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


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.

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  1. Abbas UL, Anderson RM, Mellors JW (2007) Potential impact of antiretroviral chemo- prophylaxis on HIV-1 transmission in resource-limited settings. PLoS ONE 2:e875

    Article  Google Scholar 

  2. AIDS (2014) Steping up the pace global fact sheet: HIV/AIDS, 20th International AIDS Conference. Melbourne, Australia

  3. Anderson RM (1988) The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS. J AIDS 1:241–256

    Google Scholar 

  4. Avert (2017) Avert: HIV around the World. Accessed 17 Jan 2019

  5. Avert (April, 2019) HIV and AIDS in Nigeria. Accessed 23 Apr 2019

  6. Berker MH, Joseph JG (1998) AIDS and behavioral change to reduce risk: a review. Am J Public Health 78:394–410

    Article  Google Scholar 

  7. Blayneh K, Cao Y, Kwon H (2009) Optimal control of vector-borne diseases: treatment and prevention. Discrete Contin Dyn Syst Ser B 11(3):587

    MathSciNet  MATH  Google Scholar 

  8. Blower SM, Gershengorn HB, Grant RM (2000) A tale of two futures: HIV and antiretroviral therapy in San Francisco. Science 287:650–654

    Article  Google Scholar 

  9. Castillo-Chavez C, Song B (2004) Dynamical model of tuberclosis and their applications. Math Biosci Eng 1(2):361–404

    MathSciNet  MATH  Article  Google Scholar 

  10. Central Intelligence Agency (CIA) (2019) The world fact book on country comparism by GDP. Accessed 16 Jan 2019

  11. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. Mc-Graw Hill Co., Inc., New York

    Google Scholar 

  12. Connell Gumel AB, McCluskey C, van den Driessche P (2006) Mathematical analysis of a staged-progression HIV model with imperfect vaccine. Bull Math Biol 68(8):2105–2128

    MathSciNet  MATH  Article  Google Scholar 

  13. Cristiana JS, Delfim FMT (2017) Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete Contin Dyn Syst Ser S.

  14. Elbasha EH, Gumel AB (2006) Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits. Bull Math Biol 68:577–614

    MATH  Article  Google Scholar 

  15. Eze JI (2009) Modelling HIV/AIDS in Nigeria. Ph.D. dissertation, University of Glasgow

  16. Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Google Scholar 

  17. Garba SM, Gumel AB (2010) Mathematical recipe for HIV elimination in Nigeria. J Niger Math Soc 29:1–66

    MathSciNet  MATH  Google Scholar 

  18. Gumel AB, Mccluskey CC, Van Den Driessche P (2006) Mathematical study of a staged-progression HIV model with imperfect vaccine. Bull Math Biol 68:2105–2128

    MathSciNet  MATH  Article  Google Scholar 

  19. Hale JK, Lunel SMV (1993) Introduction to functional differential equations. Springer, Berlin

    Google Scholar 

  20. Hove-Musekwa DS, Nyabadza F, Mambili-Mamboundou H, Chiyaka C, Mukandavire Z (2014) Cost-effectiveness analysis of hospitalization and home-based care strategies for people living withHIV/AIDS: the case of Zimbabwe. Int Sch Res Not 2014:1–13

    Article  Google Scholar 

  21. Hussaini N, Winter M, Gumel AB (2011) Qualitative assessment of the role of public health education program on HIV transmission dynamics. Math Med Biol J IMA 28:245–270

    MathSciNet  MATH  Article  Google Scholar 

  22. Hyman JM, Stanley EA (1988) Using mathematical models to understand the AIDS Epidemic. Math Biosci 90:415–473

    MathSciNet  MATH  Article  Google Scholar 

  23. Karrakchou J, Rachik M, Gourari S (2006) Optimal control and infectiology: application to an HIV/AIDS model. Appl Math Comput 177(2006):807–818

    MathSciNet  MATH  Google Scholar 

  24. Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability analysis of nonlinear systems. Basel, New York

