Modeling the Effects of Augmentation Strategies on the Control of Dengue Fever With an Impulsive Differential Equation

Abstract

Dengue fever has rapidly become the world’s most common vector-borne viral disease. Use of endosymbiotic Wolbachia is an innovative technology to prevent vector mosquitoes from reproducing and so break the cycle of dengue transmission. However, strategies such as population eradication and replacement will only succeed if appropriate augmentations with Wolbachia-infected mosquitoes that take account of a variety of factors are carried out. Here, we describe the spread of Wolbachia in mosquito populations using an impulsive differential system with four state variables, incorporating the effects of cytoplasmic incompatibility and the augmentation of Wolbachia-infected mosquitoes with different sex ratios. We then evaluated (a) how each parameter value contributes to the success of population replacement; (b) how different release quantities of infected mosquitoes with different sex ratios affect the success of population suppression or replacement; and (c) how the success of these two strategies can be realized to block the transmission of dengue fever. Analysis of the system’s stability, bifurcations and sensitivity reveals the existence of forward and backward bifurcations, multiple attractors and the contribution of each parameter to the success of the strategies. The results indicate that the initial density of mosquitoes, the quantities of mosquitoes released in augmentations and their sex ratios have impacts on whether or not the strategies of population suppression or replacement can be achieved. Therefore, successful strategies rely on selecting suitable strains of Wolbachia and carefully designing the mosquito augmentation program.

Appendices

Appendix 1: The Expression of Each Element of $D_XN(\bar{\theta }_1,\bar{X})$

First from (26) and derivative of a multivariate compound function, we have the expression of each element of $D_XN(\bar{\theta }_1,\bar{X})$ as follows

$$\begin{aligned} \left\{ \begin{array}{l} a_{ij}=-\sum \limits _{k=1}^{4}\frac{\partial {I_i(\theta _0+\bar{\theta }_1,X_0+\bar{X})}}{\partial {X_k}}\frac{\Phi _k(X_0+\bar{X})}{\partial {X_j}},i\ne j,\\ a_{ii}=1-\sum \limits _{k=1}^{4}\frac{\partial {I_i(\theta _0+\bar{\theta }_1,X_0+\bar{X})}}{\partial {X_k}}\frac{\Phi _k(X_0+\bar{X})}{\partial {X_i}},i= j.\\ \end{array} \right. \end{aligned}$$

According to (27), we have the differential equations of derivatives of $\Phi =(\Phi _1,\Phi _2,\Phi _3,\Phi _4)$ for $(\bar{\theta }_1,\bar{X})=(0,0)$ with respect to $X=(X_1,X_2,X_3,X_4).$ Note that

$$\begin{aligned} \left\{ \begin{array}{l} \frac{d}{{d}t}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_1}}=(b_1(1-q)-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_1}},\ \frac{\partial {\Phi _3}}{\partial {X_1}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_1}}=b_1(1-q)\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_1}}-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_1}},\ \frac{\partial {\Phi _4}}{\partial {X_1}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_2}}=(b_1(1-q)-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_2}},\ \frac{\partial {\Phi _3}}{\partial {X_2}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_2}}=b_1(1-q)\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_2}}-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_2}},\ \frac{\partial {\Phi _4}}{\partial {X_2}}(0,X_0)=0.\\ \end{array} \right. \end{aligned}$$

So we obtain

$$\begin{aligned} \frac{\partial {\Phi _i}}{\partial {X_j}}(t,X_0)\equiv 0,i=3,4,j=1,2, \end{aligned}$$

(41)

for $0\leqslant t<T.$ Then we have

$$\begin{aligned} \left\{ \begin{array}{l} \frac{d}{{d}t}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_1}}=(b_2-(d+D)(2\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_1}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_1}},\ \frac{\partial {\Phi _1}}{\partial {X_1}}(0,X_0)=1,\\ \frac{d}{{d}t}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_2}}=(b_2-(d+D)(2\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_2}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_2}},\ \frac{\partial {\Phi _1}}{\partial {X_2}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_1}}=(b_2-(d+D)\tilde{M}_I)\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_1}}-(d+D)(\tilde{F}_I+2\tilde{M}_I)\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_1}},\ \frac{\partial {\Phi _2}}{\partial {X_1}}(0,X_0)=0,\\ \frac{d}{\hbox {d}t}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_2}}=(b_2-(d+D)\tilde{M}_I)\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_2}}-(d+D)(\tilde{F}_I+2\tilde{M}_I)\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_2}},\ \frac{\partial {\Phi _2}}{\partial {X_2}}(0,X_0)=1.\\ \end{array} \right. \end{aligned}$$

