A dynamic model and cost-effectiveness on screening coverage and treatment of syphilis included MSM population in the United States

Abstract

Syphilis is a major sexually transmitted disease, causing a significant public health burden for countries all over the world. Since 2000, there has been a new outbreak of the syphilis epidemic in the United States. Therefore, the prevention and control of syphilis have important research significance. We have established a sex structure and ordinary differential equation model that includes men who have sex with men (MSM). Its epidemiological and biological parameters were obtained by fitting with regional monitoring data from the Centers for Disease Control and Prevention from 1984 to 2014, and the basic reproduction number (${\mathcal{R}_{0}}$) of syphilis is 1.3876. Through cost-effectiveness analysis, we have found that the most cost-effective strategies in the cases of sufficient and insufficient funds are conducting syphilis screening for 50% of sexually active susceptible individuals and conducting syphilis screening for 30% of sexually active susceptible individuals while increasing the treatment rate, respectively. Therefore, in the prevention and control strategies of syphilis, measures such as increasing the coverage rate of syphilis screening for susceptible individuals and simultaneously increasing both the screening coverage rate and the treatment rate are valuable control strategy measures for reference.

1 Introduction

Syphilis is a chronic disease, and the only known natural host of treponema pallidum is humans [1, 2]. When a person is infected with syphilis, sores, blisters, or ulcers may appear on his or her genitals, anus (bottom) [3, 4], or mouth. The disease is primarily transmitted from one person to another when a susceptible person has sex without a condom or shares sex toys with an infected person [5]. Although sores spread the disease, most sores will go away without being detected [5]. In addition, mother-to-child transmission has been observed several years after infection with syphilis, especially in untreated cases [1, 6].

Syphilis has four distinct stages: primary, secondary, latent, and tertiary. In the initial stages, symptoms may appear as an isolated, painless chancre at the inoculation site. However, the primary chancre may disappear without the infected person noticing it. If the disease is not treated in the primary stage, it will progress to the secondary stage. The symptoms in the secondary stage of syphilis are more pronounced than those in the primary stage. It includes mucosal lesions affecting the skin, mucous membranes, and lymph nodes [1, 6]. Most often, in the second stage, an infected person develops a rash on the palms and soles of the feet, which can mimic other infectious and non-communicable diseases. It is important to note that the rash on the palms and feet of people with secondary syphilis may disappear even without treatment. If the disease is left untreated, the infected person will progress to the latent stage [1, 6]. The latent syphilis is asymptotic, described by positive syphilis serology without any clinical manifestations [1, 6]. At this stage, syphilis is usually divided into two parts: early latent syphilis and late latent syphilis. Early latent syphilis is characterized by an infection of less than two years, while late latent syphilis, on the other hand, is an infection of two years or more. Tertiary syphilis refers to gummas, cardiovascular syphilis, psychiatric manifestations (e.g., memory loss or personality changes), or late neurosyphilis and it developed from the primary, secondary, and latent stages of syphilis due to untreated or failed treatment. Transmission of syphilis occurs during primary, secondary, or early latent syphilis. Syphilis is more hidden and more likely to cause transmission because symptoms disappear inadvertently during the primary and secondary stages of syphilis.

Syphilis can be treated with antibiotics like penicillin. Following treatment and recovery from the infection, individuals may develop transitory immunity to re-infection before becoming susceptible again [1]. Although there are targeted treatment programs for syphilis, studies in recent years have found that syphilis is not well controlled. This is particularly evident in areas with poor medical conditions and a high prevalence of MSM (Men who have Sex with Men). Therefore, the dynamic model has been widely used to simulate the pathogenesis and evolution of syphilis to prevent and control the transmission and prevalence of syphilis and to understand the transmission dynamics of syphilis in the population deeply.

Several mathematical modeling studies have been carried out to understand the transmission dynamics of syphilis. For example, Garnett et al. [7] were the first to study syphilis using a mathematical model that includes all stages of the disease described by Doherty et al. ’s article [8] outlines recent changes in the epidemiology of infectious syphilis in England, including the features of several outbreaks. Grassley et al. [9] fitted accurate life data to a SIRS model and estimated an epidemic period of 8 to 11 years. Milner et al. [2] proposed a new mathematical model based on the assumption that secondary and later infections conferred partial immunity. Based on the immune loss aspect of syphilis, Iboi et al. [10] proposed a new mathematical model of syphilis and found that the reversal of early latent cases to the stage of infection affected the control of syphilis.

We found that earlier models of syphilis dynamics did not take into account homosexual and heterosexual transmission and, therefore, ignored the importance of MSM in syphilis. In 2016, Saad-Roy et al. [11] focused their modeling on gay men, where syphilis was making a comeback. An ODE model was developed for the MSM population that includes all stages of syphilis (including exposure, early and late latency), as well as treatment of latency and infection. In 2017, Gumel et al. [12] proposed a new two-group sex structure model to better assess the impact of treatment and condom use on the dynamics and control of syphilis transmission at the community level. In 2021, Momoh et al. [13] developed a sex structured model to study the dynamics of syphilis transmission, including the control of syphilis, that was, the prevention and treatment of infection in men and women. In terms of syphilis control and treatment, also referred to in the 2011 study by Brewer et al. [14], successful treatment of infectious syphilis failed to produce protective immunity in patients. Thus, individuals will be at risk of re-infection with syphilis if they continue to engage in high-risk behavior. In this article, it is hypothesized that, compared with susceptible individuals without a history of infection, recovered infected individuals have no significantly comparable risk of syphilis re-infection.

In the U.S. CDC’s 2020 report [15], we got information about the rising incidence of syphilis in both men and women. However, from the primary and secondary stages of syphilis, 77% of the patients were males. Among the male patients in the primary and secondary stages of syphilis, 63% were bisexual and MSM, and 37.9% were MSM. In 2000, MSM accounted for only 4% [16] of primary and secondary syphilis cases in the United States. However, nowadays, it has reached more than half of syphilis cases in the United States [15]. From the detailed data reported by the CDC, we can see that the number of men with syphilis has increased rapidly since 2002. Between 2020 and 2021, the incidence of primary and secondary syphilis (the most contagious stage of the disease) increased by a disturbing 32%, while syphilis reported by MSM increased by 7%. Therefore, we hypothesize that the increase in MSM leads to the rapid spread of syphilis and also to the rapid increase in syphilis patients. So, it is of great significance to the prevention, control, and treatment of syphilis.

In light of the contents mentioned above, we learned that the prevention, treatment, and control of syphilis are of great significance. Therefore, we have established a different model from the previous ones - the model takes into account the men who have sex with men (MSM) for modeling and incorporates the cost-effectiveness analysis to study the prevention and control of syphilis. In order to search for the most cost-effective prevention and control strategies for syphilis, we will conduct the study with the aid of DALYs and ICER. This research has specific reference values for the prevention and control of syphilis. Moreover, syphilis infection can also increase a person’s risk of getting HIV or transmission to others [17]. Therefore, when we built the model, we focused on the sex structure and the representation of MSM in the model and included the compartment of tertiary syphilis for study. The model’s validity was determined by fitting eight sets of data, 248 actual data points from CDC’s 2020 report [15] on the United States. As the current burden of syphilis is becoming more and more severe without further control of MSM, especially in countries and regions with poor medical development and low awareness of disease control, it is considered that limited funds should be more effectively invested in the prevention and treatment of syphilis. Based on the cost of STD Prevention and Control for Health Departments (STDPCHD) in the United States [18], the paper analyzes the transmission and prevalence, prevention, control, and cost-effectiveness of syphilis and hopes that this study will bring useful references and suggestions for the research of sexually transmitted diseases such as syphilis in other countries or regions.

The structure of this paper is as follows: The model is formulated and analyzed in Section 2. Section 3 provides some data fitting of the model. Section 4 provides cost-effectiveness. Finally, we conclude in Section 5.

2 The syphilis model and basic reproduction number

2.1 Model formulation

The total sexually active population at time t is denoted by $N\left ( t\right )$, which can be subdivided into mutually exclusive classes. Firstly, susceptible males ($S_{m}$) are classified as heterosexual ($S_{a} \left ( t\right )$) and homosexual ($S_{b}\left ( t\right )$). However, susceptible females do not account for lesbians, so it is represented by $S_{f} \left ( t\right )$. At the same time, the male infected individuals in the primary stage of syphilis infection are represented by $I_{1m}$ ($I_{1m} =I_{1a}+I_{1b}$) that are subdivided into heterosexual ($I_{1a} \left ( t\right )$) and homosexual ($I_{1b} \left ( t\right )$). Moreover, females with primary stage syphilis infection are represented by $I_{1f}(t)$. In secondary stage syphilis infection, males and females are represented by $I_{2m}\left ( t\right )$ and $I_{2f}\left ( t\right )$, respectively. In the same way, males and females infected individuals in the early latent stage are denoted by $L_{m}\left ( t\right )$ and $L_{f} \left ( t\right )$, respectively. Besides, infected men in the late latent stage or tertiary syphilis are denoted by $T_{m}\left ( t\right )$; infected women in the late latent stage or tertiary syphilis are denoted by $T_{f}\left ( t\right )$; recovered males are denoted by $R_{m}\left ( t\right )$, and recovered females are denoted by $R_{f} \left ( t\right )$. In all epidemiological classes, the natural mortality rate is μ. This paper does not consider mother-to-child transmission because it only considers sexually active people.

