МОДЕЛИРОВАНИЕ ВИРУСНОЙ ПЕРЕДАЧИ ПАПИЛЛОМЫ ЧЕЛОВЕКА Текст научной статьи по специальности «Математика»

MSC 92B05

DOI: 10.14529/ mmp200207

MODELLING THE HUMAN PAPILLOMA VIRUS TRANSMISSION IN A BISEXUALLY ACTIVE HOST COMMUNITY

0.M. Ogunmiloro, Ekiti State University, Ado Ekiti, Nigeria, oluwatayo.ogunmiloro@eksu.edu.ng

In this article, we construct a mathematical model describing the transmission dynamics of Human Papilloma Virus (HPV) in a bisexually active host community. Comprehensive mathematical techniques are used to qualitatively and quantitatively analyze the model. We analyze the local and global stabilities of the model's equilibria and show that if the basic reproduction number is less than unity, then the model is locally and globally asymptotically stable at the HPV-free static states. Also, if the basic reproduction number is less than unity, then the HPV-endemic static state is globally asymptotically stable. Numerical simulations are carried out and graphical illustrations are presented to validate the theoretical results.

Keywords: HPV; basic reproduction number; local stability; global stability.

Introduction

Human papilloma virus (HPV) is a viral infection of the genitals of sexually active male and female. Penetrative sex and skin to skin genital contact serves as the mode of transmission of this disease. HPV have been known to be the cause of cervical cancer in women, and other types of HPV causes cancer of the anus, vulva, vagina, penis and genital warts, respiratory human papillomatosis [1]. According to the Center for Disease Control (CDC) [2], 79 million Americans are infected in their early 20s. HPV is prevented by taking appropriate vaccines, surgical removal, sexual abstinence and condom usage, respectively [3]. Mathematical models are important tools used in exploring epidemic breakout and health consequences of interventions in human and environmental host populations with time [4-6]. S.L. Lee and A.M. Tameru [7] worked on mathematical model of HPV in United States of America with its special impact on cervical cancer. Also, E.J. Dasbach, E.H. Elbasha and R.P. Insinga [8] discussed the epidemiologic and economic impact of vaccination against HPV. Recently published works of A. Omame, R.A. Umana, D. Okounghae [9], A.E. Sado [10], N. Ziyadi [11], M. Dyser, P. E. Granitt, E.R. Myers [12], H.F. Brower [13], O. Sharomi, T. Malik [14], E.H. Elbasha [15], proved effective to this study. Having consulted the aforementioned literature, this work extends the existing HPV models by considering the infectious transmission of HPV in an active bisexually intimate human host community, see [16,17]. The paper is organized as follows. Section 1 discusses the model formulation, mathematical analysis and the basic reproduction number. Section 2 involves the local and global analysis of the model at the HPV-free steady-state solutions. Finally, Section 3 uses mathematical computational software MAPLE 18 for numerical simulation of the theoretical results with the data available in recent literature.

1. Model Formulation

In this section, the model system of equations describing the transmission of HPV is based on the system of ordinary differential equations of the first order. The total bisexually

active human host is divided into sub-populations as follows. Let Ms be a number of sexually active males who are prone to acquiring HPV, Mj be a sub-population of sexually active males who are infected with HPV, Fs be a number of sexually active females who are prone to acquiring HPV, Fj be a number of sexually active HPV infected females, Rmf be a number of individuals recovered from HPV. The number of total bisexually active host population is denoted by

Nh = Nm.Nf = Ms + Mi + Fs + Fi + Rmf.

Sexually active males who are prone to acquiring HPV are recruited into the bisexual host population at the rate Am. There is an effective infectious sexual intimacy between sexually active male individuals who are prone to acquiring HPV and sexually active infected males and females denoted by the quantity

PiMs(Mj + Fj)

Nh U

where is the effective transmission rate. Also, ^ denotes the natural mortality rate applied to all sub-population. The sub-population of sexually active infected male is increased by and decreased by the quantity 7QMi} where 7,, denotes

the progression rate of sexually active infected males to the recovered sub-population. Furthermore, the sub-population of sexually active females who are prone to acquiring HPV is recruited into the bisexual host population at the rate Af, while there is an effective infectious sexual intimacy between sexually active female individuals who are prone to acquiring HPV and sexually active HPV infected male and female individuals given by the quantity