So the solution $\frac{\partial {\Phi _i}}{\partial {X_j}}(t,X_0)(i,j=1,2)$ of the above system for $0\leqslant t<T$ can be solved which is not identically equal to zero. Further, we have

$$\begin{aligned} \left\{ \begin{array}{lllllll} \frac{d}{{d}t}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_3}} &{}=(b_2-(d+D)(2\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_3}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_3}}\\ &{}\quad -(d+D)\tilde{F}_I\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_3}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_3}},\ \frac{\partial {\Phi _1}}{\partial {X_3}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_4}} &{}=(b_2-(d+D)(2\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_4}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_4}}\\ &{}\quad -(d+D)\tilde{F}_I\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_4}}-(d+D)\tilde{F}_I\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_4}},\ \frac{\partial {\Phi _1}}{\partial {X_4}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_3}} &{}=(b_2-(d+D)\tilde{M}_I)\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_3}}-(d+D)(\tilde{F}_U+2\tilde{M}_U)\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_3}}\\ &{}\quad -(d+D)\tilde{M}_I\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_3}}-(d+D)\tilde{M}_I\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_3}},\ \frac{\partial {\Phi _2}}{\partial {X_3}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_4}} &{}=(b_2-(d+D)\tilde{M}_I)\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_4}}-(d+D)(\tilde{F}_U+2\tilde{M}_U)\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_4}}\\ &{}\quad -(d+D)\tilde{M}_I\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_4}}-(d+D)\tilde{M}_I\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_4}},\ \frac{\partial {\Phi _2}}{\partial {X_4}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_3}}&{}=(b_1(1-q)-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_3}},\ \frac{\partial {\Phi _3}}{\partial {X_3}}(0,X_0)=1,\\ \frac{d}{{d}t}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_4}}&{}=(b_1(1-q)-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_4}},\ \frac{\partial {\Phi _3}}{\partial {X_4}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_3}}&{}=b_1(1-q)\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_3}}-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_3}},\ \frac{\partial {\Phi _4}}{\partial {X_3}}(0,X_0)=0,\\ \frac{d}{{d}t}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_4}}&{}=b_1(1-q)\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_4}}-d(\tilde{F}_I+\tilde{M}_I))\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_4}},\ \frac{\partial {\Phi _4}}{\partial {X_4}}(0,X_0)=1.\\ \end{array} \right. \end{aligned}$$

We can solve the above equations and give the expression of solutions for the last four equations as follows:

$$\begin{aligned} \frac{\partial {\Phi _3}}{\partial {X_3}}(t,X_0)= & {} {\mathrm e}^{\int _{0}^t(b_1(1-q)-d(\tilde{F}_I+\tilde{M}_I))\mathrm{d}\eta }\triangleq \beta (t),\ \frac{\partial {\Phi _3}}{\partial {X_4}}(t,X_0)\equiv 0,\nonumber \\ \frac{\partial {\Phi _4}}{\partial {X_3}}(t,X_0)= & {} {\mathrm e}^{b_1(1-q)\int _{0}^t\beta (s) ds},\ \frac{\partial {\Phi _4}}{\partial {X_4}}(t,X_0)={\mathrm e}^{\int _{0}^t(-d(\tilde{F}_I+\tilde{M}_I))\mathrm{d}\eta }. \end{aligned}$$

(42)

Thus, the expression of each element of $D_XN(0,O)$ is as follows:

