ISSN 1 746-7233, England, UK
World Journal of Modelling and Simulation
Vol. 9 (2013) No. 4, pp. 302-320
‘Transmission dynamics of trichomoniasis in bisexuals’ without the ‘E’
Phillip Mafuta, Josiah Mushanyu, Steady Mushayabasa, Claver P. Bhunu ∗
Department of Mathematics, Box MP 167, Mount Pleasant, Harare, Zimbabwe
(Received January 10 2013, Revised June 11 2013, Accepted September 22 2013)
Abstract. A deterministic model for transmission dynamics of Trichomonas vaginalis in a population with
bisexuals is formulated and analysed. The disease free equilibrium point has been shown to be globally
asymptotically stable when the reproduction number is less than a unity. Furthermore, the endemic equilibrium point has been shown to be locally asymptotically stable and globally asymptotically stable for when
the reproduction number is greater than a unity, using the Centre manifold theory and Liapunovs functional
approach, respectively. Analysis of the reproduction number has shown that an increase in the number of
infected bisexuals result in an increase in the number of infectives among heterosexuals and vise-versa. This
suggests that straight women are turning into bisexuals already infected and that bisexuals are linked to
straight females indirectly by males. Simulations results has shown that treatment is the major parameter in
controlling the spread of the infection.
Keywords: reproduction number, bisexuality, treatment.
1
Introduction
Trichomoniasis is abbreviated as TV, it is a sexually transmitted Infection (STI) or a sexually transmitted disease (STD)[10] . A single celled protozoan (microscopic parasite) called trichomonas vaginalis is the
causative agent, usually this parasite is found in the vagina and urethral tissues[4, 5, 10, 14] . Trichomoniasis is
diagnosed by visually observing the trichomonands via a microscope, this is so, because, trichomonands are
too small to be seen by a naked eye[6, 10, 17] . The trichomonands are pear shaped and have several flagella
(whip-like tails) at one end. In women, TV is detected through inserting a speculum into the vagina, followed
by collecting a sample of vaginal discharge using a cotton-tipped applicator. The collected sample will then be
placed onto a microscopic slide and sent to a laboratory to be analyzed. Results on TV tests may also reveal
small red ulcerations on the vaginal wall or cervix[9] . It is a far more sexually transmitted infection than either
Clamydia trachomatis or Neisseria gonorrhoea[2, 19] .
Some researchers have shown that trichomoniasis is more prevalent in females (67-100 percent of female sexual partners of an infected male get infected) than males (14-60 percent of male sexual partners of
an infected female get infected)[2, 10, 11, 19] . The reason why females are more infected is poorly understood,
although some researchers have the idea that the pro-static fluid contains zinc and other substances that are
harmful to the pathogen[2, 21, 22] . The WHO has estimated that 160 million cases of infection are acquired annually worldwide[3] . The estimates for North America alone are between five and eight million new infections
each year, with estimated rate of asymptomatic cases as high as fifty percent[3–5, 23] . Some researchers have
also shown that trichomonas vaginalis is more prevalent in industrialised countries[2, 6, 12] .
In females, the parasite usually affects the vagina, urethra, cervix, bladder and glands in genital
areas[15, 17, 18] . In males, the parasite infects the urethra or under the foreskin of the penis if it is not circumcised. Females usually reveal symptoms, while infection in males are usually asymptomatic. Signs and
∗
Corresponding author.
E-mail address: [email protected].
Published by World Academic Press, World Academic Union
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
303
symptoms in females includes: soreness, vaginal discharge, inflammation and vaginal itching, instability during sexual intercourse, strange unpleasant smell and pain when passing urine[7, 9, 11] . Although, males do not
usually reveal symptoms, when they appear (usually after a week of infection), they are associated with discharge from the penis that may be thin and whitish, pain or burning sensation when passing urine, swelling of
the scrotum and inflammation of the foreskin (though not common)[2, 8, 12] .
Trichomoniasis infection is treated and cured with metronidazole or tinidazole except in the first trimester
of pregnancy, when clotrimazole is used topically. Usually the treatment is given as a single-dose therapy and
should be prescribed to any sexual partner(s) as well because they may be asymptomatic carriers[2, 6, 9, 13] .
The medicine is usually taken by mouth as pills, tablets or capsules. Medicine given in the vagina will not
cure the infection[2] . It is important not to drink alcohol while taking metronidazole or tinidazole, since the
combination can lead to abdominal pain and vomiting[2, 6, 14] .
Left untreated trichomoniasis may leads to complications especially in women and these includes:pelvic
inflammatory of the disease, pre-term delivery, premature rapture of membranes, low birth weight infants
and predisposing to HIV infection and cervical cancer[8, 9, 19] . Trichomoniasis can also leads to infertility in
men[2] . People are encouraged to prevent infections than to cure. Since TV is a sexually transmitted infection,
abstinence is the most preferable measure to avoid infection. Further, people are encouraged to practise safe
sex and hygiene, for instance, use of condoms. Condoms are effective at reducing, but not wholly preventing
transmission[6] . Individuals are encouraged to wash before and after sex, not to share swimsuits and towels,
since, the trichomonands can survive for up to forty-five minutes outside the body. Individuals should also
shower immediately after swimming in a public pool[2, 6] .
This work extends the work done by Bhunu and Mushayabasa[1] , to incorporate the aspect of bisexuality,
a common phenemenon in African homosexuals.
2
Model description
The model subdivides the total population into the following sub-population: susceptible males (non
homosexuals) Sm (t), Trichomanas vaginalis infected males Im (t), susceptible straight females (non homosexuals) Sfs (t), infected straight females Ifs (t), susceptible bisexual females Sfb (t), infected bisexual female
Ifb (t).
The total population N (t) is given by: N (t) = Nm (t) + Nf (t), Nm = Sm + Im (t), Nf (t) = Nfs (t) +
Nfb (t), Nfs (t) = Sfs (t) + Ifs (t), Nfb (t) = Sfb (t) + Ifb (t), where Nm (t), Nf (t), Nfs (t) and Nfb (t) are the
total number of males, females, straight females and bisexual females respectively. In this model we assumed
that there are no male homosexuals (gays), females are either straight or bisexuals and that homosexuality
in females is a result of environmental factors[20] . Further, no person is born being a bisexual, some straight
females becomes bisexuals due to environmental factors at a rate γ. Susceptible humans enter the population
through birth at a rate Λ, a proportion ρ being males and a complementary proportion 1 − ρ being straight
females. Susceptible males acquire trichomonas vaginalis infection following sexual contact with an infected
female (straight or bisexual) at a rate λf which is given by:
λf =
βf (Ifs + Ifb )
,
N fs + N fb
(1)
with βf being the effective contact rate for trichomonas vaginalis transmission from female to male. Susceptible straight females acquire trichomonas vaginalis infection following contact with an infected straight male
at a rate λm which is given by:
βm Im
λm =
,
(2)
Nm
with βm being the effective contact rate for trichomonas vaginalis transmission from male to female. Susceptible bisexual females acquire trichomonas vaginalis infection following sexual contact with an infected
straight male or a bisexual female at a rate λb which is given by :
λb =
βfl Ifb
βm Im
+
,
N fb
Nm
(3)
WJMS email for subscription: [email protected]
