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Analysis of a Mathematical Model Incorporating Dual Protection and ART Adherence for a

High Risk HIV Population

I. S. Oriedo1, G. O. Lawi2, J. O. Bonyo3,∗
1Department of Pure and Applied Mathematics, Maseno University, P.O. Box 333-40105, Maseno, Kenya

samsonoriedo@gmail.com
2Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 190-50100,

Kakamega, Kenya
glawi@mmust.ac.ke

3Department of Mathematics, Multimedia University of Kenya, P.O. Box 15653-00503, Nairobi, Kenya
jbonyo@mmu.ac.ke

∗Correspondence: jobbonyo@maseno.ac.ke

Abstract. In this paper, a mathematical model for dual protection, incorporating PrEP and Condomuse, and ART adherence is formulated, based on a system of ordinary differential equations andanalyzed. The results obtained from stability analysis indicate that provided the basic reproductivenumber is less than unity, the disease free equilibrium point is both locally and globally asymptoticallystable, while provided the basic reproductive number is greater than unity, the endemic equilibriumpoint exists and is locally asymptotically stable. Sensitivity analysis is undertaken to establish themost sensitive model parameter. The most sensitive parameter to the value of R0 is β1, the meancontact rate with undiagnosed infectives. This implies that in order to control the spread of HIVin a high risk population, efforts should be geared towards reducing the undiagnosed by testingand enrolling them on ART treatment. This in turn lowers their infectivity as well as chances ofprogressing to the AIDS class.

1. Introduction
Numerous efforts have been made in an attempt to control the spread of HIV, with the aim ofreducing its effects. According to the UNAIDS fact sheet 2019, at least 1.7 million new HIVinfections were reported by the end of the year 2018 [13].Scientific as well as public health interventions such as testing and counseling, circumcision,use of PrEP (Pre-Exposure Prophylaxis), PeP (Post-Exposure Prophylaxis), condom use, and an-tiretroviral therapy have been proposed and utilized. Consistent use of condoms can result to 80%reduction in HIV incidence among the heterosexual population [2], while the effectiveness of condom

Received: 1 May 2023.
Key words and phrases. mathematical modeling; PrEP; dual protection; HIV/AIDS; stability analysis.1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 2
use for men who have sex with men is 70% [3]. Proper use (correctly and consistently) as well asquality concerns have been directly attributed to the success of this approach.In 2012, the U.S Food and Drug Administration (FDA) approved the use of Truvada for PrEP asan oral pill taken once a day [14]. Numerous efficacy trials( by iPrEX, Partners PrEP, TDF2,e.t.c) have since been conducted to ascertain the potential of PrEP to prevent HIV infection. TheiPrEX trial demonstrated that PrEP has the potential of reducing the risk of HIV infection amongtransgender women, bisexual men, as well as men who have sex with men [8]. Two major studies;Partners PrEP, and TDF2 demonstrated the effectiveness of PrEP among heterosexual men andwomen. Out of all these studies, none displayed a 100% effectiveness [11]. Adherence has beenfound to be directly correlated with the effectiveness of PrEP [11]. In the absence of adherence,which guarantees efficacy, PrEP failures have been characterized by; system failures, people fail-ures, Doctor failures, drug failures, as well as assay failures [9]. These failures expose PrEP usersto the risk of HIV infection hence the need for additional protection whenever PrEP has beenutilized.The nature of storage, date of manufacture, religious as well as socio-cultural beliefs also influencehow each HIV prevention venture is utilized. The challenges experienced when various approachesare employed in an attempt to control the spread of HIV infection in a high risk population form thebasis for the need to use dual protection in order to achieve maximum protection. A combinationprevention approach as proposed by [6], based on proven efficacy interventions, provides one withthe best opportunity to curb the spread of HIV among the high risk population.In this study, we propose a mathematical model of dual protection against HIV infection by the useof condom and PrEP, and adherence to ART treatment, while focusing on the high risk populationcollectively. Earlier studies have either narrowed down to a particular category of persons at highrisk of infection [3], or have used a combination of prevention techniques where one techniqueacts as a supplement to the other [4], [5]. The study will focus on the impact of dual protectionon reducing the number of new infections, and that of ART adherence in ensuring those who areinfected remain less infectious.

