# Understanding Neutrophil Dynamics during COVID-19 Infection
## 3 Results
### Asymptotic Analysis Results
Proposition 2: _The no-inflammation equilibrium \(S_{1}\) exists and is locally asymptotically stable for all parameters._Proof. : The Jacobian associated with system Equation (3) at equilibrium point \(S=(\bar{n},\bar{a},\bar{m},\bar{c},\bar{g},\bar{e})\) is  
\[J(S)= \begin{bmatrix}-\
u\bar{g}&0&0&\alpha_{c}&-\
u\bar{n}&0\\ \
u\bar{g}&-\gamma_{a}-\phi\bar{m}&-\phi\bar{a}&0&\
u\bar{n}&0\\ 0&0&-\gamma_{m}&\alpha_{m}&0&0\\ 0&2\gamma_{a}k_{a}\beta_{a}^{2}\frac{\bar{a}}{(\bar{a}^{2}+\beta_{a}^{2})^{2} }&0&-\gamma_{c}&0&0\\ 0&k_{g}\phi\bar{m}&k_{g}\phi\bar{a}&0&-\gamma_{g}&0\\ 0&0&\frac{\alpha_{e}}{1+\beta_{e}\bar{g}}&0&-\frac{\alpha_{e}\beta_{e}\bar{m} }{(1+\beta_{e}\bar{g})^{2}}&-\gamma_{e}\end{bmatrix} . \tag{12}\]  
The corresponding characteristic equation at the no-inflammation equilibrium point  
\[S_{1}=(\bar{n}_{1},\bar{a}_{1},\bar{m}_{1},\bar{c}_{1},\bar{g}_{1},\bar{e}_{1 })=(0,0,0,0,s_{g}/\gamma_{g},s_{e}/\gamma_{e})\]  
is  
\[\det(J(S_{1})-\lambda I_{6})=(\lambda+\gamma_{e})(\lambda+\gamma_{g})(\lambda+ \gamma_{c})(\lambda+\gamma_{m})(\lambda+\gamma_{a})(\lambda+\frac{s_{g}\
u}{ \gamma_{g}})=0,\]  
with eigenvalues \(\lambda_{1}=-\gamma_{a}<0\), \(\lambda_{2}=-\gamma_{c}<0\), \(\lambda_{3}=-\gamma_{e}<0\), \(\lambda_{4}=-\gamma_{g}<0\), \(\lambda_{5}=-\gamma_{m}<0\), and \(\lambda_{6}=-\frac{s_{g}\
u}{\gamma_{g}}<0\). Therefore, the no-inflammation equilibrium is always locally asymptotically stable. One of the chronic inflammation equlibria is always unstable and the other is locally asymptotically stable when condition (6) holds. Proof. : The characteristic equation for the chronic-inflammation equilibria \(S_{i}\), \(i\in\{2,3\}\) is  
\[\det(J(S_{i})-\lambda I_{6})=(\lambda+\gamma_{e})(\lambda^{5}+A_{1}\lambda^{4 }+A_{2}\lambda^{3}+A_{3}\lambda^{2}+A_{4}\lambda+A_{5})=0,\]  
where  
\[ \begin{split} A_{1}&=\bar{g}_{i}\
u+\bar{m}_{i}\phi+ \gamma_{a}+\gamma_{c}+\gamma_{g}+\gamma_{m},\\ A_{2}&=\bar{m}_{i}\
u\phi(\bar{g}_{i}-k_{g}\bar{n}_{i})+ \bar{g}_{i}(\gamma_{a}+\gamma_{c}+\gamma_{g}+\gamma_{m})\
u+(\bar{m}_{i}\phi+ \gamma_{a})(\gamma_{c}+\gamma_{g}+\gamma_{m})+\gamma_{c}(\gamma_{g}+\gamma_{ m})+\gamma_{g}\gamma_{m},\\ A_{3}&=\bar{m}_{i}\
u\phi(\bar{g}_{i}-k_{g}\bar{n}_{i})+ \bar{g}_{i}(\gamma_{a}+\gamma_{c}+\gamma_{g}+\gamma_{m})\
u+\bar{m}_{i}\phi( \gamma_{c}+\gamma_{g}+\gamma_{m})+\gamma_{a}(\gamma_{c}+\gamma_{g}+\gamma_{ m})\\ &+\gamma_{c}(\gamma_{g}+\gamma_{m})+\gamma_{g}\gamma_{m},\\ A_{4}&=\bar{g}_{i}\bar{m}_{i}(\gamma_{c}\gamma_{g}+ \gamma_{c}\gamma_{m}+\gamma_{g}\gamma_{m})\
u\phi+\bar{m}_{i}\gamma_{c}\gamma_{ m}(\gamma_{g}\phi-k_{g}\
u\phi\bar{n}_{i})+\bar{g}_{i}(\gamma_{a}\gamma_{c}( \gamma_{g}+\gamma_{m})+\gamma_{m}\gamma_{g}(\gamma_{a}+\gamma_{c})\
u\phi\\ &+\gamma_{a}\gamma_{c}\gamma_{g}\gamma_{m}-\frac{\beta_{a}^{2}}{( \bar{a}_{i}+\beta_{a}^{2})^{2}}(2\alpha_{m}\gamma_{a}(\bar{g}_{i}\
u-k_{g}\bar {n}_{i}\
u+\gamma_{g})\phi+\bar{g}_{i}\gamma_{c}\gamma_{g}\bar{m}_{i}\
u\phi+ \bar{g}_{i}(\gamma_{c}\gamma_{m}+\gamma_{g}\gamma_{m})\
u\phi\\ &-\gamma_{c}\gamma_{m}k_{g}\bar{m}_{i}\bar{n}_{i}\
u\phi+\bar{g}_{i}( \gamma_{a}\gamma_{c}\gamma_{g}+\gamma_{a}\gamma_{c}\gamma_{m}+\gamma_{a} \gamma_{g}\gamma_{m}+\gamma_{c}\gamma_{g}\gamma_{m})\
u+\bar{m}_{i}\gamma_{c} \gamma_{g}\gamma_{m}\phi+\gamma_{a}\gamma_{c}\gamma_{g}\gamma_{m}),\\ A_{5}&=\bar{g}_{i}\gamma_{c}\gamma_{g}\gamma_{m}\
u(\bar{m}_{i} \phi+\gamma_{a})+2\frac{\bar{a}_{i}\bar{g}_{i}\beta_{a}^{2}}{(\beta_{a}^{2}+ \bar{a}_{i}^{2})^{2}}(\bar{a}_{i}\alpha_{m}\gamma_{g}\phi-\alpha_{c})\gamma_{ a}\
u.\end{split}  \tag{13}\]  
By Routh-Hurwitz conditions, we have eigenvalues with negative real parts (and hence local asymptotoc stability for \(S_{2,3}\)) if and only if the Hurwitz matrices \(H_{i}\) for \(i=\{1,\ldots,5\}\)\[H_{1} =[A_{1}],\] \[H_{2} = \begin{bmatrix}A_{1}&A_{3}\\ 1&A_{2}\end{bmatrix} ,\] \[H_{3} = \begin{bmatrix}A_{1}&A_{3}&A_{5}\\ 1&A_{2}&A_{4}\\ 0&A_{1}&A_{3}\end{bmatrix} ,\] \[H_{4} = \begin{bmatrix}A_{1}&A_{3}&A_{5}&0\\ 1&A_{2}&A_{4}&0\\ 0&A_{1}&A_{3}&A_{5}\\ 0&1&A_{2}&A_{4}\end{bmatrix} , \tag{14}\] \[H_{5} =