304
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
with βfl being the effective contact rate for trichomonas vaginalis transmission from female to female.
Infected individuals are treated at a rate α and move back into their corresponding susceptible classes.
Individuals experience natural death at a rate µ. Furthermore, since trichomoniasis infection does not kill,
there is no disease induced death rate.
The structure of the model is given in Fig. 1.
Fig. 1. Structure of model
Based on the given assumptions the following system of differential equations describes the model:
0 (t)
Sm
0
Im (t)
Sf0 s (t)
If0 s (t)
Sf0 b (t)
If0 b (t)
3
=
=
=
=
=
=
ρΛ − λf Sm − µSm + αIm ,
λf Sm − (α + µ)Im ,
(1 − ρ)Λ − λm Sfs − γSfs − µSfs + αIfs ,
λm Sfs − (µ + α + γ)Ifs ,
γSfs − λb Sfb − µSfb + αIfb ,
λb Sfb + γIfS − (µ + α)Ifb .
(4)
Invariant region
In this section, we study some basic results of the model system (4) which will be useful in the proofs of
stability and persistence results. We start by showing that the solutions of the model system (4) are positive
and defined on (0, ∞).
0 , I 0 , S 0 , I 0 , S 0 , I 0 > 0 there exist S , I , S , I , S , I such that: (0, ∞) →
Theorem 1. For all Sm
m m
fb fb
fs fs
m
fb fb
fs fs
(0, ∞) which solve the model system (4) with initial conditions
0 , I = I 0 , S = S0 , I = I 0 , S = S0 , I = I 0 .
S m = Sm
m
fs
fb
m
fs fs
fs
fb fb
fb
0 (t), F (x) = I 0 (t), F (x) = S 0 (t), F (x) = I 0 (t),
Proof. Let F1 (x) = Sm
F5 (x) = Sf0 b (t), and
2
3
4
m
fs
fs
0
F6 (x) = Ifb (t) where x = (x1 , x2 , x3 , x4 , x5 , x6 ) with x1 = Sm , x2 = Im , x3 = Sfs ,
x4 = Ifs , x5 = Sfb , and x6 = Ifb .
In this case: Nm = x1 + x2 , Nfs = x3 + x4 , Nfb = x5 + x6 , Nf = x3 + x4 + x5 + x6 and the forces of
infection becomes:
(x4 + x6 )βf
x6 βfl
x2 βm
x2 βm
λf =
, λm =
, λb =
+
.
x3 + x4 + x5 + x6
x1 + x2
x1 + x2 x5 + x6
It follows that:
(x4 + x6 )x1 βf
(x4 + x6 )x1 βf
F1 (x) = ρΛ −
− µx1 + αx2 , F2 (x) =
− k1 x2 ,
(5)
x3 + x4 + x5 + x6
x3 + x4 + x5 + x6
x2 x3 βm
x2 x3 βm
F3 (x) = (1 − ρ)Λ −
− k4 x3 + αx4 , F4 (x) =
− k2 x4 ,
(6)
x1 + x2
x1 + x2
x2 x5 βm x5 x6 βfl
x2 x5 βm x5 x6 βfl
F5 (x) = γx3 −
−
− µx5 − αx6 , F6 (x) =
+
+ γx4 − k1 x6 .
(7)
x1 + x2
x5 + x6
x1 + x2
x5 + x6
WJMS email for contribution: [email protected]
305
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
Using the properties of differentiable functions, Fi (x) is differentiable for each i = 1, 2, 3, 4, 5, 6 and hence
Fi (x) are continuous functions. It follows that:
(x4 + x6 )βf
(x4 + x6 )x1 βf
∂F1
∂F1
∂F1
=
, Nf , 0;
= α,
=
, Nf , 0;
2
∂x1
∂x2
∂x3
Nf
Nf2
(−Nf + x4 + x6 )x1 βf
∂F1
∂F1
∂F1
Nf , 0;
=
=
(by symmetry), and
2
∂x4
∂x5
∂x3
Nf
∂F1
∂F1
=
(by symmetry).
∂x6
∂x4
We see that the partial derivatives of the function F1 (x) exists and are continuous.
∂Fj
In a similar manner, it can be shown that the partial derivatives
, i = 1, 2, 3, 4, 5, 6 and
∂xi
j = 2, 3, 4, 5, 6 exist and are continuous, hence
F (x) = (F1 (x), F2 (x), F3 (x), F4 (x), F5 (x), F6 (x)) is a locally Lipschitz continuous function.
Let x1 = 0 with x2 > 0, x3 > 0, x4 > 0, x5 > 0, and x6 > 0, then
F1 (x) = ρΛ + αx2 > 0.
(8)
Let x2 = 0 with x1 > 0, x3 > 0, x4 > 0, x5 > 0, and x6 > 0, then
F2 (x) =
(x4 + x6 )x1 βf
> 0.
Nf
(9)
Let x3 = 0 with x1 > 0, x2 > 0, x4 > 0, x5 > 0, and x6 > 0, then
F3 (x) = (1 − ρ)Λ + αx4 > 0.
(10)
Let x4 = 0 with x1 > 0, x2 > 0, x3 > 0, x5 > 0, and x6 > 0, then
F4 (x) =
x2 x3 βm
> 0.
Nm
(11)
Let x5 = 0 with x1 > 0, x2 > 0, x3 > 0, x4 > 0, and x6 > 0, then
F5 (x) = γx3 + αx6 > 0.
(12)
Let x6 = 0 with x1 > 0, x2 > 0, x3 > 0, x4 > 0, and x5 > 0, then
F6 (x) = γx4 +
x2 x5 βm
> 0.
Nm
(13)
0 , I 0 , S 0 , I 0 , S 0 , I 0 ) ∈ R6 , there exists a unique solution
It follows by theorem 2 that, for every x0 = (Sm
m
+
fs fs
fb fb
0
6
of x = F (x), x(0) = x0 with values in R+ which is defined in some interval (0, b] with b ∈ (0, ∞) or
b = ∞. If b < ∞, then
sup N (t) = ∞.
(14)
0≤t≤b
Using a theorem on differential
inequality[30] ,
it follows that:
N 0 = Λ − µN.
(15)
It follows that, as t tends to infinity, that is t → ∞,
0 < N (t) <
Λ
,
µ
(16)
so N (t) is bounded which is a contradiction, hence, b = ∞. Thus the solutions of the model system (4) are
positive and are defined on (0, ∞) and this completes the proof.
WJMS email for subscription: [email protected]
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
306
Theorem 2. The system (4) is dissipative that is all the solutions are uniformly bounded on Ω3 ⊆ R6+ .
Proof. Let (Sm , Im , Sfs , Ifs , Sfb , Ifb ) ∈ R6+ be any solutions with non negative initial conditions as shown
in the above theorem. Using a theorem on differential inequality[30] , it follows that
lim (Sm + Sfs + Sfb ) ≤
t→∞
Λ
.
µ
(17)
Taking the derivative of N along a solution path of the system (4) gives
0
0
N 0 = Sm
+ Im
+ Sf0 s + If0 s + Sf0 b + If0 b ,
(18)
N 0 = Λ − µN.
(19)
that is
All parameters and state variables of the system (4) are assumed to be non-negative for t ≥ 0 since it monitors
human population, hence 0 < N (t). Now by solving the above linear first order differential Eq. (19), we
obtain the following result: 0 < N (t) < Λ
µ as t → ∞, hence, all feasible solutions of the system (4) are
6
bounded in Ω3 ⊆ R+ , where
Ω3 =
(
Sm , Im , Sfs , Ifs , Sfb Ifb
)
Λ
: N (t) ≤
.
µ
(20)
The argument above shows that Ω3 is positively invariant and it is sufficient to consider solutions in Ω3 .
Existence, uniqueness and continuation holds in Ω3 .
4
Equilibria points and stability analysis
In this section we look at the disease free equilibrium point, reproduction number of the system and its
analysis and the endemic equilibrium point. To calculate the disease free equilibrium point, we equate the
system (4) to the zero vector and solve for the unknown variables Sm , Im , Sfs , Ifs , Sfb , Ifb .
Let
0
0
0
0
0
0
(21)
Q0 = Sm , Im , Sfs , Ifs , Sfb , Ifb .
be the disease free equilibrium point.
At the disease free equilibrium point, we have:
Im = Ifs = Ifb = 0.
(22)
Now by solving the system (4) as mentioned above we have:
Q0 =
!
(1 − ρ)Λ
γ(1 − ρ)Λ
ρΛ
, 0,
, 0,
,0 .
µ
µ+γ
µ(µ + γ)
The disease free equilibrium point Q0 attracts the region:
(
)
0
0
0
Ω0 =
Sm
, Im
, Sf0s , If0s , Sf0b , If0b ∈ Ω3 : Im
= If0s = If0b = 0 .
(23)
(24)
The disease free equilibrium point assist us in calculating the effective reproduction number which we
shall call RT , which is the average number of secondary new cases of trichomoniasis produced by each
infected individual in a totally susceptible population of heterosexuals and homosexuals in the presence of
treatment.Following van den Driessche and Watmough[24] , the reproduction number is given by:
WJMS email for contribution: [email protected]
307
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
2
1
1
p
βf
2 3 (h2 + h22 − 4k26 k46 h31 ) 3
2 3 k2 k4 h1
.
RT = l +
p
1 +
3k1 3k1 (h2 + h2 − 4k 6 k 6 h3 ) 3
6k1 k2 k4
2 4 1
2
(25)
where
h1 = βf2l + 3βf βm , h2 = k22 k42 [(−18µk1 + 9µγ + 9γk2 )βf βfl βm + 2k2 k4 βf3l ], k1 = α + µ > 0,
ρ
k2 = α + µ + γ, k3 =
, and k4 = µ + γ,
1−ρ
throughout the manuscript.
Theorem 3 follows from van den Driessche and Watmough[24] .
Theorem 3. The disease free equilibrium point Q0 of the system (4) is locally asymptotically stable for RT <
1 and unstable otherwise.
Theorem 4. The disease free equilibrium point Q0 of the system (4) is globally asymptotically stable provided
RT < 1.
Proof. Following Castillo-Chavez et al.[25] , we write the system (4) in the form
X 0 (t) = F (X, Y ), Y 0 (t) = G(X, Y ), G(X, 0) = 0,
(26)
where the components of X ∈ R3+ denotes the number of uninfected individuals and the components of
Y ∈ R3+ denotes the number of infected individuals. In this case the disease free equilibrium point is denoted
by Q0 = (X0 , 0), where
!
ρΛ (1 − ρ)Λ γ(1 − ρ)Λ
X0 =
,
,
.
(27)
µ
µ+γ
µ(µ + γ)
It suffices to prove the two conditions that: For X 0 (t) = F (X, 0), X0 is globally asymptotically stable.
Secondly
G(X, Y ) = U Y − G∗ (X, Y ), G∗ (X, Y ) ≥ 0 X, Y ∈ Ω.
(28)
Now consider
−k1



