## Abstract

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

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## Acknowledgements

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

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## Appendices

### Appendix A: Derivation of Model Equations

The susceptible population *S*(*t*) is generated by the recruitment of individuals (assumed susceptible) into the population at a rate \(\Lambda \). The population of individuals in the *S*(*t*) compartment is reduced due to HIV and syphilis infections at the rates \(\lambda _H\), \(\lambda _S\), \(\lambda _{SH}\) and \(\lambda _{HS}\). The population is further reduced by natural death (at a rate \(\mu \); natural death occurs in all epidemiological compartments at this rate). So that

The population of individuals exposed to syphilis is increased by those who acquire syphilis infection at the rates \(\lambda _S\) and \(\lambda _{SH}\) and those reinfected with syphilis after treatment, at the rates \(\alpha _1\lambda _S\) and \(\alpha _2\lambda _{SH}\), where \(\alpha _1\) and \(\alpha _2\) are modification parameters which account for the susceptibility of singly infected individuals with syphilis who recovered from syphilis. This population is reduced by progression to primary syphilis, at the rate \(\gamma _1\), those who get infected with HIV at the rates \(\lambda _H\) and \(\lambda _{HS}\) and then by natural death. Hence

The population of infected individuals with primary syphilis is increased by those who progressed from the exposed stage and reduced by those who get infected with HIV at the rates \(\phi _1\lambda _H\) and \(\phi _2\lambda _{HS}\) (where \(\phi _1\) and \(\phi _2\) are modification parameters which account for their increased susceptibility to HIV infection). This population is further reduced by those who recover at the rate \(r_1\), those who progress to the secondary stage of syphilis, at the rate \(\gamma _2\), and by natural death. So that we have

The population of infected individuals with secondary syphilis is increased by those who progressed from the primary stage and reduced by those who get infected with HIV at the rates \(\phi _3\lambda _H\) and \(\phi _4\lambda _{HS}\) (where \(\phi _3\) and \(\phi _4\) are modification parameters which account for their increased susceptibility to HIV infection). This population is further reduced by those who recover at the rate \(r_2\), those who progress to the early latent stage of syphilis, at the rate \(\gamma _3\), and by natural death. Hence

The population of infected individuals with early latent syphilis is increased by those who progressed from the secondary stage and reduced by those who get infected with HIV at the rates \(\phi _5\lambda _H\) and \(\phi _6\lambda _{HS}\) (where \(\phi _5\) and \(\phi _6\) are modification parameters which account for their increased susceptibility to HIV infection). This population is further reduced by those who recover at the rate \(r_3\), those who progress to the late latent stage of syphilis, at the rate \(\gamma _4\), and finally by natural death. Hence

The population of infected individuals with late latent syphilis is increased by those who progressed from the early latent stage and reduced by those who get infected with HIV at the rates \(\phi _7\lambda _H\) and \(\phi _8\lambda _{HS}\) (where \(\phi _7\) and \(\phi _8\) are modification parameters which account for their increased susceptibility to HIV infection). This population is further reduced by those who recover at the rate \(r_4\), those who progress to the tertiary stage of syphilis, at the rate \(\gamma _5\), and by natural death. Hence

The population of infected individuals with tertiary syphilis is increased by those who progress from the late latent stage and diminished by those who get infected with HIV at the rates \(\phi _9\lambda _H\) and \(\phi _{10}\lambda _{HS}\) (where \(\phi _9\) and \(\phi _{10}\) are modification parameters which account for their increased susceptibility to HIV infection). This population is further reduced by those who recover, at the rate \(r_5\), and natural mortality. We therefore have that

The population of those recovered from syphilis is increased by all singly infected individuals who recover from syphilis at the rates \(r_1\), \(r_2\), \(r_3\), \(r_4\) and \(r_5\) and reduced by the those who get reinfected with syphilis, those who get infected with HIV, at the rates \(\lambda _H\) and \(\lambda _{HS}\), and also by natural death. We then have that

