Mathematical Analysis of an Industrial HIV/AIDS Model that Incorporates Carefree Attitude Towards Sex

Abstract

A nonlinear differential equation model is proposed to study the dynamics of HIV/AIDS and its effects on workforce productivity. The disease-free equilibrium point of the model is shown to be locally asymptotically stable when the associated basic reproduction number \(\mathcal{{R}}_{0}\) is less than unity. The model is also shown to exhibit multiple endemic states for some parameter values when \(\mathcal{{R}}_{0}<1\) and \(\mathcal{{R}}_{0}>1\). Global asymptotic stability of the disease-free equilibrium is guaranteed only when the fractions of the Susceptible subclass populations are within some bounds. Optimal control analysis of the model revealed that the most cost effective strategy that should be adopted in the fight against HIV/AIDS spread within the workforce is one that seeks to prevent infections and the treatment of infected individuals.

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Correspondence to Christopher S. Bornaa.

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Appendix: Coefficients of Eq. (6) and Other Definitions

Appendix: Coefficients of Eq. (6) and Other Definitions

$$\begin{aligned} \eta _{4}= & {} \left[ k_{{7}}k_{{6}}k_{{5}} \left( k_{{8}}+\delta _{{1}} \right) \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) + k_{{7}}\left( k_{{6}}\xi _{{1}}\pi _{{1}}+k_{{5}}\xi _{{3}}\pi _{{2}}\right) \right. \\&\left. +\, \left( k_{{6}}k_{{5}}k_{{4}} \left( k_{{8}}+\delta _{{4}} \right) + \sigma \,k_{{5}}\xi _{{3}}+\theta \,k_{{6}}\xi _{{1}} \right) \pi _{{3}} \right] {\tau _{{1}}}^{2};\\ \eta _{3}= & {} k_{5}k_{6}k_{7}\left[ \xi _{{6}}{\tau _{{1}}}^{2}+ \left( k_{{3 }}+k_{{2}} \right) \left( k_{{8}}+\delta _{{1}} \right) \tau _{{1}} \right] \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) \\&-\,k_{{8}}\beta {\tau _{1}}^{2} \left\{ k_{{6}}k_{{7}} \left( \sigma +k_{{4}} \right) \pi _{{1}}+k_{{5}}k_{{7}} \left( k_{{4}}\tau _{{2}}+\theta \right) \pi _{{2}}\right. \\&\left. +\, \left[ \left( k_{{4}}\tau _{{2}}+\theta \right) k _{{5}}\sigma +k_{{6}} \left( \left( \sigma +k_{{4}} \right) \theta +\tau _{{2}}k_{{5}}k_{{4}} \right) \right] \pi _{{3}}+k_{{6}}k_{{7}}k_{{5}} \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) \right\} \\&+\, k_{6}k_{7}\left[ \left( \mu \,\xi _{{1}}+k_{{5}} \xi _{{2}} \right) {\tau _{{1}}}^{2}\right. \\&\left. +\,\left( k_{{3}}+k_{{2 }} \right) \xi _{{1}}\tau _{{1}} \right] \pi _{{1}}+ k_{{5}}k_{{7 }}\left[ \xi _{{3}} \left( \mu +k_{{1}} \right) {\tau _{{1}}}^{2}+ \left( k_{{6}}\xi _{{4}}+k_{{3}}\xi _{{3}} \right) \tau _{{1}} \right] \pi _{{2}}\\&+\, \left[ \left( k_{{6}}k_{{5}}k_{{4}} \left( k_{{8}}+\delta _{{4}} \right) +\sigma \,k_{{5}}\xi _{{3}}+\theta \,k_{{6}}\xi _{{1}} \right) \left( \mu +k_{{1}} \right) {\tau _{{1}}}^{2}\right. \\&\left. +\, \left( k_{{6}} \left( \alpha \,k_{{7}}+k_{{2}}\theta \right) \xi _{{1}}+k_{{5}} \left( k_{{7}}\rho +k_{{2}}\sigma \right) \xi _{{3}}+k_{{6}}k_{{5}}k_{{ 4}}\xi _{{5}} \right) \tau _{{1}} \right] \pi _{{3}};\\ \eta _{2}= & {} -k_{{8}}\beta \tau _{{1}} \left\{ k_{6}k_{7} \left[ \left( \sigma \,\mu +k_ {{4}}\mu +\rho \,k_{{5}} \right) \tau _{{1}}\right. \right. \\&\left. +\, \left( k_{{3} }+k_{{2}} \right) \left( \sigma +k_{{4}} \right) \right] \pi _{{1}} +k_{{6}}k_{{7}}k_{{5}} \left( k_{{3}}+k_{{2}}+k_{{1 }}\tau _{{1}} \right) \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) \\&+\,k _{{5}}k_{{7}} \left[ \left( \mu +k_{{1}} \right) \left( k_{{4}}\tau _{ {2}}+\theta \right) \tau _{{1}}+\alpha \,k_{{6}}+ \left( k_{{4}}\tau _{{2 }}+\theta \right) k_{{3}} \right] \pi _{{2}}\\&+\, \left[ \left( k_{{5}}k_ {{4}} \left( k_{{6}}+\sigma \right) \left( \mu +k_{{1}} \right) \tau _{ {2}}+\theta \,\xi _{{7}} \left( \mu +k_{{1}} \right) \right) \tau _{{1}}\right. \\&\left. \left. +\, k_{{5}}k_{{4}}\xi _{{8}}\tau _{{2}}+\alpha \,k_{{6}}k_{{7}} \left( \sigma +k_{{4}} \right) +\theta \,k_{{5}}k_{{7}}\rho +k_{{2}}\theta \,\xi _{{7}} \right] \pi _{{3}} \right\} \\&+\, k_{6}k_{7}\left[ k_{{5}}k_ {{4}}k_{{8}}{\tau _{{1}}}^{2}k_{{1}}+ \left( k_{{3}}+k_{{ 2}} \right) \left( \mu \,\xi _{{1}}+k_{{5}}\xi _{{2}} \right) \tau _{{1}} +k_{{3}}k_{{2}}\xi _{{1}} \right] \pi _{{1}}\\&+\,k_{{5}}k_{{7}} \left[ \mu \,k_{{1}}\xi _{{3}}{\tau _{{1}}}^{2}+ \left( k_{{3}} \left( \mu +k_{{1}} \right) \xi _{{3}} +\delta _{{1}}k_{{6 }}\alpha \,k_{{1}} +k_{{6}}k_{{8}} \left( k_{{4}} \left( k_{{1}}+k_{{2}} \right) +\alpha \,k_{{1}} \right) \right) \tau _{{1}}\right. \\&\left. +\,k_{{3}}k_{{6}}\xi _{{4}} \right] \pi _{{2}}\\&+\, \left[ k_{{6}}k_{{7}} \left( k_{{5}} \left( k_{{4}}k_{{8}}k_{{3}}+ \left( k_{{8}}+\delta _{{1}} \right) \alpha \,\rho \right) +\alpha \,k_{{2}}\xi _{{1}} \right) + \mu \,k_{{1}} \left( k _{{6}}k_{{5}}k_{{4}} \left( k_{{8}}+\delta _{{4}} \right) \right. \right. \\&\left. \,+\sigma \,k_{{ 5}}\xi _{{3}}+\theta \,k_{{6}}\xi _{{1}} \right) {\tau _{{1}}}^{2}\\&+\, \left( k_{{5}} \left( k_{{7}}\rho +k_{{2}}\sigma \right) \left( \mu +k _{{1}} \right) \xi _{{3}}+k_{{6}} \left( \alpha \,\mu \,k_{{7}}+\left( k_{1}+\mu \right) \theta \,k_{{2}} \right) \xi _{{1}}\right. \\&\left. \left. +\,k_{{6}}k_{{5} } \left( k_{{4}}\xi _{{5}} \left( \mu +k_{{1}} \right) +k_{{7}}\xi _{{2}} \alpha \right) \right) \tau _{{1}} \right] \pi _{{3}}\\&+\,k_{{7}}k_{{6}}k_{{5}} \left[ k_{{4}}k_{{8}}{\tau _{{1}}}^{2}k_{{1} }+ \left( k_{{3}}+k_{{2}} \right) \xi _{{6}}\tau _{{1}}+k_{{2}}k_{{3}} \left( k_{{8}}+\delta _{{1}} \right) \right] \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) ;\\ \eta _{1}= & {} -\beta \,k_{8} \left\{ \left[ k_{{7}}k_{{6}} \left( k_{{3}}+k_{{2}} \right) \left( \sigma \,\mu +k_{{4}}\mu +\rho \,k_{{5}} \right) \tau _{{1}}+k_{{7} }k_{{6}}k_{{3}}k_{{2}} \left( \sigma +k_{{4}} \right) \right] \pi _{{1 }}\right. \\&+\, \left[ k_{{5}}k_{{1}}k_{{7}} \left( \left( k_{{4}}\tau _{{2}}+ \theta \right) \mu \,\tau _{{1}}+\alpha \,k_{{6}} \right) \tau _{{1}}+k_{{ 1}}k_{{3}}k_{{7}}k_{{5}} \left( k_{{4}}\tau _{{2}}+\theta \right) \tau _ {{1}}\right. \\&\left. +\,k_{{3}}k_{{7}}k_{{5}} \left( \left( k_{{4}}\tau _{{2}}+\theta \right) \mu \,\tau _{{1}}+\alpha \,k_{{6}} \right) \right] \pi _{{2}}\\&+\, \left[ \alpha \,k_{{7}}k_{{6}} \left( k_{{2 }}k_{{4}}+\rho \,k_{{5}}+k_{{2}}\sigma \right) \right. \\&+\, \left( \mu \,k_{{5}}k_{{4}}k_{{1}} \left( k_{{6}}+\sigma \right) \tau _{{2}}+\mu \,\theta \,k_{{1}}\xi _{{7}} \right) {\tau _{{1}}} ^{2}\\&+\, \left( k_{{5}}k_{{4}}\xi _{{8}} \left( \mu +k_{{1}} \right) \tau _{ {2}}+ \left( k_{{5}}\left( k_{{2}}\sigma +k_{{7}}\rho \right) \right. \right. \\&\left. \left. \left. +\,k_{{6}}k_{{2}}\left( k_{{4}}+\sigma \right) \right) \theta \, \left( \mu +k_{{1}} \right) +\alpha \,k_{{7}}k_{{6}} \left( \mu \left( k_{{4}}+\sigma \right) +\rho \,k _{{5}} \right) \right) \tau _{{1}} \right] \pi _{{3}}\\&+\,k_{{ 7}}k_{{6}}k_{{5}} \left( k_{{1}}\tau _{{1}} \left( k_{{3}}+k_{{2}} \right) +k_{{2}}k_{{3}} \right) \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) \Bigg \}\\&+\, \left[ k_{{7}}k_{{ 6}}k_{{5}}k_{{4}}k_{{8}} \left( k_{{3}}+k_{{2}} \right) k_{{1}}\tau _{{ 1}}+k_{{7}}k_{{6}}k_{{3}}k_{{2}} \left( \mu \,\xi _{{1}}+k_{{5}}\xi _{{2} } \right) \right] \pi _{{1}}\\&+\, \left[ k_{{5}}k_{{1}}k_{{7}} \left( \mu \,k_{{3}}\xi _{{3}}+k_{{6}}k_{{4}}k_{{8}}k_{{2}} \right) \tau _{{1}}+k_{ {7}}k_{{6}}k_{{5}}k_{{3}} \left( k_{{4}}k_{{8}} \left( k_{{1}}+k_{{2}} \right) +\alpha \,k_{{1}} \left( k_{{8}}+\delta _{{1}} \right) \right) \right] \pi _{{2}}\\&+\, \left[ \left( \mu \,k_{{5}}k_{{1}} \left( k_{{7}}\rho +k_{{2}}\sigma \right) \xi _{{3}}+\mu \,\theta \,k_{{6 }}k_{{2}}k_{{1}}\xi _{{1}}+k_{{6}}k_{{5}}k_{{4}} \left( \mu \,k_{{2}}k_{ {1}} \left( k_{{8}}+\delta _{{4}} \right) +k_{{1}}k_{{7}}k_{{8}}k_{{2}} \right) \right) \tau _{{1}}\right. \\&+\,k_{{7}}k_{{6}} \left( \alpha \,\mu \,k_{{2} }\xi _{{1}}+\alpha \,\rho \,k_{{5}} \left( \left( k_{{1}}+k_{{2}} \right) \left( k_{{8}}+\delta _{{1}} \right) +k_{{4}}k_{{8}} \right) \right. \\&\left. \left. +\,k_{{5}}k_{{4}}k_{{8}} \left( k_{{1}}k_{{3}}+{k_{{2}}}^{2}+\mu \,\rho \right) \right) \right] \pi _{{3}}\\&+\,k_{{7}}k_{{6}}k_{{5}} \left[ k_{ {1}}\tau _{{1}}k_{{4}}k_{{8}} \left( k_{{3}}+k_{{2}} \right) +k_{{2}}k_ {{3}}\xi _{{6}} \right] \left( 1-\pi _{1}-\pi _{2}-\pi _{3}\right) ;\\ \eta _{0}= & {} k_{{1}}k_{{2}}k_{{3}}k_{{4}}k_{{5}}k_{{6}}k_{{7}}k_{{8}} \left( 1-{\mathcal {R}}_{0}\right) ;\\ \xi _{{1}}= & {} k_{{4}}\delta _{{2}}+\delta _{{1}}\sigma +k_{{4}}k_{{8}}+k_{{8 }}\sigma ,\quad \xi _{{2}}=\delta _{{1}}\rho +k_{{4}}k_{{8}}+k_{{8}}\rho ,\quad \xi _{{3 }}=\delta _{{1}}\theta +k_{{8}}\theta +k_{{4}}\delta _{{3}}+k_{{4}}k_{{8}},\\ \xi _{{4}}= & {} \delta _{{1}}\alpha +k_{{8}}\alpha +k_{{4}}k_{{8}},\quad \xi _{{5}}=k _{{2}}k_{{8}}+k_{{7}}k_{{8}}+\delta _{{4}}k_{{2}},\quad \xi _{{6}}=k_{{4}}k_{{ 8}}+\delta _{{1}}k_{{1}}+k_{{8}}k_{{1}},\\ \xi _{{7}}= & {} \sigma \,k_{{6}}+k_{{6 }}k_{{4}}+\sigma \,k_{{5}},\quad \xi _{{8}}=k_{{7}}\rho +k_{{2}}k_{{6}}+k_{{2}}\sigma . \end{aligned}$$

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Seidu, B., Makinde, O.D. & Bornaa, C.S. Mathematical Analysis of an Industrial HIV/AIDS Model that Incorporates Carefree Attitude Towards Sex. Acta Biotheor (2021). https://doi.org/10.1007/s10441-020-09407-7

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Keywords

  • HIV/AIDS
  • Stability analysis
  • Sensitivity analysis
  • Workforce productivity
  • Optimal control