2. Model Formulation and Description
The population is subdivided into the classes; susceptible, infected, and Aids individuals. Thesusceptible class has been further subdivided into two compartments on the basis of degree ofrisk of infection. These include susceptible individuals at high risk of infection, denoted by (SH),and those at low risk, denoted by (SL). The high risk population incorporates mainly commercialsex workers, men who have sex with men (MSM), and HIV-Discordant couples [8]. The infectedclass is subdivided into two compartments; those who are unaware of their HIV status (I), andthose who have been diagnosed and consequently enrolled for treatment (TD). The individualswho are unaware of their HIV status may progress to the TD compartment after successful HIV

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 3
awareness campaigns that will persuade them to get tested, or when they develop HIV symptomsand consequently enroll for ART treatment. If ART treatment fails, the individual progresses to theAIDS compartment. This happens when there is lack of adherence to ART, which allows the virusto multiply, thus increasing the plasma viral load. This results in weakening of the immune systemand hence the AIDS symptoms begin to manifest. The AIDS compartment comprises of those whoposses full blown symptoms, and are mostly bedridden, they thus do not significantly contribute tothe spread of the disease. Exit from the AIDS class is through natural death.Thus, considering apopulation of size N(t), at a time t,

N(t) = SH(t) + SL(t) + I(t) + TD(t) + A(t). (1)
The following interventions have been incorporated in the model;(a) 0 ≤ φ1 ≤ 1 - measures PrEP effectiveness, including its awareness and proper use as a meansto prevent susceptible individuals from being infected. Thus, (1 −φ1) measures PrEP failure.(b) 0 ≤ φ2 ≤ 1- measures condom effectiveness as a result of proper use, following adequateawareness campaigns and availability. Thus, (1 −φ2) measures condom failure.(c) 0 ≤ φ3 ≤ 1 - measures the efficacy of ART treatment, including uptake with proper adherence,with the aim of reducing the plasma viral load and reconstructing the individual’s immune systemhence making them less infectious.Movement of individuals from the susceptible to infected and then to the AIDS classes is illustratedby the compartmental model shown in Figure 1.

SH

SL A

TD

I

��

(� − �)�

(� − ��)(� − ��)���

(� − ��)���

�SH

�SL

���

(� + �)�

�I

��

���

(� − ��)��

Figure 1. Compartmental Model.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 4
The following symbols will be used to represent various phenomena as described in Table 1.

Symbol Description
Λ constant rate of recruitment of susceptible upon becoming sexually active.
δ proportion of susceptible individuals at high risk of infection.
(1 −δ) proportion of susceptible population at low risk of HIV infection.
λ

rate of acquisition of an infection by susceptibles.It is given by;
λ = (

β1I+β2TD
N

), where β1, and β2 are the mean contact rates for thesusceptible individuals with I and TD respectively.
µ natural removal rate by death.
σ aids induced mortality.
α

represents the proportion of infected individuals who upon beingtested and found to be HIV positive,they enroll for ART treatment.
γ2

represents the proportion of infected individuals who do not get testedhence remain undiagnosed until they begin to exhibit AIDS symptoms.Table 1. Table showing symbols and their description
From the dynamics described above, the following system of ordinary differential equations isformulated.

dSH
dt

= δΛ − (1 −φ1)(1 −φ2)λSH −µSH
dSL
dt

= (1 −δ)Λ − (1 −φ2)λSL −µSL
dI

dt
= (1 −φ1)(1 −φ2)λSH + (1 −φ2)λSL −αI −γ2I −µI

dTD
dt

= αI − (γ3 + µ)TD
dA

dt
= γ2I + γ3TD − (µ + σ)A. (2)

3. Model Analysis
It can be shown that the solutions for the system of ordinary differential equations (2) areall positive and bounded for all t > 0, with positive initial conditions in the feasible region Γ ={

