Study of a Multi-delayed SEIR Epidemic Model with Immunity Period and Treatment Function in Deterministic and Stochastic Environment

Abstract

In this paper a multi-delayed epidemic SEIR model with immunity period and treatment function has been considered in deterministic and stochastic environment. The delays are taken as immunity delay and treatment delay. Firstly, in deterministic situation, a feasible region has been obtained in parametric space, where the solutions of the system are bounded and positive. The conditions for stability of both endemic and epidemic equilibrium have been derived. Later, in presence of stochasticity the dynamics for the SEIR model has been discussed in both presence and absence of delays. Latin Hypercube Sampling/Partial Rank Correlation Coefficient (LHS/PRCC) sensitivity analysis, which is an efficient tool often employed in uncertainty analysis is used to explore the entire parameter space of a model. To study the LHS/PRCC sensitivity analysis for our model, we have chosen a suitable set of parameter values.

Appendices

Appendix:

Appendix A

Derivation of $k_i$ and $w_i, i=1, 2,\ldots , 10$:

$$\begin{aligned} k_1= & {} k_5=2a_3+(a_5+a_1)\tau _1+a_6\tau _2,\\ k_2= & {} (a_1+a_8)\tau _1+a_3\tau _2,\\ k_3= & {} k_6=2a_2+\frac{(2a_4+\eta _{12}+\beta _1)\tau _1}{2}+\frac{(2a_3+a_1+a_7-\beta _1-\eta _{12})\tau _2}{2},\\ k_4= & {} k_7=(2a_4+2a_6+a_2)+\frac{(\beta _1+\eta _{12})\tau _2}{2}+\frac{(a_5+a_7-\beta _1-\eta _{12})\tau _1}{2},\\ k_8= & {} \frac{(a_5+2a_6+a_8)\tau _2}{2},\\ k_9= & {} k_{10}=2(\beta _1+\eta _{12})\tau _1+(a_2+a_4)\tau _2+\frac{(a_7+a_8-\beta _1-\eta _{12})(\tau _1+\tau _2)}{2}.\\ w_1(t)= & {} S^2(t),\\ w_2(t)= & {} E^2(t),\\ w_3(t)= & {} P_1^2(t)+\{a_6 \beta _1+\beta _1(a_7-\beta _1-\eta _{12})\}\\&\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds+\{a_6 \eta _{12}+\eta _{12} (a_7-\beta _1-\eta _{12})\}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_4(t)= & {} P_2^2(t)+\{a_8 \beta _1+\beta _1(\beta _1+\eta _{12})\}\\&\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds+\{a_8 \eta _{12}+\eta _{12} (\beta _1+\eta _{12})\}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_5(t)= & {} S(t)E(t),\\ w_6(t)= & {} S(t) P_1(t)+\frac{(a_1+a_2)\beta _1}{2}\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds+\frac{(a_1+a_2)\eta _{12}}{2}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_7(t)= & {} S(t) P_2(t)+\frac{(a_1+a_2)\beta _1}{2}\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds+\frac{(a_1+a_2)\eta _{12}}{2}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_8(t)= & {} E(t) P_1(t)+\frac{(a_3+a_4+a_5)\beta _1}{2}\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds\\&+\frac{(a_3+a_4+a_5)\eta _{12}}{2}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_9(t)= & {} E(t) P_2(t)+\frac{(a_3+a_4+a_5)\beta _1}{2}\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds\\&+\frac{(a_3+a_4+a_5)\eta _{12}}{2}\int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds,\\ w_{10}(t)= & {} P_1(t)P_2(t)+\frac{\beta _1\{-a_8-(\beta _1+\eta _{12})+a_6 +(a_7-\beta _1-\eta _{12})\}}{2}\int _{t-\tau _1}^{t}\int _{s}^{t}I^2(l) dl ds\\&+\frac{\eta _{12} \{-a_8-(\beta _1+\eta _{12})+a_6+(a_7-\beta _1-\eta _{12})\}}{2} \int _{t-\tau _2}^{t}\int _{s}^{t}I^2(l) dl ds. \end{aligned}$$