The population of individuals infected with HIV only is increased by HIV infection of susceptible individuals and individuals who recovered from a previous syphilis infection at the rates \(\lambda _H\) and \(\lambda _{HS}\). The population is further increased by HIV-infected individuals treated for syphilis at the primary, secondary, early latent, late latent and tertiary stages, at the rates \(r_{S1}\), \(r_{S2}\), \(r_{S3}\), \(r_{S4}\) and \(r_{S5}\), respectively. The population is however diminished by those who get infected with syphilis at the rates \(\psi _1\lambda _S\) and \(\psi _2\lambda _{SH}\) (where \(\psi _1\) and \(\psi _2\) are modification parameters which account for the increased susceptibility of HIV-infected individuals to syphilis infection), as well as through natural mortality and an additional disease-induced death, at the rate \(d_{H1}\). Hence

The population of HIV-infected individuals who are exposed to syphilis is increased by individuals with HIV who get infected with syphilis and those exposed to syphilis who get infected with HIV at the rates \(\lambda _H\) and \(\lambda _{HS}\) but diminished by the progression of these individuals to the primary stage of syphilis infection, at the rate \(\gamma _{S1}\), by natural mortality and disease-induced mortality, at the rates \(\mu \) and \(d_{H2}\), respectively. So we have

The population of HIV-infected individuals in the primary stage of syphilis is increased by those with primary syphilis who get infected with HIV at the rates \(\phi _1\lambda _H\) and \(\phi _2\lambda _{HS}\), and by individuals who progressed from the exposed stage of syphilis infection, at the rate \(\gamma _{S1}\). This population is decreased by those who progress to secondary stage of syphilis, at the rate \(\gamma _{S2}\), by those who are treated for syphilis, at the rate \(r_{S1}\) and by disease-induced mortality, at the rate \(d_{H3}\), as well as natural mortality. Hence

The population of HIV-infected individuals in the secondary stage of syphilis is increased by those with secondary syphilis who get infected with HIV, at the rates \(\phi _3\lambda _H\) and \(\phi _4\lambda _{HS}\), and by individuals who progressed from the primary stage of syphilis infection, at the rate \(\gamma _{S2}\). This population is decreased by those who progress to early latent stage of syphilis, at the rate \(\gamma _{S3}\), by those who are treated for syphilis, at the rate \(r_{S2}\), and by disease-induced mortality, at the rate \(d_{H4}\), as well as natural mortality. Hence

The population of HIV-infected individuals in the early latent stage of syphilis is increased by those with early latent syphilis who get infected with HIV, at the rates \(\phi _5\lambda _H\) and \(\phi _6\lambda _{HS}\), and by individuals who progressed from the secondary stage of syphilis infection at the rate \(\gamma _{S3}\). This population is decreased by those who progress to late latent stage of syphilis, at the rate \(\gamma _{S4}\), by those who are treated for syphilis, at the rate \(r_{S3}\), and by disease-induced mortality, at the rate \(d_{H5}\), as well as natural mortality. Hence

The population of HIV-infected individuals in the late latent stage of syphilis is increased by those with late latent syphilis who get infected with HIV, at the rates \(\phi _7\lambda _H\) and \(\phi _8\lambda _{HS}\), and by individuals who progressed from the early latent stage of syphilis infection, at the rate \(\gamma _{S4}\). This population is decreased by those who progress to the tertiary stage of syphilis, at the rate \(\gamma _{S5}\), by those who are treated for syphilis, at the rate \(r_{S4}\), and by disease-induced mortality, at the rate \(d_{H6}\), as well as natural mortality. Hence

The population of HIV-infected individuals in the tertiary stage of syphilis is increased by those with tertiary syphilis who get infected with HIV, at the rates \(\phi _9\lambda _H\) and \(\phi _{10}\lambda _{HS}\), and by individuals who progressed from the late latent stage of syphilis infection, at the rate \(\gamma _{S5}\). This population is decreased by those who are treated for syphilis, at the rate \(r_{S5}\), and by disease-induced mortality, at the rate \(d_{H7}\), as well as natural mortality. Hence

### Appendix B: Positivity and Boundedness of Solutions

The basic qualitative properties of model (5) are given below. We claim the following.