ρΛ − µSm
F (X, 0) =  (1 − ρ)Λ − (µ + γ)Sfs 
γSfs − µSfb


 µβm

and U =  k3 k4


 γβ
m
k3 k4
k3 βf
k3 βf
−k2
0
γ
βfl − k1





,



which implies that

G∗1 (X, Y )
G∗ (X, Y ) =  G∗2 (X, Y )  ,
G∗3 (X, Y )

Sf s
Sm
µ
∗
where
) = βf (Ifs + Ifb )(k3 −
), G2 (X, Y ) = βm Im
−
and G∗3 (X, Y ) =
Nf
k3 k4 Nm
Sf
Sf
γ
βm Im (
− b ) + βfl Ifb (1 − b ).
k3 k4 Nm
Nfb
G∗1 (X, Y
It suffices to show that G∗1 (X, Y ), G∗2 (X, Y ) and G∗3 (X, Y ) are all greater than or equal to zero.
ρΛ
Now, since Sfs ≤ Sf0s and Nm −→
as t −→ ∞, we have:
µ
Sf
µ
µ
(1 − ρ)Λ
G∗2 (X, Y ) = βm Im
− s ≥ βm Im
−
−→ 0 as t −→ ∞,
k3 k4 Nm
k3 k4
k4 Nm
WJMS email for subscription: [email protected]
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
308
0 and N + N −→ N 0 + N 0 as t −→ ∞, we have:
hence G∗2 (X, Y ) ≥ 0. Also, since Sm ≤ Sm
fs
fb
fs
fb
G∗1 (X, Y
) = βf (Ifs
Sm
+ Ifb ) k3 −
Nf
≥ βf (Ifs
ρΛ
+ Ifb ) k3 −
µNf
−→ 0 as t −→ ∞,
Sf b
ρΛ
≥ 0 and Nm −→
as t −→ ∞, we have:
N fb
µ
Sf b
Sf b
γ
γ
γ(1 − ρ)Λ
∗
−
+ βfl Ifb 1 −
≥ βm Im
−
−→ 0 as t −→ ∞,
G3 (X, Y ) = βm Im
k3 k4 Nm
N fb
k3 k4
µk4 Nm
hence G∗1 (X, Y ) ≥ 0. Similarly, since Sfb ≤ Sf0b , 1 −
hence G∗3 (X, Y ) ≥ 0. Therefore G∗ (X, Y ) ≥ 0 and the result follow.
4.1
Analysis of reproduction number
In this section, we look into the effect an increase in trichonomiasis infections among heterosexuals has
on bisexuals and vice-versa. Also, we look into the effect an increase in trichonomiasis infections among
heterosexuals has on the total population, as well as the effect an increase in trichonomiasis infections among
bisexuals has on the whole population. Furthermore, we analyze the effects that the parameters that defines
the reproduction number has on the average number of secondary new cases of trichonomiasis produced by
an average infected individual in a totally susceptible population. Now for us to analyze the effect that treatment has on the average number of secondary new cases of trichonomiasis produced by an average infected
individual in a totally susceptible population, we consider first the limit of the reproduction number RT as the
parameter α tends to infinity. It follows by properties of limits that lim RT is given by:
α→∞
lim
α→∞
βfl
+ lim
α→∞
3k1
1
2 3 k2 k4 h1
p
1
3k1 (h2 + h22 − 4k26 k46 h31 ) 3
2
!
+ lim
α→∞
2 3 (h2 +
p
1
h22 − 4k26 k46 h31 ) 3
6k1 k2 k4
!
= 0. (29)
This suggest that treatment has the potential to reduce infection. In other words, provision of enough treatment,
especially in the form of metronidazole or tinidazole reduces the average number of secondary new cases of
trichonomiasis produced by an average infected individual in a totally susceptible population.
We now continue to do the analysis of the reproduction number RT by considering the case where all straight
females turn to be strictly lesbians, that is the case where βm = βf = 0. In this case the reproduction number
βf
βf
becomes RT = l . In this manuscript, we shall call this reproduction number Rfl , thus Rfl = l , where
k1
k1
Rfl represents the average number of secondary trichomoniasis infections in females caused by one infected
female (non straight female) in a fully susceptible population of non straight females. It is clear that for case
where all females turn to be lesbians, the reproduction number depends on the parameters α, µ and βfl . In
this case, the analysis of the reproduction number Rfl is the same as that which has been done by Bhunu and
Mushayabasa[1] .
Similarly, as shown by Bhunu and Mushayabasa[1] the reproduction number that governs the relationship
between males and females is:
s
βm βf
Rf m =
,
(30)
(α + µ)2
where Rf m denotes the average number of secondary trichomoniasis infections caused by one infected straight
male or non lesbian female in a fully susceptible population of heterosexuals in the presence of treatment.
Hence, in the case where there is no contribution of bisexuality in trichonomiasis transmission dynamics,
the analysis of the reproduction number has been dealt with, by Bhunu and Mushayabasa[1] . Now, for us to
complete, the analysis of the reproduction number RT , it suffices to express the reproduction number RT
in terms of both Rf m and Rfl , so that we can analyze the effect that the change in either Rf m or Rfl has
on the reproduction number RT and vice-versa. In other words, this helps us to analyze the effect that the
WJMS email for contribution: [email protected]
309
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
change either in the number of trichonomiasis infected individuals amongst heterosexuals or in the number of
trichonomiasis infected individuals amongst non- heterosexuals has on the total population. It follows that
RT =
Rfl
+ H (G1 , G2 ) ,
3
(31)
where
G1 = Rf2l + 3Rf2 m ,
G2 = k22 k42 [(−18µk1 + 9µγ + 9γk2 )Rfl Rf2 m + 2k2 k4 Rf3l ],
p
2
1
1
2 3 (G2 + G22 − 4k26 k46 G31 ) 3
2 3 k2 k4 G1
H (G1 , G2 ) =
.
p
1 +
6k2 k4
3(G2 + G2 − 4k 6 k 6 G3 ) 3
2 4
2
1
It is clear that G1 > 0, G2 > 0 since, h1 > 0, h2 > 0 and k1 > 0. Using the chain rule, we have:
∂RT
∂Rfl
∂RT
∂Rf m
=
=
1 ∂H (G1 , G2 ) ∂G1
∂H (G1 , G2 ) ∂G2
+
,
+
3
∂G1
∂Rfl
∂G2
∂Rfl
∂H (G1 , G2 ) ∂G1
∂H (G1 , G2 ) ∂G2
+
.
∂G1
∂Rf m
∂G2
∂Rf m
It follows that
1
1
−1
1
−1
1
∂H (G1 , G2 )
2 3 (h33 ) + 16 3 (h33 )h4 k26 k46 G31 32 3 k26 k46 G21 h33 )h4
−
=
> 0.
2
∂G1
6k2 k4
3h 3
(32)
3
1
16 3
∂H (G1 , G2 )
=
∂G2
12k25 k45 G21
−4
3
h3 (1 +
G2 h4 )(4k26 k46 G31 )
+ h3 (1 − G2 h4 ) + h5 > 0,
2
3
(33)
where
2
h3 = G 2 +
q
G22 − 4k26 k46 G31 , h4 =
(G22
−
−1
4k26 k46 G31 ) 2
−2
2 3 (1 + G2 h4 )h33
.
and h5 =
3
Also,
∂G2
= k22 k42 [(−18µk1 + 9µγ + 9γk2 )Rf2 m + 6k2 k4 Rf2l ] > 0
∂Rfl
and
∂G2
= 2k22 k42 (−18µk1 + 9µγ + 9γk2 )Rfl Rf m > 0.
∂Rf m
Moreover,
∂G1
∂G1
= 2Rfl > 0 and
= 2Rf m > 0.
∂Rfl
∂Rf m
Hence
∂RT
∂RT
> 0 and
> 0.
∂Rfl
∂Rf m
The above mathematical arguments indicate that an increase in trichonomiasis infection either amongst heterosexual or non-heterosexuals result in an increase in the cases amongst the individuals in the population.
WJMS email for subscription: [email protected]
310
4.2
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
Existence and uniqueness of the endemic equilibrium
The are a number of theoretically feasible endemic equilibrium states:
•
•
•
the first case is where there are no homosexuals and this has been dealt with by Bhunu and
Mushayabasa[1] and the endemic equilibrium point has been shown to exist for Rf m > 1 as well as
being both locally and globally asymptotically stable for the case where it exists.
The second case is where all females becomes strictly homosexuals. We can do the analysis for this case
by simply setting βm = βf = 0.