&Fs(Mj + Fj)

Nh

(2)

where в is the transmission rate. The sub-population of infected sexually active females is increased by the quantity Ё1ЕАМ1+ЕА; anc[ decreased by the quantity 71 Mi} where 71 denotes the progression rate of sexually active infected females to the recovered subpopulation. Moreover, the quantities o1Ms and o2Fs increase the sub-population of sexually active male and female who are prone to acquiring HPV, where o1 and o2 denote the loss of immunity to HPV infection after recovery. The inclusion of the variables and parameters in combination with assumptions leads to the system of evolution equations that control the HPV transmission given by

,V , (h Ms(Ft + Mj)

Ms = Am-----fiMs + с i Ms,

Nh

Mt =ьщ±ю_(11 + ъ)Мш>

Ft + (3)

Nh

F, = +

Nh

Rmf = YoMi + YiFi - ^Rmf - 0 Ms - a2Fs,

under the initial conditions Ms(t) > 0, M,(t) > 0, Fs(t) > 0, F(t) > 0, Rm/ (t) > 0.

Normalization of (3) such that ms = = fs = jf^, fi = rmf = -jjf- leads to

mis = - ^ims(/j + mi) - + aims, rhi = ^ims(/, + m,) - (^ + 70)ms,

fs = A/ - ^2fs(/i + mi) - + a2fs,

A = ^2fs(/i + mi) - + 7l)/i,

rm/ = Yomi + Yi/i - - aims - a/s,

(4)

under the initial conditions ms(t) > 0,mi(t) > 0,/s(t) > 0,/i(t) > 0,rm/(t) > 0.

1.1. Well-Posedness and Boundedness of HPV Model System

The solutions of model system (4) with nonnegative initial conditions are bounded and remain nonnegative for time t > 0.

Theorem 1. Let ms(t) > 0,mi(t) > 0,/s(t) > 0,/i(t) > 0,rm/(t) > 0, then the solutions of model system (4) are positive for all t > 0. Also, the domain Z is positively invariant such that all solutions begin and remain in Z-

Proof. We do not consider the fifth state equation in (4), since all the first four state equations depends on the fifth one. Add up the total human host population of bisexually active male and female individuals to obtain

dN„

~dT

Am -

Integrate both sides of (5):

(5)

rN„

and obtain

hence

1

I A AfdNm

'Nm (to) Am -

dt,

~ 111 (Am - tlNm)

Nm

Nm (to )

to

t - to

1

--ln[(Am - nNm) - (Am - nNm{to))\ =t-t0

ln

Am - |UN,

m

LAm - ^Nm(t0)J Am - ^Nm

-^(t - to),

= g-Mt-to)

LAm - ^Nm(to)J

(Am - ^Nm) = (Am - Nm(t0))e^(t-to),

Am (Am ~ iVm(t0))e-^-t°)

Nm =

(6)

(7)

(8)

(9)

t

Ci

and

Following the same procedure, addition of the bisexual female population leads to

N AL_ (Af-N(t0))e~^_ Vf Vf

The domain q = qi x q2 C Re + x Re + such that

A i

(ms ,mi) £ Re + : ms + mi ill)

V J

r At1

(/„./:.)•: lie2 :/, • J)< -1 (12)

Vf

is positively invariant. It is enough to consider the dynamics of HPV model system (4) in q. In this domain, the model is meaningful in the sense of HPV transmission and mathematically well-posed.

□

1.2. Steady-State Analysis

The steady-state analysis is performed on model system (4) to obtain the steady-state solutions in the absence and presence of HPV infections in the model. HPV-free steady-state solutions implies that mi = fi = 0, while it is necessary to know that the trivial steady-state does not exist as long as the recruitment terms Am and Af are presented. Hence, the HPV-free steady-state solutions are given by

E° = (m8, mt, fs, fi, rmf) = (^,0,—,0,0). (13)

V v Vf '

Also, the HPV-endemic steady-state solutions are given by

K, ml,/;,/;) = (-

Am

E* = (m!,m*, /„*, /Л = [—-0-ttF-v

.(miPi + fi Pi + v - 0i)

—Pirns fi Af fsmi

{mspi — v — jo)' (mip2 + fi в — 02 + Vf)' (fs в2 — Yi — Vf)