So that

$$\begin{aligned} N\left ( t\right ) &=S_{f}\left ( t\right ) +S_{a}\left ( t\right ) +S_{b} \left ( t)\right ) +I_{1f}\left ( t\right ) +I_{1a}\left ( t)\right ) +I_{1b} \left ( t\right ) +I_{2f}\left ( t\right ) +I_{2m}\left ( t\right ) \\ &+L_{f}\left ( t\right ) +L_{m}\left ( t\right ) +T_{f}\left ( t \right ) +T_{m}\left ( t\right ) +R_{f}\left ( t\right ) +R_{m}\left ( t\right ) . \end{aligned}$$

This population is reduced by syphilis infection and progression to the primary stage

$$ \lambda _{f}=\alpha _{m}\psi _{0}\left ( \frac{I_{1a}+\eta _{1}(I_{2m}+L_{m})}{N}\right ), $$

where $\psi _{0}$ is the average number of sexual partners of females per unit of time. The population of susceptible males (heterosexual ($S_{a} \left ( t\right )$) and homosexual ($S_{b}\left ( t\right )$)) reduces due to the development of newly infected males with syphilis who progress to the males with primary stage syphilis by functions:

$$ \lambda _{a}=\alpha _{f}\psi _{1}\left (\frac{I_{1f}+I_{2f}+L_{f}}{N} \right ),\ \ \lambda _{b}=\alpha _{m}\psi _{2}\left ( \frac{I_{1b}+\eta _{2}(I_{2m}+L_{m})}{N}\right ), $$

where $\alpha _{f}$ ($\alpha _{m}$) represents the transmission probability of syphilis in females (male), and $\psi _{1}$ ($\psi _{2}$) is the average number of sexual partners of males heterosexual (homosexual) per unit time. The ratio of male heterosexuality (homosexual) in secondary and early latent stages of syphilis is $\eta _{1}$ ($\eta _{2}$). $\Lambda _{f}$ is the recruitment of new sexually active individuals. Recovery from syphilis infection contributes to an increase in susceptible females at rate $\varphi _{f}$. Thus

$$ \frac{dS_{f}}{dt}=\Lambda _{f}-\lambda _{f}S_{f}-\mu S_{f}+\varphi _{f}R_{f}. $$

Recruitment of new sexually active individuals into the susceptible males at time t, $S_{a}\left ( t\right )$ and $S_{b}\left ( t\right )$ is done by the increase in the number of sexually active individuals, who did not previously contact syphilis at rate $\Lambda _{m}$ because males are divided into heterosexual and homosexual, so we have to think about the ratio of gay (π). The population is also increased by the rate $\varphi _{m}$ of males who are recovered from syphilis infection, and p means the proportion of recurrent syphilis in homosexual males. So, the equations become

$$\begin{aligned}& \frac{dS_{a}}{dt}=\left ( 1-\pi \right ) \Lambda _{m}-\lambda _{a} S_{a}- \mu S_{a}+ \varphi _{m} (1-p) R_{m} ,\\& \frac{dS_{b}}{dt}=\pi \Lambda _{m}-\lambda _{b} S_{b}-\mu S_{b}+ \varphi _{m} p R_{m}. \end{aligned}$$

The population of females with primary stage syphilis at time t, $I_{1f} \left ( t\right )$, increases due to the progression of newly infected females who have contracted syphilis from susceptible female $\lambda _{f} S_{f}$ and decreases by progression to infected females with secondary stage syphilis ($I_{2f}$) at rate $\gamma _{1f}$. There are syphilis treatment using antibiotics at rates $\sigma _{1f}$ from females. Thus

$$ \frac{dI_{1f}}{dt}=\lambda _{f}S_{f}-(\gamma _{1f}+\mu +\sigma _{1f})I_{1f}. $$

The population of males with primary stage syphilis at time t, $I_{1}a \left ( t\right )$ ($I_{1}b \left ( t\right )$), increases due to the progression of newly infected syphilis individuals from the susceptible male $\lambda _{a} S_{a}$ ($\lambda _{b} S_{b}$) and reduces due to progression to infected males with secondary stage syphilis ($I_{2m}$) at rate $\gamma _{1m}$. There are syphilis treatment using antibiotics at rate $\sigma _{1m}$ from male so that

$$\begin{aligned}& \frac{dI_{1a}}{dt}=\lambda _{a} S_{a}-(\gamma _{1m}+\mu +\sigma _{1m})I_{1a},\\& \frac{dI_{1b}}{dt}=\lambda _{b} S_{b}-(\gamma _{1m}+\mu +\sigma _{1m})I_{1b}. \end{aligned}$$

The number of infected females with secondary stage syphilis population at time t, $I_{2f}\left ( t\right )$, increases due to progression of females with primary stage syphilis ($I_{1f}$) to females with secondary stage syphilis ($I_{2f}$) at rate $\gamma _{1f}$. The population reduces by the progression of females with secondary stage syphilis ($I_{2f}$) at rate $\gamma _{2f}$ to the early latent stage syphilis ($L_{f}$) and further reduced treatment using antibiotics at rate $\sigma _{2f}$ so that the equation is given by

$$ \frac{dI_{2f}}{dt}=\gamma _{1f}I_{1f}-\bigl(\gamma _{2f}+\mu +\sigma _{2f} \bigr)I_{2f} . $$

The number of infected males with secondary stage syphilis infection at time t, $I_{2m}(t)$, increases due to the progression of males with primary stage syphilis ($I_{1m}$) to males with secondary stage syphilis ($I_{2m}$) at rate $\gamma _{1m}$. The population reduces by the progression of males with secondary stage syphilis ($I_{2m}$) at rate $\gamma _{2m}$ to males with early latent stage syphilis ($L_{m}$) and further reduces by treatment using antibiotics at rate $\sigma _{2m}$ so that the equation is given by

$$ \frac{dI_{2m}}{dt}=\gamma _{1m}(I_{1a}+I_{1b})-(\gamma _{2m}+\mu + \sigma _{2m})I_{2m}. $$

The population of females with early latent syphilis at time t, $L_{f}(t)$, increases by the progression of infected females with secondary stage syphilis ($I_{2f}$) at rate $\gamma _{2f}$. The population reduces by the progression of females with early latent syphilis ($L_{f}$) at rate $\gamma _{3f}$ to the late latent stage and tertiary syphilis ($T_{f}$) and further reduces due to the presence of treatment using antibiotics at rate $\sigma _{3f}$ so that the equation is given by

$$ \frac{dL_{f}}{dt}=\gamma _{2f}I_{2f}-\left (\gamma _{3f}+\mu +\sigma _{3f} \right )L_{f}. $$

The population of males with early latent syphilis at time t, $L_{m}(t)$, is increased by the progression of infected males with secondary stage syphilis ($I_{2m}$) at rate $\gamma _{2m}$. The population is reduced by the progression of males with early latent syphilis ($L_{m}$) at rate $\gamma _{3m}$ to the late latent stage and tertiary syphilis ($T_{m}$) and further reduced due to the presence of treatment using antibiotics at rate $\sigma _{3m}$ so that the equation is given by

$$ \frac{dL_{m}}{dt}=\gamma _{2m}I_{2m}-(\gamma _{3m}+\mu +\sigma _{3m})L_{m}. $$

The population of females with late latent stage and tertiary syphilis at time t, $T_{f}(t)$, increases by the progression of infected females with early latent syphilis ($L_{f}$) at rate $\gamma _{3m}$ and reduces due to diseased death rate ν and treatment using antibiotics at rate $\sigma _{4f}$, so that the equation is given by

$$ \frac{dT_{f}}{dt}=\gamma _{3f}L_{f}-(\nu +\mu +\sigma _{4f})T_{f}. $$

The population of males with late latent stage and tertiary syphilis at time t, $T_{m}\left ( t\right )$, increases by the progression of infected males with early latent syphilis ($L_{m}$) at rate $\gamma _{3m}$ and reduces due to diseased death rate ν and treatment using antibiotics at rate $\sigma _{4m}$, so that the equation is given by

$$ \frac{dT_{m}}{dt}=\gamma _{3m}L_{m}-(\nu +\mu +\sigma _{4m})T_{m}. $$

The population of recovered males at time t, $R_{f}(t)$, increases due to progression of treated males from primary, secondary, and latent stages of syphilis ($I_{1f}$, $I_{2f}$, $L_{f}$, $T_{f}$), respectively, at rates $\sigma _{1f}$, $\sigma _{2f}$, $\sigma _{3f}$, and $\sigma _{4f}$, which are the treatment rates of syphilis infection in female population. This population is further decreased due to loss of immunity acquired due to treatment and transfer of such individuals to the susceptible male population at rate $\varphi _{f}$, so that

$$ \frac{dR_{f}}{dt}=\sigma _{1f}I_{1f}+\sigma _{2f}I_{2f}+\sigma _{3f}L_{f}+ \sigma _{4f}T_{f}-(\mu +\varphi _{f})R_{f}. $$