Appendix B

Derivation of $\Lambda _1$, $\Lambda _2$, $\Lambda _3$, $\Lambda _4$

$$\begin{aligned} \Lambda _1= & {} 2a_1k_1+a_3k_5+\frac{a_1\beta _1\tau _1+a_1\eta _{12} \tau _2}{2}(k_7-k_6) +\frac{a_3\beta _1\tau _1+a_3\eta _{12} \tau _2}{2}(k_9-k_8),\\ \Lambda _2= & {} 2a_5k_2-(a_6\beta _1\tau _1+a_6\eta _{12}\tau _2)k_3+\frac{2a_6-a_5\beta _1\tau _1 -a_5\eta _{12}\tau _2}{2}k_8\\&+\frac{a_5\beta _1\tau _1+a_5\eta _{12}\tau _2}{2}k_9 +\frac{a_6\beta _1\tau _1+a_6\eta _{12}\tau _2}{2}k_{10},\\ \Lambda _3= & {} k_3[(a_7-\beta _1-\eta _{12})(2-2\beta _1\tau _1-2\eta _{12}\tau _2) -a_6(\beta _1 \tau _1+\eta _{12} \tau _2)]\\&+k_4[(\beta _1+\eta _{12})(2\beta _1\tau _1 +2\eta _{12}\tau _2)+a_8(\beta _1\tau _1+\eta _{12}\tau _2)]\\&+k_6\left[ a_2-a_2\beta _1\tau _1 -a_2\eta _{12}\tau _2-\frac{a_1(\beta _1\tau _1+\eta _{12}\tau _2)}{2}\right] \\&+k_7\left[ a_2\beta _1 \tau _1+a_2\eta _{12}\tau _2+\frac{a_1(\beta _1\tau _1+\eta _{12}\tau _2)}{2}\right] \\&+k_8\left[ a_4-a_4\beta _1\tau _1-a_4\eta _{12}\tau _2-\frac{a_3(\beta _1\tau _1+\eta _{12}\tau _2) +a_5(\beta _1\tau _1+\eta _{12}\tau _2)}{2}\right] \\&+k_9\frac{2a_4\beta _1\tau _1+2a_4\eta _{12}\tau _2 +a_3(\beta _1\tau _1+\eta _{12}\tau _2)+a_5(\beta _1\tau _1+\eta _{12}\tau _2)}{2}\\&+k_{10}\left[ \frac{(\beta _1+\eta _{12})(2-2\beta _1\tau _1-2\eta _{12}\tau _2)}{2} +\frac{(a_7-\beta _1-\eta _{12})(2\eta _{12}\tau _2+\beta _1\tau _1)}{2}\right. \\&\left. +\frac{(\beta _1\tau _1+\eta _{12}\tau _2)(a_6-a_8)}{2}\right] ,\\ \Lambda _4= & {} k_4(2a_8+a_8\beta _1\tau _1+a_8\eta _{12}\tau _2)-k_{10}\frac{a_8\beta _1\tau _1 +a_8\eta _{12}\tau _2}{2}. \end{aligned}$$

Appendix C

Derivation of $\Gamma _{ij}, i, j=1, 2, 3, 4$.

