Modeling Co-Dynamics of Cervical Cancer and HIV Diseases

Transcription

1 Global ournal of Pure Applied Mathematics. SSN Volume 3 Number (0) pp Research ndia Publications Modeling o-dynamics of ervical ancer V Diseases Geomira G. Sanga Department of Mathematics University of Dar es Salaam Dar es Salaam Tanzania. Oluwole D. Makinde Faculty of Military Science Stellenbosch University Private Bag X Saldanha 395 South Africa Estomih S. Massawe Department of Mathematics University of Dar es Salaam Dar es Salaam Tanzania Lucy Namkinga Department of Molecular Biology Biotechnology University of Dar es Salaam Tanzania. Abstract ervical cancer is a major cause of morbidity mortality among women in sub-sahara countries while the V (uman immunodeficiency virus) epidemic still exists. The co-infection model of cervical cancer V diseases is formulated analyzed qualitatively. The analysis shows that the co-infection model is locally asymptotically stable when the reproduction is less than unit unstable otherwise. Stability of endemic equilibrium is performed using center manifold theory the results show the co-infection model has backward bifurcation under established condition otherwise it has forward bifurcation. Backward bifurcation shows that when the reproduction orresponding author

2 058 Geomira G. Sanga et al number is less than the unit is necessary but not sufficient in eradicating co-infection of cervical cancer V diseases in the community. Keywords: ervical cancer V Stability enter Manifold Theory. NTRODUTON ervical cancer is cancerous cell growth in the cervix about 0% of cervical cancer is caused by human papillomavirus (PV) types 8. ervical cancer can be screened by using conventional Pap smear visual inspection with Acetic Acid []. owever survival of infectious individuals after diagnosis of cervical cancer is much poor in most of the sub-sahara countries because the majority of women having cervical cancer seek medical assistance when the disease is already in advanced stage []. ervical cancer is the fourth among other types of cancer affected women worldwide it is estimated new cases of cervical cancer occurring every year. Globally 0% of the burden of this disease falls in a low level of development. n sub-sahara countries the rate of 3.8 new cases of cervical cancer is diagnosed annually per women.5 per women die due to cervical cancer [3]. Some epidemiological studies showed that V infectious individuals if they acquire PV infection have a high risk to develop invasive cervical cancer compared to individuals without having V infection [ 5]. owever other researchers found that even though highly active antiretroviral therapy (AART) is suitable for Vinfected individuals but did not show a clear benefit in reducing PV-related cervical cancer for women having V infection [0]. The aim of this paper is to get insight on the dynamic of co-infection of cervical cancer V disease. n [] developed a deterministic model for co-infection of PV V infections rposely to investigate the impact of PV infection on the natural history of V infection vice versa. This work differs with the work presented in [] because the co-infection model presented in this work considers the possibility of individuals to acquire both infections of PV V per sexual contact. Also the co-infection model takes into consideration the impact of PV infection leads to cervical cancer for transmission term the model uses the stard incidence function. MODEL FORMULATON n developing the co-infection model of cervical cancer V diseases we consider the sexually active polation of girls aging 5 years above plus women

3 Modeling o-dynamics of ervical ancer V Diseases 059 who mixes homogeneously such that all individuals are equally likely to be infected with PV infection or V infection or multiple infections (V PV). The total polation Nt is divided into ten compartments at time t according to the individual s status of infection as follows; The polation of Susceptible individuals S V-infected individuals no PV (uman Papilloma Virus) infection ADS individuals no PV infection D hl Unscreened PV infected individuals no V infection Screened PV infected individuals no V infection ndividuals with cervical cancer no V infection c Unscreened PV infected individuals with V infection h Screened PV infected individuals with V infection h V-infected individuals with cervical cancer hc ADS individuals with cervical cancer D. Thus the polation size is given by hlc N S h h h c hc Dhl Dhlc. PV infection V infection multiple infections spread through contact with the infected individual. Since there is no treatment of PV infection the model assumes that individuals in h h compartments show symptoms/consequences of PV infection resulted in cervical cancer. Also the model assumes that individuals in the c hc D hlc compartments they have already reached the stage where their cervical cancer status cannot be treated. According to our model susceptible individuals have three chances of acquiring these infections firstly may acquire V infection with infection force of q D N h hl where a transmission rate of V infection is a mean number of contacts is q infectivity rates of V infection are with. The second chance susceptible individuals may acquire multiple infections with infection force of q D N 3 h h 5 hc hlc where a transmission rate of multiple infections is infections are 5 3 with 5 3. infectivity rates of multiple h