$$\begin{aligned} a^{'}_{11}= & {} 1-\frac{\partial {\Phi _1}}{\partial {X_1}}(T,X_0),\ \ a^{'}_{21}=-\frac{\partial {\Phi _2}}{\partial {X_1}}(T,X_0),\nonumber \\ a^{'}_{12}= & {} -\frac{\partial {\Phi _1}}{\partial {X_2}}(T,X_0),\ \ a^{'}_{22}=1-\frac{\partial {\Phi _2}}{\partial {X_2}}(T,X_0),\nonumber \\ a^{'}_{13}= & {} -\frac{\partial {\Phi _1}}{\partial {X_3}}(T,X_0),\ \ a^{'}_{23}=-\frac{\partial {\Phi _2}}{\partial {X_3}}(T,X_0),\nonumber \\ a^{'}_{14}= & {} -\frac{\partial {\Phi _1}}{\partial {X_4}}(T,X_0),\ \ a^{'}_{24}=-\frac{\partial {\Phi _2}}{\partial {X_4}}(T,X_0),\nonumber \\ a^{'}_{31}= & {} -\frac{\partial {\Phi _3}}{\partial {X_1}}(T,X_0)=0,\ \ a^{'}_{41}=-\frac{\partial {\Phi _4}}{\partial {X_1}}(T,X_0)=0,\nonumber \\ a^{'}_{32}= & {} -\frac{\partial {\Phi _3}}{\partial {X_2}}(T,X_0)=0,\ \ a^{'}_{42}=-\frac{\partial {\Phi _4}}{\partial {X_2}}(T,X_0)=0,\nonumber \\ a^{'}_{33}= & {} 1-\frac{\partial {\Phi _3}}{\partial {X_3}}(T,X_0),\ \ a^{'}_{43}=-\frac{\partial {\Phi _4}}{\partial {X_3}}(T,X_0),\nonumber \\ a^{'}_{34}= & {} -\frac{\partial {\Phi _3}}{\partial {X_4}}(T,X_0)=0,\ \ a^{'}_{44}=1-\frac{\partial {\Phi _4}}{\partial {X_4}}(T,X_0). \end{aligned}$$

(43)

Appendix 2: The Second-Order Partial Derivatives of $\Phi _{i},i=3,4$

From (27), we have

$$\begin{aligned} \frac{d}{\hbox {d}t}\frac{\partial {\Phi _i}(t,X_0)}{\partial {X_j}}= & {} \frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_1}}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_2}}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_j}}\\&+\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_4}}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_j}}, \end{aligned}$$

with $i,j=1,2,3,4.$ Then

$$\begin{aligned} \frac{d}{\hbox {d}t}\frac{\partial ^2{\Phi _i}(t,X_0)}{\partial {X_j}\partial {X_k}}= & {} \frac{\partial ^2{F_i(\tilde{X}(t))}}{\partial {X_1}\partial {X_k}}\frac{\partial {\Phi _1}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_1}}\frac{\partial ^2{\Phi _1}(t,X_0)}{\partial {X_j}\partial {X_k}}\\&+\frac{\partial ^2{F_i(\tilde{X}(t))}}{\partial {X_2}\partial {X_k}}\frac{\partial {\Phi _2}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_2}}\frac{\partial ^2{\Phi _2}(t,X_0)}{\partial {X_j}\partial {X_k}}\\&+\frac{\partial ^2{F_i(\tilde{X}(t))}}{\partial {X_3}\partial {X_k}}\frac{\partial {\Phi _3}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X_j}\partial {X_k}}\\&+\frac{\partial ^2{F_i(\tilde{X}(t))}}{\partial {X_4}\partial {X_k}}\frac{\partial {\Phi _4}(t,X_0)}{\partial {X_j}} +\frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_4}}\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X_j}\partial {X_k}},\\ \end{aligned}$$

with $i,j,k=1,2,3,4.$ By simple calculation, it follows that

$$\begin{aligned} \frac{\partial {F_i(\tilde{X}(t))}}{\partial {X_j}}= & {} \frac{\partial {\Phi _i}(t,X_0)}{\partial {X_j}}=0,i=3,4,j=1,2,\\ \frac{\partial {F^2_i(\tilde{X}(t))}}{\partial {X^2_1}}= & {} \frac{\partial {F^2_i(\tilde{X}(t))}}{\partial {X_1}\partial {X_2}} =\frac{\partial {F^2_i(\tilde{X}(t))}}{\partial {X_j}\partial {X_k}}=0,i=3,4,j,k=1,2,\\ \frac{\partial {F_3(\tilde{X}(t))}}{\partial {X_4}}= & {} 0.\\ \end{aligned}$$

Thus

$$\begin{aligned} \frac{d}{\hbox {d}t}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_1}}= & {} \frac{\partial {F_3(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_1}},\nonumber \\ \frac{d}{\hbox {d}t}\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_1}}= & {} \frac{\partial {F_4(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_1}} +\frac{\partial {F_4(\tilde{X}(t))}}{\partial {X_4}}\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_1}},\nonumber \\ \frac{d}{\hbox {d}t}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_2}}= & {} \frac{\partial {F_3(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_2}},\nonumber \\ \frac{d}{\hbox {d}t}\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_2}}= & {} \frac{\partial {F_4(\tilde{X}(t))}}{\partial {X_3}}\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_2}} +\frac{\partial {F_4(\tilde{X}(t))}}{\partial {X_4}}\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_2}}, \end{aligned}$$

