Assessing the Potential Impact of Hormonal-Based Contraceptives on HIV Transmission Dynamics Among Heterosexuals

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.

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 

$$\begin{aligned} \left\{ \begin{array}{l} \lambda ^*_{f_n}=\displaystyle \frac{c_f\beta _m\lambda ^*_m\rho _{f_nm}}{({{\mathcal {K}}}_1+\lambda ^*_m)},\\ \lambda ^*_{f_c}=\displaystyle \frac{(1-p_{c})c_f\beta _m\lambda ^*_m\rho _{f_cm}}{({{\mathcal {K}}}_1+\lambda ^*_m)},\\ \lambda ^*_{f_h}=\displaystyle \frac{\chi c_f\beta _m\lambda ^*_m\rho _{f_hm}}{({{\mathcal {K}}}_1+\lambda ^*_m)}. \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{l} {{\mathcal {A}}}_1={{\mathcal {K}}}_3{{\mathcal {K}}}_5{{\mathcal {K}}}_8\delta _2+{{\mathcal {K}}}_4({{\mathcal {K}}}_5{{\mathcal {K}}}_6\delta _1+{{\mathcal {K}}}_3{{\mathcal {K}}}_7\delta _3),\\ {{\mathcal {A}}}_2={{\mathcal {K}}}_1({{\mathcal {K}}}_2({{\mathcal {K}}}_4{{\mathcal {K}}}_6\delta _1+{{\mathcal {K}}}_5{{\mathcal {K}}}_6\delta _1+{{\mathcal {K}}}_3{{\mathcal {K}}}_8\delta _2+{{\mathcal {K}}}_5{{\mathcal {K}}}_8\delta _2+({{\mathcal {K}}}_3+{{\mathcal {K}}}_4){{\mathcal {K}}}_7\delta _3)\\ \qquad \quad +\,{{\mathcal {K}}}_1({{\mathcal {K}}}_4{{\mathcal {K}}}_5({{\mathcal {R}}}_h-{{\mathcal {R}}}_H^2+{{\mathcal {R}}}_n+\delta _1)+{{\mathcal {K}}}_3({{\mathcal {K}}}_5({{\mathcal {R}}}_c+{{\mathcal {R}}}_h-{{\mathcal {R}}}_H^2+ \delta _2)\\ \qquad \quad +\,{{\mathcal {K}}}_4({{\mathcal {R}}}_c-{{\mathcal {R}}}_H^2+{{\mathcal {R}}}_n+\delta _3)))),\\ {{\mathcal {A}}}_3={{\mathcal {K}}}_1^2{{\mathcal {K}}}_2({{\mathcal {K}}}_1({{\mathcal {K}}}_3(1+{{\mathcal {R}}}_c-{{\mathcal {R}}}_H^2-\delta _1)-{{\mathcal {K}}}_4(-1+{{\mathcal {R}}}_H^2-{{\mathcal {R}}}_n+\delta _2)\\ \qquad \quad +\,{{\mathcal {K}}}_5(1+{{\mathcal {R}}}_h-{{\mathcal {R}}}_H^2-\delta _3)\\ \qquad \quad +\,{{\mathcal {K}}}_2({{\mathcal {K}}}_6\delta _1+{{\mathcal {K}}}_8\delta _2+{{\mathcal {K}}}_7\delta _3))),\\ {{\mathcal {A}}}_3={{\mathcal {K}}}_1^4{{\mathcal {K}}}_2^2({{\mathcal {R}}}_H^2-1), \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} {{\mathcal {K}}}_3=({{\mathcal {K}}}_2+(1-p_{c})c_f\beta _m\rho _{f_cm}),{{\mathcal {K}}}_4=({{\mathcal {K}}}_2+c_f\beta _m\rho _{f_nm}),{{\mathcal {K}}}_5=({{\mathcal {K}}}_2+\chi c_f\beta _m\rho _{f_hm}),\\ {{\mathcal {K}}}_6=({{\mathcal {K}}}_1+(1-p_{c})c_f\beta _m\rho _{f_cm}),{{\mathcal {K}}}_7=({{\mathcal {K}}}_1+\chi c_f\beta _m\rho _{f_hm}),{{\mathcal {K}}}_8=({{\mathcal {K}}}_1+c_f\beta _m\rho _{f_nm}),\\ {{\mathcal {R}}}_n=\displaystyle \frac{\delta _2c_f\beta _m\rho _{f_nm} c_m\beta _f}{{{\mathcal {K}}}_1^2}, {{\mathcal {R}}}_c=\displaystyle \frac{\delta _1(-1+p_{c})^2c_fc_m\beta _f\beta _m\rho _{f_cm}^2}{{{\mathcal {K}}}_1^2}, {{\mathcal {R}}}_h=\displaystyle \frac{\delta _3\theta \chi c_fc_m\beta _f\beta _m\rho _{f_hm}^2}{{{\mathcal {K}}}_1^2}.\end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \left. \begin{array}{l} \delta _1\phi \Lambda =\lambda ^*_{f_{c}}S^*_{f_c}+{{\mathcal {K}}}_2S^*_{f_c},\\ \delta _2\phi \Lambda =\lambda ^*_{f_{n}}S^*_{f_n}+{{\mathcal {K}}}_2S^*_{f_n},\\ \delta _3\phi \Lambda =\lambda ^*_{f_{h}}S^*_{f_h}+{{\mathcal {K}}}_2S^*_{f_h},\\ (1-\phi )\Lambda =\lambda ^*_{m}S^*_{m}+{{\mathcal {K}}}_2S^*_{m},\\ \end{array} \right\} \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} {{\mathcal {K}}}_1I^*_{f_c}=\lambda ^*_{f_{c}}S^*_{f_c},{{\mathcal {K}}}_1I^*_{f_n}=\lambda ^*_{f_{n}}S^*_{f_n},\\ {{\mathcal {K}}}_1I^*_{f_h}=\lambda ^*_{f_{h}}S^*_{f_h},{{\mathcal {K}}}_1I^*_{m}=\lambda ^*_{m}S^*_{m}, \end{array} \right. \end{aligned}$$

