Mathematical Assessment of the Role of Early Latent Infections and Targeted Control Strategies on Syphilis Transmission Dynamics

Abstract

A new multi-stage deterministic model for the transmission dynamics of syphilis, which incorporates disease transmission by individuals in the early latent stage of syphilis infection and the reversions of early latent syphilis to the primary and secondary stages, is formulated and rigorously analysed. The model is used to assess the population-level impact of preventive (condom use) and therapeutic measures (treatment using antibiotics) against the spread of the disease in a community. It is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the associated control reproduction number (denoted by $\mathcal {R}_T$) is less than unity. A special case of the model is shown to have a unique and globally-asymptotically stable endemic equilibrium whenever the associated reproduction number (denoted by ${\tilde{\mathcal {R}}}_T$) exceeds unity. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to syphilis transmission dynamics in Nigeria, show that the top three parameters that drive the syphilis infection (with respect to $\mathcal {R}_T$) are the disease transmission rate ($\beta$), compliance in condom use (c) and efficacy of condom ($\epsilon _c$). Numerical simulations of the model show that the targeted treatment of secondary syphilis cases is more effective than the targeted treatment of individuals in the primary or early latent stage of syphilis infection.

Appendices

Appendix 1: Proof of Theorem 3.1

Proof

Consider the following linear Lyapunov function

$$\begin{aligned} \mathcal {F}=g_1E+g_2I_p+g_3I_s+g_4L_e, \end{aligned}$$

where,

$$\begin{aligned} \begin{aligned} g_1&=\gamma _1\beta (1-p_c)(K_4K_7 + \gamma _3 K_8), \quad \\ g_2&=K_1\beta (1-p_c)(K_4K_7+\gamma _3K_8),\\ g_3&=K_1\beta (1-p_c)\{\gamma _3[(1-f)\alpha +\eta _l(K_{11}+\gamma _2)]+\eta _s(K_{11}+\gamma _2)K_4\},\\ g_4&=K_1\beta (1-p_c)\{\gamma _3[(1-f)\alpha +\eta _l(K_{11}+\gamma _2)]+\eta _l(K_{11}+\gamma _2)K_{10}\\&\quad +\alpha [\eta _s(\gamma _2+fK_{11})+(1+f)K_{10}]\}. \end{aligned} \end{aligned}$$

(6.1)

The Lyapunov derivative (where a dot represents differentiation with respect to time t) is given by:

$$\begin{aligned} \begin{aligned} \dot{\mathcal {F}}&=g_1\dot{E}+g_2\dot{I_p}+g_3\dot{I_s}+g_4\dot{L_e},\\&=g_1[ \lambda S_u + \sigma \lambda S_r - (\gamma _1 + \mu )E]+g_2[ \gamma _1 E-(\gamma _2+\tau _1+\mu )I_p+(1-f)\alpha L_e]\\&\quad +g_3[ \gamma _2 I_p -(\gamma _3+\tau _2+\mu )I_s + f\alpha L_e]+g_4[ \gamma _3I_s - (\gamma _4 +\tau _3+\mu )L_e-\alpha L_e],\\&=g_1\lambda [S_u+\sigma S_r]+[g_2\gamma _1-g_1(\gamma _1+\mu )]E+[g_3\gamma _2-g_2(\gamma _2+\tau _1+\mu )]I_p\\ {}&\quad + [g_4\gamma _3-g_3(\gamma _3+\tau _2+\mu )]I_s+[g_2(1-f)\alpha +f\alpha g_3-g_4(\gamma _4+\tau _3+\mu )-\alpha g_4]L_e,\\&=g_1\lambda [S_u+\sigma S_r]+[g_3\gamma _2-g_2(\gamma _2+\tau _1+\mu )]I_p+ [g_4\gamma _3-g_3(\gamma _3+\tau _2+\mu )]I_s\\ {}&\quad +[g_2(1-f)\alpha +f\alpha g_3-g_4(\gamma _4+\tau _3+\mu )-\alpha g_4]L_e,\\&=g_1\lambda [S_u+\sigma S_r]-\{\beta (1-p_c)[K_1(\gamma _2(\gamma _3 K_9 + K_4 K_{10}) \\&\quad + K_{11}(K_4 K_{10} + \gamma _3 K_{12})]\} (I_p + \eta _s I_s + \eta _l L_e),\\&=\beta (1-p_c)(I_p+\eta _s I_s+\eta _l L_e)\bigg \{\frac{g_1(S_u+\sigma S_r)}{N}-K_1[\gamma _2(\gamma _3 K_9 + K_4 K_{10}) \\&\quad +K_{11}(K_4 K_{10} + \gamma _3 K_{12})]\bigg \},\\&\le \beta (1-p_c)(I_p+\eta _s I_s+\eta _l L_e)\bigg \{g_1- K_1[\gamma _2(\gamma _3 K_9 + K_4 K_{10})\\&\quad + K_{11}(K_4 K_{10} + \gamma _3 K_{12})]\bigg \}, \left( \text {since}\quad \frac{(S_u+\sigma S_r)}{N} < 1 \quad \text {in}\quad \mathcal {D}\right) \\&=\beta (1-p_c)(I_p+\eta _s I_s+\eta _l L_e)\bigg \{ K_1[\gamma _2(\gamma _3 K_9 + K_4 K_{10})\\&\quad + K_{11}(K_4 K_{10} + \gamma _3 K_{12})]\bigg \}(\mathcal {R}_T-1). \end{aligned} \end{aligned}$$

