Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

Abstract

The aim of this paper is to study the impact of introducing a partially protective vaccine on the dynamics of infection in SIRS models where primary and secondary infections are distinguished. We investigate whether a public health strategy based solely on vaccinating a proportion of newborns can lead to an effective control of the disease. In addition to carrying out the qualitative analysis, the findings are further explained by numerical simulations. The model exhibits backward bifurcation for certain values of the parameters. In these cases the standard basic reproduction number (obtained by inspection of the uninfected state) is not significant. The key threshold is the reinfection level which depends on the relative transmissibility (susceptibility) of secondary, with respect to primary, infected (susceptible) individuals and the relative loss of immunity of vaccinated, with respect to recovered, individuals. If one or all of these ratios decrease, then the threshold increases which increases the possibility to contain the infection by vaccination. The analysis shows further that symptomatic infections can be eliminated by vaccination solely.

Appendix

Proof of Proposition 5

We write the quadratic equation (26) as

$$\begin{aligned} \kappa ^2 \mathcal A - 2\kappa \mathcal B + \mathcal C =0 \end{aligned}$$

(40)

$$\begin{aligned} \mathcal A&= B_1^2 \\ \mathcal B&= g(b\sigma \!+\!\mu )(\alpha \!+\!\mu )\{(1\!-\!p)(b\sigma \!+\!\mu )[g\mu (q\alpha \!+\!\sigma \!+\!\mu )(\mu (q\alpha \!+\!\sigma \!+\!\mu )\!+\!r\alpha \sigma )\\&+(q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma -\mu (q\alpha +\sigma +\mu ))] \\&+rpb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu (q\alpha +\sigma +\mu )] \}, \\ \mathcal C&= \{(\alpha +\mu )(b\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu (q\alpha +\sigma +\mu )]\}^2. \end{aligned}$$

We first define the following quantities

$$\begin{aligned} X_1&= b\sigma +\mu \\ Y_1&= q\alpha +\sigma +\mu \\ Z_1&= r\alpha \sigma + \mu (q\alpha +\sigma +\mu ) = r\alpha \sigma + \mu Y_1. \end{aligned}$$

Now we evaluate

$$\begin{aligned} \mathcal B ^2 -\! \mathcal AC&= g^2 X_1^2(\alpha \!+\!\mu )^2\{(1\!-\!p)^2 X_1^2\{[g\mu Y_1 Z_1 \!+\! (q\alpha \!+\!\mu )(\sigma \!+\!\mu )(Z_1\!-\!2\mu Y_1)]^2 \\&-\, Z_1^2[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]^2\} \\&+\, 2 p(1\!-\!p)rb\sigma X_1 (\alpha \!+\!\mu )(\sigma \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]\{g\mu Y_1 Z_1 \\&+\,(q\alpha +\mu )(\sigma +\mu )(Z_1-2 \mu Y_1) - Z_1[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\}\} \end{aligned}$$