(SH(t),SL(t), I(t), (TD(t),A(t)) ∈R5+ : N(t) ≤
Λ

µ

}
. It therefore suffices to study the dynamicsof the system (2) in this region.The mathematical model developed in (2) has two unique equilibrium points, that is, the DiseaseFree Equilibrium (D.F.E), and the Endemic Equilibrium (E.E). The D.F.E is obtained by setting

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 5
I = TD = A = 0 in (2) to yield

E0 =

(
δΛ

µ
,

(1 −δ)Λ
µ

, 0, 0, 0

)
. (3)

The basic reproduction number (R0) of the system (2), computed using the next generation matrixapproach [10] is given by
R0 =

β3
Q1

+
αβ4
Q1Q2

. (4)
By [10, Theorem 2], the following result is thus established.

Theorem 3.1. The Disease Free Equilibrium of the model (2), E0 = (δΛµ , (1−δ)Λµ , 0, 0, 0), is locally
asymptotically stable whenever R0 < 1 and unstable otherwise.

Proof. The proof follows immediately from the computation of R0 above and Theorem 2 of Van denDriessche and Watmough [10]. �
Mathematically, Theorem (3.1) implies that whenever there is a small perturbation on the system,the system returns to the disease free equilibrium. Epidemiologically, this implies that when a fewHIV infectious individuals are introduced in a population that is fully susceptible to HIV infection,the disease dies out whenever R0 < 1, otherwise, the disease will spread. It is therefore necessaryto show that eliminating HIV in a population is independent of the size of the initial sub-populationby proving the global asymptotic stability of the disease free equilibrium.
Theorem 3.2. The Disease Free Equilibrium E0 =

(
δΛ
µ
,

(1−δ)Λ
µ

, 0, 0, 0
)

of the system (2) is globally
asymptotically stable whenever R0 < 1.

Proof. Castillo Chavez’s theorem [1] is used to analyze the global asymptotic stability of the math-ematical model (2) such that E0 = (X∗, 0),
X = (SH,SL) and Z = (I,TD,A).Now;
F (X,0) =

(
δΛ −µSH

(1 −δ)Λ −µSL

) and G(X,Z) = PZ − G̃(X,Z).
Matrix P is given by

h1β1SH
N

+
(1 −φ2)β1SL

N
− (α + γ2 + µ)

h1β2SH
N

+
(1 −φ2)β2SL

N
0

α −(γ3 + µ) 0
γ2 γ3 −(σ + µ)

 ,
where h1 = (1 −φ1)(1 −φ2), and PZ is given by

h1β1ISH
N

+
(1 −φ2)β1ISL

N
− (α + γ2 + µ) +

h1β2TDSH
N

+
(1 −φ2)β2TDSL

N
αI − (γ3 + µ)TD

γ2I + γ3TD − (σ + µ)A

 .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 6
Moreover, G(X, Z) is given by

(1 −φ1)(1 −φ2)
(
β1I + β2TD

N

)
SH + (1 −φ2)

(
β1I + β2TD

N

)
SL − (α + γ2 + µ)I

αI − (γ3 + µ)TD
γ2I + γ3TD − (σ + µ)A

 ,

and therefore G̃(X,Z) = PZ − G(X,Z)=

G̃1(X,Z)
G̃2(X,Z)
G̃3(X,Z)

=


0
0
0

 . Hence conditions H1 and H2 are
satisfied. Also from Theorem (3.1), E0 is locally asymptotically stable whenever R0 < 1. Thereforefollowing Castillo Chavez’s theorem, E0 is globally asymptotically stable whenever R0 < 1, asdesired. �
This implies that with a large perturbation of the disease free equilibrium, solutions of the modelrepresented by the system (3.2) converge to D.F.E whenever R0 < 1. Epidemiologically, this impliesthat if a sufficiently large number of HIV infected individuals are introduced in a population thatis fully susceptible to HIV infection, the disease will die out whenever R0 < 1.
3.1. Existence of the Endemic Steady State.
Theorem 3.3. An endemic equilibrium point E1 = (S∗∗H ,S

∗∗
L , I

∗∗,T∗∗D ,A
∗∗), of the system (2) exists

whenever R0 > 1.