(44)

with the initial conditions

$$\begin{aligned} \frac{\partial ^2{\Phi _3}(0,X_0)}{\partial {X^2_1}} =\frac{\partial ^2{\Phi _4}(0,X_0)}{\partial {X^2_1}} =\frac{\partial ^2{\Phi _3}(0,X_0)}{\partial {X^2_2}} =\frac{\partial ^2{\Phi _4}(0,X_0)}{\partial {X^2_2}}=0. \end{aligned}$$

Solving (44) deduces that

$$\begin{aligned} \frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_1}} =\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_1}} =\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X^2_2}} =\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X^2_2}}=0. \end{aligned}$$

(45)

Similarly, we can obtain that

$$\begin{aligned} \frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X_1}\partial {X_2}} =\frac{\partial ^2{\Phi _3}(t,X_0)}{\partial {X_2}\partial {X_1}} =\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X_1}\partial {X_2}} =\frac{\partial ^2{\Phi _4}(t,X_0)}{\partial {X_2}\partial {X_1}}=0. \end{aligned}$$

(46)

Appendix 3: The Partial Derivatives of $\tilde{\alpha }_i, i=2,3,4$ at (0, 0)

First, we calculate the values of $\partial {\tilde{\alpha }_i}(0,0)/\partial {\alpha _1}$ and $\partial {\tilde{\alpha }_i}(0,0)/\partial {\bar{\theta }_1},i=2,3,4.$ It follows from (34) that

$$\begin{aligned} 0= & {} \left. \frac{\partial {N_j}}{\partial {\alpha _1}}\right| _{(0,0)} =\left. \frac{\partial {N_j}}{\partial {X_1}}\frac{\partial {X_1}}{\partial {\alpha _1}} +\frac{\partial {N_j}}{\partial {X_2}}\frac{\partial {X_2}}{\partial {\alpha _1}} +\frac{\partial {N_j}}{\partial {X_3}}\frac{\partial {X_3}}{\partial {\alpha _1}} +\frac{\partial {N_j}}{\partial {X_4}}\frac{\partial {X_4}}{\partial {\alpha _1}}\right| _{(0,0)}\nonumber \\= & {} \left. \frac{\partial {N_j}}{\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}}{\partial {\alpha _1}}\right) +\frac{\partial {N_j}}{\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}}{\partial {\alpha _1}}\right) +\frac{\partial {N_j}}{\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}}{\partial {\alpha _1}}\right) +\frac{\partial {N_j}}{\partial {X_4}}Y_{14}\right| _{(0,0)}.\nonumber \\ \end{aligned}$$

(47)

Since $Y_1$ is a basis of $\ker (D_XN(0,O)),$ then

$$\begin{aligned} \left. \frac{\partial {N_i}}{\partial {X_1}}Y_{11}+\frac{\partial {N_i}}{\partial {X_1}}Y_{12} +\frac{\partial {N_i}}{\partial {X_1}}Y_{13}+\frac{\partial {N_i}}{\partial {X_1}}Y_{14}\right| _{(0,O)} =0,i=1,2,3,4. \end{aligned}$$

(48)

Hence from (47) and (48), we deduce that

$$\begin{aligned} \left. a^{'}_{11}\frac{\partial {\tilde{\alpha }_2}}{\partial {\alpha _1}} +a^{'}_{12}\frac{\partial {\tilde{\alpha }_3}}{\partial {\alpha _1}} +a^{'}_{13}\frac{\partial {\tilde{\alpha }_4}}{\partial {\alpha _1}}\right| _{(0,0)}= & {} 0,\nonumber \\ \left. a^{'}_{21}\frac{\partial {\tilde{\alpha }_2}}{\partial {\alpha _1}} +a^{'}_{22}\frac{\partial {\tilde{\alpha }_3}}{\partial {\alpha _1}} +a^{'}_{23}\frac{\partial {\tilde{\alpha }_4}}{\partial {\alpha _1}}\right| _{(0,0)}= & {} 0,\nonumber \\ \left. a^{'}_{31}\frac{\partial {\tilde{\alpha }_2}}{\partial {\alpha _1}} +a^{'}_{32}\frac{\partial {\tilde{\alpha }_3}}{\partial {\alpha _1}} +a^{'}_{33}\frac{\partial {\tilde{\alpha }_4}}{\partial {\alpha _1}}\right| _{(0,0)}= & {} 0. \end{aligned}$$