and hence

$$\begin{aligned} \mathcal B ^2 -&\!\mathcal A C \\&= g^2 X_1^2(\alpha \!+\!\mu )^2\{(1\!-\!p)^2 X_1^2\{Z_1^2[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]^2\\&+\, 4 \mu ^2 Y_1^2 (q\alpha \!+\!\mu )^2(\sigma \!+\!\mu )^2 \!-\! 4\mu Y_1 Z_1 (q\alpha \!+\!\mu )(\sigma \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!+\! g\mu Y_1]\\&-\, Z_1^2[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]^2 \}\\&+\, 2 p(1-p)rb\sigma X_1 (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1][2g\mu Y_1 Z_1\\&-\, 2\mu Y_1(q\alpha +\mu )(\sigma +\mu )]\}\\&= 4 g^2 X_1^2(\alpha +\mu )^2\mu Y_1 X_1 (1-p) \{(1-p) X_1(q\alpha +\mu )(\sigma +\mu )[(Z_1 - \mu Y_1)(g Z_1 \\&-\,(q\alpha +\mu )(\sigma +\mu ))] \\&+\, rb\sigma p(\alpha \!+\!\mu )(\sigma \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1][g Z_1 \!-\! (q\alpha \!+\!\mu )(\sigma \!+\!\mu )]\}\\&= 4 g^2 X_1^2(\alpha +\mu )^2(1-p) X_1\mu Y_1(\sigma +\mu )[g Z_1 - (q\alpha +\mu )(\sigma +\mu )] \\&\times \{(1\!-\!p) X_1(q\alpha \!+\!\mu )(Z_1 \!-\! \mu Y_1) \!+\! rb\sigma p(\alpha \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]\} \\&= 4 g^2 X_1^2(\alpha +\mu )^2(1-p) X_1\mu Y_1(\sigma +\mu )[g Z_1 - (q\alpha +\mu )(\sigma +\mu )] \\&\times \{r\sigma \alpha (1-p) X_1(q\alpha +\mu ) + rb\sigma p(\alpha +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\} \\&= 4 g^2 X_1^2(\alpha +\mu )^2\{(1-p) X_1\mu Y_1[g Z_1 - (q\alpha +\mu )(\sigma +\mu )]\} \\&\times \{r\sigma (\sigma \!+\!\mu )[(1\!-\!p) X_1 \alpha (q\alpha \!+\!\mu ) \!+\! b p(\alpha \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]]\}\\&= 4 g^2 X_1^2(\alpha +\mu )^2 H_1(p) H_2(p). \end{aligned}$$

Also,

$$\begin{aligned} \mathcal B&= g(b\sigma \!+\!\mu )(\alpha \!+\!\mu )\{(1\!-\!p)(b\sigma \!+\!\mu )[g\mu (q\alpha \!+\!\sigma \!+\!\mu )(\mu (q\alpha \!+\!\sigma \!+\!\mu )\!+\!r\alpha \sigma )\\&+(q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma -\mu (q\alpha +\sigma +\mu ))] \\&+rpb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu (q\alpha +\sigma +\mu )] \}, \\&= g X_1(\alpha +\mu )\{(1-p) X_1 [g\mu Y_1 Z_1 + X_1(\sigma +\mu )(Z_1 - 2 \mu Y_1)] \\&+rpb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1] \}, \\&= g X_1(\alpha +\mu )\{(1-p)\mu X_1 Y_1[g Z_1 -(q\alpha +\mu )(\sigma +\mu )] \\&+ r\sigma (\sigma +\mu )[(1-p) X_1 \alpha (q\alpha +\mu )+bp(\alpha +\mu )((q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1)]\} \\&= g X_1(\alpha +\mu ) \{H_1(p) + H_2(p)\}. \end{aligned}$$

Recall $\mathcal A =B_1^2$. Hence,

$$\begin{aligned} \kappa _1^\star&= \frac{g X_1(\alpha +\mu ) \{H_1(p) + H_2(p)\} + \sqrt{4 g^2 X_1^2(\alpha +\mu )^2 H_1(p) H_2(p)}}{g^2 \{H_3(p)\}^2} \\&= \frac{X_1(\alpha +\mu )}{g} \times {\left(\frac{\sqrt{H_1(p)} + \sqrt{H_2(p)}}{H_3(p)}\right)}^2. \end{aligned}$$

$\square$

Direction of bifurcation

To get the direction of bifurcation at the uninfected equilibrium, we use the implicit function theorem

$$\begin{aligned} \frac{\partial \lambda }{\partial \kappa }=-\frac{F_\kappa }{F_\lambda }\vert _{\lambda =0, \kappa =\kappa _v}\end{aligned}$$

(41)