The population of recovered males at time t, $R_{m}(t)$, increases due to progression of treated males from primary, secondary, and latent stages of syphilis ($I_{1m}$, $I_{2m}$, $L_{m}$, $T_{m}$) respectively, at rates $\sigma _{1m}$, $\sigma _{2m}$, $\sigma _{3m}$, and $\sigma _{4m}$, which are the treatment rates of syphilis infection in male population. This population is further decreased due to loss of immunity acquired as a result of treatment and transfer of such individuals to the susceptible male population at rate $\varphi _{m}$, so that

$$ \frac{dR_{m}}{dt}=\sigma _{1m}(I_{1a}+I_{1b})+\sigma _{2m}I_{2m}+ \sigma _{3m}L_{m}+\sigma _{4m}T_{m}-(\mu +\varphi _{m})R_{m}. $$

Thus,

$$\begin{aligned}& N_{f}(t)=S_{f}(t)+I_{1f}(t)+I_{2f}(t)+L_{f}(t)+T_{f}(t)+R_{f}(t),\\& N_{m}(t)=S_{a}(t)+I_{1a}(t)+S_{b}(t)+I_{1b}(t)+I_{2m}(t)+L_{m}(t)+T_{m}(t)+R_{m}(t). \end{aligned}$$

Based on the above formulations and assumptions, the model for the transmission of syphilis in a sexually active population is given by the following equations (a flow diagram of the model is depicted in Fig. 1, and the associated parameters of the model (1) are tabulated in Table 1).

Figure 1

Schematic flow diagram of the model (1)

Table 1 Parameters definition of model (1)

$$ \textstyle\begin{cases} \frac{dS_{f}}{dt}=\Lambda _{f}-\lambda _{f}S_{f}-\mu S_{f}+\varphi _{f}R_{f}, \\ \frac{dI_{1f}}{dt}=\lambda _{f}S_{f}-(\gamma _{1f}+\mu +\sigma _{1f})I_{1f}, \\ \frac{dI_{2f}}{dt}=\gamma _{1f}I_{1f}-\bigl(\gamma _{2f}+\mu + \sigma _{2f}\bigr)I_{2f}, \\ \frac{dL_{f}}{dt}=\gamma _{2f}I_{2f}-\bigl(\gamma _{3f}+\mu +\sigma _{3f} \bigr)L_{f}, \\ \frac{dT_{f}}{dt}=\gamma _{3f}L_{f}-(\nu +\mu +\sigma _{4f})T_{f} , \\ \frac{dR_{f}}{dt}=\sigma _{1f}I_{1f}+\sigma _{2f}I_{2f}+\sigma _{3f}L_{f}+ \sigma _{4f}T_{f}-(\mu +\varphi _{f})R_{f}, \\ \frac{dS_{a}}{dt}=(1-\pi )\Lambda _{m}-\lambda _{a}S_{a}-\mu S_{a}+ \varphi _{m}(1-p)R_{m}, \\ \frac{dI_{1a}}{dt}=\lambda _{a}S_{a}-(\gamma _{1m}+\mu +\sigma _{1m})I_{1a}, \\ \frac{dS_{b}}{dt}=\pi \Lambda _{m}-\lambda _{b}S_{b}-\mu S_{b}+ \varphi _{m}pR_{m}, \\ \frac{dI_{1b}}{dt}=\lambda _{b}S_{b}-(\gamma _{1m}+\mu +\sigma _{1m})I_{1b}, \\ \frac{dI_{2m}}{dt}=\gamma _{1m}(I_{1a}+I_{1b})-(\gamma _{2m}+\mu + \sigma _{2m})I_{2m}, \\ \frac{dL_{m}}{dt}=\gamma _{2m}I_{2m}-(\gamma _{3m}+\mu +\sigma _{3m})L_{m}, \\ \frac{dT_{m}}{dt}=\gamma _{3m}L_{m}-(\nu +\mu +\sigma _{4m})T_{m}, \\ \frac{dR_{m}}{dt}=\sigma _{1m}(I_{1a}+I_{1b})+\sigma _{2m}I_{2m}+ \sigma _{3m}L_{m}+\sigma _{4m}T_{m}-(\mu +\varphi _{m})R_{m}, \end{cases} $$

(1)

where

$$\begin{aligned}& \lambda _{f}=\alpha _{m}\psi _{0}\left ( \frac{I_{1a}+\eta _{1}(I_{2m}+L_{m})}{N}\right ),\\& \lambda _{a}=\alpha _{f}\psi _{1}\left (\frac{I_{1f}+I_{2f}+L_{f}}{N} \right ),\\& \lambda _{b}=\alpha _{m}\psi _{2}\left ( \frac{I_{1b}+\eta _{2}(I_{2m}+L_{m})}{N}\right ). \end{aligned}$$

2.2 Basic properties

Theorem 1

We give a maximum invariant set Ω of model (1), where

$$ \begin{aligned} \Omega &=\left \{\left (S_{f}, I_{1f}, I_{2f}, L_{f}, T_{f}, R_{f}, S_{a}, I_{1a}, S_{b}, I_{1b}, I_{2m}, L_{m}, T_{m}, R_{m}\right )\in \mathbb{R}_{+}^{14},\right. \\ &\quad \left.N_{f}\leq \frac{\Lambda _{f}}{\mu},N_{m}\leq \frac{\Lambda _{m}}{\mu}\right \}. \end{aligned} $$

Proof

It is easy to check that the equations for male and female susceptible individuals in the model (1) lead to the following first-order inequality equations:

$$ \frac{dS_{f}}{dt}+\alpha _{m}\psi _{0}\left ( \frac{I_{1a}+\eta _{1}(I_{2m}+L_{m})}{N}\right )S_{f}+\mu S_{f}\geq 0. $$

Multiplying this inequality by the integrating factors:

$$ \rho _{F}(t)=\exp{(\int _{0}^{t} \frac{\alpha _{m}\psi _{0}\bigl(I_{1a}+\eta _{1}(I_{2m}+L_{m})\bigr)}{N}ds+ \mu t)}, $$

and observing that

$$ \rho _{F}\Big[\frac{dS_{f}}{dt}+\alpha _{m}\psi _{0}\Big( \frac{I_{1a}+\eta _{1}(I_{2m}+L_{m})}{N}\Big)S_{f}+\mu S_{f}\Big]= \frac{d(S_{f}\rho _{F})}{dt}. $$

Mean $S_{f}(t)\geq S_{f}(0)\exp \left \{-\left ( \int _{0}^{t} \frac{\alpha _{m}\psi _{o}(I_{1a}+\eta _{1}(I_{2m}+L_{m}))}{N}ds+\mu t \right ) \right \}>0$, $\forall t>0$. We apply a similar method to establish that $I_{1f},\ I_{2f},\ L_{f},\ T_{f},\ R_{f},\ S_{a},\ I_{1a},\ S_{b},\ I_{1b}, \ I_{2m},\ L_{m},\ T_{m},\ R_{m}>0$ remain nonnegative for all $t>0$.

