*4.4. Endemic Stability*

The endemic stability of equilibrium point *K*∗= (*I*∗, *M*∗, *U*∗ *<sup>I</sup>* ) is investigated in this section for the values of *R*<sup>0</sup> > 1 and *I*<sup>∗</sup> ≥ 0.

**Theorem 3.** *Endemic equilibrium point K*<sup>∗</sup> *is locally asymptotically stable, if R*<sup>0</sup> > 1*.*

**Proof.** Consider the function *<sup>f</sup>* : *<sup>R</sup>*<sup>3</sup> <sup>→</sup> *<sup>R</sup>*<sup>3</sup> with components and the Jacobian matrix of the system (14) as:

$$D^{\mathfrak{a}}I = f\_1(I^\*, M^\*, \mathcal{U}\_I^\*) = \frac{\beta\_1 (N^\* - I^\* - M^\*)I^\*}{2^{32}} + \frac{\beta\_2 (N^\* - I^\* - M^\*)\mathcal{U}\_I^\*}{N^\*} - \rho I^\* - r\_1 I^\*,$$

$$D^{\mathfrak{a}}M = f\_2(I^\*, M^\*, \mathcal{U}\_I^\*) = \rho I^\* - r\_1 M^\*,$$

$$D^{\mathfrak{a}}\mathcal{U}\_I = f\_3(I^\*, M^\*, \mathcal{U}\_I^\*) = \frac{\beta\_2 (\mathcal{U}^\* - \mathcal{U}\_I^\*)I^\*}{2^{32}} - r\_2 \mathcal{U}\_I^\*,$$

$$f(I^\*, M^\*, \mathcal{U}\_I^\*) = \begin{pmatrix} \frac{\partial f\_1}{\partial I} \frac{\partial f\_1}{\partial M} \frac{\partial f\_1}{\partial I} \\\frac{\partial f\_2}{\partial I} \frac{\partial f\_2}{\partial M} \frac{\partial f\_2}{\partial I} \\\frac{\partial f\_3}{\partial I} \frac{\partial f\_3}{\partial M} \frac{\partial f\_3}{\partial I} \end{pmatrix}.$$

The endemic equilibrium of system (14) is *K*∗= (*I*∗, *M*∗, *U*∗ *<sup>I</sup>* ), for the value of *α* = 1, the Jacobian matrix at endemic point is mentioned below.

$$J(K^\*) = \begin{pmatrix} \Lambda & -\frac{\mathcal{S}\_1 I^\*}{2^{32}} - \frac{\mathcal{S}\_2 \mathcal{U}\_I}{N^\*} & \frac{\mathcal{S}\_2 (N^\* - I^\* - M^\*)}{N^\*} \\ \rho & -r\_1 & 0 \\ \frac{\mathcal{S}\_2 (I^\* - \mathcal{U}\_I^\*)}{N^\*} & 0 & \frac{\mathcal{S}\_2 I^\*}{N^\*} - r\_2 \end{pmatrix},\tag{35}$$

where Λ = *<sup>β</sup>*1(*N*∗−2*I*∗−*M*∗) <sup>232</sup> <sup>−</sup> *<sup>β</sup>*2*UI* ∗ *<sup>N</sup>*<sup>∗</sup> − *ρ* − *r*1. The characteristic equation of (35) is

$$\begin{vmatrix} \lambda I - f(K^\*) \end{vmatrix} = 0,$$

$$\begin{vmatrix} \lambda - \Lambda & \frac{\beta\_1 I^\*}{2^{32}} + \frac{\beta\_2 \iota I I^\*}{N^\*} & -\frac{\beta\_2 (N^\* - I^\* - M^\*)}{N^\*} \\ -\rho & \lambda + r\_1 & 0 \\ -\frac{\beta\_2 (\iota I^\* - \iota I I^\*)}{N^\*} & 0 & \lambda + \frac{\beta\_2 I^\*}{N^\*} + r\_2 \end{vmatrix} = 0,$$

simplifies as:

$$\begin{aligned} \lambda^3 + (b\_{11} + b\_{22} + b\_{33})\lambda^2 + (b\_{11}b\_{22} + b\_{11}b\_{33} + b\_{22}b\_{33} \\ -b\_{12}b\_{21} - b\_{13}b\_{31})\lambda + b\_{11}b\_{22}b\_{33} - b\_{12}b\_{21}b\_{33} - b\_{13}b\_{31}b\_{22} = 0, \end{aligned} \tag{36}$$