We find that $F_\kappa$ at $\lambda =0$ is equal to $C_\kappa = - C_1 <0$. Hence the direction of bifurcation is given by the sign of $F_\lambda$ which is the value of $B$ at $\kappa =\kappa _v$. This expression is (see below), up to a positive factor,

$$\begin{aligned}&(1-p)(q\alpha +\mu )(b\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-rg\alpha \sigma ]\nonumber \\&\quad + prg^2b\sigma \mu (q\alpha +\sigma +\mu )(\alpha +\mu ). \end{aligned}$$

(42)

Proof of formula (42)

At $\kappa = \kappa _v$, then

$$\begin{aligned} B&= B_0 - \kappa _v B_1 \\&= X_1(\alpha +\mu )[g\mu Y_1 + (q\alpha +\mu )(\sigma +\mu )] \\&- \frac{g(q\alpha +\mu )(\alpha +\mu )X_1\{(1-p)X_1(\mu Y_1+r\alpha \sigma )+ r p b \sigma (\alpha +\mu )(\sigma +\mu )\}}{(1-p)(q\alpha +\mu )X_1 + r g p b \sigma (\alpha +\mu )}. \end{aligned}$$

Since $(1-p)(q\alpha +\mu )X_1 + r g p b \sigma (\alpha +\mu )>0$, then the sign of $B$ depends on the sign of

$$\begin{aligned}&[g\mu Y_1 + (q\alpha +\mu )(\sigma +\mu )][{(1-p)(q\alpha +\mu )X_1 + r g p b \sigma (\alpha +\mu )}]\\&\qquad -\,g(q\alpha +\mu )\{(1-p)X_1(\mu Y_1+r\alpha \sigma )+ r p b \sigma (\alpha +\mu )(\sigma +\mu )\} \\&\quad = (1-p)(q\alpha +\mu )X_1[(q\alpha +\mu )(\sigma +\mu )-rg\alpha \sigma ] + prg^2b\sigma \mu Y_1(\alpha +\mu ) \end{aligned}$$

which is formula (42). $\square$

Proof of formula (36)

$$\begin{aligned} \kappa <\kappa _1^\star \Leftrightarrow \kappa ^2 \mathcal A - 2\kappa \mathcal B + \mathcal C < 0 \end{aligned}$$

(43)

where

$$\begin{aligned}&\mathcal A = B_1^2 = g^2 (\phi _1+\phi _2 p)^2, \\&\mathcal B = g X_1(\alpha +\mu )(\eta _1+\eta _2 p), \\&\mathcal C \, {\text{ is}} \ {\text{ independent} \text{ of} } p,\\&\phi _1 = X_1 Z_1, \\&\phi _2 = r b \sigma (\alpha +\mu )(\sigma +\mu ) - X_1 Z_1, \\&\eta _1 = X_1[g\mu Y_1 Z_1 + (q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma -\mu Y_1)], \\&\eta _2 = rb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\\&\quad - X_1[g\mu Y_1 Z_1 + (q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma -\mu Y_1)]. \end{aligned}$$

Inequality (43) is rewritten as

$$\begin{aligned} p^2 \xi _1 + 2 p \xi _2 + \xi _3 < 0 \end{aligned}$$

where

$$\begin{aligned} \xi _1&= \kappa ^2 g^2 \phi _2^2, \\ \xi _2&= \kappa ^2 g^2 \phi _1\phi _2 - \kappa g X_1(\alpha +\mu )\eta _2, \\ \xi _3&= \kappa ^2 g^2\phi _1^2 - 2\kappa g X_1 (\alpha +\mu )\eta _1 + \mathcal C . \end{aligned}$$

Thus $p$ lies between $\frac{-\xi _2 \pm \sqrt{\xi _2^2 - \xi _1\xi _3}}{\xi _1}$ $\square$

i.e., $\frac{-[\kappa ^2 g^2 \phi _1\phi _2 - \kappa g X_1(\alpha +\mu )\eta _2] \pm \sqrt{\xi _2^2 - \xi _1\xi _3}}{\kappa ^2 g^2 \phi _2^2}.$

