Cervical Cancer and HIV Diseases Co-dynamics with Optimal Control and Cost Effectiveness

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1 Pure and Applied Matematics Journal 017; 6(4: ttp:// doi: /j.pamj ISS: (Print; ISS: (Online Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness Geomira George Sanga 1 Oluwole Daniel Makinde Estomi Sedrack Massawe 1 Lucy amkinga 3 1 Department of Matematics University of Dar es Salaam Dar es Salaam Tanzania Faculty of Military Science Stellenbosc University Saldana Sout Africa 3 Department of Molecular Biology and Biotecnology University of Dar es Salaam Dar es Salaam Tanzania address: gmtisi@gmail.com (G. G. Sanga makinded@gmail.com (O. D. Makinde estomimassawe@yaoo.com (E. S. Massawe odulajalucy@yaoo.com (L. amkinga Corresponding autor To cite tis article: Geomira G. Sanga Oluwole D. Makinde Estomi S. Massawe Lucy amkinga. Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness. Pure and Applied Matematics Journal. Vol. 6 o pp doi: /j.pamj Received: June 017; Accepted: July 7 017; Publised: August Abstract: Te deterministic model for co-infection of cervical cancer and HIV (Human Immunodeficiency Virus diseases is formulated and rigorously analyzed. Te optimal control teory is employed to te model to study te level of effort is needed to control te transmission of co-infection of cervical cancer and HIV diseases using tree controls; prevention screening and treatment control strategies. umerical solutions sow a remarkable decrease of infected individuals wit HPV (Human Papilloma Virus infection cervical cancer cervical cancer and HIV cervical cancer and AIDS (Acquire Immunodeficiency Syndrome HIV infection and AIDS after applying te combination of te optimal prevention screening and treatment control strategies. However Incremental Cost-Effective Ratio (ICER sows tat te best control strategy of minimizing cervical cancer among HIV-infected individuals wit low cost is to use te combination of prevention and treatment control strategies. Keywords: HPV Infection HIV Infection Cervical Cancer Optimal Control Cost-Effectiveness 1. Introduction Cervical cancer is a major cause of morbidity and mortality among women in sub-saaran countries and about 70% of cervical cancers are caused by Human Papillomavirus (HPV types 16 and 18 wic are transmitted sexually troug body contact. Some studies ave sown tat HIV-infected women after being infected wit HPV infection ave a ig risk to progress to HPV-related cervical diseases and invasive cervical cancer tan women witout aving HIV infection [1 3]. Te aim of tis work is to study te effect of incorporating tree optimal control strategies to te co-infection model of cervical cancer and HIV diseases. In [15] formulated coinfection model of cervical cancer and HIV diseases but te findings of tis paper differ from te work presented in [15] because te co-infection model incorporates tree optimal control strategies; prevention screening and treatment.. Optimal Control Analysis Here we introduce optimal control strategies to te coinfection model of cervical cancer and HIV disease as presented in [15]. Te co-infection model in [15] is developed as follows: Te total polation of individuals at any time t denoted as is categorized into ten compartments according to te individual s status of infection as follows: Susceptible S HIV-infected individuals no HPV infection individuals ( ( I AIDS individuals no HPV infection ( HPV infected individuals no HIV infection ( I D Unscreened l Screened