$$\begin{aligned} \Gamma _{11}= & {} \frac{a_2}{a_6(\lambda _1-a_1)}\Lambda _{11}e^{\lambda _1t}-\frac{a_2}{a_6(\lambda _2-a_1)}\Lambda _{12}e^{\lambda _2t}+\frac{a_2}{a_6(\lambda _3-a_1)}\Lambda _{13}e^{\lambda _3t}\\&-\frac{a_2}{a_6(\lambda _4-a_1)}\Lambda _{14}e^{\lambda _4t},\\ \Gamma _{12}= & {} -a_2^2\Lambda _{21}e^{\lambda _1t}+a_2^2\Lambda _{22}e^{\lambda _2t}-a_2^2\Lambda _{23}e^{\lambda _3t}+a_2^2\Lambda _{24}e^{\lambda _4t},\\ \Gamma _{13}= & {} -\frac{a_2^2}{a_6}\Lambda _{31}e^{\lambda _1 t}+\frac{a_2^2}{a_6}\Lambda _{32}e^{\lambda _2 t}-\frac{a_2^2}{a_6}\Lambda _{33}e^{\lambda _3 t}+\frac{a_2^2}{a_6}\Lambda _{34}e^{\lambda _4 t},\\ \Gamma _{14}= & {} -\frac{a_2^2}{a_6}\Lambda _{41}e^{\lambda _1 t}+\frac{a_2^2}{a_6}\Lambda _{42}e^{\lambda _2 t}-\frac{a_2^2}{a_6}\Lambda _{43}e^{\lambda _3 t}+\frac{a_2^2}{a_6}\Lambda _{44}e^{\lambda _4 t},\\ \Gamma _{21}= & {} \frac{\beta _1-a_7+\eta _{12}+\lambda _1}{a_6^2}\Lambda _{11}e^{\lambda _1t} -\frac{\beta _1-a_7+\eta _{12}+\lambda _2}{a_6^2}\Lambda _{12}e^{\lambda _2t}\\&+\frac{\beta _1-a_7+\eta _{12}+\lambda _3}{a_6^2}\Lambda _{13}e^{\lambda _3t}-\frac{\beta _1-a_7+\eta _{12}+\lambda _4}{a_6^2}\Lambda _{14}e^{\lambda _4t},\\ \Gamma _{22}= & {} -\frac{a_2(\lambda _1-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1)}{a_6} \Lambda _{21}e^{\lambda _1t}\\&+\frac{a_2(\lambda _2-a_1)(\beta _1-a_7+\eta _{12}+\lambda _2)}{a_6}\Lambda _{22}e^{\lambda _2t}\\&-\frac{a_2(\lambda _3-a_1) (\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6}\Lambda _{23}e^{\lambda _3t}\\&+\frac{a_2(\lambda _4-a_1)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6}\Lambda _{24}e^{\lambda _4t},\\ \Gamma _{23}= & {} -\frac{a_2 (\lambda _1-a_1)}{a_6^2}\Lambda _{31}e^{\lambda _1 t} +\frac{a_2 (\lambda _2-a_1)}{a_6^2}\Lambda _{32}e^{\lambda _2 t} -\frac{a_2 (\lambda _3-a_1)}{a_6^2}\Lambda _{33}e^{\lambda _3 t}\\&+\frac{a_2 (\lambda _4-a_1)}{a_6^2}\Lambda _{34}e^{\lambda _4 t},\\ \Gamma _{24}= & {} -\frac{a_2 (\lambda _1-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1)}{a_6^2} \Lambda _{41}e^{\lambda _1 t}\\&+\frac{a_2 (\lambda _2-a_1)(\beta _1-a_7+\eta _{12} +\lambda _2)}{a_6^2}\Lambda _{42}e^{\lambda _2 t}\\&-\frac{a_2 (\lambda _3-a_1)(\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6^2}\Lambda _{43 }e^{\lambda _3 t}\\&+\frac{a_2 (\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _4)}{a_6^2}\Lambda _{44}e^{\lambda _4 t},\\ \Gamma _{31}= & {} \frac{1}{a_6}\Lambda _{11}e^{\lambda _1t} -\frac{1}{a_6}\Lambda _{12}e^{\lambda _2t}+\frac{1}{a_6}\Lambda _{13}e^{\lambda _3t} -\frac{1}{a_6}\Lambda _{14}e^{\lambda _4t},\\ \Gamma _{32}= & {} -a_2(\lambda _1-a_1)\Lambda _{21}e^{\lambda _1t} +a_2(\lambda _2-a_1)\Lambda _{22}e^{\lambda _2t}-a_2(\lambda _3-a_1)\Lambda _{23}e^{\lambda _3t}\\&+a_2(\lambda _4-a_1)\Lambda _{24}e^{\lambda _4t},\\ \Gamma _{33}= & {} -\frac{a_2 (\lambda _1-a_1)}{a_6}\Lambda _{31}e^{\lambda _1 t} +\frac{a_2 (\lambda _2-a_1)}{a_6}\Lambda _{32}e^{\lambda _2 t} -\frac{a_2 (\lambda _3-a_1)}{a_6}\Lambda _{33}e^{\lambda _3 t}\\&+\frac{a_2 (\lambda _4-a_1)}{a_6}\Lambda _{34}e^{\lambda _4 t},\\ \Gamma _{34}= & {} -\frac{a_2 (\lambda _1-a_1)}{a_6}\Lambda _{41}e^{\lambda _1 t} +\frac{a_2 (\lambda _2-a_1)}{a_6}\Lambda _{42}e^{\lambda _2 t} -\frac{a_2 (\lambda _3-a_1)}{a_6}\Lambda _{43}e^{\lambda _3 t}\\&+\frac{a_2 (\lambda _4-a_1)}{a_6}\Lambda _{44}e^{\lambda _4 t},\\ \Gamma _{41}= & {} \frac{\beta _1+\eta _{12}}{a_6(\lambda _1-a_8)}\Lambda _{11}e^{\lambda _1t} -\frac{\beta _1+\eta _{12}}{a_6(\lambda _2-a_8)}\Lambda _{12}e^{\lambda _2t} +\frac{\beta _1+\eta _{12}}{a_6(\lambda _3-a_8)}\Lambda _{13}e^{\lambda _3t}\\&-\frac{\beta _1+\eta _{12}}{a_6(\lambda _4-a_8)}\Lambda _{14}e^{\lambda _4t},\\ \Gamma _{42}= & {} -\frac{a_2(\lambda _1-a_1)(\beta _1+\eta _{12})}{\lambda _1-a_8} \Lambda _{21}e^{\lambda _1t} +\frac{a_2(\lambda _2-a_1)(\beta _1+\eta _{12})}{\lambda _2-a_8}\Lambda _{22}e^{\lambda _2t}\\&-\frac{a_2(\lambda _3-a_1)(\beta _1+\eta _{12})}{\lambda _3-a_8}\Lambda _{23}e^{\lambda _3t} +\frac{a_2(\lambda _4-a_1)(\beta _1+\eta _{12})}{\lambda _4-a_8}\Lambda _{24}e^{\lambda _4t},\\ \Gamma _{43}= & {} -\frac{a_2 (\lambda _1-a_1)(\lambda _1-a_8)(\beta _1 +\eta _{12})}{a_6}\Lambda _{31}e^{\lambda _1 t} +\frac{a_2 (\lambda _2-a_1)(\lambda _2-a_8)(\beta _1+\eta _{12})}{a_6}\Lambda _{32}e^{\lambda _2 t}\\&-\frac{a_2 (\lambda _3-a_1)(\lambda _3-a_8)(\beta _1+\eta _{12})}{a_6}\Lambda _{33}e^{\lambda _3 t} +\frac{a_2 (\lambda _4-a_1)(\lambda _4-a_8)(\beta _1+\eta _{12})}{a_6}\Lambda _{34}e^{\lambda _4 t},\\ \Gamma _{44}= & {} -\frac{a_2 (\lambda _1-a_1)(\beta _1+\eta _{12})}{a_6 (\lambda _1-a_8)} \Lambda _{41}e^{\lambda _1 t}+\frac{a_2 (\lambda _2-a_1)(\beta _1+\eta _{12})}{a_6 (\lambda _2-a_8)} \Lambda _{42}e^{\lambda _2 t}\\&-\frac{a_2 (\lambda _3-a_1)(\beta _1+\eta _{12})}{a_6 (\lambda _3-a_8)}\Lambda _{43}e^{\lambda _3 t} +\frac{a_2 (\lambda _4-a_1)(\beta _1+\eta _{12})}{a_6 (\lambda _4-a_8)}\Lambda _{44}e^{\lambda _4 t},\\ \end{aligned}$$