(49)

Solving (49) with respect to $\partial {\tilde{\alpha }_i}/\partial {\alpha _1},i=2,3,4$ obtains

$$\begin{aligned} \frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\alpha _1}} =\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\alpha _1}} =\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\alpha _1}}=0. \end{aligned}$$

(50)

Again considering (34), we have

$$\begin{aligned}&\left\{ \begin{array}{lll} N_i(\bar{\theta }_1,\alpha _1)=X_{0i}+Y_{1i}\alpha _1 +\tilde{\alpha }_{i+1}(\bar{\theta }_1,\alpha _1)-(\Phi _i(\theta _0+\bar{\theta }_1,X_0+\bar{X}(\bar{\theta }_1,\alpha _1))+\theta _0+\bar{\theta }_1),\\ N_j(\bar{\theta }_1,\alpha _1)=X_{0j}+Y_{1j}\alpha _1 +\tilde{\alpha }_{j+1}(\bar{\theta }_1,\alpha _1)-(\Phi _j(\theta _0+\bar{\theta }_1,X_0+\bar{X}(\bar{\theta }_1,\alpha _1))+\theta _2),\\ N_1(\bar{\theta }_1,\alpha _1)=N_2(\bar{\theta }_1,\alpha _1)=N_3(\bar{\theta }_1,\alpha _1)=0, \end{array} \right. \nonumber \\ \end{aligned}$$

(51)

with $i=1,3,j=2,4,$ $\tilde{\alpha }_{5}=0,$ $X_0=(X_{01},X_{02},X_{03},X_{04})$ and $\bar{X}=(\bar{X}_1,\bar{X}_2,\bar{X}_3,\bar{X}_4).$ Hence based on (43) and (51), we can deduce that

$$\begin{aligned} 0= & {} \frac{\partial {N_1}(0,0)}{\partial {\bar{\theta }_1}} =\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} -\left( 1+\sum \limits _{i=1}^{3}\left( \frac{\partial {\Phi _1(\theta _0,X_0)}}{\partial {X_i}}\frac{\partial {X_i}(0,0)}{\partial {\bar{\theta }_1}}\right) \right) \nonumber \\= & {} \frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +(a^{'}_{11}-1)\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{12}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{13}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}-1\nonumber \\= & {} a^{'}_{11}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{12}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{13}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}-1. \end{aligned}$$

(52)

Similarly, from (43) and (51), we can obtain that

$$\begin{aligned} \frac{\partial {N_2}(0,0)}{\partial {\bar{\theta }_1}}= & {} a^{'}_{21}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{22}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{23}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}=0,\nonumber \\ \frac{\partial {N_3}(0,0)}{\partial {\bar{\theta }_1}}= & {} a^{'}_{31}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{32}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +a^{'}_{33}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}\nonumber \\= & {} a^{'}_{33}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}=0. \end{aligned}$$

(53)

Solving (52) and (53) with respect to $\partial {\tilde{\alpha }_i}(0,0)/\partial {\bar{\theta }_1},i=2,3,4$ yields that

$$\begin{aligned} \frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}}= & {} \frac{a^{'}_{22}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}},\nonumber \\ \frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}}= & {} -\frac{a^{'}_{21}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}},\nonumber \\ \frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}= & {} 0. \end{aligned}$$

(54)