2 15 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness infectious individuals sowing te impact of HPV infection witout HIV infection ( I ps Individuals wit cervical cancer no HIV infection Unscreened HPV infected individuals wit Screened infectious individuals HIV infection ( I sowing te impact of HPV infection wit HIV infection I ( I ps HIV-infected individuals wit cervical cancer ( c and AIDS individuals wit cervical cancer ( D lc. Tus ( = ( + ( + ( + ( + ( + ( + ( + ( + ( + ( t S t I t I t I t I t I t I t I t D t D t ps ps c c l lc Te rates of transferring between different compartments are as described in Table 1. Table 1. Description of parameters and teir values for te model of co-infection HIV and cervical cancer diseases. Parameter Description Value Reference µ atural mortality rate yr -1 [11] ω 1 atural recovered rate for HIV-infected individuals wit HPV infection 0.7yr -1 [1] ω atural recovered rate for HPV unscreened infected individuals 0.366yr -1 σ Treatment rate for HPV screened infected individuals wit only HPV infection yr -1 σ 1 Treatment rate for HPV screened infected individuals wit HPV and HIV infections. 0.00yr -1 Progression rate from HPV unscreened infected individuals wit HIV infection to α 1 HPV screened infected individuals wit HIV infection yr -1 Progression rate from HPV screened infected individuals wit HIV infection to α cervical cancer individuals wit HIV infection yr -1 Progression rate from cervical cancer individuals wit HIV infection to AIDS α 3 individuals wit cervical cancer yr -1 Progression rate from HPV unscreened infected individuals to HPV screened α 4 infected individuals yr -1 Progression rate from HPV screened infected individuals to cervical cancer α 5 individuals yr -1 [1] Progression rate from HPV unscreened infected individuals wit HIV infection to α 6 cervical cancer individuals wit HIV infection yr -1 α 7 Progression rate from HIV infected individuals to AIDS individuals yr -1 [4] Progression rate from HPV unscreened infected individuals to cervical cancer individuals yr -1 d Te deat rate due to HPV infection for HPV unscreened infected individuals. Close to zero [1] d ps Te deat rate due to impact of HPV infection for HPV screened infected individuals dc Deat rate due to cervical cancer 0.037yr -1 [13] d Te deat rate due to HIV and impact of HPV infection for HPV unscreened infected individuals wit HIV infection yr -1 d ps Te deat rate due to HIV and impact of HPV infection for HPV screened infected individuals wit HIV infection yr -1 dc Deat rate due to cervical cancer and HIV yr -1 dlc Deat rate due to cervical cancer and AIDS yr -1 d Deat rate due to HIV yr -1 dl Deat rate due to AIDS yr -1 [14] η1 Infectivity rate of HIV infection η Infectivity rate of AIDS η3 Infectivity rate for HPV unscreened infected individuals wit HIV infection η4 Infectivity rate for HPV screened infected individuals wit HIV infection η5 Infectivity rate for cervical cancer individuals wit HIV infection η6 Infectivity rate for cervical cancer individuals wit AIDS η7 Infectivity rate for HPV unscreened infected individuals η8 Infectivity rate for HPV screened infected individuals η9 Infectivity rate for cervical cancer individuals β H Probability of transmission HIV infection β HC Probability of transmission bot HIV and HPV infections 0 < β HC < 1 β C Probability of transmission HPV infection 0 < β C < 1 q Mean number of contacts 1-3

3 Pure and Applied Matematics Journal 017; 6(4: Te matematical model is defined by te following system of ordinary differential equations: ds ( = π + ω I + σ I µ + λ + λ + λ S ps H HC C ( = λ S + ω I + σ I µ + d + α + λ I H 1 1 ps 7 C ( = λ S + λ I + λ I α + α + ω + µ + d I HC C H ( 4 8 = λ S λ I ω + µ + d + α + α I C H ps ( = α I λ + σ + α + µ + d I (1 4 H 5 ps ps ps c ( = α I + λ I µ + d + α + σ I 1 H ps ps 1 ps ( = α I + α I µ + d + α I ps 6 c 3 c c ( = α I + α I µ + d I 8 5 ps c c dd dd l lc 7 ( = α I µ + d D 3 l l ( = α I µ + d D c lc lc were βhq λh = ( η1i + ηdl wit η > η1 βhcq λhc = ( η3i + η4ips + η5ic + η6dlc wit η6 > η5 > η3 > η4 βcq λc = ( η7i + η8ips + η9ic wit η9 > η7 > η8. Hence te system (1 is modified by introducing time-dependent control; u1 ( t represents prevention control strategy (Education campaign and ealt ygiene practice u ( t represents te treatment of early stages of cervical cancer and u3 ( t represents screening individuals sowing an impact of HPV infection leads to cervical cancer. Te following system of equations is obtained ds ( ( 1 ( = π + u + σ I + ω I u λ + λ + λ S µ S ps 1 H HC C ( 1 ( λ λ ( σ ω ( α µ = u S I + u + I + I + d + I 1 H C 1 ps 1 7 ( 1 1( λ λ λ ( 3 α1 ( α6 ω 1 µ = u S + I + I u + I + d + + I HC H C