and

$$\begin{aligned} \Lambda _{11}= & {} \frac{(\lambda _1-a_8)(\lambda _2-\lambda _3)(\lambda _2-\lambda _4)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3},\\ \Lambda _{12}= & {} \frac{(\lambda _1-\lambda _3)(\lambda _1-\lambda _4)(\lambda _2-a_8)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3},\\ \Lambda _{13}= & {} \frac{(\lambda _3-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _4)(\lambda _2-\lambda _4)}{(\beta _1+\eta _{12})^3},\\ \Lambda _{14}= & {} \frac{(\lambda _4-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _3)(\lambda _2-\lambda _3)}{(\beta _1+\eta _{12})^3},\\ \Lambda _{21}= & {} \frac{(a_1-a_8)(\lambda _1-a_8)(\lambda _2-\lambda _3)(\lambda _2-\lambda _4)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{22}= & {} \frac{(a_1-a_8)(\lambda _2-a_8)(\lambda _1-\lambda _3)(\lambda _1-\lambda _4)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{23}= & {} \frac{(a_1-a_8)(\lambda _3-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _4)(\lambda _2-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{24}= & {} \frac{(a_1-a_8)(\lambda _4-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _3)(\lambda _2-\lambda _3)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-1)},\\ \Lambda _{31}= & {} \frac{(a_1-a_8)(\lambda _1-a_8)(\lambda _2-\lambda _3)(\lambda _2-\lambda _4)(\lambda _3-\lambda _4)(a_1+a_7+a_8-\beta _1-\eta _{12}-\lambda _2-\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{32}= & {} \frac{(a_1-a_8)(\lambda _2-a_8)(\lambda _1-\lambda _3)(\lambda _1-\lambda _4)(\lambda _3-\lambda _4)(a_1+a_7+a_8-\beta _1-\eta _{12}-\lambda _1-\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{33}= & {} \frac{(a_1-a_8)(\lambda _3-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _4)(\lambda _2-\lambda _4)(a_1+a_7+a_8-\beta _1-\eta _{12}-\lambda _1-\lambda _2-\lambda _4)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{34}= & {} \frac{(a_1-a_8)(\lambda _4-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _3)(\lambda _2-\lambda _3)(a_1+a_7+a_8-\beta _1-\eta _{12}-\lambda _2-\lambda _3-\lambda _1)}{(\beta _1+\eta _{12})^3(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{41}= & {} \frac{(\lambda _1-a_8)(\lambda _2-a_8)(\lambda _3-a_8)(\lambda _4-a_8)(\lambda _2-\lambda _3)(\lambda _2-\lambda _4)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^4(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{42}= & {} \frac{(\lambda _1-a_8)(\lambda _2-a_8)(\lambda _3-a_8)(\lambda _4-a_8)(\lambda _1-\lambda _3)(\lambda _1-\lambda _4)(\lambda _3-\lambda _4)}{(\beta _1+\eta _{12})^4(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{43}= & {} \frac{(\lambda _1-a_8)(\lambda _2-a_8)(\lambda _3-a_8)(\lambda _4-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _4)(\lambda _2-\lambda _4)}{(\beta _1+\eta _{12})^4(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)},\\ \Lambda _{44}= & {} \frac{(\lambda _1-a_8)(\lambda _2-a_8)(\lambda _3-a_8)(\lambda _4-a_8)(\lambda _1-\lambda _2)(\lambda _1-\lambda _3)(\lambda _2-\lambda _3)}{(\beta _1+\eta _{12})^4(\lambda _1-a_1)(\lambda _2-a_1)(\lambda _3-a_1)(\lambda _4-a_1)}.\\ \end{aligned}$$

Appendix D

Derivation of $\sigma _{S}^2$, $\sigma _{E}^2$, $\sigma _{I}^2$, $\sigma _{R}^2$.