As the upper root involves $+\sqrt{\xi _2^2 - \xi _1\xi _3}>1$, we need to consider only the lower root. Hence

$$\begin{aligned} p_2^\star = \frac{\kappa g X_1(\alpha +\mu )\eta _2-\kappa ^2 g^2 \phi _1\phi _2 - \sqrt{\xi _2^2 - \xi _1\xi _3}}{\kappa ^2 g^2 \phi _2^2}. \end{aligned}$$

The term under the square root is

$$\begin{aligned} \Delta&= \xi _2^2 - \xi _1\xi _3 \\&= [\kappa ^2 g^2 \phi _1\phi _2 - \kappa g X_1(\alpha +\mu )\eta _2]^2 - {\kappa ^2 g^2 \phi _2^2}{[\kappa ^2 g^2\phi _1^2 - 2\kappa g X_1 (\alpha +\mu )\eta _1 + \mathcal C ]}\\&= g^2 \kappa ^2\left\{ [\kappa g \phi _1\phi _2 - X_1(\alpha +\mu )\eta _2]^2 - {\phi _2^2}{[\kappa ^2 g^2\phi _1^2 - 2\kappa g X_1 (\alpha +\mu )\eta _1 + \mathcal C ]} \right\} \\&= \kappa ^2[G_3^{\prime }+ \kappa G_4^{\prime }]\\&= \kappa ^2 g^2 X_1^2[G_3+ \kappa G_4], \end{aligned}$$

where

$$\begin{aligned} G_3^{\prime }&= g^2 [X_1^2(\alpha +\mu )^2\eta _2^2 - \phi _2^2 \mathcal C ] \\&= g^2 [X_1^2(\alpha +\mu )^2\eta _2^2 - \phi _2^2 X_1^2(\alpha +\mu )^2[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]^2]\\&= g^2 X_1^2(\alpha +\mu )^2\left\{ \eta _2^2 - \phi _2^2 [(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]^2\right\} \\&= g^2 X_1^2 G_3,\\ G_4^{\prime }&= 2 g^3\phi _2X_1(\alpha +\mu )[\phi _2\eta _1 - \phi _1\eta _2]\\&= 4 g^3 X_1^2 Y_1 r b \sigma \mu (\alpha +\mu )^2(\sigma +\mu )\phi _2 [g Z_1 - (q\alpha +\mu )(\sigma +\mu )]\\&= g^2 X_1^2 G_4,\\ G_3&= (\alpha \!+\!\mu )^2\left\{ \eta _2 \!-\! \phi _2[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]\right\} \left\{ \eta _2 \!+\! \phi _2[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1]\right\} \!,\\ G_4&= - 4 r b g\phi _2 \sigma \mu Y_1 (\alpha +\mu )^2(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu ) - gZ_1]. \end{aligned}$$

Note that

$$\begin{aligned} \kappa g X_1(\alpha +\mu )\eta _2-\kappa ^2 g^2 \phi _1\phi _2&= \kappa g X_1\left\{ (\alpha +\mu )\eta _2-\frac{\kappa g\phi _1\phi _2}{X_1}\right\} \\&= \kappa g X_1\left\{ (\alpha +\mu )\eta _2-{\kappa Z_1 g\phi _2}\right\} \\&= \kappa g X_1\left\{ G_1(\alpha +\mu ) + G_2 \kappa \right\} , \end{aligned}$$

