Analysis and optimal control of an HIV model based on CD4 count

Abstract

A non-linear mechanistic model for the transmission dynamics of HIV/AIDS is developed and analyzed. The model classified the infected individuals based on their CD4 count level. Furthermore, education campaign, voluntary testing and counseling and treatment are considered as intervention strategies for controlling the disease. The analysis of the model reveals that imperfect public enlightenment campaign can induce backward bifurcation. It has been shown that when public enlightenment campaign is $100\%$ effective, the disease free equilibrium is globally asymptotically stable for $R_{eff} \le 1$, whereas for $R_{eff} > 1$ the global stability of the endemic equilibrium is proved only in a special case. Time dependent controls of the intervention strategies mentioned above are incorporated into the model and the optimal control strategies with minimal implementation cost are identified. In addition, cost effectiveness analysis in the form of incremental cost effectiveness ratio is carried-out to identify the most cost effective strategies. The results suggest that out of the three non dominated strategies, the strategy of educating the newly entrants only or combination of newly entrants and susceptible individuals is very cost effective using per capita GDP of Nigeria as at 2018. However, the choice of which strategy to implement depends on budgetary allocation and resource availability.

Appendix

Proof of Theorem 4

To show the existenceof backward bifurcation in model (3) when ${\mathcal {R}}_{eff} = 1$, center manifold theorem is used as in Elbasha and Gumel (2006), Hussaini et al. (2011).

Consider model (3) as

$$\begin{aligned} \frac{dX}{dt}=f(X,\phi ) \end{aligned}$$

(44)

where $\phi$ is the bifurcation parameter and f is a continuously differentiable at least twice (both in x and $\phi$).

For ease of notation, let $S_u=x_{1}$, $S_e=x_{2}$, $H_u=x_{3}$, $H_a=x_{4}$, $H_{ul}=x_{5}$ , $H_t=x_{6}$, $A_u=x_{7}$, and $A_t=x_{8}$, thus $N(t)= x_{1} +x_{2} +x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}$.

$$\begin{aligned} \begin{aligned} \frac{dx_{1}}{dt}&=f_1=(1-p)\varLambda - \left( \lambda +W_0 \right) x_{1}\\ \frac{dx_{2}}{dt}&=f_2=p\varLambda +\xi x_{1}- \left( (1-k)\lambda +\mu \right) x_{2}\\ \frac{dx_{3}}{dt}&=f_3=\lambda x_{1} + (1-k)\lambda x_{2}-W_2x_{3}\\ \frac{dx_{4}}{dt}&=f_4=\tau _{{1}}x_{3}-W_3x_{4}\\ \frac{dx_{5}}{dt}&=f_5=\gamma _{1}x_{3}-W_4x_{5}\\ \frac{dx_{6}}{dt}&=f_6=\alpha \,x_{4}+\tau _{2}x_{5}-W_5x_{6}\\ \frac{dx_{7}}{dt}&=f_7=\gamma _{2}x_{5}-W_6x_{7}\\ \frac{dx_{8}}{dt}&=f_8=\rho \,x_{6}+\tau _{3}x_{7}-W_7x_{8} \end{aligned} \end{aligned}$$

(45)

with force of infection $\lambda$ defined as

$$\begin{aligned} \lambda =\beta \left( {\frac{\eta _{{1}}x_{3}+x_{5}+\eta _{{4 }}x_{7}+\theta \left( \eta _{{3}}x_{8}+\eta _{{2}}x_{4}+\eta _5x_{6} \right) }{N}} \right) \end{aligned}$$

where $\theta$ and $W_i, \quad i=0,1,2,\ldots , 6$ are defined in (9)

Now, consider the case when ${\mathcal {R}}_{eff}=1$. Suppose, that $\beta$ is choosen as a bifurcation parameter, then solving for $\beta$, we have

$$\begin{aligned} \beta = \beta ^{c} =\frac{W_{{0}}W_{{1}}W_{{2}}W_{{3}}W_{{4}}W_{{5}}W_{{6}}}{\left( \left( \varUpsilon _{{5}}+\varUpsilon _{{6}} \right) W_{{5}}+W_{{2}}W _{{4}}\gamma _{{1}}\gamma _{{2}} \left( \theta \,\eta _{{3}}\tau _{{3}}+ \eta _{{4}}W_{{6}} \right) \right) \left( \left( 1-kp \right) \mu + \left( 1-k \right) \xi \right) }.\nonumber \\ \end{aligned}$$

