Computational and Theoretical Analysis of the Association Between Gender and HSV-2 Treatment Adherence

Abstract

Herpes simplex virus type 2 (HSV-2) is the most prevalent sexually transmitted infection in the world, despite the availability of effective anti-viral treatments. A mathematical model to explore the association between gender and HSV-2 treatment adherence is developed. Threshold parameters are determined and stabilities analyzed. Sensitivity analysis of the reproduction number and the numerical simulations suggest that treatment adherence for both females and males are equally important in keeping the reproduction as low as possible. The basic model is then extended to incorporate time-dependent intervention strategies. The Pontryagin’s Maximum Principle is used to characterize the optimal level of the controls, and the resulting optimality system is solved numerically.

Appendices

Appendix 1

Proof of Theorem 6

We employ the Centre Manifold Theory (Theorem 4.1 in Castillo-Chavez and Song (2004)) in order to establish the local asymptotic stability of the endemic equilibrium. Let us make the following change of variables in order to apply the Centre Manifold Theory: $S_{m} = x_{1},~A_{m}= x_{2},~H_{m}=x_{3},~Q_{m}=x_{4},~S_{f}=x_{5},~A_{f}=x_{6},~H_{f}=x_{7}~\text{ and }~Q_{f}=x_{8}.$ 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 (3) can be written in the form $\dfrac{dX}{dt} = F = (f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},f_{7},f_{8})^{T},$ such that

$$\begin{aligned} \begin{array}{llll} x_{1}' = f_{1} = \Lambda _{m} - \dfrac{\beta _{f}c_{m}(x_{6}(1-\eta _{f}) + x_{7})x_{1}}{x_{5} + x_{6} + x_{7} + x_{8}} - \mu _{m}x_{1}, \\ x_{2}' = f_{2} = \dfrac{\beta _{f}c_{m}(x_{6}(1-\eta _{f}) + x_{7})x_{1}}{x_{5} + x_{6} + x_{7} + x_{8}} + \gamma _{m} x_{4} - (\mu _{m} + \kappa _{m} + \psi _{m} + \delta _{m}(1-\theta _{m}))x_{2}, \\ x_{3}' = f_{3} = \delta _{m}(1-\theta _{m})x_{2} + \gamma ^{0}_{m} x_{4} - (\mu _{m} + \kappa _{m})x_{3}, \\ x_{4}' = f_{4} = (\kappa _{m} + \psi _{m})x_{2} + \kappa _{m}x_{3} - (\gamma _{m} + \mu _{m} + \gamma ^{0}_{m})x_{4}, \\ x_{5}' = f_{5} = \Lambda _{f} - \dfrac{\beta _{m}c_{f}(x_{2}(1-\eta _{m}) + x_{3})x_{5}}{x_{1} + x_{2} + x_{3} + x_{4}} - \mu _{f}x_{5}, \\ x_{6}' = f_{6} = \dfrac{\beta _{m}c_{f}(x_{2}(1-\eta _{m}) + x_{3})x_{5}}{x_{1} + x_{2} + x_{3} + x_{4}} + \gamma _{f} x_{8} - (\mu _{f} + \kappa _{f} + \psi _{f} + \delta _{f}(1-\theta _{f}))x_{6}, \\ x_{7}' = f_{7} = \delta _{f}(1-\theta _{f})x_{6} + \gamma ^{0}_{f} x_{8} - (\mu _{f} + \kappa _{f})x_{7}, \\ x_{8}' = f_{8} = (\kappa _{f} + \psi _{f})x_{6} + \kappa _{f}x_{7} - (\gamma _{f} + \mu _{f} + \gamma ^{0}_{f})x_{8}. \\ \end{array} \end{aligned}$$

(38)

The method entails evaluating the Jacobian of system (38) at $\mathcal {E}^{0} = (S_{m}^{0},A_{m}^{0},H_{m}^{0},Q_{m}^{0},S_{f}^{0},A_{f}^{0},H_{f}^{0},Q_{f}^{0}),$ with $S_{m}^{0}=x_{1}^{0},~A^{0}_{m}=x_{2}^{0},~H^{0}_{m}=x_{3}^{0},~ Q^{0}_{m}=x_{4}^{0},~S^{0}_{f}=x_{5}^{0},~A^{0}_{f} =x_{6}^{0},~H^{0}_{f}=x_{7}^{0}~\text{ and }~Q_{f}^{0}=x_{8}^{0}.$ Thus,