Moreover,

$$\begin{aligned}& \begin{aligned} &\frac{dN_{f}}{dt}=\Lambda _{f}-\mu N_{f}-\nu T_{f}\leq \Lambda _{f}- \mu N_{f}, \\ &\frac{dN_{m}}{dt}=\Lambda _{m}-\mu N_{m}-\nu T_{m}\leq \Lambda _{m}- \mu N_{m}, \end{aligned} \\& \begin{aligned} &0\leq N_{f}(t)\leq N_{f}(0)\exp (-\mu t)+\frac{\Lambda _{f}}{\mu} \left (1-\exp (-\mu t)\right ), \\ &0\leq N_{m}(t)\leq N_{m}(0)\exp (-\mu t)+\frac{\Lambda _{m}}{\mu} \left (1-\exp{(-\mu t)}\right ). \end{aligned} \end{aligned}$$

In particular, we have a priori estimates

$$ \begin{gathered} 0\leq N_{f}(t)\leq \frac{\Lambda _{f}}{\mu}\ \text{if}\ N_{f}(0) \leq \frac{\Lambda _{f}}{\mu}, \\ 0\leq N_{m}(t)\leq{\frac{\Lambda _{m}}{\mu}}\ \text{if}\ N_{m}(0) \leq{\frac{\Lambda _{m}}{\mu}}. \end{gathered} $$

Combining these a priori estimates and the fact that the right-hand side of the model (1) is locally Lipschitz, we conclude that a unique global solution is in the domain Ω. Thus, the model (1) is a dynamical system on Ω. On the other hand, if a solution is outside the region Ω, that is,

$$ N_{f}(t)\geq \frac{\Lambda _{f}}{\mu} \quad \text{and}\quad N_{m}(t)\geq \frac{\Lambda _{m}}{\mu} , $$

then it follows from the above conservation law that $\frac{dN_{f}}{dt}\leq 0$ and $\frac{dN_{m}}{dt}\leq 0$. Hence, the above general a priori estimates show that $N_{f}(t)$ tends to $\frac{\Lambda _{f}}{\mu}$ and $N_{m}(t)$ tends to $\frac{\Lambda _{m}}{\mu}$ as $t\rightarrow \infty$. Thus, the region Ω is attracted. There proves the boundedness of the solutions inside Ω. □

2.3 Basic reproduction number

The disease-free equilibrium of model (1) are established by setting it to zero, given by:

$$\begin{aligned} E^{0}& =\left (S^{0}_{f}, S^{0}_{a}, S^{0}_{b}, I^{0}_{1f}, I^{0}_{1a}, I^{0}_{1b}, I^{0}_{2f}, I^{0}_{2m}, L^{0}_{f}, L^{0}_{m}, T^{0}_{f}, T^{0}_{m}, R^{0}_{f}, R^{0}_{m}\right ) \\ &=(\frac{\Lambda _{f}}{\mu}, \frac{(1-\pi )\Lambda _{m}}{\mu}, \frac{\pi \Lambda _{m}}{\mu}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). \end{aligned}$$

Following Diekmann et al. [19] and Driessche et al. [20], we can compute the basic reproduction number (see Appendix A for details).

Then the characteristic equation of $\textbf{F} \textbf{V}^{-1}$ is derived as

$$ \lambda ^{6}[(\lambda -A_{33})(\lambda ^{2}-A_{12}A_{21})-A_{13}A_{21}A_{32}]=0. $$

(2)

Where,

$$\begin{aligned}& A_{12} =\frac{T_{1}}{h_{1}}+ \frac{T_{1}\eta _{1}\gamma _{1m}}{h_{1}h_{2}}+ \frac{T_{1}\eta _{1}\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}},\quad A_{13}= \frac{T_{1}\eta _{1}\gamma _{1m}}{h_{1}h_{2}}+ \frac{T_{1}\eta _{1}\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}}, \\& A_{21}=\frac{T_{2}}{k_{1}}+\frac{T_{2}\gamma _{1f}}{k_{1}k_{2}}+ \frac{T_{2}\gamma _{1f}\gamma _{2f}}{k_{1}k_{2}k_{3}},\quad A_{32}= \frac{T_{3}\eta _{2}\gamma _{1m}}{h_{1}h_{2}}+ \frac{T_{3}\eta _{2}\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}}, \\& A_{33}= \frac{T_{3}}{h_{1}}+ \frac{T_{3}\eta _{2}\gamma _{1m}}{h_{1}h_{2}}+ \frac{T_{3}\eta _{2}\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}}, \\& k_{1}=\gamma _{1f}+\mu +\sigma _{1f},k_{2}=\gamma _{2f}+\mu +\sigma _{2f},k_{3}= \gamma _{3f}+\mu +\sigma _{3f},k_{4}=\mu +\sigma _{4f}+\nu , \\& h_{1}=\gamma _{1m}+\mu +\sigma _{1m},h_{2}=\gamma _{2m}+\mu +\sigma _{2m},h_{3}= \gamma _{3m}+\mu +\sigma _{3m},h_{4}=\mu +\sigma _{4m}+\nu . \\& T_{1}=\frac{\alpha _{m}\psi _{0}S^{0}_{f}}{N^{0}},T_{2}= \frac{\alpha _{f}\psi _{1}S^{0}_{a}}{N^{0}},T_{3}= \frac{\alpha _{m}\psi _{2}S^{0}_{b}}{N^{0}},\\& \beta _{1}=\alpha _{m}\psi _{0}, \ \beta _{2}=\alpha _{f}\psi _{1},\ \beta _{3}=\alpha _{m}\psi _{2},\ s_{f}=\frac{S^{0}_{f}}{N^{0}},\ s_{a}= \frac{S^{0}_{a}}{N^{0}},\ s_{b}=\frac{S^{0}_{b}}{N^{0}}. \end{aligned}$$

Obviously, the Eq. (2) has six zero eigenroots, leaving three eigenroots determined by the following cubic equation with real coefficients:

$$ \lambda ^{3}-A_{33}\lambda ^{2}-A_{12}A_{21}\lambda +A_{21}(A_{12}A_{33}-A_{13}A_{32})=0. $$

(3)

From the above Eq. (3), we know that the expression cannot be factorized easily, so there is no way to know the explicit expression of ${\mathcal{R}_{0}}$. From Eq. (3), we can also see that this equation has three real roots, or one real root and two conjugate complex roots. The way of finding the root of a cubic equation with one unknown is no longer described right here. It follows that the basic reproduction number of the model (1), denoted by ${\mathcal{R}_{0}}= \rho (\textbf{F} \textbf{V}^{-1})$ (where ρ denotes the spectral radius and ${\mathcal{R}_{0}}$ is the maximum spectral radius of $\textbf{F} \textbf{V}^{-1}$).

3 Data fitting and parameter estimation

3.1 Parameter estimation

Based on the biological significance of the parameters and the transmission characteristics of syphilis in the United States, we determine and assume the values of some parameters. The rest of the parameters are obtained by simulation (see Table 2 for details).

Table 2 Values of the model parameters

(1) According to the report of CDC [15], syphilis has a higher incidence over the age of 15 (0 to 14 years old is usually congenital syphilis, which is transmitted vertically from mother to baby), so we start to consider adults at 15. According to the report of the World Bank [21] and World Meter [22], the increase in the male population from 1984 to 2014 ranged from (797 697, 1 315 780), with an average value of 1 083 224. So we assume that the maximum number of males’ recruitment rate ($\Lambda _{m}$) each year is 1 083 224. Similarly, we know that the range of female population increase is (813 461, 1 394 066), and the average is 1 079 157. So we assume that the maximum recruitment rate ($\Lambda _{f}$) in the female population is 1 079 157.

(2) We infer from Garnett et al. ’s article [7], WHO [6] and CDC reports [23] that the number of sexual partners (including sex workers) ranges from 5 to 40 in the female population ($\psi _{0}$) and the heterosexual male population ($\psi _{1}$). The sexual partners’ range of male homosexuality ($\psi _{2}$) we infer from CDC reports is (1, 3) [24, 25].

(3) From references [6, 7, 23] show the average duration of the various stages of syphilis. The mean course of disease in the primary stage is 46 days, i.e., $365/46$, so the progression rate to the secondary stage of syphilis ($\gamma _{1}$) ranges from (0, 7.93). Moreover, the mean course of the secondary stage is 108 days, i.e., $365/108$, so the progression rate to early latent syphilis ($\gamma _{2}$) ranges from (0, 3.378). The mean course of the early latent period of syphilis is one year, then the rate of progression to the late latent period of tertiary stages of syphilis ($\gamma _{3}$) ranges from (0, 1).

(4) Because men are divided into homosexuality and heterosexuality, consider the proportion of homosexuality and heterosexuality in the recruitment rate of the model. We made a range of about the proportion of heterosexual men at high risk of syphilis (π) is (0.07, 0.23) [26–28] and reference the value of p. Therefore, we assumed the value of π is 0.2.

(5) From reference [29], we estimate the recovery rate ($\varphi _{m}$, $\varphi _{f}$) range at (0,1) and treatment rates ($\sigma _{ij},i=1,2,3,4;j=m,f$) for each stage of syphilis.

3.2 Data fitting

This section uses this model for the American infectious disease syphilis. The data is from the United States from 1984 to 2014 [32]. This data covers the four stages of syphilis: primary stage, secondary stage, early latent stage, late latent, and tertiary syphilis (late latent and tertiary syphilis are considered one stage). We obtain four sets of syphilis stage data corresponding to the four compartments in the model: primary stage ($I_{1}$), secondary stage ($I_{2}$), early latent stage (L), late latent and tertiary stage (T). The data is disaggregated by gender. Then we obtain eight sets of data and correspond to the primary stage ($I_{1m}$, $I_{1f}$), secondary stage ($I_{2m}$, $I_{2f}$), early latency ($L_{m}$, $L_{f}$), late latent and tertiary syphilis ($T_{m}$, $T_{f}$) in the model. We simultaneously fit eight sets of data for syphilis [33–37], as shown in Fig. 2, 3, 4, 5. The figures show the fitting results and the actual data results. The coincidence results significantly verify the model’s accuracy, and the fitting results and optimal parameters are also obtained. The model presented can fit the primary trends of eight sets and 248 real data points of data.