(46)

Then, the jacobian of $f=(f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8)^T$ evaluated around disease free equilibrium $(E_0)$ denoted by $J(E_0)_{\beta ^{c}}$ is given by:

$$\begin{aligned} J(E_0)_{\beta ^{c}}= \left[ \begin{array}{cccccccc} -W_{{0}}&{} \quad 0&{} \quad -C_0Z_{{1}}&{} \quad -C_1Z_{{1}}&{} \quad -\beta ^{c}\,Z_{{1}}&{} \quad -C_4Z_{{1}}&{} \quad -Z_{{1}}&{} \quad -C_3Z_{{1}} \\ \xi &{} \quad -\mu &{} \quad -C_0Z_{{2}}&{} \quad -C_1Z_{{2}}&{} \quad -\beta ^{c}\,Z_{{2}}&{} \quad -C_4Z_{{2}}&{} \quad -C_2Z_{{2}}&{} \quad -C_3Z_{{2}} \\ 0&{} \quad 0&{} \quad C_0Z_{{3}}-W_{{1}}&{} \quad C_1Z_{{3}}&{} \quad \beta ^{c}\,Z_{{3}}&{} \quad C_4Z_{{3}}&{} \quad C_2Z_{{3}}&{} \quad C_3Z_{{3}} \\ 0&{} \quad 0&{} \quad \tau _{{1}}&{} \quad -W_{{2}}&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad \gamma _{{1}}&{} \quad 0&{} \quad -W_{{3}}&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad 0&{} \quad \alpha &{} \quad \tau _{{2}}&{} \quad -W_{{4}}&{} \quad 0&{} \quad 0 \\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \gamma _{{2}}&{} \quad 0&{} \quad -W_{{6}}&{} \quad 0 \\ 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad 0&{} \quad \rho &{} \quad \tau _{{3}}&{} \quad -W_{{5}}\end{array} \right] \nonumber \\ \end{aligned}$$

(47)

where $C_0=\beta ^{c}\,\eta _{{1}}$, $C_1=\beta ^{c}\,\theta \,\eta _{{2}}$, $C_2=\beta ^{c}\,\eta _{{4}}$, $C_3=\beta ^{c}\,\theta \,\eta _{{3}}$, $C_4=\beta ^{c}\,\theta \,$$Z_{{1}}={\frac{ \left( 1-p \right) \mu }{\mu +\xi }}$       $Z_{{2}}={\frac{ \left( 1-k \right) \left( \mu \,p+\xi \right) }{\mu +\xi }}$       $Z_{{3}}=Z_1+Z_2$ The Jacobian $J(E_0)_{\beta ^{c}}$ of the linearized system (45) has a simple zero eigenvalue (meaning that the remaining eigenvalues have negative real part). Hence, by center manifold theorem developed in Castillo-Chavez and Song (2004), it is possible to analyze the dynamics of the system (45) within $\beta =\beta ^{c}$.

Eigenvectors of $J(E_0)_{\beta ^{c}}$:

Let $w = [w_1, w_2, w_3, w_4, w_5, w_6, w_7, w_8]^T$ and $v = [v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8]$ be the right and left eigenvectors respectively, associated with the zero eigenvalue of the Jacobian of linearized system $J(E_0)_{\beta ^{c}}$, defined by