the last case is where both heterosexuality and homosexuality exist. For this case we shall call the endemic
equilibrium point the interior equilibrium point.
Now, for the case where the contribution of bisexuality in trichonomiasis transmission dynamics is involved,
it is a daunting task to solve for the forces of infection in terms of the reproduction number RT in order for us
to show the existence of the endemic equilibrium point, but, the global stability of the disease free equilibrium
point for RT < 1 guarantees the existence of the endemic equilibrium point for RT > 1.
Alternatively, to establish the existence of the endemic equilibrium point for RT > 1, we can employ the
mean value theorem to do this job. Firstly, by equating the system (4) to the zero vector in R6 and solve for
∗ , I ∗ , S ∗ , I ∗ , S ∗ and I ∗ we have:
the unknown variables Sm
m
fs fs
fb
fb
∗ =
Sm
Sf∗s =
Sf∗b =
ρΛλ∗f
k1 ρΛ
∗
,
I
=
,
µ(k1 + λ∗f ) m µ(k1 + λ∗f )
k2 (1 − ρ)Λ
(1 − ρ)Λλ∗m
∗
=
,
I
,
k4 (k2 + λ∗m ) fs
k4 (k2 + λ∗m )
(1 − ρ)Λγ[(k1 k2 + αλ∗m )λ∗b + µλ∗m (k1 + λ∗b )]
(1 − ρ)Λγ(k1 k2 + αλ∗m )
∗
=
,
and
I
.
f
b
µk4 (k2 + λ∗m )(k1 + λ∗b )
µk1 k4 (k2 + λ∗m )(k1 + λ∗b )
(34)
Thus, the interior endemic equilibrium point is given by:
∗
∗
Q∗ = Sm
, Im
, Sf∗s , If∗s , Sf∗b , If∗b ,
(35)
∗ , I ∗ , S ∗ , I ∗ , S ∗ and I ∗ are as given in the system (34). Using the above Eq. (34), we can now
where Sm
m
fs fs
fb
fb
∗ , N ∗ and N ∗ to get:
solve for Nm
fb
f
∗
Nm
=
(1 − ρ)Λ
(1 − ρ)Λγ
ρΛ
, Nf∗ =
and Nf∗b =
.
µ
k4
µk4
(36)
Now, for us to solve for the forces of infections, we have to consider the following equations:
λ∗f =
βf (If∗s + If∗b )
Nf∗s + Nf∗b
, λ∗m =
∗
∗
βfl If∗b
βm Im
βm Im
∗
and
λ
=
+
.
b
∗
∗
Nm
Nf∗∗
Nm
b
(37)
∗ , I ∗ , N ∗ and N ∗ in (37) we obtain:
By substituting the systems (34) and (36) for Im
m
fb
fb
λ∗b = βfl
∗
k2 λ∗b + µλ∗m + λ∗b λ∗m βm Im
+
.
∗
(k2 + λ∗m )(k1 + λ∗b )
Nm
(38)
Secondly, the endemic equilibrium point exist for all values of βm , βf and βfl . Without loss of generality, we
now use the fact that the endemic equilibrium point exist for all values of βm , in particular, it exist for βm = 0,
that is the case where there is no heterosexuality, which is case (2) of the equilibrium states. By, substituting
βm = 0 in Eq. (38), we get:
WJMS email for contribution: [email protected]
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
311
λ∗b
.
k1 + λ∗b
(39)
λ∗b = βfl
Upon solving the previous Eq. (39), we obtain: λ∗b = 0 or λ∗b = βfl − k1 .
In this case λ∗b = 0 corresponds to the disease free equilibrium point and λ∗b = βfl − k1 corresponds to
the endemic equilibrium point. The endemic equilibrium point exist for λ∗b > 0, that is for:
βfl
> 1,
k1
(40)
hence, indeed the endemic equilibrium point exist for Rfl > 1. Now to show existence for the case where
RT > 1, that is the existence of the interior equilibrium point, it suffices to take the reproduction RT as
function of βm , that is, RT = RT (βm ).
Consider
p
1
2
1
βfl
2 3 (h2 + h22 − 4k26 k46 h31 ) 3
2 3 k2 k4 h1
RT (βm ) =
+
,
(41)
+
p
3k1 3k1 (h2 + h2 − 4k 6 k 6 h3 ) 13
6k1 k2 k4
2 4 1
2
it follows by properties of differentiable functions that, RT (βm ) is a differentiable function, hence, it is also
a continuous function. Also, since βm is the probability of infection, it is a non-negative number such that:
0 ≤ βm ≤ 1. Moreover, the rate of change of the function RT = RT (βm ) with respect to βm is non-negative,
hence by the mean value theorem,it follows that the the function RT = RT (βm ) is a monotonic increasing
function on the domain 0 ≤ βm ≤ 1. Moreover, since RT (βm ) is a continuous function on a closed and
bounded interval [0, 1], it implies that the function itself is bounded and attains its minimum and maximum
values.
Now by definition of infimum:
RT ≥ inf RT .
(42)
0≤βm ≤1
It follows that, since RT = RT (βm ) is a monotonic increasing function on the domain 0 ≤ βm ≤ 1, the
infimum of RT on the interval 0 ≤ βm ≤ 1 is the value of RT at the point where βm = 0, that is:
inf
0≤βm ≤1
RT = RT (0),
(43)
this implies
inf
0≤βm ≤1
RT =
βfl
βf
βf
+ l + l,
3k1 3k1 3k1
(44)
βfl
= Rfl ,
k1
(45)
hence,
inf
0≤βm ≤1
RT =
it follows that: RT ≥ Rfl .
Since at the endemic equilibrium point, Rfl > 1, it follows by the previous result that the endemic
equilibrium point exist for RT > 1 and this lead to the following theorem:
Theorem 5. The interior endemic equilibrium point Q∗ of the system (4) exist for RT > 1.
Theorem 6. The interior endemic equilibrium point Q∗ of the system (4) guaranteed by theorem (5) is locally
asymptotically stable for RT > 1.
Proof. We now employ the Centre-Manifold theory (Carr, 1981) to determine the local stability of the interior
endemic equilibrium point Q∗ . Here, we use the following substitution or change of variable in order to apply
the Centre-Manifold theory:
0 (t), F (x) = I 0 (t), F (x) = S 0 (t), F (x) = I 0 (t), F (x) = S 0 (t), and
Let F1 (x) = Sm
2
3
4
5
m
fs
fs
fb
0
F6 (x) = Ifb (t) where x = (x1 , x2 , x3 , x4 , x5 , x6 ) with x1 = Sm , x2 = Im , x3 = Sfs ,
x4 = Ifs , x5 = Sfb and x6 = Ifb .
In this case: Nm = x1 + x2 , Nfs = x3 + x4 , Nfb = x5 + x6 , Nf = x3 + x4 + x5 + x6 so that N (t) =
x1 + x2 + x3 + x4 + x5 + x6 and the forces of infection becomes:
WJMS email for subscription: [email protected]
312
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
λf =
(x4 + x6 )βf
x6 βfl
x2 βm
x2 βm
, λm =
and λb =
+
.
x3 + x4 + x5 + x6
x1 + x2
x1 + x2 x5 + x6
Using the vector notation, let X = (x1 , x2 , x3 , x4 , x5 , x6 )T , then the model system (4) can be written in the
form:
dX
= F (X),
dt
(46)
where F (X) = (F1 (x), F2 (x), F3 (x), F4 (x), F5 (x), F6 (x))T and Fi (x), i = 1, 2, 3, 4, 5, 6 are as given in
(5).
Now, since trichomoniasis affects females more than males, we have βm > βf , hence we can write
βm = k5 βf , k5 > 1. Similarly, βf = k6 βfl , k6 > 1. Thusβm = k7 βfl , k7 = k5 k6 > 1 and βm βf =
k6 k7 βf2l , k6 k7 > 1. We can now express h1 and h2 in terms of βfl , that is: h1 = k8 βf2l and h2 = k9 βf3l where
k8 = 1 + 3k6 k7 and k9 = k22 k42 [k6 k7 (−18µk1 + 9µγ + 9γk2 ) + 2k2 k4 ].
Having expressed h1 and h2 in terms of βfl , we now express the effective reproduction number RT in
terms of βfl also, that is
1
2
βf
2 3 k2 k4 k8 βfl
2 3 (k9 +
RT = l +
p
1 +
3k1 3k1 (k9 + k 2 − 4k 6 k 6 k 3 ) 3
9
2 4 8
p
1
k92 − 4k26 k46 k83 ) 3 βfl
.
6k1 k2 k4
(47)
If βfl is taken as a bifurcation point and consider RT = 1 and solve for βfl to obtain:
βf∗l =
where
1
,
σ
(48)
p
1
2
1
2 3 (k9 + k92 − 4k26 k46 k83 ) 3
1
2 3 k2 k4 k8
σ=
+
.
+
p
3k1 3k1 (k9 + k 2 − 4k 6 k 6 k 3 ) 31
6k1 k2 k4
9
2 4 8
Let J(Q0 ) be the Jacobian matrix of the system (4) evaluated at the disease free equilibrium
follows that:

0
0
−βf Sm
−βf Sm
−µ
α
0
0

Nf0s + Nf0b
Nf0s + Nf0b


0
0
βf Sm
βf Sm

0
0
 0 −(α + µ)

Nf0s + Nf0b
Nf0s + Nf0b

0

−βm Sfs
 0
−(µ + γ)
α
0
0

0
Nm

J(Q0 ) = 
βm Sf0s

 0
0
−(α + µ + γ) 0
0
0

N
m

−βm Sf0b
βfl Sf0b

 0
γ
0
−µ
α−

0
Nm
Nf0b


βm Sf0b
βfl Sf0b

0
0
γ
0
−(α
+
µ)
+
0
Nm
Nf0b
(49)
point Q0 . It












 .










(50)
The linearised system of (4) with βfl = βf∗l has a simple zero eigenvalue, hence the Centre-Manifold theory can be used to analyse the dynamics of the system (4) near the point βfl = βf∗l . The Jacobian matrix (50) of the system (4) has a right eigenvector associated with the zero eigenvalue which is given by:
u = (u1 , u2 , u3 , u4 , u5 , u6 )T , where
2βm βf
µβm α
u1 = −
+ 1 u2 < 0, u2 = u2 > 0, u3 =
− 1 u2 < 0,
k2 k4
k3 k42 k2
WJMS email for contribution: [email protected]
313
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
(α − βf∗l )u6 γu3 γβm u2
µβm u2
k1 u2
µβm
u4 =
> 0, u5 =
< 0, u6 =
+
−
+
k2 k3 k4
µ
µ
µk3 k4
k3 βf
k3 k42
1−α
k2
u2 > 0.. (51)
Similarly, by using the transpose of the Jacobian matrix (50), the left eigenvector of the Jacobian matrix (50)
is given by v = (v1 , v2 , v3 , v4 , v5 , v6 )T where v1 = 0, v2 = v2 > 0, v3 = 0.
v4 = (
γk3 βf v2
k3 βf
γk3 βf
+
)v2 > 0, f or k1 − βf∗l > 0, v5 = 0, v6 =
> 0.
∗
k2 (k1 − βfl )
k2
k1 − βf∗l
(52)
Having found the right and the left eigenvectors associated with the zero eigenvalue, we can now use CentreManifold theorem as used by Castillo-Chavez et al.[25] to do the analysis of the stability of the interior endemic
equilibrium point. It follows that:
∂ 2 Fk (0, 0)
, i, j, k = 1, 2, 3, 4, 5, 6,
∂xi ∂xj
X
∂ 2 Fk (0, 0)
, i, k = 1, 2, 3, 4, 5, 6.
=
vk ui
∂xi ∂βf∗l
a =
b
X
vk ui uj
(53)
Since, v1 = 0, v3 = 0 and v5 = 0 the above definitions for a and b (53) reduces to:
∂ 2 F2 (0, 0)
∂ 2 F4 (0, 0)
∂ 2 F6 (0, 0)
+ v4 ui uj
+ v6 ui uj
, i, j = 1, 2, 3, 4, 5, 6
∂xi ∂xj
∂xi ∂xj
∂xi ∂xj
X
∂ 2 F4 (0, 0)
∂ 2 F6 (0, 0)
∂ 2 F2 (0, 0)
+
v
u
+
v
u
, i = 1, 2, 3, 4, 5, 6.
=
v2 ui
4 i
6 i
∂xi ∂βf∗l
∂xi ∂βf∗l
∂xi ∂βf∗l
a =
b
X
v2 ui uj
(54)
Since the functions Fk k = 1, 2, 3, 4, 5, 6 are continuous, it implies that:
∂ 2 Fk
∂ 2 Fk
=
.
∂xi ∂xj
∂xj ∂xi
(55)
It follows that the non-zero second derivatives of the function F2 are:
∂ 2 F2 (0, 0)
∂xj ∂x1
∂ 2 F2 (0, 0)
∂x4 ∂x3
∂ 2 F2 (0, 0)
∂x6 ∂x3
∂ 2 F2 (0, 0)
∂x5 ∂x4
∂ 2 F2 (0, 0)
∂x6 ∂x5
∂ 2 F4 (0, 0)
∂x2 ∂x1
∂ 2 F4 (0, 0)
∂x3 ∂x2
∂ 2 F6 (0, 0)
∂x22
µβf
∂ 2 F2 (0, 0)
=
, j = 4, 6,
∂x1 ∂xj
(1 − ρ)Λ
µk3 βf
∂ 2 F2 (0, 0)
=
=−
,
∂x3 ∂x4
(1 − ρ)Λ
µk3 βf
µk3 βf
∂ 2 F2 (0, 0)
∂ 2 F2 (0, 0)
=
=−
,
= −2
,
2
∂x3 ∂x6
(1 − ρ)Λ
(1 − ρ)Λ
∂x4
µk3 βf
µk3 βf
∂ 2 F2 (0, 0)
∂ 2 F2 (0, 0)
∂ 2 F2 (0, 0)
=
=−
,
=
= −2
,
∂x4 ∂x5
(1 − ρ)Λ ∂x6 ∂x4
∂x4 ∂x6
(1 − ρ)Λ
µk3 βf
µk3 βf
∂ 2 F2 (0, 0)
∂ 2 F2 (0, 0)
=
=−
,
= −2
,
2
∂x5 ∂x6
(1 − ρ)Λ
(1 − ρ)Λ
∂x6
∂ 2 F4 (0, 0)
µ2 βm ∂ 2 F4 (0, 0)
µ2 βm
=
−2
=
=−
,
,
∂x1 ∂x2
k3 k4 ρΛ
k3 k4 ρΛ
∂x22
∂ 2 F4 (0, 0)
µβm ∂ 2 F6 (0, 0)
∂ 2 F6 (0, 0)
µγβm
=
=
,
=
=−
,
∂x2 ∂x3
ρΛ
∂x2 ∂x1
∂x1 ∂x2
k3 k4 ρΛ
∂ 2 F6 (0, 0)
µβm
µγβm ∂ 2 F6 (0, 0)
,
=
=
.
= −2
k3 k4 ρΛ ∂x5 ∂x2
∂x2 ∂x5
ρΛ
=
Following the above arguments together with the system (54), the value of the scalar a is given by:
WJMS email for subscription: [email protected]
314
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
a = k10 (u4 + u6 )
2µβf v2
k11 µ + k3 k4 u3 2µβm u2 v4 k11 γ + k3 k4 u5 2µβm u2 v6
+
+
,
(1 − ρ)Λ
k3 k4
ρΛ
k3 k4
ρΛ
(56)
where k10 = u1 + k3 u3 − k3 u4 + k3 u5 − k3 u6 < 0 and k11 = u1 − u2 < 0. We can see that a <
0, since u1 , u3 and u5 are all negative. To compute the value of the scalar b, it requires us to express the
functions Fj , j = 1, 2, 3, 4, 5, 6 in (5) in terms βfl by using the relationships: βf = k6 βfl , k6 > 1 and
βm = k7 βfl , k7 = k5 k6 > 1, that is
(x4 + x6 )x1 k6 βfl
− µx1 + αx2 ,
x3 + x4 + x5 + x6
(x4 + x6 )x1 k6 βfl
− k1 x2 ,
x3 + x4 + x5 + x6
x2 x3 k7 βfl
(1 − ρ)Λ −
− k4 x3 + αx4 ,
x1 + x2
x2 x3 k7 βfl
− k2 x4 ,
x1 + x2
x2 x5 k7 βfl
x5 x6 βfl
γx3 −
−
− µx5 − αx6 ,
x1 + x2
x5 + x6
x2 x5 k7 βfl
x5 x6 βfl
+
+ γx4 − k1 x6 .
x1 + x2
x5 + x6
F1 (x) = ρΛ −
F2 (x) =
F3 (x) =
F4 (x) =
F5 (x) =
F5 (x) =
It follows from the above system of (57) that the non-zero derivatives are:
(57)
∂ 2 F2 (0, 0)
∂ 2 F2 (0, 0)
=
k
k
,
=
3
6
∂x4 ∂βf∗l
∂x6 ∂βf∗l
∂ 2 F4 (0, 0)
µk7 ∂ 2 F6 (0, 0)
γk7
∂ 2 F6 (0, 0)
=
,
=
and
= 1. Now, it follows by the definition of b
∂x2 ∂βf∗l
k3 k4 ∂x2 ∂βf∗l
k3 k4
∂x6 ∂βf∗l
in (54) that:
µk7 u2 v4 γk7 u2 v6
b = (u4 + u6 )k3 k6 v2 + u6 v6 +
+
.
(58)
k3 k4
k3 k4
Clearly b > 0 since u2 , u4 , u6 , v2 , v4 , v6 , k3 , k4 and k7 are all strictly positive. Therefore, by using the CentreManifold theory, the interior endemic equilibrium point is locally asymptotically stable for RT > 1, since
a < 0 and b > 0. This completes the proof.
k3 k6 ,
Theorem 7. The interior endemic equilibrium point Q∗ of the system (4) guaranteed by theorem (5) is globally
asymptotically stable for RT > 1.
Proof. To prove this theorem,we make use of the Lyapunov functional theorem, to which we show that our
interior endemic equilibrium point is the only stationary point of the Lyapunov function. To do that, we change
the variable such that: Sm = x1 , Im = x2 , Sfs = x3 , Ifs = x4 , Sfb = x5 and Ifb = x6 . Define a function
V (x1 , x2 , x3 , x4 , x5 , x6 ) by:
V = x1 − x∗1 Inx1 + x2 − x∗2 Inx2 + x3 − x∗3 Inx3 + x4 − x∗4 Inx4 + x5 − x∗5 Inx5 + x6 − x∗6 Inx6 , (59)
this implies that:
∂V
x∗
= 1 − i , i = 1, 2, 3, 4, 5, 6.
(60)
∂xi
xi
By equating the partial derivatives to zero and solve for xi we get: xi = x∗i , i = 1, 2, 3, 4, 5, 6.
It is clear that the interior endemic equilibrium point is the only stationary point of the function
V (x1 , x2 , x3 , x4 , x5 , x6 ).
∂2V
x∗i
=
, i = 1, 2, 3, 4, 5, 6,
(61)
xi
∂x2i
it is clear that the second partial derivative of the function V (x1 , x2 , x3 , x4 , x5 , x6 ) with respect to xi for
all i, is positive, hence the interior endemic equilibrium point is the global minimum point of the function