4 17 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness ( 1 1 ( λ λ ( 3 α4 ( α8 ω µ = u S I u + I + d + + I C H ps ( 3 α1 ( 1 1 λ ( σ1 ( α µ = u + I + u I u + I + d + I H ps ps ps ps ps ( 3 α4 ( 1 1 λ ( σ ( α 5 µ = u + I u I u + I + d + I H ps ps ps ps c ( = α I + α I µ + d + α I ps 6 c 3 c c ( = α I + α I µ + d I 5 ps 8 c c dd l 7 ( = α I µ + d D l l dd lc 3 ( = α I µ + d D ( c lc lc Te objective functional is defined as follows were A i and weigts. Te term tf u3 J( u u u lim AI A I A I A I B B B (3 u1 u 1 3 = ps + 4 ps u1 u u 3 0 B j for i = and j = 13 are positive 1 1 Bu represents te cost of control effort on prevention strategy against HPV infection B u represents te cost of control effort on screening individuals wit or witout HIV infection aving HPV infection and 3 3 B u represents te cost of control effort on treating individuals aving cervical cancer wit or witout HIV infection. Te main goal of introducing time-dependent controls in te co-infection model is to prevent women not to acquire HPV infection wic leads to cervical cancer and to minimize infected women sowing te impact of HPV infection lead to cervical cancer wile minimizing te cost of controls u1 ( t u ( t and 3 ( to seek an optimal control tat u t as in [5 6 7]. Te goal is u 1 u and ( ( u 3 numerically suc { J u u u u u u } J u u u = min Ω ( Ω = { 1 3 suc tat 1 3 were ( u u u u u u measurable wit 0 u1 10 u 1 and 0 u1 1 for t 0 t f } is te control set. Te necessary conditions tat an optimal must satisfy come from te Pontryagin s maximum principle [8]. Tis 3 into teproblem of principle converts ( and ( minimizing point-wise Hamiltonian H wit respect to u 1 u and u 3. u1 u 1 3 ps 4 ps 1 3 u H = AI + A I + A I + A I + B + B + B ( π ( σ ω ( 1 1 ( λ λ λ µ + M + u + I + I u + + S S S ps H HC C (( 1 1 ( λ λ ( σ1 ω1 ( α 7 µ + M u S I + u + I + I + d + I I H C ps (( 1 1( λ λ λ ( 3 α1 ( α6 ω 1 µ + M u S + I + I u + I + d + + I I HC H C (( 1 1( λ λ ( 3 α4 ( α8 ω µ + M u S I u + I + d + + I I C H 3

