*4.3. Disease Free Equilibrium*

**Theorem 1.** *Disease-free equilibrium (DFE) point of a system is locally and asymptotically stable at K*0*, if R*<sup>0</sup> < 1*.*

**Proof.** The DFE point of a system is locally asymptotically stable at *K*<sup>0</sup> = (*I*, *M*, *UI*) = (0, 0, 0). The Jacobian matrix of function *<sup>f</sup>* : *<sup>R</sup>*<sup>3</sup> <sup>→</sup> *<sup>R</sup>*<sup>3</sup> with components:

$$\begin{split} D^a I &= \frac{\beta\_1 (N^\*-I-M)I}{2^{32}} + \frac{\beta\_2 (N^\*-I-M)\mathcal{U}\_I}{N^\*} - \rho I - r\_1 I, \\ D^a M &= \rho I - r\_1 \mathcal{M}, \\ D^a \mathcal{U}\_I &= \frac{\beta\_2 (\mathcal{U}^\* - \mathcal{U}\_I)I}{N^\*} - r\_2 \mathcal{U}\_I. \end{split} \tag{25}$$

Thus, the Jacobian matrix at *K*0, DFE point of integer-order model (14) is given as:

$$DFE(\mathbf{K}\_0) = \begin{pmatrix} \frac{\beta\_1 N^\*}{2^{2^\*}} - \rho - r\_1 & 0 & \beta\_2 \\ \rho & -r\_1 & 0 \\ \frac{\beta\_2 \mathbf{U}^\*}{N^\*} & 0 & -r\_2 \end{pmatrix} \tag{26}$$

System (26) characteristic equation is

$$|\lambda I - DFE(K\_0)| = \begin{vmatrix} \lambda - \frac{\beta\_1 \mathcal{U}^\*}{2^{2^\*}} + \rho + r\_1 & 0 & -\beta\_2 \\ -\rho & \lambda + r\_1 & 0 \\ -\frac{\beta\_2 \mathcal{U}^\*}{N^\*} & 0 & \lambda + r\_2 \end{vmatrix} = 0,\tag{27}$$

and simplify as:

$$(\lambda + r\_1) \left[ \left( \lambda - \frac{N^\* \beta\_1}{2^{32}} + \rho + r\_1 \right) (\lambda + r\_2) - \frac{\beta\_2^2 \mathcal{U}^\*}{N^\*} \right] = 0. \tag{28}$$

The corresponding Eigen values of the above relation are

$$\begin{cases} \lambda\_1 = -r\_{1\prime} \\ \left[ \left( \lambda - \frac{N^\* \beta\_1}{2^{32}} + \rho + r\_1 \right) (\lambda + r\_2) - \frac{\beta\_2^2 \mathcal{U}^\*}{N^\*} \right] = 0. \end{cases} \tag{29}$$

Simplifying the above expression to find the remaining Eigenvalues

$$\begin{split} r\_1(\lambda + r\_2) + \rho(\lambda + r\_2) + \lambda(\lambda + r\_2) - (\lambda + r\_2) \frac{N^\* \beta\_1}{2^{32}} - \frac{\beta\_2^{-2} \mathcal{U}^\*}{N^\*} &= 0, \\ \lambda^2 + \lambda \left(r\_1 + r\_2 + \rho - \frac{N^\* \beta\_1}{2^{32}}\right) + r\_1 r\_2 + \rho r\_2 - r\_2 \frac{N^\* \beta\_1}{2^{32}} - \frac{\beta\_2^{-2} \mathcal{U}^\*}{N^\*} &= 0, \\ \frac{\lambda^2}{r\_2(\rho + r\_1)} + \frac{\lambda \left(r\_1 + r\_2 + \rho - \frac{N^\* \beta\_1}{2^{32}}\right)}{r\_2(\rho + r\_1)} + \left(1 - \frac{N^\* \beta\_1}{2^{32}(\rho + r\_1)} - \frac{\beta\_2^{-2} \mathcal{U}^\*}{N^\* r\_2(\rho + r\_1)}\right) &= 0, \\ \frac{\lambda^2}{r\_2(\rho + r\_1)} + \frac{\lambda}{r\_2} \left(\frac{r\_2}{\rho + r\_1} + \frac{r\_1 + \rho}{\rho + r\_1} - \frac{N^\* \beta\_1}{2^{32}(\rho + r\_1)}\right) + (1 - R\_0) &= 0, \end{split}$$

and rearranging the above expression

$$\frac{\lambda^2}{r\_2(\rho + r\_1)} + \frac{\lambda}{r\_2} \left(\frac{r\_2}{\rho + r\_1} + 1 - \frac{N^\* \beta\_1}{2^{32}(\rho + r\_1)}\right) + (1 - R\_0) = 0,\tag{30}$$

and, for *R*<sup>0</sup> < 1, Equation (9) can be written as:

$$\frac{\lambda^2}{r\_2(\rho + r\_1)} + \frac{\lambda}{r\_2} \left( \frac{r\_2}{\rho + r\_1} + 1 - \frac{N^\* \beta\_1}{2^{32}(\rho + r\_1)} \right) + (1 - R\_0) = 0. \tag{31}$$

Using the expression (31) in Section 4.3, make the coefficient positive for *R*<sup>0</sup> < 1, which shows that system Section 4.3 eigenvalues are in a stable region; this confirms that the system is asymptotically stable for point *K*<sup>0</sup> when *R*<sup>0</sup> < 1. If system is stable for the value of *α* = 1, it will be stable for the value of *α* < 1, as reported in [52]. This completes the proof.