Figure 2

The comparison chart of the primary syphilis data in the United States and simulation results by model (1)

Figure 3

The comparison chart of the secondary syphilis data in the United States and simulation results by model (1)

Figure 4

The comparison chart of the early latent syphilis data in the United States and simulation results by model (1)

Figure 5

The comparison chart of the late latent and tertiary syphilis data in the United States and simulation results by model (1)

In the above, we obtain data from the CDC on syphilis by gender and stage of infection in the United States from 1984 to 2014. We select the values of 1984 from 31 annual data points as the initial values of each infection stage for simulation (see Table 3 for detailed values). In the model, we do not have data on the values of male susceptible $S_{m}(t)$, female susceptible $S_{f}(t)$, male recovered $R_{m}(t)$, and female recovered $R_{f}(t)$ ($S_{m}(t)=S_{a}(t)+S_{b}(t)$, $I_{1m}(t)=I_{1a}(t)+I_{1b}(t)$), so we obtain them through data fitting.

Table 3 Initial values for the variables

We choose the fmincon optimization algorithm in MATLAB software global optimization toolbox to estimate the 19 parameters and four initial values, respectively. The algorithm is an evolutionary algorithm that helps find the optimal parameters. Through Section 2.3, we can obtain three real eigenvalues, the largest of which is the basic reproduction number of the model, so we can get ${\mathcal{R}_{0}}$ is 1.3876. Note that the basic reproduction number ${\mathcal{R}_{0}}$ stands for the number of infected during the initial patient’s infectious (not sick) period, which is always more than the unit. This explains why syphilis is prevalent, but it has not yet been pandemic. According to the literature [7, 38], ${\mathcal{R}_{0}}$ is consistent with the reality and the model is reasonable. Hence, the above results show that parameter estimation conforms well to reality.

4 Cost-effectiveness

Since the mode of transmission of MSM is critical in syphilis, and male patients account for a large proportion of syphilis patients, the method of control of syphilis is significant. To better control syphilis, this section discusses studies using DALYs and ICER. In addition, this section will use the parameters modeled in Table 2 to predict the development of different stages of syphilis in the United States between 2015 and 2044.

DALYs were calculated following the methodology described by Rushby et al. [39]. A DALY can be thought of as one lost year of a “healthy” life. This includes mobility, self-care, participation in usual activities, pain and discomfort, anxiety and depression, and cognitive impairment. Note that disability (i.e., a state other than ideal health) may be short-term or long-term. For example, a day with a cold is a day with disability. The sum of these DALYs across the population, or the burden of disease, can be seen as a measure of the gap between current health and an ideal health situation in which the entire population lives to an old age free of disease and disability life. A DALY is a summary measure of the burden of disease that, for a given disease or conditions, combines the number of years of life lost due to premature mortality (YLLs) with the number of years lived with disability (YLDs) [40]. Thus,

$$ \mathrm{DALY=YLL+YLD ,} \quad \mathrm{YLL = Ns \cdot Le,} \quad \mathrm{YLD=I\cdot DW\cdot Ld}, $$

where Ns: number of deaths; Le: standard life expectancy at the age of death in years; I: number of incident cases; DW: disability weight; Ld: average duration of disability years. Each range estimation is displayed in Table 5. The incremental cost-effectiveness ratio (ICER) measures the change between the costs and health benefits of two different intervention strategies competing for the same limited resources. Considering that the post-intervention strategy and the current situation are two competing control intervention strategies, ICER is expressed as

$$ \text{ICER}= \frac{\mathrm{Costs_{Intervention}-Costs_{Statnsquo}}}{\mathrm{DALYs_{Intervention}-DALYs_{Statusquo}}} . $$

The model assumes that mass screening will not be carried out until the control measures are taken because people engaging in high-risk sex do not know that they are engaging in high-risk sex, so the susceptible people will not take the initiative to be screened. Measures are taken because the specific infection is not known or there is concern about other sexually transmitted diseases. Therefore, the STIs test is needed to screen for $C_{STIs}$ [41], as well as for Nontreponemal and Treponemal tests for $C_{Tre}+C_{Non}$ [1]. Nontreponemal and Treponemal tests are also required for primary, secondary, early latent, late latent, and tertiary syphilis patients because these patients are regularly examined after the disease to determine the incidence of syphilis.

For infected and susceptible patients who test positive, a doctor’s consultation and a further Treponemal test are required to verify the stage of disease progression, therefore $C_{pos}=C_{Tre}+C_{coun}^{pos}$.

For the treatment of syphilis, Penicillin G administered by injection is the preferred drug across all stages of the disease [1, 4, 6]. The choice of preparation (namely benzathine, aqueous procaine, or aqueous crystalline), dosage, and duration of the treatment depends on the stage and clinical manifestations. Treatment for late latent syphilis (>1 year’s duration) and tertiary syphilis necessitate an extended duration of therapy due to the theoretical possibility of slower organism division (the validity of this hypothesis has not been evaluated). A lengthier treatment period is necessary for individuals with latent syphilis of unknown duration to ensure effective treatment of those who have not contracted syphilis in the previous year.

In the treatment of syphilis, primary, secondary, and early latent syphilis are treated differently from late latent and tertiary stage syphilis [1]. Treatment for primary, secondary, and early latent stages of syphilis entails a single dose of 2.4 million IM units ($C_{treatment}^{E}=C_{penicillin}$). The treatment for late latent and tertiary syphilis involves administering a total of 7.2 million units of intramuscular injections, which are distributed equally into three doses of 2.4 million units each, given weekly ($C_{treatment}^{L}=3 C_{penicillin}$). Other costs, such as injection fees, are not considered. Penicillin allergy is not considered in this article. For patients diagnosed with syphilis who do not receive treatment at different stages of the article, it is assumed that they will undergo primary healthcare ($C_{treatment}^{U}$). See Table 4 for more details.

Table 4 Annual syphilis-related health care costs (US $)

Table 5 Model variables

The total costs over a period of T years are as follows: (1) the cost of screening ($C_{screen}$); (2) the cost of screening tests for individuals who tested positive for syphilis ($C_{test}^{pos}$); (3) the cost of treatment ($C_{treatment}$); and (4) the healthcare costs related to syphilis ($C_{HC}$). The discount rate for costs is assumed to be z, and its value is 3% [49–51]. The total discounted costs for the entire population are calculated by adding up the yearly healthcare and screening costs of all individuals over the intervention’s T duration as follows:

$$ Costs=C_{screen}+C_{test}^{pos}+C_{treatment}+C_{HC}, $$

where

$$\begin{aligned} &C_{screen}=\int _{0}^{T}e^{-zt}(C_{STIs}(S_{f}(t)+S_{a}(t)+S_{b}(t))+(C_{Tre}(t)+C_{Non}(t))(S_{f}(t)+S_{a}(t) \\ &{}+S_{b}(t)+I_{1f}(t)+I_{1a}(t)+I_{1b}(t)+I_{2f}(t)+I_{2m}(t)+L_{f}(t)+L_{m}(t)+T_{f}(t)+T_{m}(t) \\ &{}+R_{f}(t)+R_{m}(t)))dt, \\ &C_{test}^{pos}=\int _{0}^{T}e^{-zt}C_{pos}(\lambda _{f}(t)S_{f}(t)+ \lambda _{a}(t)S_{a}(t)+\lambda _{b}S_{b}(t)+\gamma _{1f}I_{1f}(t)+ \gamma _{2f}I_{2f}(t)+\gamma _{3f}L_{f}(t) \\ &{}+\gamma _{1m}(I_{1a}(t)+I_{1b}(t))+\gamma _{2m}I_{2m}(t)+\gamma _{3m}L_{m}(t))dt, \\ &C_{treatment}=\int _{0}^{T}e^{-zt}(C_{treatment}^{E}(\sigma _{1f}I_{1f}(t)+ \sigma _{2f}I_{2f}(t)+\sigma _{3f}L_{f}(t)+\sigma _{1m}(I_{1a}(t)+I_{1b}(t)) \\ &{}+\sigma _{2m}I_{2m}(t)+\sigma _{3m}L_{m}(t))+C_{treatment}^{L}( \sigma _{4f}T_{f}(t)+\sigma _{4m}T_{m}(t))) dt, \\ &C_{HC}=\int _{0}^{T}e^{-zt}C_{treatment}^{U}(I_{1f}(t)+I_{1a}(t)+I_{1b}(t)+I_{2f}(t)+I_{2m}(t)+L_{f}(t)+L_{m}(t) \\ &{}+T_{f}(t)+T_{m}(t))dt. \end{aligned}$$

Suppose the mean age of a syphilis patient is 46. Based on data about the age distribution of syphilis cases, it appears that individuals aged 20 to 29 are at greater risk of acquiring the infection, regardless of gender [15]. Based on the disease course estimated in the model, it can be seen that the final age of death from syphilis, taking into account all the courses of syphilis, will be between 41.5 and 50.5 years of age, so the mean age of death is assumed to be 46 years and the mean age of onset is assumed to be 25 years. According to the Macrotrends [31] and WHO [40], the life expectancy in the United States is 76.66 years. This assumes that the difference between death from disease and life expectancy is about 30.