Based on the first equation of (51), we have that

$$\begin{aligned} 0= & {} \frac{\partial ^2{N_3(0,0)}}{\partial {\bar{\theta }_1}^2} =\frac{\partial }{{\partial {\bar{\theta }_1}}}\frac{\partial {N_3(0,0)}}{\partial {\bar{\theta }_1}}\nonumber \\= & {} \frac{\partial ^2{\tilde{\alpha }_4(0,0)}}{\partial {\bar{\theta }^2_1}} -\frac{\partial }{\partial {\bar{\theta }_1}}\sum \limits _{i=1}^{3}\left( \frac{\partial {\Phi _3(\theta _0,X_0)}}{\partial {X_i}}\frac{\partial {X_i}(0,0)}{\partial {\bar{\theta }_1}}\right) \nonumber \\= & {} \frac{\partial ^2{\tilde{\alpha }_4(0,0)}}{\partial {\bar{\theta }^2_1}} -\left[ \frac{\partial {\Phi _3(\theta _0,X_0)}}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2(0,0)}}{\partial {\bar{\theta }^2_1}}\right. \nonumber \\&+\frac{\partial {\Phi _3(\theta _0,X_0)}}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3(0,0)}}{\partial {\bar{\theta }^2_1}} +\frac{\partial {\Phi _3(\theta _0,X_0)}}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4(0,0)}}{\partial {\bar{\theta }^2_1}}\nonumber \\&+\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} \left( \frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X^2_1}}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_1}\partial {X_2}}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_1}\partial {X_3}}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}\right) \nonumber \\&+\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} \left( \frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_2}\partial {X_1}}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X^2_2}}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_2}\partial {X_3}}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}}\right) \nonumber \\&+\left. \frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}} \left( \frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_3}\partial {X_1}}\frac{\partial {\tilde{\alpha }_2}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_3}\partial {X_2}}\frac{\partial {\tilde{\alpha }_3}(0,0)}{\partial {\bar{\theta }_1}} +\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X^2_3}}\frac{\partial {\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }_1}} \right) \right] .\nonumber \\ \end{aligned}$$

(55)

Substituting (41), (42), (45), (46) and (54) into (55) yields that

$$\begin{aligned} \frac{\partial ^2{\tilde{\alpha }_4}(0,0)}{\partial {\bar{\theta }^2_1}}=0. \end{aligned}$$

(56)

Then, we calculate the value of $\partial ^2{\tilde{\alpha }_4(0,0)}/\partial {\bar{\theta }_1}\partial {\alpha _1}.$ From the first equation of (51) and combination with (31), (41), (42), (45), (46), (50) and (54), we have

$$\begin{aligned} 0= & {} \frac{\partial ^2{N_3\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} =\frac{\partial }{{\partial {\alpha _1}}}\frac{\partial {N_3\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\\= & {} \frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} -\frac{\partial }{\partial {\alpha _1}}\sum \limits _{i=1}^{3}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_i}}\frac{\partial {X_i}\left( 0,0\right) }{\partial {\bar{\theta }_1}}\right) \\= & {} \frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} - \left[ \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} +\frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} +\frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}}\right. \\&+\left. \frac{\partial {\tilde{\alpha }_2\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}}\right) +\frac{\partial {\tilde{\alpha }_3\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}}\right) \right. \\&\left. +\frac{\partial {\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_3}}\right) \right] \\= & {} \frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} -\left[ \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} +\frac{\partial {\tilde{\alpha }_2\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}}\right) \right. \\&\left. +\frac{\partial {\tilde{\alpha }_3\left( 0,0\right) }}{\partial {\bar{\theta }_1}}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}}\right) \right] \\= & {} \frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} -\left[ \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}}\right. \\&+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\bar{\theta }_1}} \left( \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X^2_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \\&+\left. \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_4}}Y_{14}\right) \\&+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\bar{\theta }_1}} \left( \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X^2_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \\&+\left. \left. \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_4}}Y_{14}\right) \right] \\= & {} \frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} -\left[ \frac{\partial {\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4\left( 0,0\right) }}{\partial {\bar{\theta }_1}\partial {\alpha _1}} +\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\bar{\theta }_1}} \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_4}}\right. \\&\left. +\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\bar{\theta }_1}} \frac{\partial ^2{\Phi _3\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_4}}\right] .\\ \end{aligned}$$

Thus we have

$$\begin{aligned} \frac{\partial ^2{\tilde{\alpha }_4(0,0)}}{\partial {\bar{\theta }_1}\partial {\alpha _1}} =\frac{1}{a^{'}_{33}} \left( \frac{a^{'}_{22}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}} \frac{\partial ^2{\Phi _3(X_0)}}{\partial {X_1}\partial {X_4}} -\frac{a^{'}_{21}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}} \frac{\partial ^2{\Phi _3(X_0)}}{\partial {X_2}\partial {X_4}}\right) .\nonumber \\ \end{aligned}$$

(57)

