Investigating Alcohol Consumption as a Risk Factor for HIV Transmission in Heterosexual Settings in Sub-Saharan African Communities

Abstract

Alcohol consumption and abuse is widespread in sub-Saharan Africa where most HIV infections occur and has been associated with risky sexual behaviors. It may therefore be one of the most common, potentially modifiable HIV risk factors in this region. A deterministic system of ordinary differential equations incorporating heterogeneity and biased sexual preferences is formulated to assess the effects of alcohol consumption on the transmission dynamics of the disease in heterosexual settings. Extensive qualitative analysis of the model is carried out and epidemic threshold such as the alcohol-induced reproductive number $({\mathcal{R}}_{A})$, and equilibria are derived and their stabilities examined. The disease-free equilibrium is found to be globally attracting whenever the reproductive number is less than unity. In the model, heterosexuality is the source of transmissions, and therefore, targeting a reduction of the basic reproductive number $({\mathcal{R}}_{0})$ should be primary objective for any intervention programme. We show that the preference to form partnerships amongst the heterogeneous groups influences the severity of disease and its evolution, and consequently the rate of partnership formation between females and alcohol consumers and their relative infectiousness over nondrinkers has a huge positive correlation with the alcohol-induced reproductive number and hence the epidemic. The proportion or absolute number of drinkers is shown to have minimal influence on the disease dynamics, and in a community with alcohol consumers, it is more prudent to reduce their risk sexual behavior rather than to fight the spread of alcohol consumption. Thus, intervention measures targeted at reducing heterogeneous group interactions and behavior change are the key to disease control in these settings.

Appendices

Appendix A

We now prove result in Lemma 4.

Proof

We make the following change of variables in order to use the Center Manifold theory (Carr 1981) as described in Castillo-Chavez and Song (2004) (Theorem 4.1) to establish the local asymptotic stability of the endemic equilibrium $S_{f}=x_{1}, I_{f}=x_{2}, A_{f}=x_{3}, S_{m_{1}}=x_{4}, S_{m_{2}}=x_{5}, I_{m_{1}}=x_{6}, I_{m_{2}}=x_{7},\ \mbox{and}\ A_{m}=x_{8}$, so that $N=\sum^{8}_{n=1}x_{n}$. We now use the vector notation X=(x ₁,x ₂,x ₃,x ₄,x ₅,x ₆,x ₇,x ₈)^T. Then, model system (8) can be written in the form $\frac{dX}{dt}=F=(f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, f_{8})^{T}$ such that

$$ \left \{ \begin{array}{l} x'_1(t)=f_1= \pi\varLambda-\beta_m c_f\frac{x_6\rho_{fm_1}+\vartheta x_7\rho_{fm_2}}{\sum^{7}_{n=4}x_n}x_1-(\alpha+\mu)x_1,\\ \noalign{\vspace*{6pt}} x'_2(t)=f_2= \beta_m c_f\frac{x_6\rho_{fm_1}+\vartheta x_7\rho_{fm_2}}{\sum^{7}_{n=4}x_n}x_1-(\gamma+\alpha+\mu)x_2,\\ \noalign{\vspace*{6pt}} x'_3(t)=f_3= \gamma x_2 - (\nu+\mu)x_3,\\ \noalign{\vspace*{6pt}} x'_4(t)=f_4= (1-\pi)\varLambda-\beta_f c_m\rho_{m_1f}\frac{x_2}{\sum^{2}_{n=1}x_n}x_4-(\alpha+\delta+\mu)x_4,\\ \noalign{\vspace*{6pt}} x'_5(t)=f_5= \delta x_4-\beta_f c_m\rho_{m_2f}\vartheta\frac{x_2}{\sum^{2}_{n=1}x_n}x_5-(\alpha+\mu)x_5,\\ \noalign{\vspace*{6pt}} x'_6(t)=f_6= \beta_f c_m\rho_{m_1f}\frac{x_2}{\sum^{2}_{n=1}x_n}x_4-(\gamma+\alpha+\delta+\mu)x_6,\\ \noalign{\vspace*{6pt}} x'_7(t)=f_7= \delta x_6+ \beta_f c_m\rho_{m_2f}\vartheta\frac{x_2}{\sum^{2}_{n=1}x_n}x_5-(\gamma+\alpha+\mu)x_7,\\ \noalign{\vspace*{6pt}} x'_8(t)=f_8= \gamma(x_6+x_7)-(\nu+\mu)x_8. \end{array} \right . $$

(18)

The Jacobian matrix of transformed system (18) at the disease-free equilibrium is given by

(19)

where $C_{1}=\frac{\pi c_{f}\beta_{m}\rho_{fm_{1}}}{(1-\pi)}\ \mbox{and}\ C_{2}=\frac{(1- \pi)(\alpha+\mu)c_{m}\beta_{f}\rho_{fm_{1}}}{\pi(\alpha+\delta+\mu)}$, and the K _i ’s are as defined before. We deduce from the Jacobian that