The paper states that the development and spread of syphilis occur without a human control strategy. This is done by adding controls to observe changes in patients. The usual screening strategy for syphilis, according to the articles by Kahn et al. [51], Tuite et al. [52, 53], and Tuli et al. [54], is to increase the extent and frequency of syphilis screening. This paper considers the case for increasing the scope of syphilis screening in susceptible individuals ($S_{j}\ (j=f, a, b)$). Suppose that the screening population is 10% and only $\lambda _{j}^{\prime }\cdot S_{j}\ (j=f, a, b)$ individuals are infected. Because the infected population is detected with $\sigma _{ij}(i=1,2,3,4;j=f,m)$ and treated, so in the screened population, people with $\sigma _{ij}$ are removed, meaning they no longer transmit syphilis. The effect of screening on the transmission of syphilis is then assumed to be $\sigma _{ij}$. From the pathology of syphilis, we know that the first stage for a susceptible person after infection with syphilis is the primary stage.

Hence, by screening susceptible individuals, we can identify patients with $\sigma _{1j}\ (j=f, m)$ who have been treated and are no longer capable of transmitting syphilis. This enables us to obtain the two necessary parameter values of $\sigma _{1f}=0.6700$ and $\sigma _{1m}=0.2456$ from Table 2, representing the influence of syphilis screening on female and male individuals at risk, respectively. According to the CDC syphilis report, between 2017 and 2018, just 60% of MSM individuals who took part in the community survey declared that they had undergone a syphilis screening in the last twelve months [55]. As the distinction between the proportions of gay and straight men in the male populations was not made during screening, we are able to establish the proportions of gay and straight men as 33% and 67% respectively from Table 2. Emphasis is placed on the assumption that both homosexuals and heterosexuals are evenly distributed in the population of males susceptible to the issue. Hence, the proportion of gay men screened must be taken into account.

In this section, we explore strategies to enhance the screening coverage rate, categorizing it into four levels: 10%, 30%, 50%, and 100%. Utilizing the parameters $\sigma _{1f}=0.67$ and $\sigma _{1m}=0.2456$, we can compute the number of susceptible individuals under different coverage rates. Subsequently, we assess the variations in the number of individuals within each subgroup $\lambda _{j}^{\prime }\cdot S_{j}\ (j=f, a, b)$. For instance, when the screening scope extends to 10%, the number of susceptible female undergoes a change of $93.3\%\cdot S_{f}$ $((1-10\% \cdot \sigma _{1f})\cdot S_{f})$, the number of susceptible heterosexual male experiences a change of $97.54\%\cdot S_{a}$ $((1-10\% \cdot \sigma _{1m})\cdot S_{a})$, and the number of susceptible homosexual male changes to $98.52\%\cdot S_{b}$ $((1-10\% \cdot 60\% \cdot \sigma _{1m} )\cdot S_{b})$. Similarly, with a screening range of 30%, the alterations are $79.9\%\cdot S_{f}$, $92.54\%\cdot S_{a}$ and $95.57\%\cdot S_{b}$ respectively. Moreover, with a screening range of 50%, the alterations are $66.5\%\cdot S_{f}$, $87.72\%\cdot S_{a}$ and $92.63\%\cdot S_{b}$ respectively. Finally, with a screening range 100%, the alterations are $33\%\cdot S_{f}$, $75.44\%\cdot S_{a}$ and $85.26\% \cdot S_{b}$ respectively.

The total costs and loss of DALYs for the four strategies are displayed in Table 6. Depending on the range used, the estimated total cost of all inspection strategies varies from $3 million to $24.5 million. However, strategic syphilis screening has significantly reduced DALYs; the number of DALYs has ranged from about 250 000 in the absence of a screening program to between 38 000 and 218 000 after the implementation of a screening program.

Table 6 Benefits and costs of expanding screening coverage from 2015 to 2044

In the base case (without screening for susceptible persons), our model estimates 1 517 951 new cases of syphilis, 1744 deaths attributed to the disease, a cost of $352.2 million, and 255 339.9 DALYs. A 10% screening coverage of susceptible individuals would reduce the number of new syphilis infections by 215 825, and DALYs could be reduced by 37 811.5 compared to the base case. Additionally, the number of deaths could be reduced by 185. However, the cost will increase by $572 million (over 30 years), which is nearly 1.5 times the cost of the base case. Expanding the screening of susceptible individuals to 30% is estimated to reduce the number of new syphilis infections by 704 851, and the DALYs would be reduced by 117 954.5 compared to the base case, potentially preventing 551 deaths, but resulting in an increase in costs of $854.8 million. With 50% screening, the estimated number of new infections would be reduced by 1 071 967, the number of deaths would be reduced by 809, the exact relative cost would increase to $1534.1 million, and the DALYs averted would be 175 048.9. Finally, suppose that the screening rate is increased to 100% and the detection effect is maintained. In this case, the number of new cases will fall by 1 336 380, the cumulative number of new cases will be 181 571, only 12% of the base case (1 517 951), and the number of deaths from the disease will also fall by 1029. However, costs would rise to $2449.3 million, almost seven times the base case, and the number of DALYs averted would increase to 216 837.6.

Table 6 shows that increasing screening coverage without screening individuals susceptible to syphilis is cost-effective. The ICER for expanding the screening scope to 10%, 30%, 50%, and 100% is 15 126.6, 7246, 6751.2, and 9671.4, respectively. According to the CDC report [18], the Strengthening STD Prevention and Control for Health Departments (STDPCHD) offers 5-year funding to all states, nine cities, and territories to prevent and control STIs. In 2021, the CDC allocated a total of $95.6 million in funding to PCHD. Of this amount, $8 million was allocated to HIV funding. The remaining $87.6 million was allocated to the prevention and control of sexually transmitted diseases other than HIV. Based on the budget, the optimal strategy is to screen susceptible individuals with 10% coverage if the annual budget is less than $30 million. If the annual budget is less than $51.1 million, the best approach is to expand screening to 50% at a cost of $6751.2 per DALY.

Fig. 6 shows the projected trends in the total number of infected patients from 2015 to 2044. The graph illustrates a decrease in the number of syphilis patients as the screening coverage increased to 10%. However, the decrease was less significant compared to when the screening coverage was 30%, 50%, and 100%. Notably, a screening coverage of 30% had a more pronounced effect on the number of syphilis cases. At 100% screening coverage, the number of syphilis cases gradually decreases between 2039 and 2044 but does not reach zero.

Figure 6

The number of syphilis infections (Only expanding syphilis screening)

The above analysis highlights the significant screening effect. Next, we will study treatment rates as the effect of screening. Assuming that the susceptible individuals are screened and the test results are positive, receiving treatment implies that the screening has impacted the model. Watson-Jones et al.’ research article on Tanzania [56] reveals that control measures for syphilis include screening and treatment. Therefore, we will analyze the impact of treatment rates on the model [57]. Shen et al. [58] and Jin et al. [41] has been made to HIV research aimed at controlling HIV disease by expanding access to antiretroviral therapy.

It is assumed that the hospitals and the communities will increase patient treatment rate to 80% ($\sigma _{1j}=0.8$, $j=f, m$) through persuasion and other means. This is because changes in treatment rate not only affect the effectiveness of screening on the model but also directly affect the number of patients treated in the overall model. Additionally, the text states changes in the number of syphilis patients and associated costs at different levels of syphilis screening coverage: 10%, 30%, 50%, and 100%.

After increasing the treatment rate for primary stage syphilis in males and females to 0.8, we observe variations in outcomes for different strategies. When the screening scope extends to 10%, the number of susceptible females changes by $92\%\cdot S_{f}$, the number of susceptible heterosexual males changes by $97.54\%\cdot S_{a}$, and the number of susceptible homosexual males changes by $98.52\%\cdot S_{b}$. Similarly, when the screening range is 30%, the number of susceptible females changes by $76\%\cdot S_{f}$, while the number of susceptible heterosexual males changes by $76\% \cdot S_{a}$, and the number of susceptible homosexual males changes by $85.6\%\cdot S_{b}$. Furthermore, when the screening range is 50%, the number of susceptible females decreases by $60\%\cdot S_{f}$, while the number of susceptible heterosexual males decreases by $60\%\cdot S_{a}$ and the number of susceptible homosexual males decreases by $76\%\cdot S_{b}$. Finally, when the screening range is 100%, the number of susceptible females decreases by $20\%\cdot S_{f}$, the number of susceptible heterosexual males decreases by $20\%\cdot S_{a}$, and the number of susceptible homosexual males decreases by $52\%\cdot S_{b}$.