(6.2)

Thus, $\dot{\mathcal {F}}\le 0$ whenever $\mathcal {R}_T\le 1$. Hence, $\dot{\mathcal {F}}\le 0$ if $\mathcal {R}_T \le 1$ with $\dot{\mathcal {F}}=0$ if and only if $I_p$ = $I_s$ = $L_e =$ 0. Therefore, $\mathcal {F}$ is a Lyapunov function in $\mathcal {D}$. Thus,

$$\begin{aligned} (E(t),I_p(t),I_s(t),L_e(t),L_l(t),I_t(t),R(t))\rightarrow (0,0,0,0,0,0,0,0) \end{aligned}$$

as $t\rightarrow \infty$. Substituting ${E(t)=I_p(t)=I_s(t)=L_e(t)=L_l(t)=I_t(t)=R(t)=0}$ into the first two equation of the model (2.2) gives ($S_u(t),S_r(t))\rightarrow (\frac{\Lambda }{\mu },0)$, as $t\rightarrow \infty$. Hence, it follows from the LaSalle’s Invariance Principle (LaSalle and Lefschetz 1976), that

$$\begin{aligned} (S_u(t),S_r(t),E(t),I_p(t),I_s(t),L_e(t),L_l(t),I_t(t),R(t))\rightarrow \left( \frac{\Lambda }{\mu },0,0,0,0,0,0,0,0\right) , \end{aligned}$$

as $t\rightarrow \infty$, for $\mathcal {R}_T\le 1$. $\square$

Appendix 2: Proof of Theorem 3.2

Consider the model (2.2) with $\omega = \alpha = 0$ and $\eta _l = \eta _s = 1$, given by (3.2). Recall that $\tilde{\mathcal {R}}_T = \mathcal {R}_T|_{\omega =\alpha =0,\eta _l=\eta _s=1}$. The infection rate ($\lambda$), given by (2.1), can be expressed at endemic steady-state as:

$$\begin{aligned} \lambda ^{**} = \frac{\beta (1-p_c)(I_p^{**}+ L_e^{**} + I_s^{**})}{N^{**}},\quad \text {with}\quad N^{**} = \frac{\Lambda }{\mu }. \end{aligned}$$

(7.1)

Solving the variables of the model (in this case, model (3.2)) at steady state gives:

$$\begin{aligned} \begin{aligned} S_u^{**}&=\frac{\Lambda }{\lambda ^{**} +\mu },\quad E^{**}=\frac{G_1}{M},\quad I_p^{**}=\frac{G_2}{M},\quad I_s^{**}=\frac{G_3}{M},\quad L_e^{**}=\frac{G_4}{M},\\ L_l^{**}&=\frac{G_5}{M},\quad I_t^{**}=\frac{G_6}{M},\quad R^{**} =\frac{G_7}{M}, \end{aligned} \end{aligned}$$