$$\begin{aligned} \sigma _{S}^2= & {} <S^2(t)>-(<S(t)>)^2\\= & {} -\frac{1}{\Delta ^2}\left[ \left\{ \frac{a_2^2}{a_6^2(\lambda _1-a_1)^2}\Lambda _{11}^2+a_2^4\Lambda _{21}^2+\frac{a_2^4}{a_6^2}\Lambda _{31}^2+\frac{a_2^4}{a_6^2}\Lambda _{41}^2\right\} \frac{1-e^{2\lambda _1 t}}{2\lambda _1}\right. \\&+\left\{ \frac{a_2^2}{a_6^2(\lambda _2-a_1)^2}\Lambda _{12}^2+a_2^4\Lambda _{22}^2+\frac{a_2^4}{a_6^2}\Lambda _{32}^2+\frac{a_2^4}{a_6^2}\Lambda _{42}^2\right\} \frac{1-e^{2\lambda _2 t}}{2\lambda _2}\\&+\left\{ \frac{a_2^2}{a_6^2(\lambda _3-a_1)^2}\Lambda _{13}^2+a_2^4\Lambda _{23}^2+\frac{a_2^4}{a_6^2}\Lambda _{33}^2+\frac{a_2^4}{a_6^2}\Lambda _{43}^2\right\} \frac{1-e^{2\lambda _3 t}}{2\lambda _3}\\&+\left\{ \frac{a_2^2}{a_6^2(\lambda _4-a_1)^2}\Lambda _{14}^2+a_2^4\Lambda _{24}^2+\frac{a_2^4}{a_6^2}\Lambda _{34}^2+\frac{a_2^4}{a_6^2}\Lambda _{44}^2\right\} \frac{1-e^{2\lambda _4 t}}{2\lambda _4}\\&+2\frac{1-e^{\overline{\lambda _1+\lambda _2}t}}{\lambda _1+\lambda _2}\left\{ \frac{a_2^2}{a_6^2(\lambda _1-a_1)(\lambda _2-a_1)}\Lambda _{11}\Lambda _{12}+a_2^4\Lambda _{21}\Lambda _{22}+\frac{a_2^4}{a_6^2}\Lambda _{31}\Lambda _{32}+\frac{a_2^4}{a_6^2}\Lambda _{41}\Lambda _{42}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _3}t}}{\lambda _1+\lambda _3}\left\{ \frac{a_2^2}{a_6^2(\lambda _1-a_1)(\lambda _3-a_1)}\Lambda _{11}\Lambda _{13}+a_2^4\Lambda _{21}\Lambda _{23}+\frac{a_2^4}{a_6^2}\Lambda _{31}\Lambda _{33}+\frac{a_2^4}{a_6^2}\Lambda _{41}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _4}t}}{\lambda _1+\lambda _4}\left\{ \frac{a_2^2}{a_6^2(\lambda _1-a_1)(\lambda _4-a_1)}\Lambda _{11}\Lambda _{14}+a_2^4\Lambda _{21}\Lambda _{24}+\frac{a_2^4}{a_6^2}\Lambda _{31}\Lambda _{34}+\frac{a_2^4}{a_6^2}\Lambda _{41}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _3}t}}{\lambda _2+\lambda _3}\left\{ \frac{a_2^2}{a_6^2(\lambda _2-a_1)(\lambda _3-a_1)}\Lambda _{12}\Lambda _{13}+a_2^4\Lambda _{22}\Lambda _{23}+\frac{a_2^4}{a_6^2}\Lambda _{32}\Lambda _{33}+\frac{a_2^4}{a_6^2}\Lambda _{42}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _4}t}}{\lambda _2+\lambda _4}\left\{ \frac{a_2^2}{a_6^2(\lambda _2-a_1)(\lambda _4-a_1)}\Lambda _{12}\Lambda _{14}+a_2^4\Lambda _{22}\Lambda _{24}+\frac{a_2^4}{a_6^2}\Lambda _{32}\Lambda _{34}+\frac{a_2^4}{a_6^2}\Lambda _{42}\Lambda _{44}\right\} \\&\left. +2\frac{1-e^{\overline{\lambda _3+\lambda _4}t}}{\lambda _3+\lambda _4}\left\{ \frac{a_2^2}{a_6^2(\lambda _3-a_1)(\lambda _4-a_1)}\Lambda _{13}\Lambda _{14}+a_2^4\Lambda _{23}\Lambda _{24}+\frac{a_2^4}{a_6^2}\Lambda _{33}\Lambda _{34}+\frac{a_2^4}{a_6^2}\Lambda _{43}\Lambda _{44}\right\} \right] , \end{aligned}$$