(14)

1.3. Basic Reproduction Number of HPV Model System

Let x = (x1, ...,xn)T be a vector of the numbers of human individuals in each subpopulation, xi > 0, therefore the state of the model is bounded by the closed positive cone x £ X = Re n+. Assume that the sub-populations are constructed such that the first m sub-populations are associated with infected individuals. Then Xs = {x £ 0|xi = 0,i = 1, ...,m}, is the set of disease free steady states. The system of differential equations that control disease transmission is given on X such that

x = f (x), (15)

where the components of f are fi(x) = Fi(x).Vi(x) for i = 1,...,n. Here Fi(x) denotes the rate of appearance of clinical manifestations of symptoms in the i-th sub-population, while Vi (x) = V- (x) — V+ (x), where Vi+(x) is the rate of individuals moving into the i-th sub-population by all other means, and V~(x) is the rate of movement of individuals out of the i-th sub-population. Assume that each function is at least twice continuously differentiable in each variable. Other assumptions are stated below.

If x > 0, then Fi(x), V+(x), V~ (x) > 0 for i = 1,...,n.

If x, = 0, then V,-(x) = 0. Therefore, if x G Xs, then ^-(x) = 0 for i =1, ...,m. F, (x) = 0 if i > m.

If x G Xs, then F, (x) = 0 and (x) = 0 for i = 1,

, m.

Note that the first two points together with the assumption on smoothness of the functions involved guarantee that the non-negative cone (x, > 0,i = 1,...,n) is forward invariant and there exists a unique non-negative solution for each non-negative initial condition. Therefore, from model system (4),

F

0 0 0 0 0 0+ + ß 0 0 0 0

0 ßl Am ß 0 ßl Am ß 0 0 ß + Yo 0 0 0

0 0 0 0 0 , V = 0 0 ^2 + ß/ 0 0

0 ßiAf ß 0 ßiAf ß 0 0 0 0 ß/ + Yi 0

0 0 0 0 0 Yo ^2 Yi ß

(16)

such that

V

-i

(cti + ß) 0 0 0

-i

and

FV

-i

0 0 0 0

0 -

0

(ß + Yo) 0

0

lo

-i

0 0

+ ß/) 0

-i

(ß+lo)ß

0

ßlAm (ß+7 o)ß

0

ß2Af (ßf +Yl)ß foßiAm___71 ß^Af

+ßf)t-

0 0 0 0

0 0 0

(ß/ + Yi)

71

-i

(ß/+7i)i

ß

0 0 0

0

-i

(17)

0

ßlAm

(ß+7 o)ß 0 ß2Af

(ß+7o)ß2

(ßf +71 )ß

0-

(ßf +71 )ß

7 oßiAm___lißiAf

(ßf +71 )ß

(ß+7o )ß2

0 0 0 0

0

(18)

The largest eigenvalue of (18) is the basic reproduction number of model system (4) given by

R (FV-1) =__

2. Stability Analysis of HPV Model System

Theorem 2. Let 0 be a critical point of x = /(x), be positive definite function on the neighborhood U of 0.

2

2

If 4 < 0 for x e U - {0} If 4 < 0 for X e U - {0} If 4 > 0 for x e U - {0}

0 is stable.

0 is asymptotically stable. 0 is unstable.

is a Lyapunov function if is positive definite and < 0.

Theorem 3. Let be a Ci (Rn) real valued function, U = {x G Re n|Ly (x) < k} , k G Re , and Ly(x) < 0. If P is the largest invariant set in D = |x G |Ly(x) = 0 j, then the solution trajectories begin in and remain there, for all time t > 0.

2.1. Local Stability Analysis of HPV-Free Steady-State Solutions

Theorem 4. The HPV-free steady-state of model system (4) is locally asymptotically stable whenever R* < 1.