(7.2)

where,

$$\begin{aligned} \begin{aligned} G_1&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma ) (\gamma _2 (\gamma _3 (\mu +\gamma _4+\tau _3)+(\mu +\tau _2) (\mu +\gamma _4+\tau _3))\\ {}&\quad + (\mu +\tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))) (\mu +\gamma _5+\tau _4) (\mu +\tau _5),\\ G_2&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma )\gamma _1 ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)\\&\quad +\gamma _3 (\mu +\gamma _4+\tau _3)) (\mu +\gamma _5+\tau _4) (\mu +\tau _5),\\ G_3&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma ) \gamma _1 \gamma _2 \left( \mu +\gamma _4+\tau _3\right) \left( \mu +\gamma _5+\tau _4\right) \left( \mu +\tau _5\right) ,\\ G_4&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma )\gamma _1 \gamma _2 \gamma _3 \left( \mu +\gamma _5+\tau _4\right) \left( \mu +\tau _5\right) ,\\ G_5&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma ) \gamma _1 \gamma _2 \gamma _3 \gamma _4 \left( \mu +\tau _5\right) ,\\ G_6&=\lambda ^{**} \mu \Lambda (\mu +\lambda ^{**} \sigma ) \gamma _1 \gamma _2 \gamma _3 \gamma _4 \gamma _5,\\ G_7&=\lambda ^{**} \Lambda (\mu +\lambda ^{**} \sigma ) \gamma _1 (\tau _1 ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)\\&\quad +\gamma _3 (\mu +\gamma _4+\tau _3)) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _2 (\tau _2 (\mu +\gamma _4+\tau _3) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad + \gamma _3(\tau _3 (\mu +\gamma _5+\tau _4) (\mu +\tau _5)+\gamma _4 (\gamma _5 \tau _5+\tau _4 (\mu +\tau _5))))), \end{aligned} \end{aligned}$$

(7.3)

and,

$$\begin{aligned} \begin{aligned} M&=\mu (\lambda ^{**} +\mu ) (\mu (\mu +\lambda ^{**} \sigma ) (\gamma _2 (\gamma _3 (\mu +\gamma _4+\tau _3)+(\mu +\tau _2) (\mu +\gamma _4+\tau _3))\\&\quad +(\mu +\tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _1 (((\mu +\lambda ^{**} \sigma )\mu +(\mu +\lambda ^{**} \sigma ) \tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)\\&\quad +\gamma _3 (\mu +\gamma _4+\tau _3))(\mu +\gamma _5+\tau _4) (\mu +\tau _5)+\gamma _2 (((\mu +\lambda ^{**} \sigma )\mu \\&\quad +(\mu +\lambda ^{**} \sigma ) \tau _2) (\mu +\gamma _4+\tau _3) (\mu +\gamma _5+\tau _4)(\mu +\tau _5)+\gamma _3 (((\mu +\lambda ^{**} \sigma ) \mu \\&\quad +(\mu +\lambda ^{**} \sigma ) \tau _3) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)+\gamma _4 (((\mu +\lambda ^{**} \sigma ) (\mu +\omega )\\&\quad +(\mu +\lambda ^{**} \sigma ) \tau _4) (\mu +\tau _5)+\gamma _5 ((\mu +\lambda ^{**} \sigma )\mu +(\mu +\lambda ^{**} \sigma ) \tau _5)))))). \end{aligned} \end{aligned}$$

(7.4)

Substituting the expressions in (7.2) into (7.1) (noting the expressions in (7.3) and (7.4)) results in the following polynomial (in $\lambda ^{**}$):

$$\begin{aligned} f(\lambda ^{**})=a_0({\lambda ^{**}})^2+ b_0\lambda ^{**} + c_0 = 0, \end{aligned}$$

(7.5)

where,

$$\begin{aligned} \begin{aligned} a_0&= \Lambda \mu \sigma [\mu (\gamma _2 (\gamma _3 (\mu +\gamma _4+\tau _3)+ (\mu +\tau _2) (\mu +\gamma _4+\tau _3))\\&\quad +(\mu +\tau _1)((\mu +\tau _2) (\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _1 ((\mu +\tau _1)((\mu +\tau _2) (\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))(\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _2 ((\mu +\tau _2) (\mu +\gamma _4+\tau _3) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)+\gamma _3 ((\mu +\tau _3) (\mu +\gamma _5+ \tau _4) (\mu +\tau _5)\\&\quad +\gamma _4 ((\mu +\tau _4) (\mu +\tau _5)+\gamma _5 (\mu +\tau _5)))))] > 0,\\ b_0&=\Lambda \mu (\mu (\sigma +1)\mu (\gamma _2 (\gamma _3 (\mu +\gamma _4+\tau _3)+(\mu +\tau _2) (\mu +\gamma _4+\tau _3))\\&\quad +(\mu +\tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _1 (((\mu (\sigma +1)-\beta \sigma (1-p_c))\mu + (\sigma \mu +\mu ) \tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)\\&\quad +\gamma _3 (\mu +\gamma _4+\tau _3)) (\mu +\gamma _5+\tau _4)(\mu +\tau _5)+\gamma _2 ((\mu ^2 (\sigma +1)\\&\quad - \beta \sigma (1-p_c) \eta _s \mu +(\sigma \mu +\mu ) \tau _2) (\mu +\gamma _4+\tau _3) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _3 ((\mu ^2 (\sigma +1) - \beta \sigma (1-p_c) \eta _l \mu +(\sigma \mu +\mu ) \tau _3) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\&\quad +\gamma _4 ((\mu ^2 (\sigma +1) +(\sigma \mu +\mu ) \tau _4) (\mu +\tau _5)+\gamma _5 (\mu ^2 (\sigma +1) + (\sigma \mu +\mu ) \tau _5)))))),\\ c_0&=\Lambda \mu ^3 \left( \mu +\gamma _5+\tau _4\right) \left( \mu +\tau _5\right) (1-\tilde{\mathcal {R}}_T). \end{aligned} \end{aligned}$$