It was shown in (16) that model system (8) is highly sensitive to changes in ${\mathcal{R}}_{A}$ and therefore its components. Taking β _m as a bifurcation point, considering the case ${\mathcal{R}}_{A}=1$, and solving for $\beta^{*}_{m}$ gives

The linearized system of the transformed equations (18) with $\beta^{*}_{m}=\beta_{m}$ has a simple zero eigenvalue. Hence, the Center Manifold Theory (Carr 1981) can be used to analyze the dynamics of system (18) near β ^∗=β _m . It can be shown that the Jacobian of (18) at $\beta^{*}_{m}=\beta_{m}$ has a right eigenvector associated with the zero eigenvalue given by u=[u ₁,u ₂,u ₃,u ₄,u ₅,u ₆,u ₇,u ₈]^T, where

$$ \left \{ \begin{array}{l} u_2=u_2>0,\qquad u_4 =\frac{(\pi-1)(\alpha +\mu) c_m u_2\beta_f}{\pi(\alpha+\delta+\mu)^2}<0,\\ \noalign{\vspace*{6pt}} u_5 =-\frac{(1-\pi)\delta c_mu_2\beta_f((\alpha+\mu)\rho_{fm_1}+(\alpha+\delta+\mu)\rho_{fm_2})}{\pi(\alpha+\mu)(\alpha+\delta+\mu)^2}<0,\\ \noalign{\vspace*{6pt}} u_7 =-\frac{(1-\pi)\delta c_mu_2\beta_f\rho_{fm_2}}{\pi(\alpha+\gamma+\mu)(\alpha+\delta+\mu)}>0,\\ \noalign{\vspace*{6pt}} u_1 = -\frac{c_f c_mu_2\beta_f\beta_m((\alpha+\mu)\rho^2_{fm_1}+\delta\vartheta^2\rho^2_{fm_2})}{(\alpha+\mu)(\alpha+\gamma+ \mu)(\alpha+\delta+\mu)}<0,\\ \noalign{\vspace*{6pt}} u_3 = \frac{\gamma u_2}{(\mu + \nu)}>0,\qquad u_8 = \frac{(1-\pi)\gamma c_m u_2\beta_f((\alpha+\mu)\rho_{fm_1}+\delta\rho_{fm_2})}{\pi(\alpha+\gamma+\mu)(\alpha+\delta+\mu)(\mu+\nu)}>0\\ \noalign{\vspace*{6pt}} u_6 = \frac{(1- \pi)(\alpha+\mu) c_m u_2\beta_f}{\pi(\alpha+\delta+\mu)(\alpha+\gamma+\delta+\mu)}>0. \end{array} \right . $$

(20)

The left eigenvector of J(ξ ₀) associated with the zero eigenvalue at β ^∗=β _m is given by v=[v ₁,v ₂,v ₃,v ₄,v ₅,v ₆,v ₇,v ₈], where

$$ \left \{ \begin{array}{l} v_1 = v_3 = v_4 = v_5 = v_8=0,\\ \noalign{\vspace*{6pt}} v_2 = v_2>0,\qquad v_6 = \frac{\pi c_fv_2\beta_m\rho_{fm_1}}{(1-\pi)(\alpha+\gamma+\mu)}>0,\\ v_7 = \frac{\pi\vartheta c_fv_2\beta_m\rho_{fm_2}}{(1-\pi)(\alpha+\gamma+\mu)}>0. \end{array} \right . $$

(21)

For system (18), the associated nonzero partial derivatives of F at the disease-free equilibrium are given by