Proof. Equating the right hand side of each equation in the system (2) to zero and simplifyingyields;
δΛ − (1 −φ1)(1 −φ2)

(
β1I
∗∗ + β2T

∗∗
D

N

)
S∗∗H −µS

∗∗
H = 0, (5)

(1 −δ)Λ − (1 −φ2)
(
β1I
∗∗ + β2T

∗∗
D

N

)
S∗∗L −µS

∗∗
L = 0, (6)

(1 −φ1)(1 −φ2)
(
β1I
∗∗ + β2T

∗∗
D

N

)
S∗∗H +

(1 −φ2)
(
β1I
∗∗ + β2T

∗∗
D

N

)
S∗∗L − (α + γ2 + µ)I

∗∗ = 0, (7)
αI∗∗ − (γ3 + µ)T∗∗D = 0, (8)

γ2I
∗∗ + γ3T

∗∗
D − (µ + σ)A

∗∗ = 0. (9)
From equation (8), T∗∗D = αQ2 I∗∗.Substituting for T∗∗tD in equation (9) and simplifying gives A∗∗ = ( γ2Q3 + αγ3Q2Q3 ) I∗∗.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 7
Using equation (5) and substituting T∗∗D gives

δΛN − (1 −φ1)(1 −φ2)
(
β1 +

β2α

Q2

)
I∗∗S∗∗H −µNS

∗∗
H = 0

⇒ S∗∗H =
δΛN

a1I
∗∗ + µN

,

where a1 = (1 − φ1)(1 − φ2) (β1 + β2α
Q2

). In a similar manner, S∗∗L is expressed as S∗∗L =
(1 −δ)ΛN
a2I
∗ + µN

, where a2 = (1 −φ2) (β1 + β2α
Q2

).Using equation (7) and substituting for S∗∗H and S∗∗L ,we obtain
a1I
∗∗δΛ

a1I
∗∗ + µN∗∗

+
a2I
∗∗(1 −δ)Λ

a2I
∗∗ + µN∗∗

−Q1I∗∗ = 0 (10)
Thus from equation (10),(

a1δΛ

a1I
∗∗ + µN∗∗

+
a2(1 −δ)Λ
a2I
∗∗ + µN∗∗

−Q1
)
I∗∗ = 0. (11)

From equation (11), I∗∗ = 0 corresponds to the disease free equilibrium point of the system (2),denoted by (E0). The other solution of (11) when I∗∗ 6= 0 corresponds to the endemic equilibriumpoint of the system such that,
a1δΛ

a1I
∗∗ + µN∗∗

+
a2(1 −δ)Λ
a2I
∗∗ + µN∗∗

−Q1 = 0. (12)
Multiplying through by (a1I∗∗ + µN∗∗)(a2I∗∗ + µN∗∗) yields

CI∗∗2 + DI∗∗ + E = 0. (13)
where: C = −Q1a1a2,D = (a1a2δΛ + a1a2(1 −δ)Λ) − (Q1a1µN + Q1a2µN), and E = a1δΛµN +
a2(1 −δ)ΛµN −Q1µNµN.The endemic equilibrium of the system exists if the roots of equation (13) are real and positive.Descarte’s rule of signs is used to check the possible number of real roots of the polynomial. Thenumber of positive real roots is equal to the number of sign changes in the coefficients of theterms of a polynomial [15]. Considering that all the parameters used are positive, the sign of C isnegative. The sign of E is then checked as follows;

