0.97	1.16	1.20	0.97
Average b	1.40	1.20	1.23	1.40	1.04	1.20	1.23	1.04

^a2023–01 is for YYYY - MM with Y (year) and M (month).

^bAverage = (P₁ × V₁ + P₂ × V₂)/(P₁ + P₂), where P_i and V_i are the periods and values that ${\mathcal{R}}_0$ belongs to.

5. Discussion and conclusion

In this study, we propose an SEILRS syphilis model with the early latent stage, in which a nonlinear saturated incidence rate is introduced to reflect the psychological inhibition effects of the infected individuals. This study focuses on the changes of the basic reproduction number and infection scale of syphilis with scenarios.

The basic reproduction number (i.e., ${\mathcal{R}}_0$) of model (1) is calculated using the next generation matrix method. It is easy to find that the expression of ${\mathcal{R}}_0$ is not affected by the psychological effect coefficients. The syphilis-free equilibrium point is locally and globally asymptotically stable if and only if ${\mathcal{R}}_0<1$. When the protective immunity loss and the psychological effects are neglected, we obtain that the endemic equilibrium point of model (1) is globally asymptotically stable if and only if ${\mathcal{R}}_0>1$. When the protective immunity loss is neglected and the psychological effects are considered, we further obtain that the existence and uniqueness of the endemic equilibrium point holds if and only if ${\mathcal{R}}_0>1$ but the global asymptotic stability of the endemic equilibrium point has not been proved. The sensitivity analyses of the basic reproduction number imply that the transmission risk of syphilis is effectively reduced by the decrease of transmission rates and the increase of treatment rates (Fig. 5).

The 2005–2022 monthly surveillance data of syphilis from Fujian CDC reveal an increasing trend in terms of the proportion of infected individuals in the latent stage (Wu et al., 2024), as shown in Fig. 2. That is to say, the risk of hidden transmission of syphilis is increased since infected individuals in the latent stage have no symptoms. Meanwhile, the awareness delay of syphilis is set as one-month and optimal fitting of 2005–2022 monthly surveillance data from Fujian Province are performed by using the least squares method, in which the basic reproduction number of syphilis ranges from 1.40 to 1.69 (Fig. 3). The value of basic reproduction number in this study is close to the ones in the recent studies. For instance, Wu et al. (2024) estimated that the value of the basic reproduction number of syphilis in Fujian Province was around 1.21, Jing et al. (2021) obtained that the value of the basic reproduction number of syphilis in Ningxia was 1.23, Chai et al. (2025) derived that the range of the basic reproduction number of syphilis of Yunnan Province was from 1.39 to 1.58. The longer awareness delay leads to the wider spread and brings the greater public health risks in Fig. 4. To put down the infection scale, the reduction of the transmission rate in the early latent stage is more effective than the transmission rates in the primary and secondary stages in Fig. 6, and the targeted treatment in the secondary stage is more effective than targeted treatment in the primary and early latent stages (Okuonghae et al., 2019) in Fig. 6. In addition, the psychological effect coefficients in the primary, secondary and early latent stages collectively minimize the infection scale of syphilis in Fig. 7, Fig. 8, and the psychological effect coefficient in the early latent stage is the more effective on infection scale compared to the primary and secondary stages in Fig. 9. Finally, based on the 2023–2030 tendency predictions of infection scale of syphilis (Fig. 10), the transmission rates have the greatest impacts on the infection scale, followed by the treatment rates and psychological effect coefficients.

This study suggests that the regions on the Chinese mainland with the similar population contact patterns and the economic backgrounds to ones of Fujian Province are encouraged to implement prevention and control measures in the following three aspects. To reduce the transmission rate in the early latent stage of syphilis, it is recommended to decrease the number of infected individuals in the early latent stage, and to shorten their duration of hidden transmission. It is recommended that high-risk groups such as sex workers and people with multiple sexual partners are encouraged to screen three times annually. This helps to identify more infected individuals in the primary and secondary stages against preventing the progression to the early latent stage, and to detect infected individuals in the early latent stage promptly to shorten their hidden transmission period. To improve the treatment rate in the secondary stage of syphilis, it is essential to ensure the effective and complete treatment for infected individuals in the secondary stage. The full-course treatment of benzathine benzylpenicillin to infected individuals should be performed in priority in the secondary stage, with follow-up visits every six months until their syphilis titer stabilizes. To enhance the intensity of psychological effect in the early latent stage of syphilis, it is recommended to vigorously publicize the asymptomatic and infectious characteristics of infected individuals in the early latent stage. For groups who have searched for syphilis-related information online, it is better to regularly push pop-up notifications and video content via health APPs. For proactive prevention and control, the posters and informational leaflets in crowded public areas are encouraged.

CRediT authorship contribution statement

Anli Yao: Writing – original draft, Visualization, Formal analysis, Data curation, Conceptualization. Fengying Wei: Writing – review & editing, Supervision, Project administration, Methodology. Jianfeng Xie: Supervision, Project administration.

Ethics approval and consent to participate

The ethical approval and individual consents were exempted as the aggregated data were used in this study.

Data and material availability

The data that supported this article may be available upon reasonable request to the corresponding author.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Handling Editor: Dr Daihai He