(7.6)

It follows from (7.6) that, when $\tilde{\mathcal {R}}_T > 1$, the coefficient $c_0$ is negative. Since $a_0 > 0$, regardless of the sign of $b_0$, the polynomial (7.5) will have only one positive root when $\tilde{\mathcal {R}}_T > 1$. Hence, the reduced model (3.2) has a unique endemic equilibrium whenever $\tilde{\mathcal {R}}_T > 1$.

Remark 7.1

It can be seen that the polynomial (7.5) will not have a positive root when $\tilde{\mathcal {R}}_T < 1$. In particular, it follows from (7.5), with (7.6), that when $\tilde{\mathcal {R}}_T < 1$, the coefficient $c_0$ is positive. For the sign of the coefficient $b_0$, when $\tilde{\mathcal {R}}_T < 1$, it is convenient to solve for $b_0$ when $\tilde{\mathcal {R}}_T < 1$. It should, first of all, be noted that, the requirement $\tilde{\mathcal {R}}_T < 1$ implies that

$$\begin{aligned} \beta = \beta _c <\frac{K_1[\gamma _2(\gamma _3 K_9 + K_4 K_{10}) + K_{11}(K_4 K_{10} + \gamma _3 K_{12})]}{\gamma _1(1-p_c)(K_4K_7 + \gamma _3 K_8)}. \end{aligned}$$

(7.7)

Substituting (7.7) into the expression for $b_0$ in (7.6), and simplifying, gives

$$\begin{aligned} \begin{aligned} b_0&= \Lambda \mu [\mu ^2 (\gamma _2 (\gamma _3 (\mu +\gamma _4+\tau _3)+(\mu +\tau _2) (\mu +\gamma _4+\tau _3))+(\mu +\tau _1) ((\mu +\tau _2)\\ {}&(\mu +\gamma _4+\tau _3)+\gamma _3 (\mu +\gamma _4+\tau _3))) (\mu +\gamma _5+\tau _4) \\ {}&(\mu +\tau _5)+\gamma _1 ((\mu ^2 + \mu \tau _1) ((\mu +\tau _2) (\mu +\gamma _4+\tau _3)\\ {}&+\gamma _3 (\mu +\gamma _4+\tau _3)) (\mu +\gamma _5+\tau _4) (\mu +\tau _5)\\ {}&+\gamma _2 ((\mu ^2 + \mu \tau _2) (\mu +\gamma _4+\tau _3) (\mu +\gamma _5+\tau _4) \\ {}&(\mu +\tau _5)+\gamma _3 ((\mu ^2 + \mu \tau _3)\\ {}&(\mu +\gamma _5+\tau _4) (\mu +\tau _5)+\gamma _4 ((\mu ^2 + \mu \tau _4) (\mu +\tau _5)+\gamma _5 (\mu ^2 \\ {}&+\mu \tau _5)))))]>0 \quad \text {since}\quad 0<\sigma <1. \end{aligned} \end{aligned}$$

(7.8)

Thus, for the case with $\tilde{\mathcal {R}}_T < 1$, the quadratic (7.5) has no positive root when $\tilde{\mathcal {R}}_T < 1$ (since the coefficients $a_0$, $b_0$ and $c_0$ are positive in this case); hence, there is no endemic equilibrium point when $\tilde{\mathcal {R}}_T < 1$.

