Stability analysis of HIV/AIDS dynamics: Modelling the tested and untested populations

Abstract

In this manuscript, the dynamic behaviour of a nonlinear HIV/AIDS dynamics model using HIV infected population is proposed. Here, we divide the HIV infected population into two subclasses: the tested and untested HIV infected populations. The novel part of the model is that when susceptible population interacts with the untested HIV infected population, the susceptible population is shifted to untested HIV infected population. Otherwise, it is transferred to tested HIV infected population, while many researchers have taken this infection as tested HIV infected population. For infection-free equilibrium point, we determine the basic reproduction number and explore the existence and local stability of equilibrium point. For two endemic equilibrium states, we investigate the positivity and stability of equilibrium points and determine the conditions where it exists. The Routh–Hurwitz criterion and Bellman and Cooke’ theorem are used to establish the stability of the non-infected and two endemic equilibrium states. Numerical simulations for all equilibria are also carried out to examine the behaviour of the system in different dynamical regimes.

Appendix

The characteristic equation of the Jacobian matrix at the endemic state $E_2$ for model (6)–(10) is

$$\begin{aligned}&H(\lambda )= \bigg (\lambda ^{4}+\lambda ^{3}\left( k_1+k_2+k_3+\frac{N_2(4d-\sigma +\tau )+(2I_{12}-N_2)\gamma _1\delta _1+(2I_{22}-N_2)\gamma _2\delta _2}{N_2}\right) \nonumber \\&~~~~~~~~~~~~~ +\lambda ^{2} \bigg ( \frac{1}{N_{2}^{2}}(3dN_{2}^{2}(2d-\sigma +\tau ) +3dN_{2}^{2}k_3-N_{2}^{2}\sigma k_3+N_{2}^{2}\tau k_3+6dI_{12}N_2\gamma _1\delta _1 -3dN_{2}^{2}\gamma _1\delta _1\nonumber \\&~~~~~~~~~~~~~ -AI_{12}\sigma \gamma _1\delta _1-I_{12}^{2}\sigma \gamma _1\delta _1 -I_{12}I_{22}\sigma \gamma _1\delta _1+I_{12}N_2\tau \gamma _1\delta _1 +2I_{12}N_2k_3\gamma _1\delta _1-N_{2}^{2}k_3\gamma _1\delta _1 \nonumber \\&~~~~~~~~~~~~~ +\gamma _2((2I_{22} -N_2)N_2(3d-\sigma +\tau +k_3)+(3I_{12}I_{22} -2(I_{12}+I_{22})N_2+N_{2}^{2})\gamma _1\delta _1)\delta _2 \nonumber \\&~~~~~~~~~~~~~ +N_2k_2(N_2(3d-\sigma +\tau )+N_2k_3+(2I_{22}-N_2)\gamma _1\delta _1+I_{22}\gamma _2\delta _2)+N_2k_1(3dN_2+N_2k_2+N_2k_3 \nonumber \\&~~~~~~~~~~~~~ +I_{12}\gamma _1\delta _1+(2I_{22}-N_2)\gamma _2\delta _2))\bigg ) +\lambda \bigg (\frac{1}{N_{2}^{3}}(d^{2}N_{2}^{3}(4d-3\sigma -3\tau ) +3d^{2}N_{2}^{3}k_3-2dN_{2}^{3}\sigma k_3\nonumber \\&~~~~~~~~~~~~~ +2dN_{2}^{3}\tau k_3 +6d^{2}N_{2}^{2}I_{12}\gamma _1\delta _1-3d^{2}N_{2}^{3}\gamma _1\delta _1 -2AI_{12}N_2\sigma \gamma _1\delta _1-2dI_{1}^{2}N_1\sigma \gamma _1\delta _1 -2dI_{12}I_{22}\sigma \gamma _1\delta _1 \nonumber \\&~~~~~~~~~~~~~ +2dI_1N_{1}^{2}\tau \gamma _1\delta _1 +4dI_1N_{1}^{2}k_3\gamma _1\delta _1-2dN_{1}^{3}k_3\gamma _1\delta _1 -AI_1N_1\sigma k_3\gamma _1\delta _1-I_{12}^{2}N_2\sigma k_3\gamma _1\delta _1\nonumber \\&~~~~~~~~~~~~~ -I_{22}I_{12}N_2\sigma k_3\gamma _1\delta _1+I_{12}N_{2}^{2}\tau k_3\gamma _1\delta _1 +\gamma _2((2I_{22}-N_2)N_{2}^{2}(d(3d-2\sigma +2\tau ) +(2d-\sigma +\tau )k_3))\nonumber \\&~~~~~~~~~~~~~ +(2dN_2(3I_{22}I_{12} -2(I_{22}+I_{12})N_2+N_{2}^{2})+I_{12}(A+I_{22}+I_{12})(-I_{12}+N_2)\sigma +N_2(3I_{22}I_{12}\nonumber \\&~~~~~~~~~~~~~ -2(I_{22}+I_{12})N_2+N_{2}^{2})k_3)\gamma _1\delta _1)\delta _2 +N_2k_1(dN_2+(3dN_2+2I_{12}\gamma _1\delta _1) +\gamma _2(2d(2I_{22}-N_2)N_2\nonumber \\&~~~~~~~~~~~~~ +I_{12}(I_{22}-N_2)\gamma _1\delta _1)\delta _2 +N_2k_2(2dN_2+N_2k_3+I_{12}\gamma _1\delta _1 +I_{22}\gamma _2\delta _2)\nonumber \\&~~~~~~~~~~~~~+k_3(2dN_{2}^{2}+I_{12}(A+I_{22}+I_{12})\gamma _1\delta _1 +(2I_{22}-N_2)N_2\gamma _2\delta _2)) \nonumber \\&~~~~~~~~~~~~~ +N_2k_2(k_3(N_{2}^{2}(2d-\sigma +\tau )+\gamma _1\delta _1(2I_{22}-N_2)N_2 +I_{22}(A+I_{22}+I_{12})\gamma _2\delta _2) \nonumber \\&~~~~~~~~~~~~~ +\gamma _1\delta _1(2d(2I_{12}-N_2)N_2-I_{12}(A+I_{22}+I_{12})\sigma +I_{12}N_2\tau +I_{22}(I_{12}-N_2)\gamma _2\delta _2) \nonumber \\&~~~~~~~~~~~~~ +N_2(dN_2(3d-2\sigma +2\tau )+I_{22}(2d-\sigma +\tau )\gamma _2\delta _2)\bigg )\bigg )+\frac{1}{N_{2}^{3}}\bigg ( k_1(N_2(d+k_2)(dN_{2}^{2}(d +k_3) \nonumber \\&~~~~~~~~~~~~~ +I_{12}(dN_2+(A+I_{22}+I_{12})k_3)\gamma _1\delta _1) +\gamma _2(d(2I_{22}-N_2)N_{2}^{2}(d+k_3)\nonumber \\&~~~~~~~~~~~~~ +I_{22}N_2k_2(dN_2+(A+I_{22}+I_{12})k_3)+I_{12}(I_{12}-N_2)(dN_2+(A+I_{22} +I_{12})k_3)\gamma _1\delta _1)\delta _2) \nonumber \\&~~~~~~~~~~~~~ +(d+k_3)(dN_{2}^{2}(d-\sigma +\tau )(dN_2+(2I_{12}-N_2)\gamma _2\delta _2) +\gamma _1\delta _1(-dN_2(dN_2(-2I_{12}+N_2) \nonumber \\&~~~~~~~~~~~~~ +I_{12}(A+I_{22}+I_{12})\sigma -I_{12}N_2\tau )+(dN_2(3I_{22}I_{12}-2(I_{22}+I_{12})N_2+N_{2}^{2}) +I_{12}(A+I_{22}+I_{12})\nonumber \\&~~~~~~~~~~~~~ (-I_{12}+N_2)\sigma )\gamma _2\delta _2)) +k_2(k_3(dN_{2}^{3}(d-\sigma +\tau )+N_2(d(2I_{12}-N_2)N_2\nonumber \\&~~~~~~~~~~~~~ -I_{12}(A+I_{22}+I_{12})(\sigma -\tau ))\gamma _1\delta _1 +I_{22}(A+I_{22}+I_{12})\gamma _2(N_2(d-\sigma +\tau )\nonumber \\&~~~~~~~~~~~~~ +(I_{12}-N_2)\gamma _1\delta _1)\delta _2) +dN_2(N_2(d-\sigma +\tau )(dN_2+I_{22}\gamma _2 \delta _2)\gamma _1\delta _1\nonumber \\&~~~~~~~~~~~~~ \times (d(2I_{12}-N_2)N_2-I_{12}(A+I_{22}+I_{12})\sigma +I_{12}N_2\tau +I_{22}(I_{12}-N_2)\gamma _2\delta _2)))\bigg ). \end{aligned}$$