where

$$\begin{aligned} G_1&= \eta _2 \\&= rb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\\&- X_1[g\mu Y_1 Z_1 + (q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma -\mu Y_1)]\\&= 2\mu Y_1(\sigma +\mu )[(q\alpha +\mu )X_1 - rgb\sigma (\alpha +\mu )] + \\&[rb\sigma (\alpha +\mu )(\sigma +\mu )-X_1Z_1][g\mu Y_1 +(q\alpha +\mu )(\sigma +\mu )]\\&= 2\mu Y_1(\sigma \!+\!\mu )[(q\alpha \!+\!\mu )X_1 \!-\! rgb\sigma (\alpha \!+\!\mu )] \!+\! \phi _2[g\mu Y_1 \!+\!(q\alpha \!+\!\mu )(\sigma \!+\!\mu )]\\&= 2\mu Y_1(\sigma \!+\!\mu )[(q\alpha \!+\!\mu )X_1 \!-\! rgb\sigma (\alpha \!+\!\mu )] \!-\! \frac{G_2}{g Z_1} [g\mu Y_1 \!+\!(q\alpha \!+\!\mu )(\sigma \!+\!\mu )],\\ G_2&= - g \phi _2 Z_1. \end{aligned}$$

Thus

$$\begin{aligned} p_2^\star&= \frac{\kappa g X_1\left\{ G_1(\alpha +\mu ) + G_2 \kappa - \sqrt{G_3 + \kappa G_4}\right\} }{\kappa ^2 g^2 \phi _2^2}\\&= \frac{ g X_1 Z_1^2 \left\{ G_1(\alpha +\mu ) + G_2 \kappa - \sqrt{G_3 + \kappa G_4}\right\} }{\kappa G_2^2}. \end{aligned}$$

Now, we evaluate $G_3$. We have

$$\begin{aligned}&\eta _2 - \phi _2[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\\&\quad = rb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1] \\&\qquad - X_1 [g\mu Y_1 (r\alpha \sigma + \mu Y_1) + (q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma - \mu Y_1)] \\&\qquad - [rb\sigma (\alpha +\mu )(\sigma +\mu )-X_1(r\alpha \sigma + \mu Y_1)][(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1] \\&\quad = 2X_1\left[\mu Y_1(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1 (r\alpha \sigma + \mu Y_1)\right] \\&\quad = 2\mu X_1 Y_1\left[(q\alpha +\mu )(\sigma +\mu ) - g(r\alpha \sigma + \mu Y_1)\right], \end{aligned}$$

and

$$\begin{aligned}&\eta _2 + \phi _2[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1]\\&\quad = rb\sigma (\alpha +\mu )(\sigma +\mu )[(q\alpha +\mu )(\sigma +\mu )-g\mu Y_1] \\&\qquad - X_1 [g\mu Y_1 (r\alpha \sigma + \mu Y_1) + (q\alpha +\mu )(\sigma +\mu )(r\alpha \sigma - \mu Y_1)] \\&\qquad + [rb\sigma (\alpha +\mu )(\sigma +\mu )-X_1(r\alpha \sigma + \mu Y_1)][(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1] \\&\quad = 2rb\sigma (\alpha \!+\!\mu )(\sigma \!+\!\mu ){\left[(q\alpha \!+\!\mu )(\sigma \!+\!\mu ) \!-\! g\mu Y_1 \right]} \!-\! 2 r \alpha \sigma X_1(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\\&\quad = 2r(\sigma +\mu )\left\{ b\sigma (\alpha +\mu ){\left[(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1 \right]} - \alpha \sigma X_1(q\alpha +\mu )\right\} . \end{aligned}$$

Hence,

$$\begin{aligned} G_3&= 4\mu Y_1(\sigma \!+\!\mu )(\alpha \!+\!\mu )^2\{r X_1[(q\alpha \!+\!\mu )(\sigma \!+\!\mu ) \!-\! g(r\alpha \sigma \!+\! \mu Y_1) ]\}\{\!-\! \alpha \sigma X_1 (q\alpha \!+\!\mu ) \\&+ b\sigma (\alpha +\mu )[(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1 ] \}. \end{aligned}$$