Table 7 shows that increasing the treatment rate in the primary stage of syphilis alone increases the cost of the syphilis control strategy by $55.7 million. However, this increase in costs could help avoid 118 443.6 DALYs and reduce the number of deaths caused by syphilis. The ICER calculated for the current situation without any control measures is 470.27. By increasing the treatment rate of primary syphilis and expanding the screening scope of syphilis to 10%, the spread of syphilis could be further controlled. Compared with the base case without any control measures, the cost will increase by $17.7 million, while the number of DALYs caused by syphilis will decrease by 162 844.1. The calculated ICER is 108.4, compared to the base case without any control measures. It has been observed that expanding syphilis screening to 30%, 50%, and 100% is cost-effective. Figure 7 shows that expanding the treatment rate has a noticeable effect on reducing the number of people infected with syphilis compared to taking no action. However, after the treatment rate increase, expanding syphilis screening no longer has as significant an impact as shown in Fig. 6. We can make a clearer comparison with the picture shown below in Fig. 8.

Figure 7

The number of syphilis infections (Treatment rate increased)

Figure 8

A comparison chart of all strategies (Expanding syphilis screening and increasing treatment rate)

Table 7 The benefits and costs of expanding screening coverage with improved treatment rates from 2015 to 2044

As we can see from the annual number of new infections under different strategies (Table 9 in Appendix B), the number of new infections only increases the treatment rate, which is well controlled compared with no strategy. The control situation is already similar to expanding screening coverage to 30% only. After increasing the treatment rate and increasing the screening coverage to 10%, the number of new infections falls further. When the screening coverage increases to 30% after increasing the treatment rate, the control effect of the number of new infections is better than that of increasing the screening coverage only to 50%. In short, after increasing the treatment rate, the control of the number of new infections is obviously more effective. Therefore, increasing the treatment rate is a more effective control strategy. In this paper, 10%, 30%, 50%, and 100% are good reference values. The most cost-effective strategies are likely to be around these reference values.

From Table 8, we can see the value of ${\mathcal{R}_{0}}$ under different strategies. Because ${\mathcal{R}_{0}}$ indicates the spread of the disease, a smaller ${\mathcal{R}_{0}}$ indicates a weaker spread of the disease. We can see a significant decrease in ${\mathcal{R}_{0}}$ after only increasing the treatment rate compared to no strategy. After increasing the treatment rate and screening coverage increases to 30%, the ${\mathcal{R}_{0}}<1$ this shows that the disease is well controlled.

Table 8 The value of ${\mathcal{R}_{0}}$ under different strategy

5 Results and discussion

5.1 Results

When only the treatment rate was increased, the ${\mathcal{R}_{0}}$ value decreased significantly, from 1.3866 to 1.2355. Later, when syphilis screening was expanded with increased treatment rates, the effect on ${\mathcal{R}_{0}}$ values was less pronounced.

Syphilis screening can significantly reduce new infections of syphilis, and in the base case (no syphilis screening was performed on susceptible individuals in 2014), we estimate that the cumulative number of new syphilis virus infections from 2015 to 2044 will be 1 517 951. Expanding syphilis screening coverage to 10%, 30%, 50%, and 100% of susceptible individuals compared to the base case would reduce these numbers by 215 825, 704 851, 1 071 967, and 1 336 380, respectively, over the next three decades. Moreover, it can be concluded by ICER that the control of syphilis is the most cost-effective strategy when the screening coverage is expanded to 50%. Then, when we looked at the expansion of the syphilis screening rate in conjunction with the treatment rate, we found that there was a significant decrease in the number of syphilis patients after only increasing the treatment rate by 705 911. By expanding the screening rate in addition to increasing the treatment rate, we can see a more significant reduction in the number of new infections of syphilis. Expanding syphilis screening coverage to 10%, 30%, 50%, and 100% of susceptible individuals compared to the base case would reduce this number by 992 751, 126 784, 1 352 885, and 1 420 074, respectively, over the next three decades. These numbers are significantly higher than strategies that expand screening rates for syphilis, suggesting that finding ways to increase treatment rates is a more effective strategy at a limited cost.

In a cost-effectiveness analysis of the treatment rate of patients in the primary stage of syphilis in the syphilis model (this assumes no other costs), we obtained the following results. When treatment rates for syphilis alone (men and women) were increased, the total cost over three decades (without screening for susceptible individuals) increased by $55.7 million, and 118 443.55 DALYs were avoided. Compared with the status quo without increasing the treatment rate, each DALY avoided would cost $470.44 more. By increasing the treatment rate of syphilis in the primary stage and increasing the coverage of syphilis screening in the susceptible stage to 10%, we found that the cost would increase by $17.7 million and 162 844.13 DALYs would be avoided relative to no intervention. Each DALY avoided would cost an additional $108.43. Increasing syphilis screening in susceptible individuals to increasing treatment rates of 30%, 50%, and 100% was a cost-saving and cost-effective measure. We conclude that increasing screening coverage to 50% is the most cost-effective option in the case of only increasing screening coverage. In combination with increased treatment rates in primary patients and expanded coverage of syphilis screening in susceptible individuals, it is cost-effective to extend screening beyond 30%. Moreover, increasing the rate of treatment in the primary stage of syphilis is more sensitive to the model and the most cost-effective to analyze. In this paper, 10%, 30%, 50%, and 100% are good reference values. The most cost-effective strategies are likely to be around these reference values.

5.2 Discussion

Let us consider controlling the treatment rate at a high level or maintaining its continuous growth over some time in order to control syphilis. That is challenging to achieve, at least in underdeveloped areas. There are also different discussions about the testing methods for syphilis, such as the test sequence [44], the number [52], the validity [56]. Further research is needed on several specific components of syphilis screening programs. The simple and inexpensive RPR test was recommended as a suitable screening test for syphilis. Different testing sites have different effects [49]. Despite these uncertainties, policymakers and the CDC must recognize that screening for susceptible persons and improving treatment rates for those with syphilis are essential components of disease control. There are many cheap and simple screening tests and effective treatments, and the savings in DALY costs are considerable. In addition, combined with published studies, increasing screening coverage is an effective way to control the spread and progression of syphilis [52]. These have reduced the number of cases because the treatment rate treats those who are susceptible after testing positive so that those who are susceptible will no longer transmit syphilis. In the populations involved in the model, strategies that focus on broader screening in susceptible populations are more effective in reducing the number of cases of syphilis. It can also be seen from Table 4 that the detection and treatment costs for each person are equivalent. However, screening measures need to be completed by everyone, while treatment is only for patients. Therefore, considering the results of cost-effectiveness, the increase in the treatment rate is an optimal control strategy under the condition of limited costs. Significantly, when the treatment rate is increased without increasing the cost, for example, when hospitals, communities, and doctors persuade patients to receive treatment, this method is more obvious for the control of syphilis.

The shortcoming of this model is that it cannot accurately describe reality. The two sets of data, $I_{2m}$ and $T_{m}$, deviate significantly from the actual situation. Similarly, $L_{m}$ also has specific deviations, especially in the data comparison after 2004. The fitting results of the other five sets of data are satisfactory. It is not easy to obtain ideal fitting results when fitting the existing eight sets of real data. As stated in literature [16], the incidence rate of men (MSM) was the highest in 2004, accounting for more than 60% of primary and secondary syphilis cases in the United States in 2004, while it accounted for only 4% of cases in the United States in 2000. Therefore, the rapid growth in the number of patients after 2000 may be caused by the growth in the proportion of MSM. However, our current model does not reflect the sharp increase in male patients after 2000. For this problem, a more complex model, such as a non-autonomous dynamics model, may be needed for a more accurate description.

Our model still has some other limitations. Firstly, during data fitting, real-world data may have led to recurrent or increased cases of syphilis at specific time points, which could not be reflected in our model and might introduce bias into some model parameters. Secondly, this paper did not conduct further analysis on the situation existing in the United States within its cost-effectiveness analysis because there is significant variation in syphilis incidence across states as well as its correlation with GDP levels. Therefore, it would be necessary to determine whether local epidemic levels and economic conditions should serve as a basis for such analyses [13]. Although we included results from many of model realizations for each strategy, we did not explicitly evaluate the impact of uncertainty on model results within each parameter set.