Proof. The Jacobian matrix of model system (4) at HPV-free steady-state solutions (13) is given by

-^ + CTi — 0 0

Pi Am V

- Yo

0

ihAf Vf ihAf Vf

0 0

-^ + a 2 0

V

fil Am V

ihAf Vf

+ Yi

(20)

The characteristics polynomial of (20) is given by

A4A4 + A3 A3 + A2A2 + AiA + Ao. Let Gi = —^ + a1, G2 = —^ — Yo, G3 = —^ + a2, G4 = —^ + y1, where

A4 = 1,

A3 = Gi + G2 + G3 + G4, A2 = (Gi + G2XG3 + G4) + Gi G2 + G3G4, Ai = (Gi + G2)G3G4 + (G3 + G4)GiG2, Ao = GiG2G3G4 — R*.

Also, the determinant of (20) is given by

(^ — a2) (^V/ — (Yi — Yo) — ^YiYo^/ — ^2AfftAm) (^ — ai)

(21)

(22)

> 0,

while the trace

-4^ + ai Yo + a 2 + Yi < 0.

The Routh-Hurwitz criteria states that all roots of characteristics polynomial (20) have negative real parts if and only if the coefficients A, are positive and the Hurwitz matrices Hi > 0 for i = 0,1, 2, 3, 4. From (22), it is easy to see that Ai > 0, A2 > 0, A3 > 0, A4 > 0,

since all Gi are positive. Moreover, if R* < 1, then A0 > 0. Construction of the positive Hurwitz matrices yields

H = A3 > 0, H2

H3

A3 Ai 0

A4 A2 Ao

0

A3

Ai

> 0, H4

Ai A4 Ai A2 A3 A4 Ai A2 0 Ao 00

> 0,

00

A3 A4

Ai A2 0 Ao

(23)

> 0.

Therefore, all the eigenvalues of the Jacobian matrix J(Eo)(20) have negative real parts if R* < 1, this implies that the HPV-free steady-state is locally asymptotically stable. Moreover, if R* > 1, then A0 < 0. Applying the Descartes' rule of signs, we see that there is only one sign change in A4, A3, A2, Ai, A0, involving the coefficients of (20), which implies that there exists the unique eigenvalue with positive real part. Hence the HPV-free steady-state is unstable.

□

2.2. Global Stability Analysis of HPV-Free Steady-State Solutions

Theorem 5. The HPV-free steady-state (13) of model system (4) is globally asymptotically stable whenever R* < 1.

Proof. Let {Ly : Q+ ^ Re , ||Q+ = {(ms,mi, fs, fi) E Q\ms > 0,mi > 0,fs > 0, fi > 0}}, then

Ly (ms,mi ,fs,fi) = Ly (mi, fi), (24)

such that

Ly (mi, fi) = ßims(fi + mi) - (ß + Yo)mi + ß2fs (fi + mi) - (ß + Yi)fi, (25)

and

(Yo + ß)[Ä*ms - 1]mi + (yi + ßf)[Rfs - 1]fi < 0

(26)

for R* < 1. Note that R* < 1 : Ly = 0 ^ m-i = 0 and f = 0, and R* = 1 : Ly = 0 ^ ms = 1 and fs = 1. It can be observed that all trajectories in Q+ approach the HPV-free steady-state solutions Eo (13). Since the vector on the right hand side of model system (4) points to the interior of Q. Hence, the HPV-free steady-state of model system (4) is globally asymptotically stable.

□

2.3. Global Stability of HPV-Endemic Steady-State Solutions

Theorem 6. If R* > 1, then the unique HPV-endemic steady-state solution E* of model system (14) is globally asymptotically stable in the interior of q.

Proof. Define the Lyapunov function Ly : {(ms,mi,fs, fi) E q : ms,mi, fs, fi} ^ R+ so that

Ly= (ms-m*s-m*s ln ^ ) + (rrii-m*-m* In ^ ) + (j8-- f* -f* In -j^J + (^fi~f*-f* In ^

'таЬ ГГЦГПз

ш..

ш..

ш:/;

шя

ш: ш:

Ш;

* \

- шг )

в1

т3тг

I ш:

ш:

ш ^

:/* ш >*-

ш* ш:-

ш* ш:

Л = (^2 -

Лшг , /г/

+

/; /

2 — — — — /; /;

/;

/;

+

- /) в

ь /: / /: / / * / ]

-О

87

3. Numerical Simulations

The numerical simulations were carried out using the in-built fourth order Runge-Kutta method in MAPLE 18 computational software. Table gives the values of parameters obtained in the cited literature which are involved in the computation/simulation.