Consider the candidate Lyapunov function V, such that

$$\begin{aligned} V= & {} \displaystyle S_{f_c}-S^*_{f_c}-S^*_{f_c}\ln {\frac{S_{f_c}}{S^*_{f_c}}}+S_{f_n}-S^*_{f_n}\nonumber \\&-S^*_{f_n}\ln {\frac{S_{f_n}}{S^*_{f_n}}}+S_{f_h}-S^*_{f_h}-S^*_{f_h}\ln {\frac{S_{f_h}}{S^*_{f_h}}}+S_{m}-S^*_{m}-S^*_{m}\ln {\frac{S_m}{S^*_m}}\nonumber \\&+\displaystyle I_{f_c}-I^*_{f_c}-I^*_{f_c}\ln {\frac{I_{f_c}}{I^*_{f_c}}}+I_{f_n}-I^*_{f_n}-I^*_{f_n}\ln {\frac{I_{f_n}}{I^*_{f_n}}}+I_{f_h}-I^*_{f_h}\nonumber \\&-I^*_{f_h}\ln {\frac{I_{f_h}}{I^*_{f_h}}}+I_{m}-I^*_{m}-I^*_{m}\ln {\frac{I_m}{I^*_m}}. \end{aligned}$$

(22)

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

$$\begin{aligned} \left\{ \begin{array}{l} V(S_{f_c},S_{f_n},S_{f_h},S_{m},I_{f_c},I_{f_n},I_{f_h},I_{m})>0, S_{f_c}>S^*_{f_c},S_{f_n}>S^*_{f_n},S_{f_h}>S^*_{f_h},\\ \quad S_{m}>S^*_{m},I_{f_c}>I^*_{f_c},\\ I_{f_n}>I^*_{f_n},I_{f_h}I^*_{f_h},I_{m}>I^*_{m},\\ V(S^*_{f_c},S^*_{f_n},S^*_{f_h},S^*_{m},I^*_{f_c},I^*_{f_n},I^*_{f_h},I^*_{m})=0 \end{array} \right. \end{aligned}$$