6 Conclusion

In this study, we developed a new dynamic model that takes into account the sex structure of syphilis while also taking into account MSM. The model was fitted with the reported cases of syphilis in the United States from 1984 to 2014. The model’s parameters were estimated under pathological conditions to confirm that the ${\mathcal{R}_{0}}$ calculated in the model was 1.3876 and in line with reality and to ensure the rationality and accuracy of the model. The real transmission of syphilis is more complex and has more factors. We then used this model to predict the future development from 2015 to 2044. We conducted a cost-effectiveness analysis to study whether expanding syphilis screening in susceptible populations can effectively control syphilis transmission and the most effective means. In this regard, our study shows that expanding syphilis screening to 50% susceptible individuals is cost-effective. Each reduction in DALY costs only $6751.2 more. In 2021, total CDC PCHD funding for preventing and controlling sexually transmitted diseases other than HIV was $87.6 million [18]. Depending on the budget, which includes other sexually transmitted diseases, the best strategy was to screen susceptible individuals with 10% coverage if the annual budget was under $30 million. The best strategy was to expand screening to 50% if the annual budget was under $ 51.1 million, at the cost of $6751.2 per DALY when it was most cost-effective. After implementing a 50% coverage screening, the transmission rate of syphilis decreased significantly, and the control measures were particularly effective. This finding aligns with a previous study [53], which reached a similar conclusion but did not consider the role of female transmission in syphilis.

In conclusion, in our model, when funds are sufficient, the most cost-effective measure is to increase the screening coverage rate for sexually active susceptible individuals. In the case of only expanding the screening rate, screening 50% of the people can effectively control and prevent syphilis. When funds are insufficient, screening 30% of the people while increasing the treatment rate is the most cost-effective measure. The control of syphilis patients is particularly obvious, especially reflected in the change of the ${\mathcal{R}_{0}}$ value in Table 8. When the treatment rate is only increased, good results are achieved from the cost-effectiveness perspective. Therefore, we can also focus on the strategy of increasing the treatment rate in the prevention and control of syphilis. In this paper, the ways to increase the treatment rate, such as through communities, hospitals, and doctors to increase the treatment acceptance rate of patients, assume that these measures do not incur costs. Increase the screening coverage rate while increasing the treatment rate because the screening of sexually active susceptible individuals is a large-scale group screening, which can effectively prevent and control the spread of syphilis. Increasing the treatment rate is only a strategy for patients who have already shown symptoms. Its preventive effect on syphilis is not as significant as expanding the screening coverage rate. However, considering the combination of the two, the performance of syphilis prevention and control is more cost-effective. Therefore, from a long-term perspective, for a country to consider the prevention and control of syphilis, increasing the screening coverage rate for sexually active susceptible individuals is a valuable control strategy measure. At the same time, measures to increase both the screening coverage rate and the treatment rate can also be considered. This measure has an excellent effect on the prevention and control of syphilis.

Appendices

Appendix A

The next-generation matrix approach in [19, 20] is applied to calculate the basic reproduction number ${\mathcal{R}_{0}}$. For this purpose, we can write the right-hand side of model (2.1) as $\mathcal{F}$- $\mathcal{V}$ with

$$\begin{aligned}& \mathcal{F}= \begin{pmatrix} \left (\frac{I_{1a}+\eta _{1}(I_{2m}+L_{m})}{N}\right )\beta _{1}S_{f} \\ \left (\frac{I_{1f}+I_{2f}+L_{f}}{N}\right )\beta _{2}S_{a} \\ \left (\frac{I_{1b}+\eta _{2}(I_{2m}+L_{m})}{N}\right )\beta _{3}S_{b}\\ 0\\ 0\\ 0\\ 0\\ {0}\\ {0}\end{pmatrix}, \\& \mathcal{V}= \begin{pmatrix} (\gamma _{1f}+\mu +\sigma _{1f})I_{1f} \\ (\gamma _{1m}+\mu +\sigma _{1m})I_{1a} \\ (\gamma _{1m}+\mu +\sigma _{1m})I_{1b} \\ (\gamma _{2f}+\mu +\sigma _{2f})I_{2f}-\gamma _{1f}I_{1f} \\ (\gamma _{2m}+\mu +\sigma _{2m})I_{2m}-\gamma _{1m}(I_{1a}+I_{1b}) \\ (\gamma _{3f}+\mu +\sigma _{3f})L_{f}-\gamma _{2f}I_{2f} \\ (\gamma _{3m}+\mu +\sigma _{3m})L_{m}-\gamma _{2m}I_{2m} \\ (\nu +\mu +\sigma _{4f})T_{f}-\gamma _{3f}L_{f} \\ (\nu +\mu +\sigma _{4m})T_{m}-\gamma _{3m}L_{m}\end{pmatrix}. \end{aligned}$$

The disease-free equilibrium (DFE) of model (1) is $E^{0}=(\frac{\Lambda _{f}}{\mu}, \frac{(1-\pi )\Lambda _{m}}{\mu}, \frac{\pi \Lambda _{m}}{\mu}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$.

Calculating the Jacobian matrices F and V at the DFE, we have

$$\begin{aligned}& \textbf{F}= \begin{pmatrix} 0&\beta _{1}s_{f}&0&0&\eta _{1}\beta _{1}s_{f}&0&\eta _{1}\beta _{1}s_{f}&0&0 \\ \beta _{2}s_{a}&0&0&\beta _{2}s_{a}&0&\beta _{2}s_{a}&0&0&0 \\ 0&0&\beta _{3}s_{b}&0&\eta _{2}\beta _{2}s_{b}&0&\eta _{2}\beta _{2}s_{b}&0&0 \\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\end{pmatrix}, \\& \textbf{V}= \begin{pmatrix} k_{1}&0&0&0&0&0&0&0&0\\ 0&h_{1}&0&0&0&0&0&0&0\\ 0&0&h_{1}&0&0&0&0&0&0\\ -\gamma _{1f}&0&0&k_{2}&0&0&0&0&0\\ 0&-\gamma _{1m}&-\gamma _{1m}&0&h_{2}&0&0&0&0\\ 0&0&0&-\gamma _{2f}&0&k_{3}&0&0&0\\ 0&0&0&0&-\gamma _{2m}&0&h_{3}&0&0\\ 0&0&0&0&0&-\gamma _{3f}&0&k_{4}&0\\ 0&0&0&0&0&0&-\gamma _{3m}&0&h_{4}\end{pmatrix}, \end{aligned}$$

and

$$ \textbf{V}^{-1}= \begin{pmatrix} \frac{1}{k_{1}}&0&0&0&0&0&0&0&0 \\ 0&\frac{1}{h_{1}}&0&0&0&0&0&0&0 \\ 0&0&\frac{1}{h_{1}}&0&0&0&0&0&0 \\ \frac{\gamma _{1f}}{k_{1}k_{2}}&0&0&\frac{1}{k_{2}}&0&0&0&0&0 \\ 0&\frac{\gamma _{1m}}{h_{1}k_{2}}&\frac{\gamma _{1m}}{h_{1}k_{2}}&0& \frac{1}{h_{2}}&0&0&0&0 \\ \frac{\gamma _{1f}\gamma _{2f}}{k_{1}k_{2}k_{3}}&0&0& \frac{\gamma _{2f}}{k_{2}k_{3}}&0&\frac{1}{k_{3}}&0&0&0 \\ 0&\frac{\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}}& \frac{\gamma _{1m}\gamma _{2m}}{h_{1}h_{2}h_{3}}&0& \frac{\gamma _{2m}}{h_{2}h_{3}}&0&\frac{1}{h_{3}}&0&0 \\ \frac{\gamma _{1f}\gamma _{2f}\gamma _{3f}}{k_{1}k_{2}k_{3}k_{4}}&0&0& \frac{\gamma _{2f}\gamma _{3f}}{k_{2}k_{3}k_{4}}&0& \frac{\gamma _{3f}}{k_{3}k_{4}}&0&\frac{1}{k_{4}}&0 \\ 0&\frac{\gamma _{1m}\gamma _{2m}\gamma _{3m}}{h_{1}h_{2}h_{3}h_{4}}& \frac{\gamma _{1m}\gamma _{2m}\gamma _{3m}}{h_{1}h_{2}h_{3}h_{4}}&0& \frac{\gamma _{2m}\gamma _{3m}}{h_{2}h_{3}h_{4}}&0& \frac{\gamma _{3m}}{h_{3}h_{4}}&0&\frac{1}{h_{4}}\end{pmatrix} . $$

Hence,

$$ \textbf{F} \textbf{V}^{-1}= \begin{pmatrix} 0&A_{12}&A_{13}&0&A_{15}&0&A_{17}&0&0 \\ A_{21}&0&0&A_{24}&0&A_{26}&0&0&0 \\ 0&A_{32}&A_{33}&0&A_{35}&0&A_{37}&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \\ 0&0&0&0&0&0&0&0&0 \end{pmatrix} . $$

Appendix B

Table 9 The annual number of new infections