$$\begin{aligned} \sigma _{E}^2= & {} <E^2(t)>-(<E(t))>^2\\= & {} -\frac{1}{\Delta ^2}\left[ \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _1)^2}{a_6^4}\Lambda _{11}^2+\frac{a_2^2(\lambda _1-a_1)^2}{a_6^2}\Lambda _{21}^2+\frac{a_2^2(\lambda _1-a_1)^2}{a_6^4}\Lambda _{31}^2\right. \right. \\&\left. +\frac{a_2^2(\lambda _1-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _1)^2}{a_6^4}\Lambda _{41}^2\right\} \frac{1-e^{2\lambda _1t}}{2\lambda _1}\\&+\left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _2)^2}{a_6^4}\Lambda _{12}^2+\frac{a_2^2(\lambda _2-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _2)^2}{a_6^2}\Lambda _{22}^2+\frac{a_2^2(\lambda _2-a_1)^2}{a_6^4}\Lambda _{32}^2\right. \\&\left. +\frac{a_2^2(\lambda _2-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _2)^2}{a_6^4}\Lambda _{42}^2\right\} \frac{1-e^{2\lambda _2t}}{2\lambda _2}\\&+\left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _3)^2}{a_6^2}\Lambda _{13}^2+\frac{a_2^2(\lambda _3-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _3)^2}{a_6^2}\Lambda _{23}^2+\frac{a_2^2(\lambda _3-a_1)^2}{a_6^4}\Lambda _{33}^2\right. \\&\left. +\frac{a_2^2(\lambda _3-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _3)^2}{a_6^4}\Lambda _{43}^2\right\} \frac{1-e^{2\lambda _3t}}{2\lambda _3}\\&+\left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _4)^2}{a_6^4}\Lambda _{14}^2+\frac{a_2^2(\lambda _4-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _4)^2}{a_6^2}\Lambda _{24}^2+\frac{a_2^2(\lambda _4-a_1)^2}{a_6^4}\Lambda _{34}^2\right. \\&\left. +\frac{a_2^2(\lambda _4-a_1)^2(\beta _1-a_7+\eta _{12}+\lambda _4)^2}{a_6^4}\Lambda _{44}^2\right\} \frac{1-e^{2\lambda _4t}}{2\lambda _4}\\&+2\frac{1-e^{\overline{\lambda _1+\lambda _2}t}}{\lambda _1+\lambda _2} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _1)(\beta _1-a_7+\eta _{12}+\lambda _2)}{a_6^4} \Lambda _{11}\Lambda _{12}+\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)}{a_6^4}\Lambda _{31}\Lambda _{32}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1) (\beta _1-a_7+\eta _{12}+\lambda _2)}{a_6^2}\Lambda _{21}\Lambda _{22}\\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1) (\beta _1-a_7+\eta _{12}+\lambda _2)}{a_6^4}\Lambda _{41}\Lambda _{42}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _3}t}}{\lambda _1+\lambda _3} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _1)(\beta _1-a_7+\eta _{12}+\lambda _3) }{a_6^4}\Lambda _{11}\Lambda _{13}+\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)}{a_6^4}\Lambda _{31}\Lambda _{33}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1) (\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6^2}\Lambda _{21}\Lambda _{23}\\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)(\beta _1-a_7+\eta _{12}+\lambda _1) (\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6^4}\Lambda _{41}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _4}t}}{\lambda _1+\lambda _4} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _1)(\beta _1-a_7+\eta _{12} +\lambda _4)}{a_6^4}\Lambda _{11}\Lambda _{14}+\frac{a_2^2(\lambda _1-a_1) (\lambda _4-a_1)}{a_6^4}\Lambda _{31}\Lambda _{34}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _1)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^2}\Lambda _{21}\Lambda _{24}\\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _1)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^4}\Lambda _{41}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _3}t}}{\lambda _2+\lambda _3} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _3 )}{a_6^4}\Lambda _{12}\Lambda _{13}+\frac{a_2^2(\lambda _2-a_1) (\lambda _3-a_1)}{a_6^4}\Lambda _{32}\Lambda _{33}\right. \\&+\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)(\beta _1-a_7+\eta _{12} +\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6^2}\Lambda _{22}\Lambda _{23}\\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)(\beta _1-a_7+\eta _{12} +\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _3)}{a_6^4}\Lambda _{42}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _4}t}}{\lambda _2+\lambda _4} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^4}\Lambda _{12}\Lambda _{14}+\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1) }{a_6^4}\Lambda _{32}\Lambda _{34}\right. \\&+\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^2}\Lambda _{22}\Lambda _{24}\\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _2)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^4}\Lambda _{42}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _3+\lambda _4}t}}{\lambda _4+\lambda _3} \left\{ \frac{(\beta _1-a_7+\eta _{12}+\lambda _3)(\beta _1-a_7+\eta _{12} +\lambda _4)}{a_6^4}\Lambda _{13}\Lambda _{14}+\frac{a_2^2(\lambda _3-a_1) (\lambda _4-a_1)}{a_6^4}\Lambda _{33}\Lambda _{34}\right. \\&+\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _3)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^2}\Lambda _{23}\Lambda _{24}\\&\left. \left. +\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1)(\beta _1-a_7+\eta _{12} +\lambda _3)(\beta _1-a_7+\eta _{12}+\lambda _4)}{a_6^4}\Lambda _{43}\Lambda _{44}\right\} \right] , \end{aligned}$$