4 00 Geomira G. Sanga et al The third chance susceptible individuals may acquire PV infection with infection force of q N 8 9 c where a transmission rate of PV infection is infectivity rates of PV infection which cause cervical cancer are 8 9 with 9 8. The model assumed that screening is only offered for individuals who show symptoms of PV infection thus individuals in the h compartments may be considered for PV screening move to the h compartments with the screening rates respectively. ndividuals in the h compartments may recover PV infection naturally with the rates move to the h S compartments respectively because PV infection may be cleared with the help of body s immune system. ndividuals in the compartments may acquire V infection with a rate move to the h h compartments respectively. ndividuals having V infection h may acquire PV infection move to the h compartment with infection force. ndividuals in the h compartments after having symptoms of PV infection may be treated move to the with the treatment rates respectively. ndividuals in the h S compartments h h compartments may develop cervical cancer move to the hc compartment with the progression rates respectively. Also individuals having cervical cancer V infection may develop ADS move to the D hlc compartment with progression rate 3. ndividuals in the compartment may develop cervical cancer move to the c compartment with progression rate 5 while individuals in the h compartment may develop ADS diseases move to the Dhl compartment with progression rate. Moreover unscreened infected individuals with PV infection or both PV V infections h may develop cervical cancer move to the c the hc compartment with progression rates 8 respectively. owever all individuals in ten compartments may die naturally with mortality rate. The total polation is not constant because there is death rate due to PV infection V infection multiple infections ADS cervical cancer. Therefore the death rate due to PV infection is given as d the death rate due to multiple infections for p

5 Modeling o-dynamics of ervical ancer V Diseases 0 unscreened screened infected individuals are given as d h d h respectively the death rate due PV infection for unscreened screened infected individuals are given as d d respectively the death rate due V cervical cancer is given as d hc the death rate due to cervical ADS is given as d hlc the death rate due to cervical cancer is given as d c the death rate due to ADS diseases is given as d hl. People progress from one compartment to another as their infection status changes. The mathematical model that describes the dynamic of the co-infection of cervical cancer V diseases is given by a system of ordinary differential equations for t 0 as follows ds d d d h S h d d h d hc dc dd dd S d h h h h S d h h h S d 8 5 d d h h h d h h hc 3 hc d hl hlc 8 5 c c d D h hl hl 3 hc dhlc Dhlc. 3. POSTVTY OF SOLUTONS AND NVARANT REGON Lemma (): All the solutions of the system are positive for all time t 0

6 0 Geomira G. Sanga et al Proof: We want to show that variables that S h h h hc ch c D hl D hlc of the model system 3. are positively nonnegative for all nonnegative initial conditions. So at any time t S N N N N N N N N D N Dhlc h h h hc c hl N From the first equation of the system 3. we have ds S q q 3 5 q 8 9 Since S h h h hc c Dhl Dhlc. N N N N N N N N N N Upon integration we obtain q q 3 5 q 8 9 t S t S e for all time t 0. 0 Similarly it can be shown that h t 0 h t 0 t 0 h 0 t 0 hc t 0 c t 0 Dhl t 0 Dhlc t 0 t ence the solutions of the model will remain nonnegative for all non-negative initial condition Lemma (): A region is positively invariant attracts all solutions in 0 0 S h h h c hc Dhl Dhlc S h h h c hc Dhl Dhlc 0 N. Proof: For the invariance region we add all equations of the system obtain dn. N d d d d d d d d D d D Thus can be deduced by h h h h h h ch ch c c hl hl hlc hlc