To prove the global asymptotic stability of the unique endemic equilibrium of the reduced model (3.2), consider the following nonlinear Lyapunov function (of Goh-Volterra type) defined in $\mathcal {D}_r\backslash \mathcal {D}_0$.

$$\begin{aligned} \begin{aligned} V&= S_u - S_u^{**}-S_u^{**} \ln \frac{S_u}{S_u^{**}} + E - E^{**} -E^{**}\ln \frac{E}{E^{**}} \\&+ \frac{\hat{\beta }S_u^{**}[K_5(\gamma _3+\tau _2 + \mu ) + \gamma _2(K_5 + \gamma _3)]}{K_5(\gamma _2 + \tau _1 + \mu )(\gamma _3 + \tau _2 + \mu )}\left( I_p - I_p^{**} -I_p^{**}\ln \frac{I_p}{I_p^{**}}\right) \\&+ \frac{\hat{\beta }S_u^{**}[K_4 + \gamma _3]}{K_5(\gamma _3 +\tau _2 +\mu )}\left( I_s - I_s^{**}-I_s^{**}\ln \frac{I_s}{I_s^{**}}\right) +\frac{\beta S_u^{**}}{K_5}\left( L_e - L_e^{**} -L_e^{**} \ln \frac{L_e}{L_e^{**}}\right) \end{aligned} \end{aligned}$$

(7.9)

where, now, $K_5 = \gamma _4 + \tau _3 + \mu$. The Lyapunov derivative is

$$\begin{aligned} \begin{aligned} \dot{V}&= \left( 1-\frac{S_u^{**}}{S_u}\right) \dot{S_u} + \left( 1 - \frac{E^{**}}{E}\right) \dot{E}\\ {}&\quad + \frac{\hat{\beta }S_u^{**}[K_5(\gamma _3 + \tau _2 + \mu )+\gamma _2(K_5 + \gamma _3)]}{K_5(\gamma _2 + \tau _1 +\mu )(\gamma _3 + \tau _2 +\mu )}\left( 1-\frac{I_p^{**}}{I_p}\right) \dot{I}_p\\&\quad + \frac{\beta S_u^{**}[K_4 + \gamma _3]}{K_5(\gamma _2 + \tau _2 + \mu )}\left( 1 - \frac{I_s^{**}}{I_s}\right) \dot{I}_s+ \frac{\beta S_u^{**}}{K_5}\left( 1 - \frac{L_e^{**}}{L_e}\right) \dot{L}_e. \end{aligned} \end{aligned}$$

(7.10)

Substituting the derivatives in (3.2) into (7.10) gives

$$\begin{aligned} \begin{aligned} \dot{V}&= \left( 1-\frac{S_u^{**}}{S_u}\right) \left( \Lambda -\hat{\beta } S_u (I_p + I_s + L_e)-\mu S_u\right) \\&\quad + \left( 1 - \frac{E^{**}}{E}\right) \left( \hat{\beta } S_u (I_p + I_s + L_e) - (\gamma _1 + \mu )E \right) \\ {}&\quad + \frac{\hat{\beta }S_u^{**}[K_5(\gamma _3 + \tau _2 + \mu )+\gamma _2(K_5 + \gamma _3)]}{K_5(\gamma _2 + \tau _1 +\mu )(\gamma _3 + \tau _2 +\mu )}\left( 1-\frac{I_p^{**}}{I_p}\right) ( \gamma _1 E-(\gamma _2+\tau _1+\mu )I_p)\\ {}&\quad + \dfrac{\beta S_u^{**}[K_4 + \gamma _3]}{K_5(\gamma _2 + \tau _2 + \mu )}\left( 1 - \dfrac{I_s^{**}}{I_s}\right) \left( \gamma _2 I_p -(\gamma _3+\tau _2+\mu )I_s \right) \\&\quad + \frac{\beta S_1^{**}}{K_5}\left( 1 - \frac{L_e^{**}}{L_e}\right) \left( \gamma _3I_s - (\gamma _4 +\tau _3+\mu )L_e\right) . \end{aligned} \end{aligned}$$

(7.11)

It follows from the equations of the reduced model (3.2), at steady-state, that

$$\begin{aligned} \begin{aligned} \Lambda&= \hat{\beta } S_u^{**} (I_p^{**} + I_s^{**} + L_e^{**}) + \mu S_u^{**},&\\ (\gamma _1 + \mu )E^{**}&= \hat{\beta } S_u^{**} (I_p^{**} + I_s^{**} + L_e^{**}),&\\ (\gamma _2 + \tau _1 + \mu )I_p^{**}&= \gamma _1 E^{**},&\\ (\gamma _3 + \tau _2 + \mu )I_s^{**}&= \gamma _2 I_p^{**},&\\ K_5 L_e^{**}&= \gamma _3 I_s^{**}. \end{aligned} \end{aligned}$$