Also, Fig. 1 shows behavior of the subpopulation of sexually active males who are prone to acquiring HPV overtime in the absence of health intervention strategies.

Fig. 2 describes behavior of sexually active males infected with HPV. As time increases, the sharp rise depicts that infected male individuals rise in the community in the absence of health intervention strategies. Fig. 3 displays behavior of the sub-population of sexually active females who are prone to acquiring HPV overtime in the absence of health intervention strategies. Fig. 4 shows behavior of sexually active females who are directly infected with HPV. Medical strategies should be adopted in order to minimize infection in this sub-population.

Table

Variables and Parameter Values of Model System (4)

Parameter Values Source

ms 0,50 [13]

m,i 0,20 [13]

Is 0,45 [H]

f* 0,20 [H]

A 0,13 [9]

Af 0,02 [9]

ßi 0,28 [9]

ß 0,12 [9]

(7l 0,123 [9]

(?2 0,142 [9]

Ä 0,011 [9]

lo 0,11 [9]

7i 0,000136 [9]

Fig. 1. Graph of sexually active males who Fig. 2. Graph of sexually active males are prone to acquiring HPV infected with HPV

In turn, Fig. 5 describes variation of the parameter a2 (0,900 - 0,930). As time increases, sexually active female individuals lose their immunity. Fig. 6 depicts behavior of the parameter (0,2 - 0,6). As time increases, effective infectious contact increases when there are no health intervention policies to stop HPV spread. Fig. 7 describes variation of the parameter o\ (0,89 - 0,93). As time increases, sexually active male individuals lose their immunity. Also, Fig. 8 depicts variation of the recovery rate Yo (0,49 - 0,53). As time increases, sexually active infected males recover with compliance to health intervention strategies.

Fig. 3. Graph of sexually active infected females who are prone to acquiring HPV

Fig. 4. Graph of sexually active females infected with HPV

Fig. 9 describes variation of Af (0,4 - 0,6). As time increases, more sexually active males are recruited into host community of bisexuals. Fig. 10 describes the rate at which sexually active male individuals are recruited into the host population varying Am (0,48 - 0,53) as time increases. Moreover, the effective infectious contact rate is displayed by varying (0,48 - 0,53) in Fig. 11, which shows that more individuals become infected as time increases.

Fig. 6. Graph of variation of ß (0,2 - 0,6) as time increases

Fig. 5. Graph of variation of a2 (0,900 -0,930) as time increases

Finally, Fig. 12 depicts variation of y1 (0,4 - 0,6). The gradual decline shows that more sexually active infected females recover from HPV in active compliance to health intervention strategies.

Conclusion and Recommendations

The deterministic model of HPV transmission in a bisexually active human host population is considered. The model is analyzed in a feasible domain and shown to be positive, well-posed and realistic in the sense of HPV transmission. The basic reproduction number R is obtained and the stability of the model's steady-state solutions at the HPV-

Fig. 7. Graph of variation of ai (0,89 0,93) as time increases

Fig. 8. Graph of variation of Yo (0,49 0,53) as time increases

Fig. 9. Graph of variation of A/ (0,4-0,6) as time increases

Fig. 10. Graph of variation of Am (0,480,53) as time increases

Fig. 11. Graph of the variation of (0,480,53) as time increases

Fig. 12. Graph of variation of Yi (0,4-0,6) as time increases

free and endemic is investigated locally and globally. It was shown that if R < 1, then the

HPV-free steady-state is locally and globally asymptotically stable. Also, if R > 1, then

the HPV-endemic steady-state is globally stable. Further, we intend to consider the impact

of optimal control strategies and cost effective analysis involving HPV transmission.

References / Литература

1. World Health Organization (WHO), WHO Fact Sheet. Available at: https://www.who.int (accessed 2019).

2. Center for Disease Control (CDC), STD Facts-Human Papilloma Virus (HPV). Available at: https://www.medicalnewstoday.com (accessed 2019).

3. Medical News Today, Human Papilloma Virus (HPV): Treatment, Symptoms and Causes. Available at: https://www.medicalnewstoday.com/articles/246670 (accessed 2019).

4. Ogunmiloro O.M., Fadugba S.E., Ogunlade T.O. Stability Analysis and Optimal Control of Vaccination and Treatment of a SIR Epidemiological Deterministic Model with Relapse. International Journal of Mathematical Modelling and Computations, 2018, vol. 8, no. 1, pp. 39-51.