(A.1)

As a result of Theorem 3.3, first we substitute $\lambda =iq$ into eq. (A.1), and we write $H(iq)=F(q)+iG(q)$, where F(q) and G(q) are the real and imaginary parts of H(iq). To define zeros, substitute $q=0$ and calculate $F(0), G(0), F'(0)$ and $G'(0)$, as follows:

$$\begin{aligned} F(0)&=\, \frac{1}{N_{2}^{3}}(k_1(N_2(d+k_2)(dN_{2}^{2}(d+k_3)\nonumber \\&\quad +I_{12}(dN_2+(A+I_{22}+I_{12})k_3)\gamma _1\delta _1) \nonumber \\&\quad +\gamma _2(d(2I_{22}-N_2)N_{2}^{2}(d+k_3)\nonumber \\&\quad +I_{22}N_2k_2(dN_2+(A+I_{22}+I_{12})k_3) \nonumber \\&\quad +I_{12}(I_{22}-N_2)(dN_2+(A+I_{22}+I_{12})k_3)\nonumber \\&\quad \times \gamma _1\delta _1)\delta _2) +(d+k_3)(dN_{2}^{2}(d-\sigma +\tau )(dN_2 \nonumber \\&\quad +(2I_{12}-N_2)\gamma _2\delta _2)+\gamma _1\delta _1(-dN_2(dN_2(-2I_{12}\nonumber \\&\quad +N_2)+I_{12}(A+I_{22}+I_{12})\sigma \nonumber \\&\quad -I_{12}N_2\tau )+(dN_2(3I_{22}I_{12}-2(I_{22}+I_{12})N_2\nonumber \\&\quad +N_{2}^{2})+I_{12}(A+I_{22}+I_{12})(-I_{12} \nonumber \\&\quad +N_2)\sigma )\gamma _2\delta _2))+k_2(k_3(dN_{2}^{3}\nonumber \\&\quad \times (d-\sigma +\tau )+N_2(d(2I_{12}-N_2)N_2 \nonumber \\&\quad -I_{12}(A+I_{22}+I_{12})(\sigma -\tau ))\gamma _1\delta _1\nonumber \\&\quad +I_{22}(A+I_{22}+I_{12})\gamma _2(N_2(d-\sigma +\tau ) \nonumber \\&\quad +(I_{12}-N_2)\gamma _1\delta _1)\delta _2)\nonumber \\&\quad +dN_2(N_2(d-\sigma +\tau )(dN_2+I_{22}\gamma _2\delta _2)\nonumber \\&\quad +\gamma _1\delta _1(d(2I_{12}-N_2)N_2 \nonumber \\&\quad -I_{12}(A+I_{22}+I_{12})\sigma \nonumber \\&\quad +I_{12}N_2\tau +I_{22}(I_{12}-N_2)\gamma _2\delta _2))) \end{aligned}$$