$$ \left \{ \begin{array}{l} \frac{\partial f^2_2}{\partial x_1\partial x_6}=\frac{\partial f^2_2}{\partial x_6 \partial x_1}=\frac{(-(\alpha +\mu)c_f\beta_m)}{(\pi-1)\varLambda} ,\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_1\partial x_7}=\frac{\partial f^2_2}{\partial x_7 \partial x_1}=\frac{\vartheta(\alpha+\mu) c_f\beta_m\rho_{fm_2}}{(1- \pi)\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_4\partial x_6}=\frac{\partial f^2_2}{\partial x_6\partial x_4}=\frac{\partial f^2_2}{\partial x_5\partial x_6}= \frac{\partial f^2_2}{\partial x_6\partial x_5} =\frac{-\pi(\alpha+\mu) c_f\beta_m\rho_{fm_1}}{(\pi-1)^2\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_4\partial x_7}=\frac{\partial f^2_2}{\partial x_7\partial x_4}=\frac{\partial f^2_2}{\partial x_5\partial x_7}= \frac{\partial f^2_2}{\partial x_7\partial x_5}=\frac{-\pi\vartheta(\alpha +\mu) c_f\beta_m\rho_{fm_1}}{(\pi-1)^2\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_6\partial x_6}=\frac{-2\pi(\alpha+\mu) c_f\beta_m\rho_{fm_1}}{(\pi-1)^2\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_6\partial x_7}=\frac{\partial f^2_2}{\partial x_7\partial x_7}=\frac{-\pi(\alpha+\mu)c_f\beta_m(\rho_{fm_1}+\vartheta \rho_{fm_1})}{(\pi-1)^2\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_2}{\partial x_7\partial x_7}=\frac{-2\pi\vartheta(\alpha+\mu)c_f\beta_m\rho_{fm_1}}{(-1+\pi)^2\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_6}{\partial x_1\partial x_2}= \frac{\partial f^2_6}{\partial x_2\partial x_1}=\frac{(\pi-1)(\alpha+\mu)^2 c_m\beta_f}{\pi^2\varLambda(\alpha +\delta+\mu)},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_6}{\partial x_2\partial x_2}=\frac{2(\pi-1)(\alpha+\mu)^2 c_m\beta_f}{\pi^2\varLambda(\alpha +\delta+\mu)},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_6}{\partial x_2\partial x_4}=\frac{\partial f^2_6}{\partial x_4\partial x_2}=\frac{(\alpha+\mu)c_m\beta_f}{\pi\varLambda},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_7}{\partial x_1\partial x_2}=\frac{\partial f^2_7}{\partial x_2\partial x_1}=\frac{(\pi-1)\delta(\alpha+\mu)c_m\beta_f}{\pi^2\varLambda(\alpha+\delta+\mu)},\\ \noalign{\vspace*{2pt}} \frac{\partial f^2_7}{\partial x_2\partial x_2}=\frac{(\pi-1)\delta (\alpha+\mu)c_m\beta_f}{\pi^2\varLambda(\alpha+\delta+\mu)} \quad \mbox{and}\quad \frac{\partial f^2_7}{\partial x_2\partial x_5}=\frac{\partial f^2_7}{\partial x_5\partial x_2}=\frac{(\alpha+\mu) c_m\beta_f}{\pi\varLambda}. \end{array} \right . $$

(22)

The bifurcation coefficient a is calculated as follows:

(23)

where

$$ \left \{ \begin{array}{l} B_1= - \frac{2c_f c^2_mu^2_2v_2\beta^2_f\beta_m((\alpha+\mu)\rho^2_{fm_1} + \delta\vartheta^2\rho^2_{fm_2})(\alpha+\mu)\rho_{fm_1}}{\pi\varLambda(\alpha+\gamma+\mu)^2(\alpha+\delta+\mu)^2}\\ \noalign{\vspace*{2pt}} \qquad {}\times {(\gamma-c_f\beta_m\rho_{fm_1}+\gamma\delta\rho_{fm_2}-\delta\vartheta^2 c_f\beta_m\rho^2_{fm_2})} ,\\ \noalign{\vspace*{2pt}} B_2=- \frac{2(\alpha+\mu)c_fc_mu^2_{v_2}\beta_f\beta_m\rho^2_{fm_1}}{\pi\varLambda(\alpha+\gamma+\mu)^2(\alpha+\delta+\mu)^2}\\ \noalign{\vspace*{2pt}} \qquad {}\times ((\alpha+\mu) (\alpha+\gamma+\mu)(\alpha+\delta+\mu)+c_m\beta_f ((\alpha+\mu)\rho_{fm_1}\\ \qquad {}\times (\alpha+\gamma+\mu-c_f\beta_m\rho_{fm_1})-\delta\vartheta^2 c_f\beta_m\rho^2_{fm_2})) \\ \noalign{\vspace*{2pt}} B_3= - \frac{2(1-\pi)\delta\vartheta(\alpha+\mu)c_fc_mu^2_2v_2\beta_f\beta_m\rho^2_{fm_2}}{(1-\pi)\pi\varLambda(\alpha+\gamma+\mu)(\alpha+\delta+\mu)^2}\\ \noalign{\vspace*{2pt}} \qquad {}\times [(\alpha+ \delta+\mu)+ c_m\beta_f(\rho_{fm_1}+(\alpha+\delta+\mu)\rho_{fm_2})\\ \qquad {}-\frac{c_fc_m\beta_f\beta_m(\rho^2_{fm_1}+\delta\vartheta^2\rho^2_{fm_2})}{(\alpha+\gamma+\mu)} ] . \end{array} \right . $$

(24)

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Appendix B

(25)

where $C_{1}=\vartheta\rho_{fm_{2}}((\alpha+\mu)(\alpha+\gamma+\mu)\rho_{fm_{1}}+(\alpha+\gamma+\delta+\mu)^{2}\rho_{fm_{2}})$ and $C_{2}=\rho_{fm_{1}}((\alpha+\gamma+\mu)(2\alpha+\gamma+2\delta+\mu)\rho_{fm_{1}}+\delta^{2}\vartheta\rho_{fm_{2}})$.