where

$$\begin{array}{ll} b\_{11} = -\frac{\beta\_1 N^\*}{2^{32}} + \frac{\beta\_1 (2I^\* + M^\*)}{2^{32}} + \frac{\beta\_2 I I\_1^\*}{N^\*} + \rho + r\_{1\prime} \\ b\_{12} = \frac{\beta\_1 I^\*}{2^{32}} + \frac{\beta\_2 I I\_1^\*}{N^\*}, \\ b\_{21} = -\rho, b\_{23} = 0, b\_{22} = r\_{1\prime} \quad b\_{13} = -\frac{\beta\_2 (N^\* - I^\* - M^\*)}{N^\*}, \\ b\_{31} = -\frac{\beta\_2 (ll^\* - II\_1^\*)}{N^\*}, \quad b\_{33} = \frac{\beta\_2 I^\*}{N^\*} + r\_{2\prime} b\_{32} = 0. \end{array}$$

For stability analysis, Hurwitz criteria may be used, as reported in [53,54] for system (36). Equating the Equation (36) coefficient with the general characteristics equation, we have

$$\begin{aligned} b\_0 &= 1, \\ b\_1 &= b\_{11} + b\_{22} + b\_{33}, \\ b\_2 &= b\_{11}b\_{22} + b\_{11}b\_{33} + b\_{22}b\_{33} - b\_{12}b\_{21} - b\_{13}b\_{31}, \\ b\_3 &= b\_{11}b\_{22}b\_{33} - b\_{12}b\_{21}b\_{33} - b\_{13}b\_{31}b\_{22}. \end{aligned}$$

Determinants (*D*1, *D*<sup>2</sup> and *D*3) of the Equation (36) are stated in Hurwitz as:

$$\begin{aligned} D\_1 &= b\_1 = b\_{11} + b\_{22} + b\_{33\prime} \\ &= -\frac{\beta\_1 N^\*}{2^{32}} + \frac{\beta\_1 (2I^\* + M^\*)}{2^{32}} + \frac{\beta\_2 II\_I^\*}{N^\*} \\ &+ \rho + r\_1 + r\_1 + \frac{\beta\_2 I^\*}{N^\*} + r\_2 \end{aligned}$$

using the value of Equation (20) for *R*<sup>0</sup> > 1 as:

$$\begin{aligned} \frac{\beta\_1 N^\*}{2^{32}} + \frac{\beta\_2^2 L^\*}{r\_2 N^\*} &> \rho + r, \text{we have} \\ D\_1 = -\frac{\beta\_1 N^\*}{2^{32}} + \frac{\beta\_1 (2I^\* + M^\*)}{2^{32}} + \frac{\beta\_2 L\_I^\*}{N^\*} + \frac{\beta\_1 N^\*}{r\_2 N^\*} + r\_1 + \frac{\beta\_2 I^\*}{N^\*} + r\_2, \\ D\_1 = \frac{\beta\_1 (2I^\* + M^\*)}{2^{32}} + \frac{\beta\_2 L\_I^\*}{N^\*} + \frac{\beta\_2^2 L^\*}{r\_2 N^\*} + r\_1 + \frac{\beta\_2 I^\*}{N^\*} + r\_2, \\ D\_1 > 0, \end{aligned}$$