$$\begin{aligned} J(\mathcal {E}^{0})=\begin{bmatrix} -\mu _{m} &{} 0 &{} 0 &{} 0 &{} 0 &{} -\dfrac{\beta _{f}(1-\eta _{f})\Lambda _{m} \mu _{f}}{\mu _{m} \Lambda _{f}} &{} -\dfrac{\beta _{f} \Lambda _{m} \mu _{f}}{\mu _{m} \Lambda _{f}} &{} 0 \\ 0 &{} -k_{1} &{} 0 &{} \gamma _{m} &{} 0 &{} \dfrac{\beta _{f}(1-\eta _{f})\Lambda _{m} \mu _{f}}{\mu _{m} \Lambda _{f}} &{} \dfrac{\beta _{f} \Lambda _{m} \mu _{f}}{\mu _{m} \Lambda _{f}} &{} 0 \\ 0 &{} \delta _{m}(1-\theta _{m}) &{} -k_{2} &{} \gamma _{m}^{0} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} (\mu _{m} + \kappa _{m}) &{} \kappa _{m} &{} -k_{3} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\dfrac{\beta _{m}(1-\eta _{m})\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}} &{} -\dfrac{\beta _{m}\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}} &{} 0 &{} -\mu _{f} &{} 0 &{} 0 &{} 0 \\ 0 &{} \dfrac{\beta _{m}(1-\eta _{m})\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}} &{} \dfrac{\beta _{m}\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}} &{} 0 &{} 0 &{} -k_{4} &{} 0 &{} \gamma _{f} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \delta _{f}(1-\theta _{f}) &{} -k_{5} &{} \gamma ^{0}_{f} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \kappa _{f} + \mu _{f} &{} \kappa _{f} &{} -k_{6} \\ \end{bmatrix}, \end{aligned}$$

(39)

from which it can be shown that

$$\begin{aligned} R_{mf} = \sqrt{R_{m}R_{f}}, \end{aligned}$$

(40)

where

$$\begin{aligned} R_{m} = \dfrac{\beta _{m} \mu _{m} c_{f} \Lambda _{f} [(1-\eta _{m})(\mu _{m} \gamma _{m}^{0} + k_{2}(\mu _{m} + \gamma _{m})) + \gamma ^{0}_{m}(\kappa _{m} + \psi _{m}) + \delta _{m}(1-\theta _{m})k_{3}]}{\Lambda _{m} \mu _{f} [\mu _{m} k_{1} k_{2}k_{3} + \mu _{m} \gamma _{m} k_{2} + \gamma _{m} \kappa _{m} \delta _{m} (1-\theta _{m})]} \end{aligned}$$

(41)

and

$$\begin{aligned} R_{f} = \dfrac{\beta _{f} \mu _{f} c_{m} \Lambda _{m} [(1-\eta _{f})(\mu _{f} \gamma _{f}^{0} + k_{5}(\mu _{f} + \gamma _{f})) + \gamma ^{0}_{f}(\kappa _{f} + \psi _{f}) + \delta _{f}(1-\theta _{f})k_{6}]}{\Lambda _{f} \mu _{m} [\mu _{f} k_{4} k_{5}k_{6} + \mu _{f} \gamma _{f} k_{5} + \gamma _{f} \kappa _{f} \delta _{f} (1-\theta _{f})]}. \end{aligned}$$

(42)

Now let us consider $\beta _{f} = \rho \beta _{m},$ regardless of whether $\rho ~\in ~(0,1)$ or $\rho ~\ge ~1.$ Taking $\beta _{f}$ as a bifurcation parameter and considering the case $R_{mf}=1$ and solving for $\beta _{m},$ we obtain

$$\begin{aligned} \beta _{m} =\beta ^{*} = \sqrt{H_{m}H_{f}}, \end{aligned}$$

(43)

where

$$\begin{aligned} H_{m} = \dfrac{\Lambda _{m} \mu _{f} [\mu _{m} k_{1} (\mu _{m} + \kappa _{m})(\mu _{m} + \kappa _{m} + \gamma _{m}^{0}) + \mu _{m} \gamma _{m} (\mu _{m} + \kappa _{m}) + \gamma _{m} \kappa _{m} \delta _{m} (1-\theta _{m})]}{\mu _{m} c_{f} \Lambda _{f} [(1-\eta _{m})(\mu _{m} \gamma _{m}^{0} + (\mu _{m} + \kappa _{m})(\mu _{m} + \gamma _{m})) + \gamma ^{0}_{m}(\kappa _{m} + \psi _{m}) + \delta _{m}(1-\theta _{m})(\gamma _{m} + \gamma ^{0}_{m} + \mu _{m})]} \end{aligned}$$