5 Pure and Applied Matematics Journal 017; 6(4: were M S M I M I M I ps (( 3 α1 ( 1 1 λ ( σ1 ( α µ + M u + I + u I u + I + d + I I H ps ps ps ps ps (( 3 α4 ( 1 1 λ ( σ ( α 5 µ + M u + I u I u + I + d + I I H ps ps ps ps c ( α α 6 ( µ α3 + M I + I + d + I I ps c c c ( α5 α 8 ( µ + M I + I + d I I ps c c ( α 7 ( µ + M I + d D M I ps Teorem: For te optimal control triples M S M I M M I ps M I I c M I c M I M Ips M D l and MD lc Dl l l ( α 3 ( µ + M I + d D (5 I ps Dlc c lc lc M M u 1 satisfying I c M I c u and 3 M D l and MD lc u tat minimize ( are te co-state variables or te adjoint variables. J u u u over Ω tere exist adjoint variables 1 3 dm j H = (6 j were j = S I I I I ps I c I c D l Dlc and wit transversality condition ( = ( = ( = ( = ( = ( = ( = ( = ( = MD ( tf = 0 M t M t M t M t M t M t M t M t M t and S I I I Ips Ips Ic Ic Dl f lc 1 u 1 max 0 min I 1 psλh ( MI M ( ps I Sλ ps H MI M H S IλH ( MI M = + + I B 1 SλHC ( MI M S SλC ( MI M S λcih ( MI M I ( ( ps ( ps ps I I ps I S 1 u = max 0min 1 I M M + I M M B ( ( ps ( ps I I I I 1 u3 = max 0min 1 I M M + I M M B3 (7 (8 (9 Proof. Corollary 4.1 of Freming and Rices [9] gives te existence of an optimal control due to te convexity of te integrand J wit respect to u 1 u and u3 a priori boundedness of te state solutions and te Lipcitz property of te state system wit respect to te state variables. Differentiating Hamiltonian functions wit respect to state variables gives differential equations governing te adjoint variables as follows; ( 1 1( IλH ( 1 u1 ( MI M I ps ps dm IpsλH u MI MI S = MS µ + + ( 1 1 ( IλC u MI MI + λ ( 1 u ( M M + + λ 1 u M M ( ( HC 1 S I H 1 S I

6 19 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness ( 1 u1 ( M M ps ( 1 1 λ ( ( 1 H I S u S 1 λhc ( MI M S + λ + +. C S I dm I S u M M ( ( 1 1 β η1 ( λ ( 1 ps ps 1( ps ps I q u M M I u M M ps H I I ps H I I = α 7MD + M l I µ + d + + ( 1 ( ( 1 ( ( 1 ps ( Sβ qη u M M I β qη u M M S u λ M M H 1 1 S I H 1 1 I I 1 HC I S ( 1 1( ( 1 1( ( 1 1( I λ u M M I λ u M M Sλ u M M ps H I I C I I H I S ( 1 1( Sλ u M M + λc ( 1 u1 ( MI M I + C I S ( I I ( ( c I ( I Ips ( I I dmi = A1 + α 6 M M + M µ + d + u3 + α1 M M + ω1 M M ( 1 1 λ ( Sλ ( 1 1 ( ( 1 1 ps ps H u MI M β η S 3 ( I u M M Sq u M M ps H I I HC S I ( 1 1 λ ( λ ( 1 1( λ ( 1 1( I u M M I u M M S u M M H I I C I I HC I S ( 1 1( SλC u MI MS +. dm I 8 ( I Ic = A + α M M + + ( 1 1( λ ( 1 ps ps 1( I λ u M M I u M M ps H I I H I I ( 1 1 β η7 ( I u q C MI MI + + M + d + u M M + M M ( ( ( µ λ ( 1 1( ω ( I H I I I S ( u3 α4 ( MI MIps ( λ ( I λ 1 u M M S 1 u M M + C 1 I I 1 HC I S ( 1 1 λc ( I ( 1 1 ( ( 1 1 S S u λh MI M β S Cη7 ( S I S u M M Sq u M M ( 1 1 λ ( λ ( 1 ps 1( dm I u M M S u M M = + ( I ps Ic + + I ps H I I ps I S A3 α M M ( λ ( ( ( I 1 u M M I λ 1 u M M + + ( u3 + σ 1( MI M ( ps I + M I µ + d ps ps + 1 C I I H 1 I I