What is left is to prove that

$$\begin{aligned} {\dot{V}}<0. \end{aligned}$$

The time derivative of V along the solution path is given by

$$\begin{aligned} {\dot{V}}= & {} \displaystyle \dot{S_{f_c}}\left[ 1-\displaystyle \frac{S^*_{f_c}}{S_{f_c}}\right] +\dot{S_{f_n}}\left[ 1-\displaystyle \frac{S^*_{f_n}}{S_{f_n}}\right] +\dot{S_{f_h}}\left[ 1-\displaystyle \frac{S^*_{f_h}}{S_{f_h}}\right] +\dot{S_{m}}\left[ 1-\displaystyle \frac{S^*_{m}}{S_{m}}\right] \nonumber \\&+\dot{I_{f_c}}\left[ 1-\displaystyle \frac{I^*_{f_c}}{I_{f_c}}\right] +\displaystyle \dot{I_{f_n}}\left[ 1-\displaystyle \frac{I^*_{f_n}}{I_{f_n}}\right] +\dot{I_{f_h}}\left[ 1-\displaystyle \frac{I^*_{f_h}}{I_{f_h}}\right] +\dot{I_{m}}\left[ 1-\displaystyle \frac{I^*_{m}}{I_{m}}\right] . \qquad \end{aligned}$$

(23)

Substituting equilibrium values and rewriting expressions for the derivatives of the state variables, we obtain the following system

$$\begin{aligned} \left. \begin{array}{l} \dot{S_{f_c}}=\lambda ^*_{f_c}S^*_{f_c}+{{\mathcal {K}}}_2S^*_{f_c}-\lambda ^*_{f_c}S^*_{f_c}\displaystyle \frac{\lambda _{f_c}S_{f_c}}{\lambda ^*_{f_c}S^*_{f_c}}-{{\mathcal {K}}}_2S^*_{f_c}\displaystyle \frac{S_{f_c}}{S^*_{f_c}},\\ \dot{I_{f_c}}=\lambda ^*_{f_c}S^*_{f_c}\displaystyle \frac{\lambda _{f_c}S_{f_c}}{\lambda ^*_{f_c}S^*_{f_c}}-{{\mathcal {K}}}_1I^*_{f_c}\displaystyle \frac{I_{f_c}}{I^*_{f_c}},\\ \dot{S_{f_n}}=\lambda ^*_{f_n}S^*_{f_n}+{{\mathcal {K}}}_2S^*_{f_n}-\lambda ^*_{f_n}S^*_{f_n}\displaystyle \frac{\lambda _{f_n}S_{f_n}}{\lambda ^*_{f_n}S^*_{f_n}}-{{\mathcal {K}}}_2S^*_{f_n}\displaystyle \frac{S_{f_n}}{S^*_{f_n}},\\ \dot{I_{f_n}}=\lambda ^*_{f_n}S^*_{f_n}\displaystyle \frac{\lambda _{f_n}S_{f_n}}{\lambda ^*_{f_n}S^*_{f_n}}-{{\mathcal {K}}}_1I^*_{f_n}\displaystyle \frac{I_{f_n}}{I^*_{f_n}},\\ \dot{S_{f_h}}=\lambda ^*_{f_h}S^*_{f_h}+{{\mathcal {K}}}_2S^*_{f_h}-\lambda ^*_{f_h}S^*_{f_h}\displaystyle \frac{\lambda _{f_h}S_{f_h}}{\lambda ^*_{f_h}S^*_{f_h}}-{{\mathcal {K}}}_2S^*_{f_h}\displaystyle \frac{S_{f_h}}{S^*_{f_h}},\\ \dot{I_{f_h}}=\lambda ^*_{f_h}S^*_{f_h}\displaystyle \frac{\lambda _{f_h}S_{f_h}}{\lambda ^*_{f_h}S^*_{f_h}}-{{\mathcal {K}}}_1I^*_{f_h}\displaystyle \frac{I_{f_h}}{I^*_{f_h}},\\ \dot{S_m}=\lambda ^*_{m}S^*_{m}+{{\mathcal {K}}}_2S^*_{m}-\lambda ^*_{m}S^*_{m}\displaystyle \frac{\lambda _{m}S_{m}}{\lambda ^*_{m}S^*_{m}}-{{\mathcal {K}}}_2S^*_{m}\frac{S_{m}}{S^*_{m}},\\ \dot{I_{m}}=\lambda ^*_{m}S^*_{m}\displaystyle \frac{\lambda _{m}S_{m}}{\lambda ^*_{m}S^*_{m}}-{{\mathcal {K}}}_1I^*_{m}\frac{I_{m}}{I^*_{m}}. \end{array} \right\} \end{aligned}$$