(7.12)

Substituting the expressions in (7.12) into (7.11) gives

$$\begin{aligned} \begin{aligned} \dot{V}&= \mu S_u^{**} \left( 2 - \frac{S_u^{**}}{S_u} - \frac{S_u}{S_u^{**}}\right) + \hat{\beta }S_u^{**} I_p^{**} \left( 3 - \frac{S_u^{**}}{S_u} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u I_p}{E S_u^{**} I_p^{**}} \right) \\&\quad + \hat{\beta }S_u^{**} I_p^{**} \left( 4 - \frac{S_u^{**}}{S_u} - \frac{I_p I_s^{**}}{I_s I^{**}_p} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u I_s}{E S_u^{**} I_s^{**}} \right) \\&\quad + \hat{\beta }S_u^{**} L_e^{**} \left( 5 - \frac{S_u^{**}}{S_u} - \frac{I_p I_s^{**}}{I_s I^{**}_p} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u L_e}{E S_u^{**} L_e^{**}} - \frac{L_e^{**}I_s}{I_S^{**} L_e} \right) , \end{aligned} \end{aligned}$$

(7.13)

from which it follows that, since the arithmetic mean exceeds the geometric mean,

$$\begin{aligned} \left( 2 - \frac{S_u^{**}}{S_u} - \frac{S_u}{S_u^{**}}\right) \le 0 , \left( 3 - \frac{S_u^{**}}{S_u} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u I_p}{E S_u^{**} I_p^{**}} \right)&\le {} 0 ,\\ \left( 4 - \frac{S_u^{**}}{S_u} - \frac{I_p I_s^{**}}{I_s I^{**}_p} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u I_s}{E S_u^{**} I_s^{**}} \right)&\le {} 0, \,\hbox {and}\\ \left( 5 - \frac{S_u^{**}}{S_u} - \frac{I_p I_s^{**}}{I_s I^{**}_p} - \frac{E I_p^{**}}{E^{**}I_p} - \frac{E^{**}S_u L_e}{E S_u^{**} L_e^{**}} - \frac{L_e^{**}I_s}{I_S^{**} L_e} \right)&\le {} 0. \end{aligned}$$

Thus, $\dot{V} \le 0$ for $\tilde{\mathcal {R}}_T > 1$. Hence, V is a lyapunov function for the reduced model (3.2) in $\mathcal {D}_r\backslash \mathcal {D}_0$. The proof is concluded as in the proof of Theorem 3.1. $\square$

Appendix 3: Cumulative Number of New Cases When $\eta _s < 1$

Here, we present some numerical simulations of the model (2.2) showing the cumulative number of new syphilis cases when $\eta _s < 1$. See Sect. 4.3 for discussions on the figures shown in this “Appendix”.

Fig. 10

Simulations of the model (2.2) for the treatment-only strategy, showing the cumulative number of new cases, as a function of time, where treatment is applied to only one stage of syphilis infection while there are no treatments for the other stages of infection, in the absence of condom use. a Low treatment rate and b high treatment rate. Parameter values used are as in Table 2, with $\beta = 0.0069$, $c = \varepsilon _c = 0$ and $\eta _s = 0.5$

Fig. 11

Simulations of the model (2.2) for the treatment-only strategy, showing the cumulative number of new cases, as a function of time, in the absence of condom use, varying treatment rate for the targeted infection group while the treatment rates for the other four infected groups are set to zero. Parameter values used are as in Table 2, with $\beta = 0.0069$, $c = \varepsilon _c = 0$ and $\eta _s = 0.5$. a$\tau _1$, b$\tau _2$ and c$\tau _3$

Fig. 12

Simulations of the model (2.2) for the universal strategy, showing the cumulative number of new cases, as a function of time, in the presence of condom use, varying treatment rate for the targeted infection group, while treatment rates for the other four infected groups are set to zero. Parameter values used are as in Table 2, with $\beta = 0.0069$, $c = 0.54$, $\varepsilon _c = 0.8$ and $\eta _s = 0.5$. a$\tau _1$, b$\tau _2$ and c$\tau _3$