V (x1 , x2 , x3 , x4 , x5 , x6 ), for all v ∈ R6+ and V 0 = 0 only at the interior endemic equilibrium point.
Further, the function V (x1 , x2 , x3 , x4 , x5 , x6 ) is continuous everywhere by properties of continuous functions and has continuous first order partial derivatives. Therefore to conclude that the the function
WJMS email for contribution: [email protected]
315
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
V (x1 , x2 , x3 , x4 , x5 , x6 ) is a Lyapunov function, it requires us to prove that, the rate of change of the function
V (x1 , x2 , x3 , x4 , x5 , x6 ) with respect to time is less than or equal to zero as t approaches infinity. Now
V 0 (t) =
X
(xi − x∗i )
x0i (t)
, i = 1, 2, 3, 4, 5, 6.
xi (t)
(62)
Since the set of solutions for the system of (4) is bounded, there exist a positive real number k such that:
k = min xi , i = 1, 2, 3, 4, 5, 6, hence
X
(xi − x∗i )
x0i (t)
1X
≤
(xi − x∗i )x0i (t), i = 1, 2, 3, 4, 5, 6.
xi (t)
k
(63)
Also, since
N 0 (t) = Λ − µN,
where 0 < N ≤
(64)
Λ
, =⇒ 0 ≤ N 0 (t) < Λ hence, the derivatives are also bounded, that is x0i (t) ∈ L∞ . Thus
µ
ΛX
1X
(xi − x∗i )x0i (t) <
(xi − x∗i ), i = 1, 2, 3, 4, 5, 6.
k
k
(65)
Therefore
ΛX
(xi − x∗i ), i = 1, 2, 3, 4, 5, 6.
k
Using the result derived from Barbalat lemma (Barbalat 1959),
X
(xi − x∗i ) −→ 0, i = 1, 2, 3, 4, 5, 6 as t −→ ∞,
V 0 (t) <
(66)
(67)
hence V 0 (t) < 0, this implies that the function V (x1 , x2 , x3 , x4 , x5 , x6 ) is a strict Lyapunov function with the
endemic equilibrium point only as its stationary point, which is a global minimum point. Thus the endemic
equilibrium point Q∗ is globally asymptotically stable for RT > 1.
5
Numerical simulations
In this chapter, we carry out detailed numerical simulations using Matlab programming language to assess
the effect that certain parameters have on the population size. Using the data in Tab. 1 above we asses the effect
Table 1. Model parameters
Model parameter
Λ
µ
βm
βf
βfl
γ
α
ρ
Value (Range)
0.029yr−1 ×200000
0.02yr−1
0.850yr−1
0.450yr−1
0.00001yr−1
0.000125yr−1
0.90yr−1
0.505yr−1
Source
Bhunu et al.[1]
Bhunu et al.[1]
Catterall,[26]
Weston and Nicol,[29]
Assumed
Assumed
Gulmezoglu and Garner,[27]
NationMaster,[28]
of parameters in the absence of treatment subject to the following initial conditions: Sm (0) = 100000.0,
Im (0) = 1000.0, Sfs (0) = 110000.0, Ifs (0) = 1300, Sfb = 1.0 and Ifb (0) = 0.0. Further we will also
assess the effect that trichomoniasis have in the community in the presence of treatment. Finally we will do
the sensitivity analysis of the reproduction numbers to certain parameters using Monte Carlo Simulations.
Fig. 2 shows the effect of varying the female to male contact rate on infected males, infected straight
females and infected bisexual females. The Figure has shown that as we increased the value of βf the number
WJMS email for subscription: [email protected]
316
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
Fig. 2. Effects of varying the female to male contact rate starting from βf = 0 to βf = 1 with a step size of 0.25 .
Parameter values used are in Tab. 1
Fig. 3. Effects of varying the male to female contact rate starting from βm = 0 to βm = 1 with a step size of 0.25 .
Parameter values used are in Tab. 1
of infected individuals increases as shown by the arrows pointing upwards on Fig. 2 (a), (b) and (c). However,
by comparing Fig. 2 (a), (b) and (c), we can see that straight females are more affected by the female to male
contact rate whereas bisexual females are less affected by the same contact rate. This might be the reason that
bisexuals has no direct sexual relationship with straight females but they are linked indirectly by males.
Fig. 3 shows the effect of varying the female to male contact rate on infected males, infected straight
females and infected bisexual females. The Figure has shown that as we increased the value of βf the number
of infected individuals increases as shown by the arrows pointing upwards on Fig. 3 (a), (b) and (c). However,
by comparing Fig. 3 (a), (b) and (c), we can see that straight females are more affected by the female to male
contact rate whereas bisexual females are less affected by the same contact rate. This might be the reason that
bisexuals has no direct sexual relationship with straight females but they are linked indirectly by males.
Fig. 4 has shown that as the value of βm increases, the number of infected individuals increases as shown
by the arrows pointing upwards on Fig. 4 (a), (b) and (c). As shown in Fig. 4 (a), (b) and (c), the increase
in the number of infected straight females leads to the increase in the number of infected males more than
bisexual infected females. This might be the fact that straight females have a direct sexual relationship with
males.
WJMS email for contribution: [email protected]
317
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
Fig. 4. Effects of varying the female to female contact rate starting from βfl = 0 to βfl = 1 with a step size of 0.25 .
Parameter values used are in Tab. 1
The parameter βfl has a negligible effect on males and straight females whereas bisexual females are
more affected as shown on Fig. 4 (a), (b) and (c). Also as the female to female contact rate increases the
number of infected individuals increases as shown on Fig. 4 (a), (b) and (c).
5.1
Sensitivity analysis
In this section, we look at the sensitivity of the reproduction number to certain parameters. We start by
representing graphically the relationships between the parameters that defines the reproduction number and
the reproduction number RT using the Monte Carlo Simulations as shown in Fig. 5.
Fig. 5. Latin Hypercube Sampling graphs
Fig. 5 (a) shows the correlation between the variable βm and RT .It reveals a strong positive relationship
between male to female probability of infection and the average number of secondary infected individuals.
That is, as the male to female contact rate increases the reproduction also increases.