and

*D*<sup>2</sup> = *b*1*b*<sup>2</sup> − *b*3*b*0, *D*<sup>2</sup> = (*b*<sup>11</sup> + *b*<sup>22</sup> + *b*33)(*b*11*b*<sup>22</sup> + *b*11*b*<sup>33</sup> + *b*22*b*<sup>33</sup> − *b*12*b*<sup>21</sup> − *b*13*b*31) − *b*11*b*22*b*<sup>33</sup> + *b*12*b*21*b*<sup>33</sup> + *b*13*b*31*b*22, = *b*<sup>2</sup> <sup>11</sup>*b*<sup>22</sup> + *<sup>b</sup>*<sup>2</sup> <sup>11</sup>*b*<sup>33</sup> + *b*11*b*22*b*<sup>33</sup> − *b*11*b*12*b*<sup>21</sup> − *b*11*b*13*b*<sup>31</sup> + *b*11*b*<sup>2</sup> <sup>22</sup> + *<sup>b</sup>*11*b*22*b*<sup>33</sup> + *<sup>b</sup>*<sup>2</sup> <sup>22</sup>*b*<sup>33</sup> − *b*22*b*12*b*<sup>21</sup> <sup>−</sup> *<sup>b</sup>*22*b*13*b*<sup>31</sup> <sup>+</sup> *<sup>b</sup>*11*b*22*b*<sup>33</sup> <sup>+</sup> *<sup>b</sup>*11*b*<sup>2</sup> <sup>33</sup> + *<sup>b</sup>*22*b*<sup>2</sup> 33 − *b*33*b*12*b*<sup>21</sup> − *b*33*b*13*b*<sup>31</sup> − *b*11*b*22*b*<sup>33</sup> + *b*33*b*12*b*<sup>21</sup> + *b*22*b*13*b*31, *D*<sup>2</sup> = *b*<sup>2</sup> <sup>11</sup>*b*<sup>22</sup> + *<sup>b</sup>*<sup>2</sup> <sup>11</sup>*b*<sup>33</sup> + *<sup>b</sup>*11*b*<sup>2</sup> <sup>22</sup> + *<sup>b</sup>*22*b*<sup>2</sup> <sup>33</sup> + *<sup>b</sup>*11*b*<sup>2</sup> <sup>33</sup> + *<sup>b</sup>*<sup>2</sup> <sup>22</sup>*b*<sup>33</sup> + 2*b*11*b*22*b*<sup>33</sup> − *b*11*b*12*b*<sup>21</sup> − *b*11*b*13*b*<sup>31</sup> − *b*22*b*12*b*<sup>21</sup> − *b*33*b*13*b*31.

The above expressions remain positive, except for −*b*13*b*31(*b*<sup>11</sup> + *b*33), *D*2, which, if positive for *R*<sup>0</sup> > 1, is simply represented as:

$$\begin{aligned} D\_2 &= +\text{vertex} + (b\_{11}b\_{33} - b\_{13}b\_{31})(b\_{11} + b\_{33}), \\ D\_2 &= D\_{2-1} + D\_{2-2\prime} \end{aligned}$$

Here, *D*2−<sup>1</sup> represent the positive terms in *D*2, while, for the remaining terms, represented with *D*2−2, we have

$$\begin{aligned} D\_2 &= +\text{vertex} + (b\_{11}b\_{33} - b\_{13}b\_{31})(b\_{11} + b\_{33}), \\ D\_2 &= D\_{2-1} + D\_{2-2}. \end{aligned}$$

*D*2−<sup>2</sup> = (*b*11*b*<sup>33</sup> − *b*13*b*31)(*b*<sup>11</sup> + *b*33) = ⎧ ⎨ ⎩ *β*1(*N*∗−*I*∗−*M*∗) <sup>2</sup><sup>32</sup> + *<sup>ρ</sup>* + *<sup>r</sup>*<sup>1</sup> *r*2 −*β*2 <sup>2</sup>(*N*∗−*I*∗−*M*∗)(*U*∗−*UI* ∗) *N*∗<sup>2</sup> ⎫ ⎬ ⎭ (*b*<sup>11</sup> + *b*33), = ⎧ ⎨ ⎩ ⎛ ⎝ *β*1(*N*∗−*I*∗−*M*∗) <sup>2</sup><sup>32</sup> + *<sup>ρ</sup>* + *<sup>r</sup>*<sup>1</sup> −*β*2 <sup>2</sup>(*N*∗−*I*∗−*M*∗)(*U*∗−*UI* ∗) *r*2*N*∗<sup>2</sup> ⎞ <sup>⎠</sup>*r*<sup>2</sup> ⎫ ⎬ ⎭ (*b*<sup>11</sup> + *b*33), = ⎧ ⎪⎪⎨ ⎪⎪⎩ ⎛ ⎜⎜⎝ *β*1(*N*∗−*I*∗−*M*∗) <sup>2</sup><sup>32</sup> + *<sup>ρ</sup>* + *<sup>r</sup>*<sup>1</sup> −*β*2 2(*N*∗−*I*∗−*M*∗)*U*<sup>∗</sup> *<sup>r</sup>*2*N*∗<sup>2</sup> <sup>+</sup> *β*2 <sup>2</sup>(*N*∗−*I*∗−*M*∗)*UI* ∗ *r*2*N*∗<sup>2</sup> ⎞ ⎟⎟⎠*r*2 ⎫ ⎪⎪⎬ ⎪⎪⎭ (*b*<sup>11</sup> + *b*33), = ⎧ ⎪⎪⎨ ⎪⎪⎩ ⎛ ⎜⎜⎝ *β*1(*N*∗−*I*∗−*M*∗) <sup>232</sup> + *<sup>ρ</sup>* + *<sup>r</sup>*1− *β*2 2*U*<sup>∗</sup> *<sup>r</sup>*2*N*<sup>∗</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> <sup>2</sup> *I*∗*U*<sup>∗</sup> *<sup>r</sup>*2*N*∗<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> 2*M*∗*U*<sup>∗</sup> *r*2*N*∗<sup>2</sup> +*β*<sup>2</sup> <sup>2</sup>(*N*∗−*I*∗−*M*∗)*UI* ∗ *r*2*N*∗<sup>2</sup> ⎞ ⎟⎟⎠*r*2 ⎫ ⎪⎪⎬ ⎪⎪⎭ (*b*<sup>11</sup> + *b*33),