5. La-Salle J., Lefschetz S. Stability by Lyapunov's Direct Method with Applications. N.Y., Academic Press, 1961.

6. Anderson R.M., May R.M. Population Biology of Infectious Diseases. Berlin, Springer, 1982.

7. Lee S.L., Tameru A.M. A Mathematical Model of Human Papilloma Virus (HPV) in the United States (US) and its Impact on Cervical Cancer. Journal of Cancer, 2012, vol. 3, pp. 262-268. DOI: 10.2150/jca.4161, 2012

8. Dasbach E.J., Elbasha E.H., Insinga R.P. Predicting the Epidemiologic Impact of Vaccination Against Human Papilloma Virus Infection and Disease. Epidemiologic Reviews, 2006, vol. 28, no. 1, pp. 88-100.

9. Omame A., Umana R.A., Okounghae D., Inyama S.C. Mathematical Analysis of a Two-Sex Human Papilloma Virus (HPV) Model. International Journal of Biomathematics, 2018, vol. 11, no. 7, article ID: 1850092. DOI: 10.1142/S1793626810000397

10. Sado A.E. Mathematical Modeling of Cervical Cancer with HPV Transmission and Vaccination. Science Journal of Applied Mathematics and Statistics, 2019, vol. 7, no. 2, pp. 21-25. DOI: 1011648/j.sjams.20190702

11. Najat Ziyadi. A Male-Female Mathematical Model of Human Papilloma Virus (HPV) in African American Population. American Institute of Mathematical Sciences, 2017, vol. 14, no. 1, pp. 339-358. DOI: 10.3934/mbe.2017022

12. Ryser M.D., Gravitt P.E., Myers E.R. Mechanistic Mathematical Models: An Underused Platform for HPV Research. Papilloma Research, 2017, vol. 3, pp 46-49.

13. Brower A.F. Models of HPV as an Infectious Disease and as an Etiological Agent. PhD Thesis, University of Michigan, 2015.

14. Sharomi O., Malik T. A Model to Assess the Effect of Vaccine Compliance on Human Papilloma Virus Infection and Cervical Cancer. Applied Mathematical Modelling, 2017, vol. 47, pp. 528-550.

15. Elbasha E.H. Global Stability of Equilibria in a Two-Sex HPV Vaccination Model. Bulletin of Mathematical Biology, 2008, vol. 70, pp. 894-909.

16. Dietz C.A., Nyberg C.R. Genital, Oral and Anal Human Papilloma Virus Infection in Men Who Have Sex with Men. The Journal of the American Osteopathic Association, 2011, vol. 111, pp. 19-25.

17. Independent News, Lesbians and Bisexual Women at Risk from Dangerous Myth "They Cannot Get Cervical Cancer" Warns Nhs. Available at: https://www.independent.co.uk (accessed 2019).

МОДЕЛИРОВАНИЕ ВИРУСНОЙ ПЕРЕДАЧИ ПАПИЛЛОМЫ ЧЕЛОВЕКА

О.М. Огунмилоро, Государственный университет Экити, г. Адо Экити, Нигерия

В этой статье сформулирована математическая модель, описывающая динамику передачи вируса папилломы человека (ВПЧ) в бисексуально активном сообществе носителей. Комплексные математические методы были использованы для качественного и количественного анализа модели. Локальная и глобальная устойчивость равновесий модели была проанализирована, и показано, что если Д* меньше единицы, то модель локально и глобально асимптотически устойчива в статических состояниях, свободных от ВПЧ. Также, если Д* больше единицы, ВПЧ-эндемичное статическое состояние является глобально асимптотически устойчивым. Было проведено численное моделирование и представлены графические иллюстрации.

Ключевые слова: ВПЧ; базовый 'репродуктивный номер; локальная устойчивость; глобальная устойчивость.

Олуватайо Майкл Огунмилоро, PhD, Государственный университет Экити (г. Адо Экити, Нигерия), oluwatayo.ogunmiloro@eksu.edu.ng.

Received September 20, 2019

УДК 57:51-76

DOI: 10.14529/ mmp200207

Поступила в редакцию 20 сентября 2019 г.