7 Modeling o-dynamics of ervical ancer V Diseases 03 dn N. Using integration factor we get N t N e e 0 t t Since the right-h side of the equation dn 0 for Nt. n particular Nt if N 0 is bounded by N it follows that. Thus is positively invariant which means that all solutions with initial conditions in remain in for t 0. owever N 0 initial conditions in attracts all solutions in then as t either every solution of the system with 0 enters or N t 0. asymptotically. ence region Therefore the co-infection model of cervical cancer V diseases is well posed has biologically meaningful. By the fundamental theory of ordinary differential equations the global existence continuity of solutions are guaranteed when initial values in.. EXSTENE OF EQULBRUM PONTS omting the steady state of the system we equate the right-h side of the system equal to zero 0 S 0 S h h dh h 0 S h dh h 0 S d d 3

8 0 Geomira G. Sanga et al where 0 h dh h 0 h h dch 3 hc dc c 0 h dhl Dhl 0 3hc dhlc Dhlc q N D 3 h h 5hc Dhlc h hl q 8 9c. N Solving nonlinear equations 3 we get q N 3 M M M M MM S M3M M M M h A A M c 8M 5 M dc = B + B h A M A dh M dh h D D A D A 3 3 hlc dhlc dhc 3 dhlc dhc 3 D A D A hc dhc 3 dhc 3 D B B hl dhl dhl where

9 Modeling o-dynamics of ervical ancer V Diseases 05 B B M h 5 K d M M dh dh K M M K M dh dh K M 3 dh B M M5 A K M K K M M M M M 3 dh B dh A K K d M 5 d M 8 5 M M d c Exping the expression of we obtain h MM M5 M M d 8 where 5 q B B N dhl Exping expression of we get q N B B d 0or P P 0 where q 3 5 P A D E L N P q 3A D 5E L N Exping the expression of we have hl

10 0 Geomira G. Sanga et al 0 or M 8 N q 9M M M Taking 0from the equations substitute to the equation 5 we get 0. Disease-free equilibrium Substituting free equilibrium E0 as follows into the equations we get disease E0 S h h h hc c Dhl Dhlc Endemic equilibrium M 8 Exping N q 9M M M where T 3 T T T3 T T5 0 from the equation we obtain 8 N a 3 T q a N q N a T q a a a N q a a N q a N q a N a a3 N a q N a N T N a a 3

11 Modeling o-dynamics of ervical ancer V Diseases 0 where ( a a N q a q N a a q N N q ( q a a a a q a N a a a a a q N a a T a q N a a a a q N q N a d 5 a d 8 a3 dc N S D D. h h h hc c hl hlc The equation 8 shows that there exist the non-linear relationship among. Thus the system exhibits multiple endemic equilibriums [8]. 5. LOAL STABLTY OF DSEASE-FREE EQULBRUM n comting the threshold value which is known as reproduction number of the system we use the techniques presented in [] we get the transmission matrix f transition matrix v respectively as follows f h h dh h S d S d 8 S 5 d 0 0 v h dh h 0 h h dhc 3 hc dc c 0 h dhl Dhl 0 3 hc dhlc Dhlc h h.

12 08 Geomira G. Sanga et al We comte derivative of matrices f v with respect to infected classes at disease-free equilibrium we obtain acobian of F V respectively so that q q 0 q3 0 0 q q5 0 q q q8 0 0 q F dh P P P V P dhc dc dhl d P d h P d 8 P3 d 5 P d h. The dominant eigenvalue of matrix where R max R R R FV is given by hlc R R 0 0 q q dh dhl dh q 3 q d d d h h h

13 Modeling o-dynamics of ervical ancer V Diseases 09 q 5 + dhc 3 dh dhp s dh q 3 dhlc dhc 3 dh dh dh R 0 q q8 d 8 d 5 d 8 q dc d 8 d 5 d 8. Therefore we the result can be summarized as follows Theorem : The disease-free equilibrium of the system is locally asymptotically stable if R0 unstable if R0.. DETERMNATON OF BAKWARD BFURATON AND LOAL STABLTY OF ENDEM EQULBRUM n determining backward bifurcation due to the existence of the multiple endemic equilibria local stability of endemic equilibrium point we use center manifold theory as illustrated in [9]. For more simplification of the system we change the state variables as follows: S x x x 3 x x 5 x hc x x 8 D x 9 D x. 0 c hlc We can thus rewrite the system as follows dx dx hl f x x ( ) x h h 5 f x x x ( d ) x 3 h dx3 f x x x ( d ) x 3 h 3 h