(44)

and

$$\begin{aligned} H_{f} = \dfrac{\Lambda _{f} \mu _{m} [\mu _{f} (\mu _{f} + \kappa _{f} + \psi _{f} + \delta _{f} (1-\theta _{f})) (\mu _{f} + \kappa _{f})(\mu _{f} + \kappa _{f} + \gamma _{f}^{0}) + \mu _{f} \gamma _{f} (\mu _{f} + \kappa _{f}) + \gamma _{f} \kappa _{f} \delta _{f} (1-\theta _{f})]}{\rho \mu _{f} c_{m} \Lambda _{m} [(1-\eta _{f})(\mu _{f} \gamma _{f}^{0} + (\mu _{f} + \kappa _{f})(\mu _{f} + \gamma _{f})) + \gamma ^{0}_{f}(\kappa _{f} + \psi _{f}) + \delta _{f}(1-\theta _{f})(\gamma _{f} + \gamma ^{0}_{f} + \mu _{f})]}. \end{aligned}$$

(45)

Note that the linearised system of the transformed equation (38) with the bifurcation point $\beta ^{*}$ has a simple zero eigenvalue. Hence the Centre Manifold Theory (Castillo-Chavez and Song 2004) can be used to analyze the dynamics of system (38) near $\beta _{m} = \beta ^{*}$. It can be shown that the Jacobian of system (38) has a right eigenvector associated with the following zero eigenvalue given by $u=(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6},u_{7},u_{8})^{T},$ where

$$\begin{aligned} \left\{ \begin{array}{llll} u_{1} = -\beta _{f}c_{m}\dfrac{\Lambda _{m} \mu _{f}}{\mu ^{2}_{m}\Lambda _{f}}(u_{6} (1-\eta _{f}) + u_{7})<0,~ u_{2}= \dfrac{[\kappa _{m}(\gamma _{m}+\mu _{m}) + \mu _{m}(\gamma _{m} + \mu _{m} +\gamma ^{0}_{m})]}{\delta _{m}(1-\theta _{m})[k_{2}(\mu _{m} + \kappa _{m}) + \kappa _{m}\delta _{m}(1-\theta _{m})]}u_{4}>0,~~\\ u_{3} = \dfrac{[\gamma _{m}^{0}(\mu _{m} + \kappa _{m}) + k_{3}\delta _{m}(1-\theta _{m})]}{k_{2}(\mu _{m}+\kappa _{m}) + \kappa _{m} \delta _{m} (1-\theta _{m})}u_{4}> 0,~u_{4}>0,~u_{5} = -\beta _{m}c_{f}\dfrac{\Lambda _{f} \mu _{m}}{\mu ^{2}_{f}\Lambda _{m}}(u_{2} (1-\eta _{m}) + u_{3})<0, \\ u_{6} = \dfrac{[\kappa _{f}(\gamma _{f}+\mu _{f}) + \mu _{f}(\gamma _{f} + \mu _{f} +\gamma ^{0}_{f})]}{\delta _{f}(1-\theta _{f})[k_{5}(\mu _{f} + \kappa _{f}) + \kappa _{f}\delta _{f}(1-\theta _{f})]}u_{8}>0,~u_{7} = \dfrac{[\gamma _{f}^{0}(\mu _{f} + \kappa _{f}) + k_{3}\delta _{f}(1-\theta _{f})]}{k_{5}(\mu _{f}+\kappa _{f}) + \kappa _{f} \delta _{f} (1-\theta _{f})}u_{8}> 0, \\ u_{8} > 0. \end{array} \right. \end{aligned}$$

(46)

The left eigenvector of $J(\mathcal {E}^{0})$ associated with the zero eigenvalue at $\beta _{m} = \beta ^{*}$ is given by