E = a1δΛµN + a2(1 −δ)ΛµN −Q1µNµN

= (1 −φ1)(1 −φ2)
(
β1 +

β2α

Q2

)
δΛµN +

(1 −φ2)
(
β1 +

β2α

Q2

)
(1 −δ)ΛµN −Q1µNµN

= (1 −φ1)(1 −φ2)(β1 + β2α)δΛµN +

(1 −φ2)(β1 + β2α)(1 −δ)ΛµN −Q1Q2µNµN

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 8
Using R0 = β3

Q1
+

αβ4
Q1Q2

and the limiting value of N = Λ
µ

, we obtain E = (R0 − 1)Λ2. Thus E >0iff R0 > 1. Since C is negative, and E is positive, we see that there is at least one sign changeregardless of the sign of D. This implies that equation (13) has at least one positive real root.Hence an endemic equilibrium point of the system (2) exists whenever R0 > 1. �
3.2. Local Stability of the Endemic Equilibrium. At the endemic equilibrium, there is persistenceof HIV infection in the population.
Theorem 3.4. The endemic equilibrium point E1 = (S∗∗H ,S

∗∗
L , I

∗∗,T∗∗D ,A
∗∗) of system (2) is locally

asymptotically stable if R0 > 1.

Proof. The Jacobian matrix of the system (2) evaluated at endemic equilibrium is

J(E1) =


−b1 0 −b2 −b3 0

0 −b4 −b5 −b6 0
b7 b8 b9 −Q1 b10 0
0 0 α −Q2 0
0 0 γ2 γ3 −Q3


where
b1 =

(1−φ1)(1−φ2)(β1Q2+β2α)µI∗∗+Q2µΛ
Q2Λ

,b2 =
(1−φ1)(1−φ2)β1a1δΛI∗∗

a1I∗∗+Λ
,b3 =

(1−φ1)(1−φ2)β2a1δΛI∗∗
a1I∗∗+Λ

b4 =
(1−φ2)(β1Q2+β2α)µI∗∗+Q2µΛ

Q2Λ
,b5 =

(1−φ2)(1−δ)Λβ1a2I∗∗
a2I∗∗+Λ

,b6 =
(1−φ2)(1−δ)Λβ2a2I∗∗

a2I∗∗+Λ

b7 =
(1−φ1)(1−φ2)(β1Q2+β2α)µI∗∗+Q2µΛ

Q2Λ
,b8 =

(1−φ2)(β1Q2+β2α)µI∗∗+Q2µΛ
Q2Λ

b9 =
(1−φ1)(1−φ2)β1a1δΛI∗∗

a1I∗∗+Λ
+

(1−φ2)(1−δ)Λβ1a2I∗∗
a2I∗∗+Λ

,b10 =
(1−φ1)(1−φ2)β2a1δΛI∗∗

a1I∗∗+Λ
+

(1−φ2)(1−δ)Λβ2a2I∗∗
a2I∗∗+ΛClearly, −Q3 is an eigenvalue of the Jacobian matrix J(E1). The other eigenvalues can be computedby finding the solution to the equation

P (λ) =

∣∣∣∣∣∣∣∣∣∣∣
λ + b1 0 −b2 −b3

0 λ + b4 −b5 −b6
b7 b8 λ− (b9 + Q1) b10
0 0 α λ + Q2

∣∣∣∣∣∣∣∣∣∣∣
=0

The characteristic equation of J(E1)is then given by;
P (λ) = λ4 + c0λ

3 + c1λ
2 + c2λ + c3 = 0 (14)

where;
c0 = b1 + b4 −b9 −Q1 + Q2
c1 = b1b4 + b2b7 + b5b8 −b1b9 −b4b9 −αb10 −b1Q1 −b4Q1 + b1Q2 + b4Q2 −b9Q2 −Q1Q2
c2 = −αb3b7 +b2b4b7 +b1b5b8−αb6b8−b1b4b9−alphab1b10−alphab4b10−b1b4Q1 +b1b4Q2 +
b2b7Q2 + b5b8Q2 −b1b9Q2 −b4b9Q2 −b1Q1Q2 −b4Q1Q2
c3 = −αb3b4b7 −αb1b6b8 −αb1b4b10 + b2b4b7Q2 + b1b5b8Q2 −b1b4b9Q2 −b1b4Q1Q2The number of negative zeros of equation (14) depends on the signs of c0,c1,c2 and c3. Descarte’s

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 9
Rule of Signs is applied to study the number of negative real roots of the polynomialP (λ1) com-prising of the coefficients c0,c1,c2 and c3 given by;