Fig. 5 (b) is a scatter graph that illustrate the correlation between the female to male contact rate βf and the
reproduction number RT . It shows that the is a positive correlation between these variables. That is, increase
in the value of the female to male contact rate leads to an increase in the average number of trichomoniasis
infected individuals in a totally susceptible population. However, Fig. 5 (a) and (b) reveals that the reproduction number is more sensitive to the male to female contact rate as compared to the female to male contact
rate. This reflects that females are more prone to trichomoniasis infection as compared to males.
Similarly, just like as shown on Fig. 5 (a) and (b) , the Scatter graph Fig. 5 (c) also reveals a positive relationship between the female to female contact rate βfl and the reproduction number RT . However, the rate at
which βfl increases the value of the reproduction number is negligible as compared to βm and βf as shown on
the graph Fig. 5. Thus in general an increase in the in the probability of being infected results in an increase in
the number of people infected in a totally susceptible population. This might be the case that individuals are
WJMS email for subscription: [email protected]
318
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
falling into sexual contact with asymptomatic carriers who are not seeking treatment.
The relationship between the treatment rate α and the reproduction number RT is shown on Fig. 5 (d). The
Figure indicates that an increase in the treatment rate decrease the value of the reproduction number thereby
reducing the number of TV infected individuals. Fig. 5 (d) clearly indicates that at least 77 percent of treatment is needed to reduce the value of the reproduction number to less than a unity. This indicates that the TV
infection can die out if treatment is increased to at least 77 percent thereby controlling TV prevalence. Hence
it is advisable for individuals to visit a nearest hospital or medical clinic to seek for their status before they
fall into sexual contact. Thus treatment plays a vital role in controlling the rate at which the infection spreads.
Fig. 6. Sensitivity of Rfl to all parameters using partial rank correlation coefficients
In Fig. 6, partial rank correlation coefficients (P RCC) were calculated to estimate the correlation between the values of female to female reproduction number Rfl and the three parameters α, µ and βfl . In this
Fig. 6 a large P RCC is an indicative of high sensitivity to parameter estimate (P RCC > 0 will increase
Rfl when the parameters are increased). In contrast, small P RCC reflects low sensitivity (P RCC < 0 will
decrease the value of Rfl when the parameters are increased). Also, Fig. 6 and shows that the reproduction
number Rfl is more sensitive to treatment rate, in this case the P RCC is negative, clearly suggesting that
infected individuals should be encouraged to seek treatment so as to reduce TV prevalence. In contrast, the
female to female transmissibility rate has a great effect in increasing the rate at which the female to female
reproduction numbers increases.
Fig. 7. Sensitivity of Rfl to all parameters using partial rank correlation coefficients
Fig. 7 shows the correlation between the values of Rf m and the four parameters α, µ, βm and βf . On
Fig. 7 a large P RCC is an indicative of high sensitivity to parameter estimate (P RCC > 0 will increase
Rf m when the parameters are increased). In contrast, small P RCC reflects low sensitivity (P RCC < 0
will decrease the value of Rf m when the parameters are increased). Also,Fig. 7 shows that the reproduction
WJMS email for contribution: [email protected]
World Journal of Modelling and Simulation, Vol. 9 (2013) No. 4, pp. 302-320
319
number Rf m is more sensitive to the treatment rate, this is seen by clearly observing that the P RCC is
negative. This shows that the treatment rate has a vital role in reducing the value of the reproduction Rf m
when it is increased. This also suggest that infected individuals should be encouraged to seek treatment so as
to reduce TV prevalence. Since the reproduction number RT depends on both Rfl and Rf m , it follows directly
from the above argument that the reproduction number RT is also more sensitive to the treatment rate.
6
Conclusion
A mathematical model for the transmission dynamics of trichomoniasis in the presence of bisexuality has
been presented and its analysis has been dealt with using both numerical and analytical methods. It has been
shown that increase in the infection among bisexuals can lead to an increase number of infected individuals
amongst heterosexuals. With the help of the Centre-Manifold theorem and the Lyapunov theorem, we managed
to show both the local and global stability of the endemic equilibrium point when the reproduction number
is greater than a unity. The disease free equilibrium point is shown to be globally asymptotically stable when
the reproduction number is less than a unity. Thus we managed to show that the infection dies out when the
reproduction number is less than a unity. In contrast the infection will persist if the reproduction number is
greater than a unity.
The analysis of the reproduction number together with numerical simulations revealed that the reproduction number is more sensitive to the treatment rate and contact rates as compared to other parameters. The
contacts rates have been shown to have negative effect on the population since they lead to an increase the
number of infected individuals in a totally susceptible population if they increases. However, we do not have
control over this transmissibilities, so it is better to seek treatment to avoid the spread of the infection.
Treatment rate has been shown to have a positive effect on the total population if it is increased especially
if it is at least 77 percent. This so since the sensitivity analysis reveals that the treatment rate can reduce the