(24)

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

$$\begin{aligned} {\dot{V}}= \left\{ \begin{array}{l} \displaystyle \lambda ^*_{f_c}S^*_{f_c}+{{\mathcal {K}}}_2S^*_{f_c}-\lambda ^*_{f_c}S^*_{f_c}\frac{\lambda _{f_c}S_{f_c}}{\lambda ^*_{f_c}S^*_{f_c}}-{{\mathcal {K}}}_2S^*_{f_c}\frac{S_{f_c}}{S^*_{f_c}}-\lambda ^*_{f_c}S^*_{f_c}\frac{S^*_{f_c}}{S_{f_c}}-{{\mathcal {K}}}_2S^*_{f_c}\frac{S^*_{f_c}}{S_{f_c}}\\ \quad +\lambda ^*_{f_c}S^*_{f_c}\frac{\lambda _{f_c}}{\lambda ^*_{f_c}}+{{\mathcal {K}}}_2S^*_{f_c}\\ +\displaystyle \lambda ^*_{f_c}S^*_{f_c}\frac{\lambda _{f_c}S_{f_c}}{\lambda ^*_{f_c}S^*_{f_c}}-\lambda ^*_{f_c}S^*_{f_c}\frac{I_{f_c}}{I^*_{f_c}}-\lambda ^*_{f_c}S^*_{f_c}\frac{\lambda _{f_c}S_{f_c}I^*_{f_c}}{\lambda ^*_{f_c}S^*_{f_c}I_{f_c}}+\lambda ^*_{f_c}S^*_{f_c}\\ +\displaystyle \lambda ^*_{f_n}S^*_{f_n}+{{\mathcal {K}}}_2S^*_{f_n}-\lambda ^*_{f_n}S^*_{f_n}\frac{\lambda _{f_n}S_{f_n}}{\lambda ^*_{f_n}S^*_{f_n}}-{{\mathcal {K}}}_2S^*_{f_n}\frac{S_{f_n}}{S^*_{f_n}}-\lambda ^*_{f_n}S^*_{f_n}\frac{S^*_{f_n}}{S_{f_n}}-{{\mathcal {K}}}_2S^*_{f_n}\frac{S^*_{f_n}}{S_{f_n}}\\ \quad +\lambda ^*_{f_n}S^*_{f_n}\frac{\lambda _{f_n}}{\lambda ^*_{f_n}}+{{\mathcal {K}}}_2S^*_{f_n}\\ +\displaystyle \lambda ^*_{f_n}S^*_{f_n}\frac{\lambda _{f_n}S_{f_n}}{\lambda ^*_{f_n}S^*_{f_n}}-\lambda ^*_{f_n}S^*_{f_n}\frac{I_{f_n}}{I^*_{f_n}}-\lambda ^*_{f_n}S^*_{f_n}\frac{\lambda _{f_n}S_{f_n}I^*_{f_n}}{\lambda ^*_{f_n}S^*_{f_n}I_{f_n}}+\lambda ^*_{f_n}S^*_{f_n}\\ +\displaystyle \lambda ^*_{f_h}S^*_{f_h}+{{\mathcal {K}}}_2S^*_{f_h}-\lambda ^*_{f_h}S^*_{f_h}\frac{\lambda _{f_h}S_{f_h}}{\lambda ^*_{f_h}S^*_{f_h}}-{{\mathcal {K}}}_2S^*_{f_h}\frac{S_{f_h}}{S^*_{f_h}}-\lambda ^*_{f_h}S^*_{f_h}\frac{S^*_{f_h}}{S_{f_h}}-{{\mathcal {K}}}_2S^*_{f_h}\frac{S^*_{f_h}}{S_{f_h}}\\ \quad +\lambda ^*_{f_h}S^*_{f_h}\frac{\lambda _{f_h}}{\lambda ^*_{f_h}}+{{\mathcal {K}}}_2S^*_{f_h}\\ +\displaystyle \lambda ^*_{f_h}S^*_{f_h}\frac{\lambda _{f_h}S_{f_h}}{\lambda ^*_{f_h}S^*_{f_h}}-\lambda ^*_{f_h}S^*_{f_h}\frac{I_{f_h}}{I^*_{f_h}}-\lambda ^*_{f_h}S^*_{f_h}\frac{\lambda _{f_h}S_{f_h}I^*_{f_h}}{\lambda ^*_{f_h}S^*_{f_h}I_{f_h}}+\lambda ^*_{f_h}S^*_{f_h}\\ +\displaystyle \lambda ^*_{m}S^*_{m}+{{\mathcal {K}}}_2S^*_{m}-\lambda ^*_{m}S^*_{m}\frac{\lambda _{m}S_{m}}{\lambda ^*_{m}S^*_{m}}-{{\mathcal {K}}}_2S^*_{m}\frac{S_{m}}{S^*_{m}}-\lambda ^*_{m}S^*_{m}\frac{S^*_{m}}{S_{m}}-{{\mathcal {K}}}_2S^*_{m}\frac{S^*_{m}}{S_{m}}\\ \quad +\lambda ^*_{m}S^*_{m}\frac{\lambda _{m}}{\lambda ^*_{m}}+{{\mathcal {K}}}_2S^*_{m}\\ +\displaystyle \lambda ^*_{m}S^*_{m}\frac{\lambda _{m}S_{m}}{\lambda ^*_{m}S^*_{m}}-\lambda ^*_{m}S^*_{m}\frac{I_{m}}{I^*_{m}}-\lambda ^*_{m}S^*_{m}\frac{\lambda _{m}S_{m}I^*_{m}}{\lambda ^*_{m}S^*_{m}I_{m}}+\lambda ^*_{m}S^*_{m}. \end{array} \right. \end{aligned}$$

(25)