Next, we calculate the value of $\partial ^2{\tilde{\alpha }_i(0,0)}/\partial {\alpha ^2_1},i=2,3,4.$ From (51) as $i=1$, we have

$$\begin{aligned} 0= & {} \frac{\partial ^2{N_1\left( 0,0\right) }}{\partial {\alpha ^2_1}} =\frac{\partial }{\partial {\alpha _1}}\frac{\partial {N_1\left( 0,0\right) }}{\partial {\alpha _1}} \nonumber \\= & {} \frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \nonumber \\&+\left. \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_4}}Y_{14}\right) \nonumber \\= & {} \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha ^2_1}} +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha ^2_1}}\\&+\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha ^2_1}} \nonumber \\&+ \left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) \frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\right) \\&+\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\right) \nonumber \\&+\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) \frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\right) +Y_{14}\frac{\partial }{\partial {\alpha _1}}\left( \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\right) \nonumber \\= & {} \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha ^2_1}} +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha ^2_1}}\\&+\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha ^2_1}} \nonumber \\&+\frac{\partial }{\partial {\alpha _1}}\left( Y_{11}\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}+Y_{12}\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\right. \\&\left. +Y_{13}\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}} +Y_{14}\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_4}}\right) \nonumber \\= & {} \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha ^2_1}} +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha ^2_1}}\\&+\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha ^2_1}} \nonumber \\&+Y_{11}\left[ \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X^2_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \\&+\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \nonumber \\&+\left. \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_1}\partial {X_4}}Y_{14}\right] \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+Y_{12}\left[ \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X^2_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \nonumber \right. \\&\left. +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_2}\partial {X_4}}Y_{14}\right] \nonumber \\&+Y_{13}\left[ \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_3}\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_3}\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \nonumber \\&\left. +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X^2_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_3}\partial {X_4}}Y_{14}\right] \nonumber \\&+Y_{14}\left[ \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_4}\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_4}\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha _1}}\right) \right. \nonumber \\&\left. +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_4}\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha _1}}\right) +\frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X^2_4}}Y_{14}\right] \nonumber \\= & {} \frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_1}}\frac{\partial ^2{\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\alpha ^2_1}} +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_2}}\frac{\partial ^2{\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\alpha ^2_1}} +\frac{\partial {N_1\left( \theta _0,X_0\right) }}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\alpha ^2_1}} \nonumber \\&+\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{N_1\left( \theta _0,X_0\right) }}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}. \end{aligned}$$

(58)

Submitting (30) into (58) gives

$$\begin{aligned}&a^{'}_{11}\frac{\partial ^2{\tilde{\alpha }_2}(0,0)}{\partial {\alpha ^2_1}} +a^{'}_{12}\frac{\partial ^2{\tilde{\alpha }_3}(0,0)}{\partial {\alpha ^2_1}} +a^{'}_{13}\frac{\partial ^2{\tilde{\alpha }_4}(0,0)}{\partial {\alpha ^2_1}}\nonumber \\&\quad =-\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{N_1(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}\nonumber \\&\quad =\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{\Phi _1(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}. \end{aligned}$$

(59)

Similarly, we can obtain from (51) as $i=2,3$ that

$$\begin{aligned} a^{'}_{21}\frac{\partial ^2{\tilde{\alpha }_2}(0,0)}{\partial {\alpha ^2_1}} +a^{'}_{22}\frac{\partial ^2{\tilde{\alpha }_3}(0,0)}{\partial {\alpha ^2_1}} +a^{'}_{23}\frac{\partial ^2{\tilde{\alpha }_4}(0,0)}{\partial {\alpha ^2_1}}= & {} \sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{\Phi _2(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}, \nonumber \\ a^{'}_{33}\frac{\partial ^2{\tilde{\alpha }_4}(0,0)}{\partial {\alpha ^2_1}}= & {} \sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}. \end{aligned}$$

(60)

Hence, we can solve the roots of equations (59) and (60) with respect to $\partial ^2{\tilde{\alpha }_i(0,0)}/\partial {\alpha ^2_1},i=2,3,4,$ and submit them with $i=4$ into (68) in Appendix 5.

Appendix 4: The First-Order Partial Derivatives of $N_4(\bar{\theta }_1,\beta _1)$