using the value of *R*<sup>0</sup> > 1, and after simplification, the above expression becomes

$$\begin{split} D\_{2-2} &> \left\{ \begin{aligned} & \left( \begin{array}{\$\delta\_{1}\$ (N^{\*}-"{\mathcal{I}}^{\star}-M^{\*})\\ \hline & \frac{\delta\_{1}\$ (N^{\*}-"{\mathcal{I}}^{\star}-M^{\*})}{2\cdot 2^{2}}\\ \hline & \frac{\delta^{2}\_{1}\circ \mathcal{I}^{\star}}{2\cdot N^{\*}}+\frac{\delta^{2}\_{2}\circ \mathcal{I}^{\star}\circ}{r\_{2}N^{\*}}\\ \hline & +\frac{\delta^{2}\_{2}\circ \mathcal{N}^{\star}\circ}{r\_{2}N^{\*}}+\\ \hline & \frac{\delta^{2}\_{2}(\mathcal{N}^{\star}-\mathcal{I}^{\star}-M^{\*})\mathcal{U}\_{\ast}}{r\_{2}N^{\*}}\\ \hline & \frac{\delta\_{1}(N^{\*}-\mathcal{I}^{\star}-M^{\*})}{2\cdot 2}\\ \hline & +\frac{\delta\_{1}\circ \mathcal{N}^{\star}}{2\cdot 2^{2}}+\frac{\delta^{2}\_{2}\circ \mathcal{I}^{\star}\circ\mathcal{I}^{\star}}{r\_{2}N^{\*}}\\ \hline & +\frac{\delta\_{2}\circ \mathcal{N}^{\star}\circ\mathcal{I}^{\star}}{r\_{2}N^{\*2}}+\\ \hline & \frac{\delta^{2}\_{2}\circ (\mathcal{N}^{\star}-\mathcal{I}^{\star}-M^{\*})\mathcal{U}\_{\ast}}{r\_{2}N^{\*2}}\\ \hline & \frac{\delta^{2}\_{2}(\mathcal{N}^{\star}-\mathcal{I}^{\star}-M^{\*})\mathcal{U}\_{\ast}}{r\_{2}N^{\*2}}\\ \end{aligned} \right\}(b\_{11}+b\_{33}),$$

as a result

$$\begin{aligned} D\_2 &> 0. \\ D\_3 &= b\_3 (b\_1 b\_2 - b\_0 b\_3), \\ D\_3 &= b\_3 (D\_2), \\ &= (b\_{11} b\_{22} b\_{33} - b\_{12} b\_{21} b\_{33} - b\_{13} b\_{31} b\_{22}) ((b\_{11} + b\_{22}) \\ &+ b\_{33}) (b\_{11} b\_{22} + b\_{11} b\_{33} + b\_{22} b\_{33} - b\_{12} b\_{21} - b\_{13} b\_{31}), \\ &- b\_{11} b\_{22} b\_{33} + b\_{12} b\_{21} b\_{33} + b\_{13} b\_{31} b\_{22}) \\ &= (b\_{11} b\_{22} b\_{33} - b\_{12} b\_{21} b\_{33} - b\_{13} b\_{31} b\_{22}) D\_{2'} \\ &> (b\_{11} b\_{33} - b\_{13} b\_{31}) b\_{22} D\_{2'} \end{aligned}$$

The positivity of the expression *b*11*b*<sup>33</sup> − *b*13*b*<sup>31</sup> for *R*<sup>0</sup> > 1 is already proved for the case *D*2; therefore, *D*<sup>3</sup> > 0.

Thus, all the values of *D*1, *D*<sup>2</sup> and *D*<sup>3</sup> are positive, so all the eigenvalues of the Equation (36) are negative, for *R*<sup>0</sup> > 1. This proves that the endemic equilibrium point *K*<sup>∗</sup> is locally asymptotically stable. The proof of theorem is completed.
