## 4 Material and Methods
### Mathematical Formulations and Solution of the Problem  
Based on the state transition diagram, the following set of Chapman-Kolmogorov differential equations, Equations (1)-(10), are generated, which govern the present mathematical model. Letting the system make the transition at time \(t+\Delta t\) and letting \(\Delta t\to 0\), the following set of differential equations is obtained:  
\[ \begin{array}{c}P_{0}(t+\Delta\ t)=(1-\lambda_{mic}\Delta t)(1-\lambda_{mix }\Delta t)\big{(}1-\lambda_{amp}\Delta t\big{)}\big{(}1-2\lambda_{sp}\Delta t \big{)}\\ +\mu_{mic}P_{1}(t)\Delta t++\mu_{mix}P_{2}(t)\Delta t++\mu_{amp}P_{3}(t) \Delta t+\mu_{sp}P_{4}(t)\Delta t\end{array} \]  
\[ \begin{array}{c}\frac{P_{0}(t+\Delta\ t)-P_{0}(t)}{\Delta\ t}+\big{(} \lambda_{mix}+\lambda_{mic}+\lambda_{amp}+2\lambda_{sp}\big{)}=\mu_{mic}P_{1 }(t)+\mu_{mix}P_{2}(t)\\ +\mu_{amp}P_{3}(t)+\mu_{sp}P_{4}(t)\end{array} \]  
\[ \begin{array}{c}lim_{\Delta t\to 0}\frac{P_{0}(t+\Delta\ t)-P_{0}(t)}{ \Delta\ t}+\big{(}\lambda_{mix}+\lambda_{mic}+\lambda_{amp}+2\lambda_{sp} \big{)}=\mu_{mic}P_{1}(t)+\mu_{mix}P_{2}(t)\\ +\mu_{amp}P_{3}(t)+\mu_{sp}P_{4}(t)\end{array}  \tag{1}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\lambda_{mic}+\lambda_{mix}+\lambda_{ amp}+2\lambda_{sp}\Big{]}P_{0}(t)=\mu_{mic}P_{1}(t)\\ +\mu_{mix}P_{2}(t)+\mu_{amp}P_{3}(t)+\mu_{sp}P_{4}(t)\end{array}  \tag{2}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\mu_{mic}\Big{]}P_{1}(t)=\lambda_{mic}P_ {0}(t)\\ +\mu_{mix}P_{2}(t)=\lambda_{mix}P_{0}(t)\end{array}  \tag{3}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\mu_{amp}\Big{]}P_{3}(t)=\lambda_{amp}P_ {0}(t)\end{array}  \tag{4}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\lambda_{mic}+\lambda_{mix}+\lambda_{amp }+\lambda_{sp}+\mu_{sp}\Big{]}P_{4}(t)\\ =\mu_{mic}P_{5}(t)+\mu_{mix}P_{6}(t)+\mu_{amp}P_{7}(t)+\mu_{sp}P_{8}(t)+2\lambda _{sp}P_{0}(t)\end{array}  \tag{5}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\mu_{mic}\Big{]}P_{5}(t)=\lambda_{mic}P_ {4}(t)\\ +\mu_{mix}P_{6}(t)=\lambda_{mix}P_{4}(t)\end{array}  \tag{6}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\mu_{amp}\Big{]}P_{7}(t)=\lambda_{amp}P_ {4}(t)\\ +\mu_{sp}P_{8}(t)=\lambda_{sp}P_{4}(t)\end{array}  \tag{7}\]  
\[ \begin{array}{c}\Big{[}\frac{d}{dt}+\mu_{sp}\Big{]}P_{8}(t)=\lambda_{sp}P_{4 }(t)\end{array}  \tag{8}\]With the initial condition  
\[P_{0}(t)= \begin{cases}1,&t=0\\ 0,&t\
eq 0\end{cases}  \tag{10}\]  
Upon performing Laplace transformations of Equations (1)-(9), one can obtain the following set of equations:  
\[ \begin{split}&\left[s+\lambda_{mic}+\lambda_{mix}+\lambda_{amp}+2 \lambda_{sp}\right]\overline{P}_{0}(s)=1+\\ &\mu_{mic}\overline{P}_{1}(s)+\mu_{mix}\overline{P}_{2}(s)+\mu_{ amp}\overline{P}_{3}(s)+\mu_{sp}\overline{P}_{4}(s)\end{split}  \tag{11}\]  
\[ \begin{split}&[s+\mu_{mic}]\overline{P}_{1}(s)=\lambda_{mic} \overline{P}_{0}(s)\end{split}  \tag{12}\]  
\[ \begin{split}&[s+\mu_{mix}]\overline{P}_{2}(s)=\lambda_{mix} \overline{P}_{0}(s)\end{split}  \tag{13}\]  
\[ \begin{split}&[s+\mu_{amp}]\overline{P}_{3}(s)=\lambda_{ amp}\overline{P}_{0}(s)\end{split}  \tag{14}\]  
\[ \begin{split}&\left[s+\lambda_{mic}+\lambda_{mix}+\lambda_{ amp}+\lambda_{sp}+\mu_{sp}\right]\overline{P}_{4}(s)=\\ &\mu_{mic}\overline{P}_{5}(s)+\mu_{mix}\overline{P}_{6}(s)+\mu_ {amp}\overline{P}_{7}(s)+2\lambda_{sp}\overline{P}_{0}(s)+\mu_{sp}\overline {P}_{8}(s)\end{split}  \tag{15}\]  
\[ \begin{split}&[s+\mu_{mic}]\overline{P}_{5}(s)=\lambda_{mic} \overline{P}_{4}(s)\end{split}  \tag{16}\]  
\[ \begin{split}&[s+\mu_{mix}]\overline{P}_{6}(s)=\lambda_{mix} \overline{P}_{4}(s)\end{split}  \tag{17}\]  
\[ \begin{split}&[s+\mu_{amp}]\overline{P}_{7}(s)=\lambda_{ amp}\overline{P}_{4}(s)\end{split}  \tag{18}\]  
\[ \begin{split}&\left[s+\mu_{sp}\right]\overline{P}_{8}(s)= \lambda_{sp}\overline{P}_{4}(s)\end{split}