Similar to (47), (52) and (53), we can calculate from (26) that

$$\begin{aligned} \left. \frac{\partial {N_4}}{\partial {\alpha _1}}\right| _{\left( 0,0\right) }= & {} \left. \frac{\partial {N_1}}{\partial {X_1}}\left( Y_{11}+\frac{\partial {\tilde{\alpha }_2}}{\partial {\alpha _1}}\right) +\frac{\partial {N_4}}{\partial {X_2}}\left( Y_{12}+\frac{\partial {\tilde{\alpha }_3}}{\partial {\alpha _1}}\right) +\frac{\partial {N_4}}{\partial {X_3}}\left( Y_{13}+\frac{\partial {\tilde{\alpha }_4}}{\partial {\alpha _1}}\right) +\frac{\partial {N_4}}{\partial {X_4}}Y_{14}\right| _{\left( 0,0\right) },\nonumber \\ \left. \frac{\partial {N_4}}{\partial {\bar{\theta }_1}}\right| _{\left( 0,0\right) }= & {} \left. -\frac{\partial {\Phi _4\left( X_0\right) }}{\partial {X_1}}\frac{\partial {\tilde{\alpha }_2}\left( 0,0\right) }{\partial {\bar{\theta }_1}} -\frac{\partial {\Phi _4\left( X_0\right) }}{\partial {X_2}}\frac{\partial {\tilde{\alpha }_3}\left( 0,0\right) }{\partial {\bar{\theta }_1}} -\frac{\partial {\Phi _4\left( X_0\right) }}{\partial {X_3}}\frac{\partial {\tilde{\alpha }_4}\left( 0,0\right) }{\partial {\bar{\theta }_1}} \right| _{\left( 0,0\right) }. \end{aligned}$$

(61)

Substituting (48), (50) and (54) into (61) obtains

$$\begin{aligned} \frac{\partial {N_4}(0,0)}{\partial {\alpha _1}}=\frac{\partial {N_4}(0,0)}{\partial {\bar{\theta }_1}}=0. \end{aligned}$$

(62)

Appendix 5: The Second-Order Partial Derivatives of $N_4(\bar{\theta }_1,\alpha _1)$

(i) Calculation the value of A.

According to the second equation of (51) as $j=4,$ we can easily get that

(63)

Substituting (41), (42), (45), (46) (50), (54) and (56) into (63) obtains

$$\begin{aligned} A=\frac{\partial ^2{N_4(0,0)}}{\partial {\bar{\theta }_1}^2}=0. \end{aligned}$$

(64)

(ii) Calculation the value of B.

Similarly, based on the second equation of (51), it follows that

(65)

Then substituting (54) and (57) into the above equation yields that

$$\begin{aligned} B= & {} \frac{\partial ^2{N_4(0,0)}}{\partial {\bar{\theta }_1}\partial {\alpha _1}}\nonumber \\= & {} -\left[ \frac{a^{'}_{22}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}} \left( \frac{\partial ^2{\Phi _4(X_0)}}{\partial {X_1}\partial {X_4}} -\frac{a^{'}_{43}}{a^{'}_{33}}\frac{\partial ^2{\Phi _3(X_0)}}{\partial {X_1}\partial {X_4}}\right) \right. \nonumber \\&\quad \left. -\frac{a^{'}_{21}}{a^{'}_{11}a^{'}_{22}-a^{'}_{12}a^{'}_{21}} \left( \frac{\partial ^2{\Phi _4(X_0)}}{\partial {X_2}\partial {X_4}} -\frac{a^{'}_{43}}{a^{'}_{33}}\frac{\partial ^2{\Phi _3(X_0)}}{\partial {X_2}\partial {X_4}}\right) \right] . \end{aligned}$$

(66)

(iii) Calculation of the value of C.

Similarly to (58), we have that

$$\begin{aligned} \frac{\partial ^2{N_4(0,0)}}{\partial {\alpha ^2_1}}= & {} \frac{\partial {N_4(\theta _0,X_0)}}{\partial {X_3}}\frac{\partial ^2{\tilde{\alpha }_4}(0,0)}{\partial {\alpha ^2_1}} +\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{N_4(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}\nonumber \\= & {} a^{'}_{43}\frac{\partial ^2{\tilde{\alpha }_4(0,0)}}{\partial {\alpha ^2_1}} -\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4} \frac{\partial ^2{\Phi _4(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}Y_{1i}Y_{1j}. \end{aligned}$$

(67)

Submitting the roots of equations (59) and (60) with $i=4$ into (68), we can obtain that

$$\begin{aligned} C=\frac{\partial ^2{N_4(0,0)}}{\partial {\alpha ^2_1}} =\sum \limits _{i=1}^{4}\sum \limits _{j=1}^{4}\left( \frac{a^{'}_{43}}{a^{'}_{33}}\frac{\partial ^2{\Phi _3(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}} -\frac{\partial ^2{\Phi _4(\theta _0,X_0)}}{\partial {X_i}\partial {X_j}}\right) Y_{1i}Y_{1j}. \end{aligned}$$

(68)