Collecting like terms, the derivative reduces to

$$\begin{aligned} {\dot{V}}= \left\{ \begin{array}{l} \displaystyle {{\mathcal {K}}}_2S^*_{f_c}\left[ 2-\frac{S_{f_c}}{S^*_{f_c}}-\frac{S^*_{f_c}}{S_{f_c}}\right] + \lambda ^*_{f_c}S^*_{f_c}\left[ 1-\frac{I_{f_c}}{I^*_{f_c}}-\frac{S^*_{f_c}}{S_{f_c}}\right] +\lambda ^*_{f_c}S^*_{f_c}\frac{\lambda _{f_c}}{\lambda ^*_{f_c}}\left[ 1-\frac{S_{f_c}I^*_{f_c}}{S^*_{f_c}I_{f_c}}\right] \\ +\displaystyle {{\mathcal {K}}}_2S^*_{f_n}\left[ 2-\frac{S_{f_n}}{S^*_{f_n}}-\frac{S^*_{f_n}}{S_{f_n}}\right] + \lambda ^*_{f_n}S^*_{f_n}\left[ 1-\frac{I_{f_n}}{I^*_{f_n}}-\frac{S^*_{f_n}}{S_{f_n}}\right] \\ \quad +\lambda ^*_{f_n}S^*_{f_n}\frac{\lambda _{f_n}}{\lambda ^*_{f_n}}\left[ 1-\frac{S_{f_n}I^*_{f_n}}{S^*_{f_n}I_{f_n}}\right] \\ +\displaystyle {{\mathcal {K}}}_2S^*_{f_h}\left[ 2-\frac{S_{f_h}}{S^*_{f_h}}-\frac{S^*_{f_h}}{S_{f_h}}\right] + \lambda ^*_{f_h}S^*_{f_h}\left[ 1-\frac{I_{f_h}}{I^*_{f_h}}-\frac{S^*_{f_h}}{S_{f_h}}\right] \\ \quad +\lambda ^*_{f_h}S^*_{f_h}\frac{\lambda _{f_h}}{\lambda ^*_{f_h}}\left[ 1-\frac{S_{f_h}I^*_{f_h}}{S^*_{f_h}I_{f_h}}\right] \\ +\displaystyle {{\mathcal {K}}}_2S^*_{m}\left[ 2-\frac{S_m}{S^*_{m}}-\frac{S^*_{m}}{S_{m}}\right] + \lambda ^*_{m}S^*_{m}\left[ 1-\frac{I_{m}}{I^*_{m}}-\frac{S^*_{m}}{S_{m}}\right] +\lambda ^*_{m}S^*_{m}\frac{\lambda _{m}}{\lambda ^*_{m}}\left[ 1-\frac{S_{m}I^*_{m}}{S^*_{m}I_{m}}\right] . \end{array} \right. \end{aligned}$$

(26)

Substituting equilibrium values in Theorem 4,

$$\begin{aligned} {\dot{V}}= \left\{ \begin{array}{l} \displaystyle -{{\mathcal {K}}}_2S^*_{f_c}\frac{\left[ S_{f_c}-S^*_{f_c}\right] ^2}{S^*_{f_c}S_{f_c}}+\lambda ^*_{f_c}S^*_{f_c}\left[ 1-\frac{I_{f_c}}{I^*_{f_c}}-\frac{S^*_{f_c}}{S_{f_c}}\right] \\ -\displaystyle {{\mathcal {K}}}_2S^*_{f_n}\frac{\left[ S_{f_n}-S^*_{f_n}\right] ^2}{S^*_{f_n}S_{f_n}}+\lambda ^*_{f_n}S^*_{f_n}\left[ 1-\frac{I_{f_n}}{I^*_{f_n}}-\frac{S^*_{f_n}}{S_{f_n}}\right] \\ -\displaystyle {{\mathcal {K}}}_2S^*_{f_h}\frac{\left[ S_{f_h}-S^*_{f_h}\right] ^2}{S^*_{f_h}S_{f_h}}+\lambda ^*_{f_h}S^*_{f_h}\left[ 1-\frac{I_{f_h}}{I^*_{f_h}}-\frac{S^*_{f_h}}{S_{f_h}}\right] \\ -\displaystyle {{\mathcal {K}}}_2S^*_{m}\frac{\left[ S_{m}-S^*_{m}\right] ^2}{S^*_{m}S_{m}}+\lambda ^*_{m}S^*_{m}\left[ 1-\frac{I_{m}}{I^*_{m}}-\frac{S^*_{m}}{S_{m}}\right] . \end{array} \right. \end{aligned}$$

(27)

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

$$\begin{aligned} (S^*_{f_c},S^*_{f_n},S^*_{f_h},S^*_{m},I^*_{f_c},I^*_{f_n},I^*_{f_h},I^*_m)>0\in {{\mathcal {D}}} \end{aligned}$$

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. $\square$