7 Pure and Applied Matematics Journal 017; 6(4: ( 1 ( ( ( ( ( 1 β HC η 4 S I λ C I S 1 λ HC I S Sq u M M S u M M S u M M ( I ( ( ps Ic Ips ps ( Ips S dmips = A4 + α 5 M M + M µ + d + u + σ1 M M ( ps ps IpsλH (1 u1 MI MI + + (1 u λ M M + (1 u I M M ( λ ps ps ( 1 H I I 1 H I I ( ( ( S(1 u β η q M M Sλ (1 u M M I λ (1 u M M 1 C 8 S I C 1 I S C 1 I I ( S(1 u1 λ H ( MI MS λ ( I (1 u β η q M M S(1 u M M 1 C 8 I I 1 HC I S dm I ( µ α3 ( ( 1 1( λ ( 1 1( I λ u M M S u M M ps H I I H I S c ps ps = MI + d c c + MI M c D + + lc ( ( 1 ( ( 1 ( Sqβ η M M Sλ u M M Sλ u M M HC 5 S I HC 1 I S C 1 I S ( 1 1( λ ( 1 1( I λ u M M I u M M + + H I I C I I ( 1 ( ( 1 ( ( 1 ( Sβ qη u M M I λ u M M Sλ u M M C 9 1 S I H 1 I I C 1 I S ( 1 ( ( 1 ( I β qη u M M I λ u M M + + C 9 1 I I C 1 I I dm D ( µ ( 1 ( ( 1 ps ps ( ps ps I β qη u M M I u λ M M ps 1 I I ps 1 H I I l = + dl MD + + l ( 1 ( ( 1 ( λhc ( 1( I S Sβ η q u M M Sλ u M M S 1 u M M H 1 S I H 1 I S ( 1 1( ( 1 1( ( 1 1( I λ u M M I λ u M M I β η q u M M C I I H I I H I I dm ( µ d ( Sλ ( 1 1 ( ( 1 ps ps H u M I M λ S 1 ( dm I λ M M S u M M Ic = MI ( µ + d c c ps H I I HC I S ( 1 u1 IpsλH ( MI MI S ( 1 u1 λh ( MI MS ( 1 1( Sλ u M M + C I S Dlc ps ps = + lc MD + + (10 lc Optimality equations ( 7 - ( 9 are obtained by comting partial derivative of te Hamiltonian equation ( 5 wit

8 131 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness respect to eac control variables as follows Solving for u 1 H 0 = for i = 13 u i u and u 3 subject to te constraints provides te caracterization equations ( 7 - ( 9. In te next part te numerical solutions of optimality system are discussed. 3. umerical Solutions In order to obtain optimal control solutions te optimality system wic consists of two systems namely; te state system and te adjoint system is solved. In solving state equations using forward fourt order Runge-Kutta an initial guess of all controls over time are made and te initial value of state variables are introduced. Having te solution of state functions and te value of optimal controls te adjoint equations are solved using backward fourt order Runge- Kutta by using transversality condition. In tis simulation te weigts are cosen be A 1 = 80 A = 75 A 3 = 50 A 4 = 45 B 1 = 100 B = 110 andb 3 = 10. Oter parameter descriptions and values used in getting numerical results of te co-infection model of cervical cancer and HIV diseases are presented in Table 1. Control strategies are formed and studied numerically as follows 3.1. Strategy I: Combination of Prevention and Treatment Control Strategies Te combination of prevention control strategy u1 and treatment control u are used to optimize objective functional wile setting screening control u 3 equal to zero. Te results sow tat applying optimal prevention and treatment control strategies; te polation of susceptible individuals increases wereas tepolation of all infected compartments decreases as illustrated in Figure1: A and Figure1: B-J respectively.

9 Pure and Applied Matematics Journal 017; 6(4: Figure 1. Te use of prevention and treatment control strategies. Figure 1: A - J are time series plot of different polation wereq 1 = u 1 0 u 0 u 3 = 0 wit control and Q ( u 0 u 0 u 0 = 1 = = 3 = witout control. 3.. Strategy II: Combination of Prevention Control Strategy and Screening Strategy Te combination of prevention control strategy u 1 and screening control strategy u 3 are used to optimize objective functional wile setting treatment control strategy u equal to zero. Results illustrate tat te polation of susceptible individuals increases (see Figure : A wile te polation of infected individuals decreases (see Figure : B-J.