14 00 Geomira G. Sanga et al dx dx 5 dx dx dx 8 dx 9 dx 0 f x ( d ) x 8 f x ( d ) x f x x ( d ) x h f x x ( d ) x 3 hc 3 f x x ( d ) x c 8 f9 x ( d ) x hl 9 f0 3x ( dhlc) x0 where q x x 0 M q x x x M We choose bifurcation parameters q x x x x q M after setting R 0 R0 R0 we get dhl dh dhl 0 x. i i M q 3 where B B B 3 d d d q h h h 5 dh dhc 3 dh dh 3 dh dhlc dhc 3 dh dh 3 3

15 Modeling o-dynamics of ervical ancer V Diseases 0 B d c 8 d d d d 8 8 c d 5 8 dc d 5 d 8 We linearize the system 8 at disease equilibrium we obtain. E0 3 b b a a a a a a dc dhl d hlc where h h a d a d 3 a d 5 a d a d h 5 a d hc 3 b b The right eigenvector associated with zero eigenvalue of the matrix W w w w w w w w w w w where T E 0 is given by w w w0 w3 3 w w 5 w9 3 w w5 8 w8 9

16 0 Geomira G. Sanga et al w 5 w w w3 w3 w3 0 w w 0 w w w 5 w w d 5 w 3 3 dh d dc d w w 9 3 dhl w0 3w3 where d w w 3 h dh ( dhlc)( dhc 3) dh dh d d d 3 hc 3 h hc 3 3 dhlc dhc 3 dh d. hc 3 Left eigenvector associated with zero eigenvalue of the matrix V v v v v v v v v v v where E 0 is given by v v v v3 v3 0 v v 0 v 0 3 d 3 8 c v dc d 5

17 Modeling o-dynamics of ervical ancer V Diseases 03 v v 5 v3 v dh 5 v3 9 v3 dhl v 0 3 v where v v dc 5 vw v w dhl dh dh 5 3 d d d hc 3 hlc hc 3 dhlc dhl dh dh 8 9 d 3 8 c d 5 dc d 5 5 d d h h d dc dc d 0 d d hl hl. d h The value of is chosen so as to satisfy the condition of W.V as its value indicated above.

18 0 Geomira G. Sanga et al Following [9] we comte the value of a b as follows omtation of a : For the system 8 we obtain the following nonzero second partial derivative at DFE f f x x x x x x f x x x x 3 f f f x x x x x x x x x x f x x x x f x x x x f x x x 9 f x x 3 0 x f x x x 0 f f f f x x x x x x x x x x x f f x x x x x x f x x x x 3 3 f x x x 3 5 f x x x f x x x 3 f x x f x x x f x x x 3 5 f x x x x 3 5 f x x x f x x x f x x x f x x x f x x x 3 f x x x f x x x x x f x x x 3 5 f x x x f x x f x x x f x x x f x x x 8 0 x f x x x f x x x 3 9 f x x x x f x x x f x x x 3 0 f x x x x 3 9 f x x x x 3 5 9

19 Modeling o-dynamics of ervical ancer V Diseases 05 f x x x f x x x f x x x f f 8 3 f 9 3 x x x x x x x x x x f x x 3 3 x f x x x f x x f x x x x x f x x x 3 8 f x x x x f x x x 3 9 f f 8 3 x x x x x x f x x x x f x x x f f 9 3 x x x x x x f x x x f x x x Using the expression of W V we obtain a 0 k wi wjvk i j k xixj a m n where f f x x x 3 f x x x x f x x x f x x 3 0 f x x x x x f x x x 5 f x x x f. x x x 5 0 m w w0 v3w vw5 3 wv3 w 8w5 9w8 x n v w w y w w w w v w w w x v w w w w y v w w w y w w w x y w w3 w w5 w w w8 w9 w0.

