Abstract
HIV susceptibility linked to hormonal contraception (HC) has been studied before, but with mixed results. Reports from some of the recent findings have prompted the World Health Organisation to encourage women who use HC to concurrently use condoms in order to prevent HIV infection in the light of possible increased HIV risk of infection associated with hormone-based contraceptives. A two-sex HIV model classifying women into three risk groups consisting of individuals who use condoms, natural methods, and hormone-based contraceptives is formulated and analysed to assess the possible effects of various birth control strategies on the transmission dynamics of the disease. Our model results showed that women who use HC could be key drivers of the epidemic and that their increased infectivity may be critical in driving the epidemic. Women who use hormone-based contraceptives potentially act as a core group from which men get infected and in turn transmit the disease to other population groups. We fitted the model to HIV prevalence data for Zimbabwe reported by UNAIDS and Zimbabwe Ministry of Health and Child Care and used the model fit to project HIV prevalence. Predictions using HIV data for Zimbabwe suggest that a hypothesised increase in susceptibility and infectivity of two-, three-, and fourfold would result in a 25, 50, and 100% increase in baseline HIV prevalence projection, respectively, thus suggesting possible increased disease burden even in countries reporting plausible HIV prevalence declines. Although a possible causal relationship between HIV susceptibility and HC use remains subject of continuing scientific probe, its inclusion as part of birth control strategy has been shown in this study, to possibly increase HIV transmission. If proven, HC use may potentially explain the inordinate spread of HIV within the sub-Saharan Africa region and therefore compel for urgent assessment with a view to reorienting birth control methods in use in settings with generalised epidemics.
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Acknowledgements
NJM and SDH-M acknowledge, with thanks, financial support from the National University of Science and Technology. The authors acknowledge Dr. Christinah Mukandavire for useful comments and discussions on the manuscript.
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Appendices
Appendix 1
Substituting the values of \(S^*_{f_c},S^*_{f_n},S^*_{f_h},S^*_{m},I^*_{f_c},I^*_{f_n},I^*_{f_h},I^*_{m}\) into \(\lambda ^*_{f_c},\lambda ^*_{f_n},\lambda ^*_{f_h} ~\hbox {and}~\lambda ^*_{m}\) gives
Using the relationship \(\lambda ^*_{f_c}=\displaystyle \frac{(1-p_{c})\rho _{fm_c}}{\rho _{fm_n}}\lambda ^*_{f_n},\lambda ^*_{f_h}=\displaystyle \frac{\chi \rho _{fm_h}}{\rho _{f_nm}}\lambda ^*_{f_n}~\hbox {and}~\lambda ^*_{f_n}=\frac{c_f\beta _m\lambda ^*_m\rho _{f_nm}}{({{\mathcal {K}}}_1+\lambda ^*_m)}\) and solving for \(\lambda ^*_{m}\) where
and
Further algebraic manipulations result in the following quatic function shown in Eq. (17) whose solution is shown in Eqs. (18), (19), (20), and (21).
Appendix 2
Global stability of endemic equilibrium.
Proof
At equilibrium, the following relations hold
and
Consider the candidate Lyapunov function V, such that
Clearly, \(S_{f_c}-S^*_{f_c}\) is an increasing polynomial and \(S^*_{f_c}\displaystyle \ln {\frac{S_{f_c}}{S^*_{f_c}}}\) is logarithmic function, and using the same argument on the rest of the variables, the polynomial will outgrow the logarithmic function and therefore
What is left is to prove that
The time derivative of V along the solution path is given by
Substituting equilibrium values and rewriting expressions for the derivatives of the state variables, we obtain the following system
Substituting into Eq. (23), the rearranged expressions for the derivatives of the state variables as well as the equilibrium values of \({{\mathcal {K}}}_1I^*_{f_c},{{\mathcal {K}}}_1I^*_{f_n},{{\mathcal {K}}}_1I^*_{f_h}~\hbox {and}~{{\mathcal {K}}}_1I^*_{m}\), we obtain the following expression
Collecting like terms, the derivative reduces to
Substituting equilibrium values in Theorem 4,
Based on the conditions for the validity of the candidate Lyapunov function \(S_{f_c}>S^*_{f_c},S_{f_n}>S^*_{f_n},S_{f_h}>S^*_{f_h},S_{m}>S^*_{m},I_{f_c}>I^*_{f_c}, {\dot{V}}\le 0\). We used the Lyapunov stability theorem to show that \({\dot{V}}<0\) for all
and the strict equality \({\dot{V}}=0\) holds only for \(S_{f_c}=S^*_{f_c},S_{f_n}=S^*_{f_n},S_{f_h}=S^*_{f_h},S_{m}=S^*_{m},I_{f_c}=I^*_{f_c},I_{f_n}=I^*_{f_n},I_{f_h}=I^*_{f_h}~\hbox {and}~I_m=I^*_m\). Then the only equilibrium state \(\xi ^*\) is the only positively invariant set of the endemic solution for model system (5) contained entirely in \({{\mathcal {D}}}\) and hence by the asymptotic stability theorem in LaSalle (1976), the endemic equilibrium state \(\xi ^*\) is a sink. This completes the proof. \(\square \)
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Malunguza, N.J., Hove-Musekwa, S.D. & Mukandavire, Z. Assessing the Potential Impact of Hormonal-Based Contraceptives on HIV Transmission Dynamics Among Heterosexuals. Bull Math Biol 79, 738–771 (2017). https://doi.org/10.1007/s11538-017-0252-y
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DOI: https://doi.org/10.1007/s11538-017-0252-y