    Google Scholar 

  25. LaSalle JP (1976) The stability of dynamical systems. Regional Conf. Ser. Appl. Math. SIAM, Philadelphia

    Google Scholar 

  26. Lenhart S, Workman JT (2007) Optimal control applied to biological problems. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  27. Li J, Yanga Y, Zhoub Y (2011) Global stability of an epidemic model with latent stage and vaccination. Nonlinear Anal Real World Appl 12:2163–2173

    MathSciNet  Article  Google Scholar 

  28. Moualeu DP, Weiser M, Ehrig R, Deuflhard P (2013) Optimal control for a tuberculosis model with undetected cases in Cameroon. ZIB-report 13–73 (November 2013)

  29. Mukandavire Z, Garirar W (2007) Effect of public health educational campaigns and the role of sex workers on the spread of HIV/AIDS among heterosexuals. Theor Popul Biol 72:346–365

    MATH  Article  Google Scholar 

  30. Naif HM (2013) Pathogenesis of HIV infection. Infect Dis J Rep 5(suppl 1):e6

    Article  Google Scholar 

  31. Naresh R, Tripathi A, Omar S (2006) Modelling the spread of AIDS epidemic with vertical transmission. Appl Math Comput 178:262–272

    MathSciNet  MATH  Google Scholar 

  32. Ogunlaran OM, Oukouomi Noutchie SC (2016) Mathematical Model for an Effective Management of HIV Infection. BioMed Res Int 2016:1–6

  33. Papa F, Binda F, Felici G, Franzetti M, Gandolfi A, Sinisgalli C, Balotta C (2018) A simple model of HIV epidemic in Italy: the role of the antiretroviral treatment. Math Biosci Eng 15(1):181

    MathSciNet  MATH  Google Scholar 

  34. Sachs JD (2001) Macroeconomics and health: investing in health for economic development. Technical report. World Health Organisation, Geneva

    Google Scholar 

  35. Seidu B, Makinde OD, Daabo MI (2016) Optimal control analysis of an HIV/AIDS model with linear incidence rate. J. Math. Comput. Sci. 6(1):58–75

    Google Scholar 

  36. Serah AA, Farida M, Muna A (2011) Stability analysis of an HIV/AIDS epidemic model with screaning. Int Math Forum 6(66):3251–3273

    MathSciNet  MATH  Google Scholar 

  37. Sharomi O, Gumel AB (2009) Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis. J Math Anal Appl 356(1):96–118

    MathSciNet  MATH  Article  Google Scholar 

  38. Sharomi O, Podder CN, Gumel AB, Elbasha EH, Watmough J (2007) Role of incidence function in vaccine-induced backward bifurcation in some HIV models. Bull Math Biol 210(2):436–463

    MathSciNet  MATH  Google Scholar 

  39. Shirazian M, Hadi MF (2010) Optimal control strategy for a fully determined HIV model. Intell Control Autom 1:15–19.

    Article  Google Scholar 

  40. Shuai Z, Van Den Driessche P (2013) Global stability of infectious disease models using Lyapunov functions. SIAM J. APPL. MATH. 73(4):1513–1532

    MathSciNet  MATH  Article  Google Scholar 

  41. Silva CJ, Torres DFM (2015) A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete Contin Dyn Syst 35(9):4639–4663

    MathSciNet  MATH  Article  Google Scholar 

  42. Smith JA, Sharma M, Levin C, Baeten JM, van Rooyen H, Celum C, Barnabas RV (2015) Cost-effectiveness of community-based strategies to strengthen the continuum of HIV care in rural South Africa: a health economic modelling analysis. The Lancet HIV 2(4):e159–e168

    Article  Google Scholar 

  43. The Department of Health (2002) Australia government, return in investiment in needle and syringe programmes in Australia: report. Accesed 17 Jan 2019