20 0 Geomira G. Sanga et al omtation of b : From the system 8 we get the following second partial derivative at DFE f x x f 0 f x 3 f 3 x f x 5 f x 9 f x 3 3 f x f x Using the expression of W V we have f b v w for j 3 x 0 k i k ik i j b v w w w w w w v w w w Sinceb 0 we can summarize the results as follows Theorem (): The co-infection model undergoes backward bifurcation whenever m notherwise undergoes forward bifurcation the endemic equilibrium is locally asymptotically stable for R 0 with R0 close to one. DSUSSON AND ONLUSON n this paper the co-infection model of cervical cancer V diseases is formulated qualitatively analyzed. The epidemiologically feasible region of the co-infection model in which mathematically well posed is determined. Stability of disease-free equilibrium is comted using next generation method obtain R max R R R where R 0 representing the reproduction number of individuals who acquire V infection only R0 representing the reproduction number of individuals who acquire multiple infections R0 representing the reproduction number of individuals who acquire PV infection only. ence we conclude the result the disease-free equilibrium is locally asymptotically stable when R0 unstable otherwise. Stability of endemic equilibrium is determined using center manifold theory due to the existence of multiple endemic equilibria the results show that the co-infection model has backward bifurcation for the established condition otherwise it has forward bifurcation. Backward bifurcation shows that the co-existence of two stable equilibria when R 0 ; stable disease-free equilibrium endemic equilibrium. Forward bifurcation means that whenever R0 only the

21 Modeling o-dynamics of ervical ancer V Diseases 0 disease-free equilibrium is present stable whenever R0 the endemic equilibrium is present stable. ABBREVATON uman immunodeficiency virus (V) uman papillomavirus (PV) Acquire immunodeficiency syndrome (ADS). AKNOWLEDGMENT Authors would like to thank Dar es Salaam nstitute of Technology for their support. REFERENES [] World ealth Organization 03 omprehensive cervical cancer prevention control : a healthier future for girls women World eal. Organ. pp.. [] American ancer Society 0 ervical cancer prevention early detection what is cervical cancer? Am. ancer Soc. ancer Soc. [3] nternational Agency for Research on ancer 03 Latest world cancer statistics Global cancer burden rises to. million new cases in 0: Marked increase in breast cancers must be addressed. nt. Agency Res. ancer World eal. Organ. December pp [] Mbulaiteye S. M. Katabira E. T. Wabinga. Parkin D. M. Virgo P. Ochai R. Workneh M. outinho A. Engels E. A. 00 Spectrum of cancers among V-infected persons in Africa: The Uga ADS-ancer Registry Match Study nt.. ancer8 () pp [5] Ng we. Lowe.. Richards P.. ause L. Wood. Angeletti P.. 00 The distribution of sexually-transmitted uman Papillomaviruses in V positive negative patients in Zambia Africa. BM nfect. Dis. pp.. [] Maregere B.0 Analysis of co-infection of human immunodeficiency virus with human papillomavirus Master thesis University of KwaZulu- Natal. [] van den Driessche P. Watmough. 00 Reproduction numbers sub-threshold endemic equilibria for compartmental models of disease transmission Mathematical Biosciences 80 pp. 9-8.

22 08 Geomira G. Sanga et al [8] Okosun K. O. Makinde O. D. 0 On a drug-resistant malaria model with susceptible individuals without access to basic amenities. Biol. Phys. 38(3) pp [9] astillo-havez. Song B.. 00 Dynamical models of tuberculosis their applications. Mathematical Biosciences Engineering () pp [0] Shrestha S. Sudenga S. L. Smith. S. Bachmann L.. Wilson. M. Kempf M.. 00 The impact of highly active antiretroviral therapy on prevalence incidence of cervical human papillomavirus infections in V-positive adolescents. BM nfect. Dis. 0() pp. 95. [] Adler D.. 00 The impact of AART on the PV-related cervical disease. urr. V Res. 8() pp