10 133 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness Figure. Te useprevention and screening control strategies. Figure : A - J are time series plot of different polation wereq 1 = u 1 0 u = 0 u 3 0wit control and Q ( u 0 u 0 u 0 = 1 = = 3 = witout control Strategy III: Combination of Screening and Treatment Control Strategies Te combination of treatment control u and screening control u 3 are used to optimize objective functional wile setting vaccination control 1 u to zero. Results indicate tat tere is unremarkable increase of susceptible individuals and te polation of infected individuals declines not muc compared wit oter combination control strategies as illustrated in Figure 3: A and Figure 3: B - J respectively.

11 Pure and Applied Matematics Journal 017; 6(4: Figure 3. Te use of screening and treatment control strategies. Figure 3: A - J are time series plot of different polation wereq 1 = u 1 = 0 u 0 u 3 0wit control and Q ( u 0 u 0 u 0 = 1 = = 3 = witout control Strategy IV: Combination of Prevention Treatment and Screening Control Strategies Te combination of prevention control strategy treatment control strategy u and screening control strategy u 3 are used to optimize objective functional. Results illustrates tat te polation of susceptible individuals increases as sown in Figure 4: A wile te polation of infected individuals decrease compared wit oter combination of control strategies as sown in Figure 4: B - J.

12 135 Geomira G. Sanga et al.: Cervical Cancer and HIV Diseases Co-dynamics wit Optimal Control and Cost Effectiveness Figure 4. Sowing impact of using all control strategies. Figure 4: A - J are time series plot of different polation wereq 1 = u 1 0 u 0 u 3 0 wit control and Q ( u 0 u 0 u 0 = 1 = = 3 = witout control. 4. Cost-effectiveness Analysis Here Incremental Cost Effectiveness Ratio (ICER is used to quantify te cost-effectiveness of different strategies. Tis approac is useful to understand wic strategy saves a lot of averted species wile spending low cost. Tis tecnique is needed to compare more tan one competing interventions strategies incrementally one intervention sould be compared wit te next less effective alternative [10]. Te ICER formula is given by Difference in intervention cost ICER= Difference in te total number of infection averted Te total number of infection averted is obtained by calculating te difference between te total number of new cases of individuals aving HPV infection witout control and te total number of new cases of individuals aving HPV infection wit control. Table. Calculation of ICER after arranging te number of total infections averted in ascending order. Strategy Total Cost ($ Total infections Averted ICER Strategy III $ Strategy II $ Strategy I $ Strategy IV $ ICER(Strategy III = = ICER(Strategy II = = ICER(Strategy I = = and ICER(Strategy IV = = Comparing strategy III and strategy II ICER of strategy II