### Theorem 7.1

System (5) preserves positivity of solutions. That is, solutions with positive initial conditions remain positive for all time *t*.

### Proof

Let

From the first equation of model (5), it follows that

when solved leads to

Using the same approach, we can equally show that all other state variable of model (5) will remain positive for all time \(t>0\). \(\square \)

We now prove the positive invariance of the region, \({\mathcal {D}}\).

### Lemma 7.1

Consider the biologically feasible region

The closed set \({\mathcal {D}}\) is positively invariant and a global attractor of all positive solution of model (5).

### Proof

Adding the equations of model (5) gives

Since the right-hand side of the above equality is bounded by \(\Lambda -\mu N\), a standard comparison theorem (Lakshmikantham et al. 1989) can be used to show that

In particular, \(N(t)\le \frac{\Lambda }{\mu }\) if \(N(0)\le \frac{\Lambda }{\mu }\) for all \(t>0\). Thus, \({\mathcal {D}}\) is a positively invariant set under the flow described by the model. The solutions with initial condition in \({\mathcal {D}}\) remain in \({\mathcal {D}}\) with respect to the model. Hence, the system is mathematically and epidemiologically well posed in \({\mathcal {D}}\) (Hethcote 2000). \(\square \)

### Appendix C: Proof of Theorem 3.4

### Proof

For system (13), let \(S=x_1\), \(E_S=x_2\), \(I_P=x_3\), \(I_S=x_4\), \(L_{S1}=x_5\), \(L_{S2}=x_6\), \(I_T=x_7\) and \(R_S=x_8\). So, the system of equations can be written as

Consider the case with \(\beta _S = \beta _S^{*}\) a bifurcation parameter. Solving for \(\beta _S = \beta _S^{*}\) from \({\mathcal {R}}_T = 1\) gives

The Jacobian of the transformed system evaluated at the DFE with \(\beta _S = \beta _S^{*}\) is given by

We obtain the right eigenvectors associated with the zero eigenvalue which is given as \({\mathbf {w}} = (\omega _1, \omega _2,...,\omega _{8})^T\), where

The components of the left eigenvector of \(J(\xi _{S0}) |_{\beta _S = \beta _S^*}\), \({\mathbf {v}} = (\nu _1, \nu _2,...,\nu _{8})\), satisfying \(\mathbf {v.w}=1\), are

It follows from Theorem 4.1 in Castillo-Chavez and Song (2004), if we compute the associated nonzero partial derivatives of *F*(*x*) (evaluated at the DFE \(\xi _{S0}\)), that the associated bifurcation coefficients a and b, defined by

are computed to be

and

with \(\nu _2\), \(\omega _2\), \(\omega _3\), \(\omega _4\), \(\omega _5\), \(\omega _6\), \(\omega _7\) and \(\omega _8\) positive.

Since the bifurcation coefficient \(b>0\), it follows from Theorem 4.1 in Castillo-Chavez and Song (2004) that system (13) will undergo a backward bifurcation if the backward bifurcation coefficient \(a>0\). This is possible if

Clearly, if \(\alpha _1=0\), then \(a<0\) and model (13) will not exhibit the backward bifurcation phenomenon at \({\mathcal {R}}_T=1\). \(\square \)

### Appendix D: Proof of Theorem 3.5

### Proof

Consider the following linear Lyapunov function

with Lyapunov derivative

It follows from (29), noting that \(S(t)\le N(t)\) and \(N(t)\le \frac{\Lambda }{\mu }\) in \({\mathcal {D}}_2\) for all \(t>0\), that

Hence, \(\dot{{\mathcal {V}}_2}\le 0\) if \({\mathcal {R}}_{T} \le 1\) with \(\dot{{\mathcal {V}}_2}= 0\) if and only if \(E_S=I_P=I_S=L_{S1}=L_{S2}=I_T=R_S=0\). Therefore, \({\mathcal {V}}_2\) is a Lyapunov function in \({\mathcal {D}}_2\), and it follows from LaSalles invariance principle Salle and Lefschetz (1976) that every solution to the equations in (13) with initial conditions in \({\mathcal {D}}_2\) converges to \(\xi _{S0}\) as \(t\rightarrow \infty \). That is,