$$\begin{aligned} w_{{1}}&= -{\frac{ \left( 1-p \right) \mu \, \left( C_1 w_{{4}}+C_3 w_{{8}}+C_4 w_{{6}}+ C_0 w_{{3}}+C_2 w_{{7}}+\beta ^{c}\,w_{{5}} \right) }{{W_{{0}}}^{2}}}<0, \nonumber \\ w_{{2}}&= {\frac{\xi \,w_{{1}}-Z_{{2}} \left( C_1 w_{{4}}+C_3 w_{{8}}+C_4 w_{{6}}+C_0 w_{{3}}+C_2 w_{{7}}+w_{{5}} \right) }{\mu }}<0,\nonumber \\ w_{3}&= w_3>0, \quad \quad w_{{4}}={\frac{\tau _{{1}}w_{{3}}}{W_{{2}}}}>0, \quad \quad w_{{5}}={\frac{W_{{6}}w_{{7}}}{\gamma _{{2}}}}>0, \quad \quad w_{{6}}={\frac{\alpha \,w_{{4}}+\tau _{{2}}w_{{5}}}{W_{{4}}}}>0, \quad \quad \nonumber \\ w_{7}&= w_{7}>0, \quad \quad w_{{8}}={\frac{\rho \,w_{{6}}+\tau _{{3}}w_{{7}}}{W_{{5}}}}>0.\end{aligned}$$

(48)

$$\begin{aligned} \text {and}&\nonumber \\ v_{1}&=0, \quad \quad v_{2}=0, \quad \quad v_{3}=v_{3}>0, \quad \quad v_{4}=\frac{\alpha v_{6}+v_{3}C_1 Z_{3}}{W_2}>0, \quad \quad \\ v_{5}&=\frac{v_{6}\tau _{2}+v_{7}\gamma _2}{W_3}>0, \quad \quad v_{6}=\frac{C_4 Z_{3}v_{3}+\rho v_{8}}{W_4}, \quad \quad v_{7}=\frac{\tau _3 v_{8}+C_2 Z_{3}v_{3}}{W_6} \quad \quad \nonumber \\ v_{8}&=\frac{v_{3}C_3 Z_3}{W_5}.\nonumber \end{aligned}$$

(49)

with $v\cdot w=1$

Following theorem 4.1 in Castillo-Chavez and Song (2004), we compute the associated backward bifurcation parameters a and b that are responsible for the direction of bifurcation,

$$\begin{aligned} \mathbf{a }=\sum \limits _{k,i,j=1}^{n}v_kw_iw_j\frac{\partial ^2f_k(0,0)}{\partial x_i\partial x_j}, \quad \quad \text {and} \quad \quad \mathbf{b }=\sum \limits _{k,i=1}^{n}v_kw_i\frac{\partial ^2f_k(0,0)}{\partial x_i\partial \beta ^{c}} \end{aligned}$$

(50)

After computing the non-zero second order partial derivatives of f as defined in (45), and evaluate around diseases free $E_0$, we obtain:

$$\begin{aligned} \mathbf{a }=\frac{2\beta \mu v_3W_{12}}{\varLambda (\mu +\xi )}\left( L_1-L_2 \right) \end{aligned}$$

(51)

and

$$\begin{aligned} \mathbf{b }= {\frac{ \left( \left( 1-kp \right) \mu +(1-k)\,\xi \right) \left( \left( \eta _{{2}}w_{{4}}+\eta _{{3}}w_{{8}}+\eta _{{5}}w_{{6}} \right) \theta +\eta _{{4}}w_{{7}}+\eta _{{1}}w_{{3}}+w_{{5}} \right) v_{{3}}}{ \mu +\xi }}>0 \end{aligned}$$

(52)

where,

• $L_1=k\left( (\xi +p\mu )W_{11}-w_2k\mu (1-p) \right)$

• $L_2=W_{11}(\mu +\xi )-(\xi +p\mu )w_1 k$

• $W_{11}=w_{{3}}+w_{{4}}+w_{{5}}+w_{{6}}+w_{{7}}+w_{{8}}$

• $W_{12}=\left( \theta \,\eta _{{2}}w_{{4}}+\theta \,\eta _{{3}}w_{{8}}+\theta \,\eta _{{5}} w_{{6}}+\eta _{{1}}w_{{3}}+\eta _{{4}}w_{{7}}+w_{{5}} \right)$

Since, $0\le k,p \le 1$, then $(1-kp)\ge 0$, implies $b>0$. Hence, backward bifurcation exist only if $a>0$. $\square$