Note that

$$\begin{aligned}&\{r X_1[(q\alpha +\mu )(\sigma +\mu ) - g(r\alpha \sigma + \mu Y_1) ]\}\{- \alpha \sigma X_1 (q\alpha +\mu ) \\&\qquad +\, b\sigma (\alpha +\mu )[(q\alpha +\mu )(\sigma +\mu )- g\mu Y_1 ] \} \\&\quad = -\,[rg\alpha \sigma -(q\alpha +\mu )(\sigma +\mu )]\{rX_1 b\sigma (\alpha +\mu )[(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1]\\&\qquad -\,r\alpha \sigma X_1^2(q\alpha +\mu )\}+ rg\mu X_1 Y_1\{b\sigma (\alpha +\mu )[(q\alpha +\mu )(\sigma +\mu ) - g\mu Y_1]\\&\qquad -\,\alpha \sigma X_1(q\alpha \!+\!\mu )\}\!-\!X_1(q\alpha \!+\!\mu )[rg\alpha \sigma \!-\!(q\alpha \!+\!\mu )(\sigma \!+\!\mu )][rb\sigma (\alpha \!+\!\mu )(\sigma \!+\!\mu )\\&\qquad -\, X_1(r\alpha \sigma \!+\!\mu Y_1)]\!-\! \mu Y_1\{X_1[rg\alpha \sigma \!-\!(q\alpha \!+\!\mu )(\sigma \!+\!\mu )][(q\alpha \!+\!\mu )X_1\!-\!rgb\sigma (\alpha \!+\!\mu )] \\&\qquad -\, rg X_1 b\sigma (\alpha \!+\!\mu )[(q\alpha \!+\!\mu )(\sigma \!+\!\mu )\!-\!g\mu Y_1] \!+\! rg\alpha \sigma (q\alpha +\mu )X_1^2\}\\&\quad = -X_1(q\alpha +\mu )[rg\alpha \sigma -(q\alpha +\mu )(\sigma +\mu )][rb\sigma (\alpha +\mu )(\sigma +\mu )-X_1(r\alpha \sigma +\mu Y_1)]\\&\qquad +\mu Y_1\{(\sigma +\mu )[(q\alpha +\mu )X_1 - rgb\sigma (\alpha +\mu )]^2 -rg^2 b\sigma (\alpha +\mu )[rb\sigma (\alpha +\mu )(\sigma +\mu ) \\&\qquad -\,X_1(r\alpha \sigma +\mu Y_1)]\}\\&\quad = \mu Y_1 (\sigma +\mu ){[(q\alpha +\mu )X_1 - rgb\sigma (\alpha +\mu )]^2}-[rb\sigma (\alpha +\mu )(\sigma +\mu ) \\&\qquad -\,X_1(r\alpha \sigma + \mu Y_1)] {[X_1(q\alpha +\mu )(rg\alpha \sigma -(q\alpha +\mu )(\sigma +\mu )) +r g^2 b\sigma \mu Y_1 (\alpha +\mu )]} \\&\quad = \mu Y_1 (\sigma +\mu ){[(q\alpha +\mu )X_1 - rgb\sigma (\alpha +\mu )]^2}\\&\qquad -\,\phi _2 {[X_1(q\alpha +\mu )(rg\alpha \sigma -(q\alpha +\mu )(\sigma +\mu )) +r g^2 b\sigma \mu Y_1 (\alpha +\mu )]}. \end{aligned}$$

Thus,

$$\begin{aligned} G_3&= 4\mu Y_1(\sigma +\mu )(\alpha +\mu )^2\{\mu Y_1 (\sigma +\mu ){[(q\alpha +\mu )X_1 - rgb\sigma (\alpha +\mu )]^2}\\&+ \frac{G_2}{g Z_1} {[X_1(q\alpha +\mu )(rg\alpha \sigma -(q\alpha +\mu )(\sigma +\mu )) +r g^2 b\sigma \mu Y_1 (\alpha +\mu )]}\}. \end{aligned}$$