$$\begin{aligned} \sigma _{I}^2= & {} <I^2(t)>-(<I(t)>)^2\\= & {} -\frac{1}{\Delta ^2}\left[ \left\{ \frac{1}{a_6^2}\Lambda _{11}^2 +a_2^2(\lambda _1-a_1)^2\Lambda _{21}^2+\frac{a_2^2(\lambda _1-a_1)^2}{a_6^2}\Lambda _{31}^2 +\frac{a_2(\lambda _1-a_1)^2}{a_6^2}\Lambda _{41}\right\} \frac{1-e^{2\lambda _1t}}{2\lambda _1} \right. \\&+\left\{ \frac{1}{a_6^2}\Lambda _{12}^2+a_2^2(\lambda _2-a_1)^2\Lambda _{22}^2 +\frac{a_2^2(\lambda _2-a_1)^2}{a_6^2}\Lambda _{32}^2 +\frac{a_2^2(\lambda _2-a_1)^2}{a_6^2}\Lambda _{42}^2\right\} \frac{1-e^{2\lambda _2t}}{2\lambda _2} \\&+\left\{ \frac{1}{a_6^2}\Lambda _{13}^2+a_2^2(\lambda _3-a_1)^2\Lambda _{23}^2 +\frac{a_2^2(\lambda _3-a_1)^2}{a_6^2}\Lambda _{33}^2+\frac{a_2^2(\lambda _3-a_1)^2 }{a_6^2}\Lambda _{43}^2\right\} \frac{1-e^{2\lambda _3t}}{2\lambda _3} \\&+\left\{ \frac{1}{a_6^2}\Lambda _{14}^2+a_2^2(\lambda _4-a_1)^2\Lambda _{24}^2 +\frac{a_2^2(\lambda _4-a_1)^2}{a_6^2}\Lambda _{34}^2+\frac{a_2^2(\lambda _4-a_1 )^2}{a_6^2}\Lambda _{44}^2\right\} \frac{1-e^{2\lambda _4t}}{2\lambda _4} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _2}t}}{\lambda _1+\lambda _2} \left\{ \frac{1}{a_6^2}\Lambda _{11}\Lambda _{12}+a_2^2(\lambda _1-a_1)(\lambda _2-a_1) \Lambda _{21}\Lambda _{22}+\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)}{a_6^2}\Lambda _{31}\Lambda _{32}\right. \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)}{a_6^2}\Lambda _{41}\Lambda _{42}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _3}t}}{\lambda _1+\lambda _3 }\left\{ \frac{1}{a_6^2}\Lambda _{11}\Lambda _{13}+a_2^2(\lambda _1-a_1) (\lambda _3-a_1)\Lambda _{21}\Lambda _{23}+\frac{a_2^2(\lambda _1-a_1) (\lambda _3-a_1)}{a_6^2}\Lambda _{31}\Lambda _{33}\right. \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)}{a_6^2}\Lambda _{41}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _4}t}}{\lambda _1+\lambda _4} \left\{ \frac{1}{a_6^2}\Lambda _{11}\Lambda _{14}+a_2^2(\lambda _1-a_1)(\lambda _4-a_1) \Lambda _{21}\Lambda _{24}+\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{31}\Lambda _{34}\right. \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{41}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _3}t}}{\lambda _2+\lambda _3} \left\{ \frac{1}{a_6^2}\Lambda _{12}\Lambda _{13}+a_2^2(\lambda _2-a_1)(\lambda _3-a_1) \Lambda _{22}\Lambda _{23}+\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)}{a_6^2}\Lambda _{32}\Lambda _{33}\right. \\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)}{a_6^2}\Lambda _{42}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _4}t}}{\lambda _2+\lambda _4} \left\{ \frac{1}{a_6^2}\Lambda _{12}\Lambda _{14}+a_2^2(\lambda _2-a_1)(\lambda _4-a_1) \Lambda _{22}\Lambda _{24}+\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{32}\Lambda _{34}\right. \\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{42}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _3+\lambda _4}t}}{\lambda _3+\lambda _4} \left\{ \frac{1}{a_6^2}\Lambda _{13}\Lambda _{14}+a_2^2(\lambda _3-a_1)(\lambda _4-a_1) \Lambda _{23}\Lambda _{24}+\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{33}\Lambda _{34}\right. \\&\left. \left. +\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1)}{a_6^2}\Lambda _{43}\Lambda _{44}\right\} \right] , \end{aligned}$$

