as sums of terms in the form of $f(x)=1-x+\ln x$. To this end, we will use the fact that the function $f(x)$ is definite negative around $x^\ast=1$. Indeed, using the properties of natural logarithm function, the expression of $\mathcal W_i$ in (\[CombinedLyap4\]) could be written as: $$\begin{aligned}
\mathcal W_i &= 6 -\frac{S_h^\ast}{S_h}-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}  -\frac{E_h}{E_h^*}\frac{I_1^\ast}{I_1}  -\frac{S_v^\ast}{S_v}-\frac{S_v}{S_v^\ast} \frac{I_i}{I_i^\ast} \frac{E_v^\ast}{E_{v}}-\frac{E_vI_v^\ast}{E_v^\ast I_v}\nonumber\\
    &= \left(1-\frac{S_h^\ast}{S_h}+\ln\frac{S_h^\ast}{S_h}\right)+\left(1-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}+\ln\frac{S_hI_vE_h^\ast}{S_h^\ast I_v^\ast E_{h}} \right)+\left(1-\frac{E_vI_v^\ast}{E_v^\ast I_v} +\ln\frac{E_vI_v^\ast}{E_v^\ast I_v} \right)\nonumber\\
&   + \left(1 -\frac{E_h}{E_h^\ast}\frac{I_1^\ast}{I_1} +\ln\frac{E_hI_1^\ast}{E_h^\ast I_1}   \right)
+\left(1-\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} +\ln\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} \right)
+\left(1 
  -\frac{S_v^\ast}{S_v}+\ln\frac{S_v^\ast }{S_v}\right)   +\ln\frac{ I_1}{I_1^\ast}\frac{ I_i^\ast }{I_i} 
      \end{aligned}$$ Noting that $$1- \frac{I_i^\ast}{I_i} \frac{I_j}{I_j^\ast}=1- \frac{I_i^\ast}{I_i} \frac{I_j}{I_j^\ast}+\ln\frac{I_i^\ast I_j}{I_iI_j^\ast} +\ln\frac{I_iI_j^\ast}{I_i^\ast I_j},$$ and substitute the expression of $\mathcal W_i$, becomes $$\begin{aligned}
\label{CombinedLyapGene2}
 \dot {\mathcal V}
 &= c_0\mu_h S_h^\ast\left(2-\frac{S_h}{S_h^\ast}-\frac{S_h^\ast}{S_h}\right)+c_{v,1}(\mu_v+\delta_v)S_v^*\left(2-\dfrac{S_v^*}{S_v}-\dfrac{S_v}{S_v^*}\right) \nonumber\\
 &+c_{v,1}aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i^\ast}{N_h}\left[
 \left(1-\frac{S_h^\ast}{S_h}+\ln\frac{S_h^\ast}{S_h}\right)+\left(1-\frac{S_h}{S_h^\ast} \frac{I_v}{I_v^\ast} \frac{E_h^\ast}{E_{h}}+\ln\frac{S_hI_vE_h^\ast}{S_h^\ast I_v^\ast E_{h}} \right)\right.\nonumber\\
 & \left.+\left(1-\frac{E_vI_v^\ast}{E_v^\ast I_v} +\ln\frac{E_vI_v^\ast}{E_v^\ast I_v} \right)
 + \left(1 -\frac{E_h}{E_h^\ast}\frac{I_1^\ast}{I_1} +\ln\frac{E_hI_1^\ast}{E_h^\ast I_1}   \right)
 +\left(1-\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} +\ln\frac{S_vI_i E_v^\ast}{S_v^\ast I_i^\ast E_{v}} \right)
 \right.\nonumber\\
  & \left.+\left(1 
  -\frac{S_v^\ast}{S_v}+\ln\frac{S_v^\ast }{S_v}\right)\right]+c_0p\pi\left(2-\frac{E_h^\ast}{E_{h}}-\frac{E_{h}}{E_h^*}  \right)
  +\sum_{i=1}^nc_ip_i\pi_h\left(2-\frac{I_i^\ast}{I_i}-\frac{I_i}{I_i^\ast}\right) \nonumber\\
  &  \sum_{i=1}^nc_i \sum_{j=1}^{n}m_{ij}I_j^\ast\left(1- \frac{I_i^\ast}{I_i} \frac{I_j}{I_j^\ast}+\ln\frac{I_i^\ast I_j}{I_iI_j^\ast}\right)\nonumber\\
 &+\underbrace{c_{v,1}aS_v^\ast\sum_{i=1}^n \frac{\beta_i I_i^\ast}{N_h} \ln\frac{ I_1}{I_1^\ast}\frac{ I_i^\ast }{I_i}   
  +\sum_{i=1}^nc_i \sum_{j=1}^{n}m_{ij}I_j^\ast \ln\frac{I_iI_j^\ast}{I_i^\ast I_j}}_{\mathcal S} 
      \end{aligned}$$

All but the last two sums in (\[CombinedLyapGene2\]) are definite negative. Let us denote by $\mathcal S$ the sum of these two 