$v=(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7},v_{8})^{T},$ where

$$\begin{aligned} \left\{ \begin{array}{llll} v_{1} = v_{5} = 0,~v_{3} = \dfrac{\kappa _{m} v_{4} + \mathcal {H}_{1} v_{6}}{k_{2}}~>0,~v_{4}> 0, \\ v_{2} = \bigg (\dfrac{\delta _{m}(1-\theta _{m})\kappa _{m}}{k_{1}k_{2}} + \dfrac{\mu _{m} + \kappa _{m}}{k_{1}} \bigg )v_{4} + \bigg (\dfrac{\delta _{m}(1-\theta _{m})\mathcal {H}_{2}}{k_{1}k_{2}} + \dfrac{\mathcal {H}_{1}}{k_{1}} \bigg )v_{6}>0,~v_{6}> 0 \\ v_{7} = \bigg ( \dfrac{\delta _{m}(1-\theta _{m})\kappa _{m}}{k_{1}k_{2}} + \dfrac{\mu _{m} + \kappa _{m}}{k_{1}} \bigg )\dfrac{k_{2} \mathcal {H}_{3}v_{4}}{M}+ \bigg ( \gamma _{f} \kappa _{f} + \dfrac{k_{2} \mathcal {H}_{1} \mathcal {H}_{3}}{k_{1}} + \dfrac{k_{2}\delta _{m}(1-\theta _{m})\mathcal {H}_{2}\mathcal {H}_{3}}{k_{1}k_{2}}\bigg ) \dfrac{v_{6}}{M}> 0, \\ v_{8} = \bigg (\dfrac{k_{2}\delta _{m}(1-\theta _{m})k_{m}\mathcal {H}_{3}}{k_{1}k_{2}} + \dfrac{k_{2}(\mu _{m}+\kappa _{m})\mathcal {H}_{3}}{k_{1}} \bigg )\dfrac{\gamma _{f}^{0}v_{4}}{M} \\ ~~~+ \bigg (\dfrac{\gamma _{f} M}{\gamma ^{0}_{f}} + \gamma _{f}\kappa _{f} + \dfrac{\mathcal {H}_{1}\mathcal {H}_{3}}{k_{1}} + \dfrac{k_{2}\delta _{m}(1-\theta _{m}) \mathcal {H}_{2}\mathcal {H}_{3}}{k_{1}k_{2}} \bigg ) \dfrac{\gamma _{f}^{0}v_{6}}{M} > 0, \end{array} \right. \end{aligned}$$

(47)

with

$$\begin{aligned} M = \kappa _{f} (\gamma _{f} + \mu _{f}) + \mu _{f}k_{6},~ \mathcal {H}_{1} = \dfrac{\beta _{m}(1-\eta _{m})\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}},~ \mathcal {H}_{2} = \dfrac{\beta _{m}\Lambda _{f}\mu _{m}}{\mu _{f} \Lambda _{m}}, ~ \mathcal {H}_{3} = \dfrac{\beta _{f}(1-\eta _{f})\Lambda _{m}\mu _{f}}{\mu _{m} \Lambda _{f}}, \mathcal {H}_{4} = \dfrac{\beta _{f}(1-\eta _{f})\Lambda _{m}\mu _{f}}{\mu _{m} \Lambda _{f}}. \end{aligned}$$

Then we use Theorem (4.1) from Castillo-Chavez and Song (2004), stated below for elucidation.

Theorem 8

Consider the following general system of ordinary differential equations with a parameter $\phi ,$

$$\begin{aligned} \displaystyle \frac{dx}{dt}=f(x,\phi ),~\text{ with }~f:\mathbb {R}^{n} \times \mathbb {R} \rightarrow \mathbb {R} ~ and ~ f \in ~C^{2}~ ~( \mathbb {R}^{n} \times \mathbb {R}), \end{aligned}$$

(48)

where 0 is an equilibrium of the system, that is $f(0,\phi )=0 ~ ~\forall \phi$, and assume

• A1) $A=D_{x}f(0,0)=\bigg (\displaystyle \frac{\partial f_{i}}{\partial x_{j}}(0,0)\bigg )$ is the linearisation matrix of system (3) around the equilibrium 0 and $\phi$ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.

• A2) Matrix A has a right eigenvector u and a left eigenvector v corresponding to the zero eigenvalue. Let $f_{k}$ be the $k^{th}$ component of f and

  $$\begin{aligned} a= & {} \sum _{k,i,j=1}^n v_{k} u_{i} u_{j} \displaystyle \frac{\partial ^{2}f_{k}}{\partial x_{i} \partial x_{j}}(0,0), \nonumber \\ b= & {} \sum _{k,i=1}^n v_{k} u_{i} \displaystyle \frac{\partial ^{2}f_{k}}{\partial x_{i} \partial \phi }(0,0). \end{aligned}$$

  (49)

The local dynamics of (48) around zero is totally governed by a and b.