  44. U.S. Department of Veterans Affairs (2018) HIV/AIDS: CD4 count. Accessed 17 Jan 2019

  45. Van den Driessche P, James W (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48

    MathSciNet  MATH  Article  Google Scholar 

  46. Weinstein M (1996) From cost-effectiveness ratios to resource allocation: Where to draw the line? In: Sloan F (ed) Valuing health care: costs, benefits, and effectiveness of pharmaceuticals and other medical technologies. Cambridge University Press, New York, pp 77–97

    Google Scholar 

  47. Weldegiorgis HB, Makinde OD, Mwangi DT (2018) Co-dynamics of measles and dysentery diarrhea diseases with optimal control and cost-effectiveness analysis. Appl Math Comput 347(2019):903–921

    MathSciNet  MATH  Google Scholar 

  48. WHO (2019) Global Health Observatory in Africa. Accessed 3 Jan 2019

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We are very thankful to the anonymous reviewers and the Handling Editor for their very constructive comments, which have, undoubtedly, significantly enhanced the quality and clarity of the manuscript.

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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

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

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 \(N(t)= x_{1} +x_{2} +x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}\).

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

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

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

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:

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

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

$$\begin{aligned} w_{{1}}&= -{\frac{ \left( 1-p \right) \mu \, \left( C_1 w_{{4}}+C_3 w_{{8}}+C_4 w_{{6}}+ C_0 w_{{3}}+C_2 w_{{7}}+\beta ^{c}\,w_{{5}} \right) }{{W_{{0}}}^{2}}}<0, \nonumber \\ w_{{2}}&= {\frac{\xi \,w_{{1}}-Z_{{2}} \left( C_1 w_{{4}}+C_3 w_{{8}}+C_4 w_{{6}}+C_0 w_{{3}}+C_2 w_{{7}}+w_{{5}} \right) }{\mu }}<0,\nonumber \\ w_{3}&= w_3>0, \quad \quad w_{{4}}={\frac{\tau _{{1}}w_{{3}}}{W_{{2}}}}>0, \quad \quad w_{{5}}={\frac{W_{{6}}w_{{7}}}{\gamma _{{2}}}}>0, \quad \quad w_{{6}}={\frac{\alpha \,w_{{4}}+\tau _{{2}}w_{{5}}}{W_{{4}}}}>0, \quad \quad \nonumber \\ w_{7}&= w_{7}>0, \quad \quad w_{{8}}={\frac{\rho \,w_{{6}}+\tau _{{3}}w_{{7}}}{W_{{5}}}}>0.\end{aligned}$$
$$\begin{aligned} \text {and}&\nonumber \\ v_{1}&=0, \quad \quad v_{2}=0, \quad \quad v_{3}=v_{3}>0, \quad \quad v_{4}=\frac{\alpha v_{6}+v_{3}C_1 Z_{3}}{W_2}>0, \quad \quad \\ v_{5}&=\frac{v_{6}\tau _{2}+v_{7}\gamma _2}{W_3}>0, \quad \quad v_{6}=\frac{C_4 Z_{3}v_{3}+\rho v_{8}}{W_4}, \quad \quad v_{7}=\frac{\tau _3 v_{8}+C_2 Z_{3}v_{3}}{W_6} \quad \quad \nonumber \\ v_{8}&=\frac{v_{3}C_3 Z_3}{W_5}.\nonumber \end{aligned}$$

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,

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

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

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


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


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

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

  • \(W_{11}=w_{{3}}+w_{{4}}+w_{{5}}+w_{{6}}+w_{{7}}+w_{{8}}\)

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

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

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Ishaku, A., Gazali, A.M., Abdullahi, S.A. et al. Analysis and optimal control of an HIV model based on CD4 count. J. Math. Biol. (2020).

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  • CD4 count
  • Stability analysis
  • Optimal control
  • Cost effectiveness analysis

Mathematics Subject Classification

  • 92B05
  • Ó4A34
  • 37G10
  • 49K15