$$\begin{aligned} \sigma _{R}^2= & {} <R^2(t)>-(<R(t)>)^2 \\= & {} -\frac{1}{\Delta ^2}\left[ \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _1-a_8)^2} \Lambda _{11}^2+\frac{a_2^2(\lambda _1-a_1)^2(\beta _1+\eta _{12})^2}{(\lambda _1-a_8)^2}\Lambda _{21}^2\right. \right. \\&\left. +\frac{a_2^2(\lambda _1-a_1)^2(\lambda _1-a_8)^2(\beta _1+\eta _{12})^2}{a_6^2} \Lambda _{31}^2+\frac{a_2^2(\lambda _1-a_1)^2(\beta _1+\eta _{12})^2}{a_6^2 (\lambda _1-a_8)^2}\Lambda _{41}^2\right\} \frac{1-e^{2\lambda _1t}}{2\lambda _1} \\&+\left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _2-a_8)^2}\Lambda _{12}^2 +\frac{a_2^2(\lambda _2-a_1)^2(\beta _1+\eta _{12})^2}{(\lambda _2-a_8)^2}\Lambda _{22}^2\right. \\&\left. +\frac{a_2^2(\lambda _2-a_1)^2(\lambda _2-a_8)^2(\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{32}^2+\frac{a_2^2(\lambda _2-a_1)^2(\beta _1+\eta _{12})^2}{a_6^2(\lambda _2-a_8)^2}\Lambda _{42}^2\right\} \frac{1-e^{2\lambda _2t}}{2\lambda _2} \\&+\left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _3-a_8)^2}\Lambda _{13}^2 +\frac{a_2^2(\lambda _3-a_1)^2(\beta _1+\eta _{12})^2}{(\lambda _3-a_8)^2}\Lambda _{23}^2\right. \\&\left. +\frac{a_2^2(\lambda _3-a_1)^2(\lambda _3-a_8)^2(\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{33}^2+\frac{a_2^2(\lambda _3-a_1)^2(\beta _1+\eta _{12})^2}{a_6^2(\lambda _3-a_8)^2}\Lambda _{43}^2\right\} \frac{1-e^{2\lambda _3t}}{2\lambda _3} \\&+\left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _4-a_8)^2}\Lambda _{14}^2 +\frac{a_2^2(\lambda _4-a_1)^2(\beta _1+\eta _{12})^2}{(\lambda _4-a_8)^2}\Lambda _{24}^2\right. \\&\left. +\frac{a_2^2(\lambda _4-a_1)^2(\lambda _4-a_8)^2(\beta _1+\eta _{12})^2 }{a_6^2}\Lambda _{34}^2+\frac{a_2^2(\lambda _4-a_1)^2(\beta _1+\eta _{12} )^2}{a_6^2(\lambda _4-a_8)^2}\Lambda _{44}^2\right\} \frac{1-e^{2\lambda _4t}}{2\lambda _4} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _2}t}}{\lambda _1+\lambda _2} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _1-a_8)(\lambda _2-a_8)} \Lambda _{11}\Lambda _{12}+\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)(\beta _1 +\eta _{12})^2}{(\lambda _1-a_8)(\lambda _2-a_8)}\Lambda _{21}\Lambda _{22}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _1-a_8)(\lambda _2-a_1)(\lambda _2-a_8 )(\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{31}\Lambda _{32} \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _2-a_1)(\beta _1+\eta _{12})^2}{a_6^2 (\lambda _1-a_8)(\lambda _2-a_8)}\Lambda _{41}\Lambda _{42}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _3}t}}{\lambda _1+\lambda _3} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _1-a_8)(\lambda _3-a_8)} \Lambda _{11}\Lambda _{13}+\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)(\beta _1+ \eta _{12})^2}{(\lambda _1-a_8)(\lambda _3-a_8)}\Lambda _{21}\Lambda _{23}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _1-a_8)(\lambda _3-a_1)(\lambda _3-a_8) (\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{31}\Lambda _{33} \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _3-a_1)(\beta _1+\eta _{12})^2}{a_6^2 (\lambda _1-a_8)(\lambda _3-a_8)}\Lambda _{41}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _1+\lambda _4}t}}{\lambda _1+\lambda _4} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _1-a_8)(\lambda _4-a_8)} \Lambda _{11}\Lambda _{14}+\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)(\beta _1 +\eta _{12})^2}{(\lambda _1-a_8)(\lambda _4-a_8)}\Lambda _{21}\Lambda _{24}\right. \\&+\frac{a_2^2(\lambda _1-a_1)(\lambda _1-a_8)(\lambda _4-a_1)(\lambda _4-a_8) (\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{31}\Lambda _{34} \\&\left. +\frac{a_2^2(\lambda _1-a_1)(\lambda _4-a_1)(\beta _1+\eta _{12})^2}{a_6^2 (\lambda _1-a_8)(\lambda _4-a_8)}\Lambda _{41}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _3}t}}{\lambda _2+\lambda _3} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _2-a_8)(\lambda _3-a_8)}\Lambda _{12} \Lambda _{13}+\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)(\beta _1+\eta _{12})^2}{(\lambda _2-a_8)(\lambda _3-a_8)}\Lambda _{22}\Lambda _{23}\right. \\&+\frac{a_2^2(\lambda _2-a_1)(\lambda _2-a_8)(\lambda _3-a_1)(\lambda _3-a_8) (\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{32}\Lambda _{33} \\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _3-a_1)(\beta _1+\eta _{12})^2}{a_6^2 (\lambda _2-a_8)(\lambda _3-a_8)}\Lambda _{42}\Lambda _{43}\right\} \\&+2\frac{1-e^{\overline{\lambda _2+\lambda _4}t}}{\lambda _2+\lambda _4} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _2-a_8)(\lambda _4-a_8)} \Lambda _{12}\Lambda _{14}+\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1) (\beta _1+\eta _{12})^2}{(\lambda _2-a_8)(\lambda _4-a_8)}\Lambda _{22}\Lambda _{24}\right. \\&+\frac{a_2^2(\lambda _2-a_1)(\lambda _2-a_8)(\lambda _4-a_1) (\lambda _4-a_8)(\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{32}\Lambda _{34} \\&\left. +\frac{a_2^2(\lambda _2-a_1)(\lambda _4-a_1)(\beta _1+\eta _{12})^2}{a_6^2(\lambda _2-a_8)(\lambda _4-a_8)}\Lambda _{42}\Lambda _{44}\right\} \\&+2\frac{1-e^{\overline{\lambda _3+\lambda _4}t}}{\lambda _3+\lambda _4} \left\{ \frac{(\beta _1+\eta _{12})^2}{a_6^2(\lambda _3-a_8)(\lambda _4-a_8) }\Lambda _{13}\Lambda _{14}+\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1) (\beta _1+\eta _{12})^2}{(\lambda _3-a_8)(\lambda _4-a_8)}\Lambda _{23}\Lambda _{24}\right. \\&+\frac{a_2^2(\lambda _3-a_1)(\lambda _3-a_8)(\lambda _4-a_1) (\lambda _4-a_8)(\beta _1+\eta _{12})^2}{a_6^2}\Lambda _{33}\Lambda _{34} \\&\left. \left. +\frac{a_2^2(\lambda _3-a_1)(\lambda _4-a_1)(\beta _1+\eta _{12})^2}{a_6^2(\lambda _3-a_8)(\lambda _4-a_8)}\Lambda _{43}\Lambda _{44}\right\} \right] . \end{aligned}$$