• i.  $a>0,~ b>0$. When $\phi < 0$ with $\mid \phi \mid<< 1$, 0 is locally asymptotically stable, there exists a positive unstable equilibrium. When $0< \phi<< 1$, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium;

• ii.   $a<0,~b<0$. When $\phi <0$ with $\mid \phi \mid<< 1$, 0 is unstable; when $0< \phi<< 1$, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium;

• iii. $a>0,~b<0$. When $\phi <0$ with $\mid \phi \mid<< 1$, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when $0< \phi<< 1$, 0 is stable, and a positive unstable equilibrium exists;

• iv. $a<0,~b>0$. When $\phi$ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

1.1 Computation of the Bifurcation Parameters a and b

For model system (38), the associated non-zero partial derivatives of F at the infection-free equilibrium are given by:

$$\begin{aligned} \begin{array}{llll} \dfrac{\partial ^{2} f_{2}}{\partial x_{1}\partial x_{6}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{6}\partial x_{1}} = -\beta _{f}c_{m}(1-\eta _{f})\dfrac{\mu _{f}}{\Lambda _{f}},~~\dfrac{\partial ^{2} f_{2}}{\partial x_{5}\partial x_{6}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{6}\partial x_{5}} = -\dfrac{\beta _{f} c_{m} (1-\eta _{f}) \Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}},\\ \dfrac{\partial ^{2} f_{2}}{\partial x_{6}^{2}} = \dfrac{-2\beta _{f}c_{m}(1-\eta _{f}) \Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}},~\dfrac{\partial ^{2} f_{2}}{\partial x_{6} \partial x_{7}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{7} \partial x_{6}} = -\dfrac{\beta _{f} c_{m} \Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}} - \dfrac{\beta _{f} c_{m} (1-\eta _{f})\Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}} \\ \dfrac{\partial ^{2} f_{2}}{\partial x_{6} \partial x_{8}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{8} \partial x_{6}} = - \dfrac{\beta _{f} c_{m} (1-\eta _{f})\Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}},~\dfrac{\partial ^{2} f_{2}}{\partial x_{1}\partial x_{7}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{7}\partial x_{1}} = -\beta _{f}c_{m}\dfrac{\mu _{f}}{\Lambda _{f}}, \\ \dfrac{\partial ^{2} f_{2}}{\partial x_{5} \partial x_{7}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{7} \partial x_{5}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{7} \partial x_{8}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{8} \partial x_{7}} = -\dfrac{\beta _{f} c_{m} \Lambda _{m} \mu _{f}^{2}}{\mu _{m} \Lambda _{f}^{2}} \\ \dfrac{\partial ^{2} f_{2}}{\partial x_{7}^{2}} = \dfrac{ -2 \beta _{f}c_{m}\mu _{f}^{2} \Lambda _{m}}{\Lambda _{f}^{2} \mu _{m}},~ \dfrac{\partial ^{2} f_{2}}{\partial x_{6}\partial x_{7}} = \dfrac{\partial ^{2} f_{2}}{\partial x_{7}\partial x_{6}} = -\dfrac{\beta _{f}c_{m}\mu _{f}^{2} \Lambda _{m}}{\Lambda _{f}^{2} \mu _{m}} - \dfrac{\beta _{f}c_{m}(1-\eta _{f})\mu _{f}^{2} \Lambda _{m}}{\Lambda _{f}^{2} \mu _{m}} \\ \dfrac{\partial ^{2} f_{6}}{\partial x_{1}\partial x_{2}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{2}\partial x_{1}} = -\dfrac{\beta _{m}c_{f}(1-\eta _{m})\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}},~~\dfrac{\partial ^{2} f_{6}}{\partial x_{1}\partial x_{3}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{3}\partial x_{1}} = -\dfrac{\beta _{m}c_{f} \Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}}, \\ \dfrac{\partial ^{2} f_{6}}{\partial x_{2}\partial x_{3}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{3}\partial x_{2}} = -\dfrac{\beta _{m}c_{f}(1-\eta _{m})\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}} - \dfrac{\beta _{m}c_{f}\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}},~\dfrac{\partial ^{2} f_{6}}{\partial x_{2}\partial x_{4}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{4}\partial x_{2}} = -\dfrac{\beta _{m}c_{f}(1-\eta _{m})\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}}, \\ \dfrac{\partial ^{2} f_{6}}{\partial x_{2}\partial x_{5}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{5}\partial x_{2}} = -\dfrac{\beta _{m}c_{f}(1-\eta _{m}) \mu _{m}}{\Lambda _{m}},~\dfrac{\partial ^{2} f_{6}}{\partial x_{3}\partial x_{4}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{4}\partial x_{3}} = -\dfrac{\beta _{m}c_{f}\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}}, \\ \dfrac{\partial ^{2} f_{6}}{\partial x_{3}\partial x_{5}} = \dfrac{\partial ^{2} f_{6}}{\partial x_{5}\partial x_{3}} = -\dfrac{\beta _{m}c_{f}\mu ^{2}}{ \Lambda _{m}},~\dfrac{\partial ^{2} f_{6}}{\partial x_{3}^{2}} = - \dfrac{2 \beta _{m} c_{f}\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}},~\dfrac{\partial ^{2} f_{6}}{\partial x_{2}^{2}} = - \dfrac{2 \beta _{m} c_{f} (1-\eta _{m})\Lambda _{f} \mu ^{2}_{m}}{\mu _{f} \Lambda _{m}^{2}}. \\ \end{array} \end{aligned}$$