(A.2)

$$\begin{aligned} G(0)=0 \end{aligned}$$

(A.3)

$$\begin{aligned} F'(0)=0 \end{aligned}$$

(A.4)

$$\begin{aligned} G(0)&=\, \Big ( \frac{1}{N_{2}^{3}}(d^{2}N_{2}^{3}(4d-3\sigma -3\tau )\nonumber \\&\quad +3d^{2}N_{2}^{3}k_3-2dN_{2}^{3}\sigma k_3+2dN_{2}^{3}\tau k_3\nonumber \\&\quad +6d^{2}N_{2}^{2}I_{12}\gamma _1\delta _1 \!-\!3d^{2}N_{2}^{3}\gamma _1\delta _1\nonumber \\&\quad -2AdI_{12}N_2\sigma \gamma _1\delta _1-2dI_{1}^{2}N_1\sigma \gamma _1\delta _1 \nonumber \\&\quad -2dI_{12}I_{22}N_1\sigma \gamma _1\delta _1 +2dI_1N_{1}^{2}\tau \gamma _1\delta _1\nonumber \\&\quad +4dI_1N_{1}^{2}k_3\gamma _1\delta _1-2dN_{1}^{3}k_3\gamma _1\delta _1\nonumber \\&\quad -AI_1N_1\sigma k_3\gamma _1\delta _1-I_{12}^{2}N_2\sigma k_3\gamma _1\delta _1 \nonumber \\&\quad -I_{22}I_{12}N_2\sigma k_3\gamma _1\delta _1+I_{12}N_{2}^{2}\tau k_3\gamma _1\delta _1\nonumber \\&\quad +\gamma _2((2I_{22}-N_2)N_{2}^{2}(d(3d-2\sigma +2\tau ) \nonumber \\&\quad +(2d-\sigma +\tau )k_3))+(2dN_2(3I_{22}I_{12}\nonumber \\&\quad -2(I_{22}+I_{12})N_2+N_{2}^{2})\nonumber \\&\quad +I_{12}(A+I_{22}+I_{12})(-I_{12} \nonumber \\&\quad +N_2)\sigma +N_2(3I_{22}I_{12}-2(I_{22}+I_{12})N_2\nonumber \\&\quad +N_{2}^{2})k_3)\gamma _1\delta _1)\delta _2+N_2k_1(dN_2+(3dN_2 \nonumber \\&\quad +2I_{12}\gamma _1\delta _1)+\gamma _2(2d(2I_{22}-N_2)N_2\nonumber \\&\quad +I_{12}(I_{22}-N_2)\gamma _1\delta _1)\delta _2+N_2k_2(2dN_2 \nonumber \\&\quad +N_2k_3+I_{12}\gamma _1\delta _1+I_{22}\gamma _2\delta _2) \nonumber \\&\quad +k_3(2dN_{2}^{2}+I_{12}(A+I_{22}+I_{12})\gamma _1\delta _1+(2I_{22} \nonumber \\&\quad -N_2)N_2\gamma _2\delta _2))+N_2k_2(k_3(N_{2}^{2}(2d-\sigma +\tau )\nonumber \\&\quad +\gamma _1\delta _1(2I_{22}-N_2)N_2+I_{22}(A+I_{22} \nonumber \\&\quad +I_{12})\gamma _2\delta _2)+\gamma _1\delta _1(2d(2I_{12}\nonumber \\&\quad -N_2)N_2-I_{12}(A+I_{22}+I_{12})\sigma \nonumber \\&\quad +I_{12}N_2\tau +I_{22}(I_{12} \nonumber \\&\quad -N_2)\gamma _2\delta _2)+N_2(dN_2(3d-2\sigma +2\tau )\nonumber \\&\quad +I_{22}(2d-\sigma +\tau )\gamma _2\delta _2))\Big ). \end{aligned}$$

(A.5)

For model (6)–(10), the characteristic eq. (A.1) shows that the eigenvalues are real and negative if it follows the inequality,

$$\begin{aligned} F(0)G^\prime (0)-F^\prime (0)G(0)>0 \end{aligned}$$

(A.6)

which is the stability condition. From eqs (A.2)–(A.5), inequality (A.6) becomes

$$\begin{aligned} J\equiv F(0)G^\prime (0)>0. \end{aligned}$$

(A.7)

In [18], Bellman and Cooke state that the equilibrium point will be stable when $J>0$.