value of the reproduction number if it is increased. Hence it is best to encourage individuals to seek treatment
so as to reduce the transmission rate of trichomoniasis infection. So the community is encouraged to provide
enough treatment especially in the form of tinidazole and metronidazole so as to reduce the average number
of infected individuals in a totally susceptible population.
The mathematical arguments have also indicated that females are more prevalent to trichomoniasis infection than males. However, it does not necessarily mean that males can not be infected, but the may be
asymptomatic carriers. Hence males should also seek for treatment in order to prevent the spread of trichomoniasis infection.
However just like an other model, we can not say the model is complete, it can be extended to include
such factors as the use of condoms, intervention of educational services and the fact that individuals can be
born naturally as homosexuals.
Acknowledgements
We are grateful to the reviewers and the handling editor for comments and suggestions which led to the
improvement of the manuscript.
References
[1] C. Bhunu, S. Mushayabasa. Transmission dynamics of Trichomonas vaginalis: A mathematical approach. Journal
of Mathematical Analysis and Applications, 2011, 379(2): 852–860.
[2] D. Soper. Trichomoniasis: under control or undercontrolled. American Journal of Obstetrics and Gynecology,
2004, 190(1): 281–90.
[3] F. Djana. Trichomoniasis: Evaluation to execution. European Journal of Obstetrics and Gynecology and Reproductive Biology, 2011, 157(1): 3–9.
[4] III. Hook, W. Edward. Trichomonas vaginalis No Longer a Minor STD. Sexually Transmitted Diseases, 1999,
26(7): 388–399.
WJMS email for subscription: [email protected]
320
P. Mafuta, J. Mushanyu, S. Mushayabasa, C. Bhunu: Transmission dynamics of trichomoniasis in bisexuals
[5] V. Johnston, D. Mabey. Global epidemiology and control of Trichomonas vaginalis. Current Opinion in Infectious
Diseases, 2008, 21 (1): 56–64.
[6] N. Nanda, R. Michel, G. Kurdgelashvili, K. Wendel. Trichomoniasis and its treatment. Expert review of antiinfective therapy, 2006, 4(1): 125–35.
[7] H. Swygard, A. Sena, M. Hobbs, M. Cohen. Trichomoniasis: clinical manifestations, diagnosis and management.
Sexually Transmitted Infections, 2004, 80(2): 91–95.
[8] J. Schwebke, D. Burgess. Trichomoniasis. Clinical Microbiology Reviews, 2004, 17(4): 794–803.
[9] R. Jennifer, G. Judson, J. Alderete. Prospective Study of Trichomonas vaginalis nfection and Prostate Cancer
Incidence and Mortality: Physicians’ Health Stud. JNCI Journal of the National Cancer Institute, 2009, 101(20):
1406–1411.
[10] C. Ohlemeyer, L. Hornberger, D. Lynch. Diagnosis of Trichomonas vaginalis in adolescent females: InPouch TV
culture versus wet-mount microscopy. Journal of Adolescent Health, 1998, 22(3): 205–208.
[11] J. Huppert, S. Jill, E. Joel. Rapid Antigen Testing Compares Favorably with Transcription-Mediated Amplification
Assay for the Detection of Trichomonas vaginalis in Young Women. Clinical Infectious Diseases, 2007, 45(2):
194–198.
[12] A. Petra, K. Irene. Trichomonas vaginalis detection using real-time TaqMan PCR. Journal of Microbiological
Methods, 2007, 68(2): 243–247.
[13] S. Cudmore, K. Delgaty, S. Hayward-Mcclelland. Trichomonas vaginalis. Clinical Microbiology Reviews, 2004,
17(4): 783–793.
[14] P. Upcroft, J. Upcroft. Drug Targets and Mechanisms of Resistance in the Anaerobic Protozoa. Clinical Microbiology Reviews, 2001, 14(1): 150–164.
[15] M. Mendoza-Lopez, C. Becerril-Garcia, L. Fattel-Facenda. CP30, a Cysteine Proteinase Involved in Trichomonas
vaginalis Cytoadherence. Infection and Immunity, 2000, 68(9): 4907–4912.
[16] J. Carlton, R. Hirt, J. Silva. Draft Genome Sequence of the Sexually Transmitted Pathogen Trichomonas vaginalis.
Science, 2000, 315(5809): 207–212.
[17] M. Lehker, J. Alderete. Resolution of Six Chromosomes of Trichomonas vaginalis and Conservation of Size and
Number among Isolates. The Journal of Parasitology, 1999, 85(5): 976–979.
[18] D. Nimisha, A. Summaiya, J. Kosambiya. Sexually transmitted infections and reproductive tract infections in
female sex workers. Indian Journal of Pathology and Microbiology, 2009, 52(5): 198–199.
[19] S. Mavedzenge, B. Van der Pol, H. Cheng. Epidemiological Synergy of Trichomonas vaginalis and HIV in Zimbabwean and South African Women. Sexually Transmitted Diseases, 2010, 37(7): 460–466.
[20] M. Frisch, A. Hviid. Childhood family correlates of heterosexual and homosexual marriages: national cohort study
of two million danes. Archives of Sexual Behavior, 2006, 35: 533–547.
[21] J. Meares. Prostatitis syndromes: new perspectives about old woes. Journal of Urology, 1980, 123: 141–147.
[22] W. Fair, J. Couch, N. Wehner. Prostatic antibacterial factor: identity and significance. Journal of Urology, 1976, 7:
169–177.
[23] W. Cates. Estimates of the incidence and prevalence of sexually transmitted diseases in the United States. Sexually
Transmitted Infections, 1999, 26: 52–57.
[24] P. Driessche, J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for the compartmental
models of disease transmission. Mathematical Biosciences, 1999, 180: 29–48.
[25] C. Castillo-Chavez, Z. Feng, W. Huang. On the computation of R0 and its role on global stability. Infectious
Diseases: An Introduction, 2002, 125: 229–250.
[26] R. Catterall. Diagnosis and treatment of trichomonal urethritis in men. British Medical Journal, 1960, 113–115.
[27] A. Gulmezoglu, P. Garner. Trichomoniasis treatment in women: a systematic review. Tropical Medicine & International Health, 2002, 3: 553–558.
[28] NationMaster.
Zimbabwean people statistics. Available at http://www.nationmaster.com/red/country/zizimbabwe/peo-people&all=1..
[29] T. Weston, C. Nicol. Natural history of trichomonal infection in males. The British Journal of Venereal Diseases,
1963, 39: 251-257.
[30] G. Birkhoff, G. Rota. Ordinary Differential Equations. Needham Heights, Ginn, 1982, 39: 251-257.
WJMS email for contribution: [email protected]