(50)

From (50) it follows that

$$\begin{aligned} a= & {} -2\beta _{m}c_{f}(1-\eta _{m})\dfrac{\mu _{m}^{2} \Lambda _{f} v_{6}}{\Lambda _{m}^{2}\mu _{f}}(u_{2}(1-\eta _{m}) + u_{3})(u_{1}+u_{2} + u_{3} + u_{4}) + 2\beta _{m}c_{f}v_{2}u_{5}\dfrac{\mu _{m}^{2}}{\Lambda _{m}^{2}}(u_{2}(1-\eta _{m}) + u_{3}) \nonumber \\&-2\beta _{f}c_{m}(1-\eta _{f})\dfrac{\mu _{f}^{6} \Lambda _{m} v_{2}}{\Lambda _{f}^{2}\mu _{m}}(u_{6}(1-\eta _{f}) + u_{7})(u_{5}+u_{6} + u_{6} + u_{8}) + 2\beta _{f}c_{m}v_{6}u_{1}\dfrac{\mu _{f}^{2}}{\Lambda _{f}^{2}}(u_{6}(1-\eta _{f}) + u_{7}) \nonumber \\&< 0.~~ \end{aligned}$$

(51)

The sign of b is associated with the following non-vanishing partial derivatives of F : 

$$\begin{aligned} \dfrac{\partial ^{2} f_{6}}{\partial x_{2}\partial {\beta ^{*}}} = \dfrac{c_{f} (1-\eta _{m}) \mu _{m} \Lambda _{f}}{\Lambda _{m} \mu _{f}},~\dfrac{\partial ^{2} f_{6}}{\partial x_{3}\partial {\beta ^{*}}} = \dfrac{c_{f} \mu _{m} \Lambda _{f}}{\Lambda _{m} \mu _{f}},~\dfrac{\partial ^{2} f_{2}}{\partial x_{6}\partial {\beta ^{*}}} = \dfrac{\rho c_{m} (1-\eta _{f}) \mu _{f} \Lambda _{m}}{\Lambda _{f} \mu _{m}},~\dfrac{\partial ^{2} f_{2}}{\partial x_{7}\partial {\beta ^{*}}} = \dfrac{\rho c_{m} \mu _{f} \Lambda _{m}}{\Lambda _{f} \mu _{m}}. \end{aligned}$$

(52)

It follows from the expressions in (52) that

$$\begin{aligned} b = \dfrac{\Lambda _{f} \mu _{m}}{\mu _{f} \Lambda _{m}} c_{m} v_{2}(u_{2}(1-\eta _{m}) + u_{3}) + \dfrac{\Lambda _{m} \mu _{f}}{\mu _{m} \Lambda _{f}} c_{f} v_{6}(u_{6}(1-\eta _{f}) + u_{7}) > 0. \end{aligned}$$

(53)

Thus, $b > 0$ and $a < 0$, and using Theorem (4.1) item (iv) from Castillo-Chavez and Song (2004) we establish Theorem 6.