Substituting \(E_S=I_P=I_S=L_{S1}=L_{S2}=I_T=R_S=0\) into the first equation in model (13) with \(\alpha _1=0\), gives \(S(t)\rightarrow \frac{\Lambda }{\mu }\) as \(t\rightarrow \infty \). Thus, \((S, E_S, I_P, I_S, L_{S1}, L_{S2}, I_T, R_S)\rightarrow (\frac{\Lambda }{\mu }, 0, 0, 0, 0, 0, 0, 0)\) as \(t\rightarrow \infty \) for \({\mathcal {R}}_{T}\le 1\), so the DFE, \(\xi _{S0}\), of model (13) with \(\alpha _1=0\), is globally asymptotically stable in \({\mathcal {D}}_2\) if \({\mathcal {R}}_{T}\le 1\). \(\square \)

### Appendix E: Proof of Theorem 3.6

### Proof

Consider model (13) (with (22) and \(\alpha _1=0\)) and \({\mathcal {R}}_{T}>1\), so that the associated unique endemic equilibrium exists. Also, consider the nonlinear Lyapunov function of the Goh–Volterra type

with Lyapunov derivative,

It can be shown from model (13) with \(\alpha _1=0\) that, at steady state,

Substituting the time derivatives of \(E_S\), \(I_P\) and \(I_S\), in (13) (with \(\alpha _1=0\)) as well as the relations in (27), into (26), we have that (after several algebraic calculations)

Finally, since the arithmetic mean is greater than the geometric mean, the following inequalities hold.

Thus we have that \(\dot{{\mathcal {F}}_2}\le 0\) for \({\mathcal {R}}_{T}>1\). Since the relevant variables in the equations for \(L_{S1}\), \(L_{S2}\), \(I_T\) and \(R_S\) are at endemic steady state, it follows that these can be substituted into the equations for \(L_{S1}\), \(L_{S2}\), \(I_T\) and \(R_S\) so that

\((L_{S1}(t), L_{S2}(t), I_T(t), R_S(t))\rightarrow (L_{S1}^{**}, L_{S2}^{**}, I_T^{**}, R_S^{**})\) as \(t\rightarrow \infty \)

Hence, \({\mathcal {F}}_2\) is a Lyapunov function in \({\mathcal {D}}_{3}|{\mathcal {D}}_{4}\) \(\square \)

### Appendix F: Proof of Theorem 4.1

### Proof

For system (5), let \(S=x_1\), \(E_S=x_2\), \(I_P=x_3\), \(I_S=x_4\), \(L_{S1}=x_5\), \(L_{S2}=x_6\), \(I_T=x_7\), \(R_S=x_8\), \(H=x_9\), \(E_H=x_{10}\), \(H_P=x_{11}\), \(H_S=x_{12}\), \(H_{S1}=x_{13}\), \(H_{S2}=x_{14}\) and \(H_T=x_{15}\). So, the system of equations can be written as

where

The Jacobian of the transformed system evaluated at the DFE with is given by

where

Without loss of generality, consider the case when \({\mathcal {R}}_{T}>{\mathcal {R}}_{H}\) and \({\mathcal {R}}_{C}=1\), so that \({\mathcal {R}}_{T}=1\). Furthermore, let \(\beta _S=\beta _S^*\) be a bifurcation parameter. Solving for \(\beta _S\) from \({\mathcal {R}}_{T}=1\) gives

We obtain the right eigenvectors associated with the zero eigenvalue which is given as \({\mathbf {w}} = (\omega _1, \omega _2,...,\omega _{15})^T\), where

The components of the left eigenvector of \(J(\xi _0) |_{\beta _S = \beta _S^*}\), \({\mathbf {v}} = (\nu _1, \nu _2,...,\nu _{15})\), satisfying \(\mathbf {v.w}=1\), are

It follows from Theorem 4.1 in Castillo-Chavez and Song (2004), if we compute the associated nonzero partial derivatives of *F*(*x*) (evaluated at the DFE \(\xi _{S0}\)), that the associated bifurcation coefficients a and b, defined by

are computed to be

and

with \(\nu _2\), \(\omega _2\), \(\omega _3\), \(\omega _4\), \(\omega _5\), \(\omega _6\), \(\omega _7\) and \(\omega _8\) positive.