13 Pure and Applied Matematics Journal 017; 6(4: is less tan ICER of strategy III. Hence strategy III is more costly and less effective tan strategy II. Tus strategy III is omitted and ICER is recalculated. Table 3. Comtation of ICER after dropping strategy III. Strategy Total Cost($ TotalInfections Averted ICER Strategy II $ Strategy I $ Strategy IV $ Comparing strategy II and strategy I ICER of strategy I is less tan ICER of strategy II. Tus strategy II is omitted and te ICER is recalculated. Table 4. Comtation of ICER after dropping strategy II. Strategy Total Cost($ TotalInfections Averted ICER Strategy I $ Strategy IV $ By comparing strategy I and strategy IV ICER of strategy I is less tan ICER of strategy IV. Terefore strategy IV is dropped and strategy I is considered. Tus according to Incremental Cost Effectiveness Ratio analysis te combination of optimal prevention and treatment control strategies is te best way of minimizing cervical cancer among women wit or witout HIV infection in our community following te combination of all control strategies. 5. Conclusion Tis paper designed and analyzed a deterministic model for co-infection of cervical cancer and HIV diseases. Te optimal control teory was employed to te main model and analyzed using Potrayagin s Maximum Principle. Te different combination of tree optimal control strategies; prevention screening and treatment were studied numerically. Also cost-effectiveness analysis was performed and te following results were obtained (a Te combination of optimal screening and treatment strategies is not muc powerful way of controlling HPV infection and cervical cancer in te community compared wit oter combination of optimal control strategies as sown in Figure 3. Also applying te combination of all control strategies (prevention screening and treatment is te best way of minimizing co-infection of cervical cancer and HIV diseases in a community as sown in Figure 4. (b In te case of Incremental Cost Effectiveness Ratio analysis te combination of optimal prevention and treatment strategies is te most cost-effective control to minimize cervical cancer among women wit or witout HIV infection. Te second is te combination of all optimal control strategies; prevention screening and treatment. Te tird is te combination of optimal prevention and screening control strategies. References [1] S. E. Hawes C. W. Critclow M. A. Faye iang M. B. Diouf A. Diop P. Toure A. Aziz Kasse B. Dembele P. Salif Sow A. M. Coll-Seck J. M. Kuypers. B. Kiviat H. S. E. C. C. W. F.. M. A. D. M. B. D. A. T. P. K. A. A. D. B. S. P. S. C.-S. A. M. K. J. M. and K.. B. Increased risk of ig-grade cervical squamous intraepitelial lesions and invasive cervical cancer among African women wit uman immunodeficiency virus type 1 and infections J. Infect. Dis. 003 vol. 188 no. 4 pp [] S. M. Mbulaiteye E. T. Katabira H. Wabinga D. M. Parkin P. Virgo R. Ocai M. Workne A. Coutino and E. A. Engels Spectrum of cancers among HIV-infected persons in Africa: Te Uganda AIDS-Cancer registry matc study Int. J. Cancer 006 vol. 118 no. 4 pp [3] C. g andwe J. J. Lowe P. J. Ricards L. Hause C. Wood and P. C. Angeletti Te distribution of sexually-transmitted uman papillomaviruses in HIV positive and negative patients in Zambia Africa. BMC Infect. Dis. 007 vol. 7 pp. 77. [4] B. Maregere Analysis of co-infection of uman immunodeficiency virus wit uman papillomavirus University of KwaZulu-atal 014. [5] K. O. Okosun and O. D. Makinde A co-infection model of malaria and colera diseases wit optimal control Mat. Biosci. 014 vol. 58 pp [6] K. O. Okosun and O. D. Makinde Optimal control analysis of malaria in te presence of non-linear incidence rate Appl. Comt. Mat. 013 vol. 1 no. 1 pp [7] K. O. Okosun O. D. Makinde and I. Takaidza Analysis of recruitment and industrial uman resources management for optimal productivity in te presence of te HIV/AIDS epidemic J. Biol. Pys. 013 vol. 39 no. 1 pp [8] S. Lenart and J. T. Workman Optimal control applied to biological models dynamic optimization 007. [9] W. H. Fleming and R. W. Risel Deterministic and stocastic optimal control Springer-Verlag Berlin Heidelberg ew York [10] K. O. Okosun O. D. Makinde and I. Takaidza Impact of optimal control on te treatment of HIV/AIDS and screening of unaware infectives Appl. Mat. Model. 013 vol. 37 no. 6 pp [11] Tanzania-life expectance at birt. [Online]. Available: ttp://countryeconomy.com/demograpy/lifeexpectancy/tanzania. [Accessed: 09-Oct-016]. [1] S. L. Lee and A. M. Tameru A matematical model of uman papillomavirus (HPV in te united states and its impact on cervical cancer J. Cancer 01 vol. 3 no. 1 pp [13] R. Federation S. Africa and S. Lanka Cervical cancer global crisis card Cerv. Cancer Free Coalit [14] HIV and AIDS in Tanzania 015. [Online]. Available: ttp:// [Accessed: 09-Oct-016]. [15] G. G. Sanga O. D. Makinde E. S. Massawe and L. amkinga Modelling co-dynamics of cervical cancer and HIV diseases Glob. J. Appl. Mat. 017 vol. 13 no. 6 pp