Since the bifurcation coefficient \(b>0\), it follows from Theorem 4.1 in Castillo-Chavez and Song (2004) that system (5) will undergo a backward bifurcation if the backward bifurcation coefficient \(a>0\). This is possible if

Clearly, if \(\alpha _1=0\), then \(a<0\) and model (5) will not exhibit the backward bifurcation phenomenon at \({\mathcal {R}}_C=1\). \(\square \)

### Appendix G: Discussion on Some Numerical Simulations

Figure 8 shows that, with increasing treatment rate for primary and secondary syphilis for both singly and dually infected individuals, there is a decrease in the cumulative incidence of HIV amongst individuals exposed to syphilis when the HIV infection is caused by dually infected individuals. This conclusion is also reached when we observe the impact of increasing treatment rates for primary and secondary syphilis (in both singly and dually infected individuals) on the cumulative incidence of HIV amongst individuals in the primary stage of syphilis infection (Fig. 9), secondary stage of syphilis infection (Fig. 10), early latent stage of syphilis infection (Fig. 11), late latent stage of syphilis infection (Fig. 12) and tertiary stage of syphilis infection (Fig. 13). However, when our interest is on investigating the impact of treatment of primary (and secondary) syphilis, for both singly and dually infected individuals, on the cumulative incidence of HIV amongst the aforementioned subpopulations, when the HIV infections are caused by HIV-only infected individuals, we observe that the relatively low treatment rates for primary and secondary syphilis (\(r_1=r_2=r_{s1}=r_{s2}=5\)) were losing its effectiveness on HIV control in the target subpopulations in the long run (see Figs. 8a, 9a, 10a, 11a, 12a, 13a). However, with higher treatment rates for primary and secondary syphilis (in both singly and dually infected individuals), i.e. \(r_1=r_2=r_{S1}=r_{S2}=10\), \(r_1=r_2=r_{S1}=r_{S2}=12\) and \(r_1=r_2=r_{S1}=r_{S2}=15\), we observe a reduction in the cumulative incidence of HIV amongst the subpopulations, just as we observed when the HIV infections were caused by dually infected individuals (as we can see in Figs. 8b, 9b, 10b, 11b, 12b, 13b).

Figure 14 reveals that increasing the treatment rates for singly infected individuals with primary and secondary syphilis infection, \(r_1\) (Fig. 14a) and \(r_2\) (Fig. 14a), respectively, results in a positive population-level impact on the burden of syphilis, as we observe a reduction in the prevalence of primary syphilis (\(I_P\)), secondary syphilis (\(I_S\)), early latent syphilis (\(L_{S1}\)), but a marginal effect on the prevalence of late latent syphilis (\(L_{S2}\)), in the long run.

Figure 15a shows that when the treatment rate of singly infected individuals with primary syphilis, (\(r_1\)), is varied, it has a positive population-level impact on the number of infected individuals with HIV and tertiary syphilis (\(H_T\)) but a marginal effect on the number of infected individuals with tertiary syphilis (\(I_T\)), in the long run, while Fig. 15b shows a decrease in the prevalence of HIV and syphilis co-infection as we increase the syphilis treatment rate of dually infected individuals with HIV and primary syphilis (\(r_{S1}\)), as expected.

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Nwankwo, A., Okuonghae, D. Mathematical Analysis of the Transmission Dynamics of HIV Syphilis Co-infection in the Presence of Treatment for Syphilis.
*Bull Math Biol* **80, **437–492 (2018). https://doi.org/10.1007/s11538-017-0384-0

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### Keywords

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