P (λ1) = c0λ
3 + c1λ

2 + c2λ + c3 = 0 (15)
Descarte’s rule of signs states that the number of negative real zeros of P (λ) is either equal tothe variations in sign of P (−λ) or less than this by an even number [15]. The possibilities ofnegative real zeros of P (λ), is as summarized in Table 2. The maximum number of variationsof signs in P (−λ) is 3, hence the characteristic polynomial (15) has three negative roots. Thus
P (−λ) = λ4 − c0λ3 + c1λ2 − c2λ + c3 = 0 has negative roots.Therefore, given that cases 1-8 inTable 1 are satisfied, model (2) is locally asymptotically stable if R0 > 1. �

Table 2. The Zeros of the characteristic equation (14)
Cases c0 c1 c2 c3 R0 > 1 Sign Change No. of - Roots1 + − − + R0 > 1 2 2,02 + − + + R0 > 1 2 2,03 − − + − R0 > 1 2 2,04 + + − − R0 > 1 1 05 − − + + R0 > 1 1 06 + + + − R0 > 1 1 07 − + − + R0 > 1 3 3,18 − − − − R0 > 1 0 0

This implies that for a small pertubation of the E1, solutions of the mathematical model representedby the system (2) always converge to E1, whenever R0 > 1. Epidemiologically, it implies that ifa few HIV infected individuals are introduces in a fully susceptible population, the disease willpersist provided R0 > 1.
4. Sensitivity Analysis

In mathematical modeling, Sensitivity refers to the degree to which a given input parameterin a mathematical model influences its output. Sensitive parameters are thus those that cause asignificant impact on the disease transmission dynamics. Sensitivity analysis will aid in identify-ing the parameters which greatly impact on the value of the basic reproductive number R0, andhence ought to be targeted when coming up with intervention strategies. The sensitivity of modelparameters is calculated using the normalized forward sensitivity index. The normalized forwardsensitivity index of the basic reproductive number is given by SR0w = ∂R0
∂w
×
w

R0
, where w is the

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 10
parameter whose sensitivity is to be determined [7]. R0 is given by

R0 =
(1 −φ1)(1 −φ2)β1δ + (1 −φ2)(1 −δ)β1

α + γ2 + µ

+
α(1 −φ1)(1 −φ2)β2δ + (1 −φ2)(1 −δ)β2

(α + γ2 + µ)(γ3 + µ)
. (16)

Forβ1,SR0β1 = β1(γ3 + µ)β1(γ3 + µ) + αβ2 .
Forβ2,SR0β2 = αβ2β1(γ3 + µ) + αβ2 .
For α,SR0α = [β2(α + γ2 + µ) − (β1(γ3 + µ) + αβ2)]α(α + γ2 + µ)(β1(γ3 + µ) + αβ2) .Forγ2,SR0γ2 = (αγ2 + γ22 + µγ2) ln |α + γ2 + µ|.
Forγ3,SR0γ3 = −αβ2γ3(β1(γ3 + µ)2 + αβ2(γ3 + µ).
For δ,SR0

δ
=

−φ1δ
1 −δφ1

For µ,SR0µ = [(α + γ2 + µ)(γ3 + µ)β1 + ((γ3 + µ)β1 + αβ2)(α + γ2 + γ3 + 2µ)]µ(α + γ2 + µ)(γ3 + µ)((β1(γ3 + µ) + αβ2)) .
Based on the sensitivity indices in Table 3, the most sensitive parameter to the value of R0 is β1,

Table 3. Sensitivity Indices for the Model Parameters
Parameter Description Sensitivity Index

δ Proportion of high risk sussceptibles −0.36986
φ1 PreP effectiveness −0.041095
φ2 Condom effectiveness −0.11111
γ3 ART Failure −0.27182
β1 Mean contact rate with I 0.72345
β2 Mean contact rate with TD 0.27654
α Progression from I to TD −0.40225
γ2 Progression from I to A −0.05123
µ Natural mortality rate −0.02969

the mean contact rate with undiagnosed infectives. This implies that in order to control the spreadof HIV in a high risk population, efforts should be geared towards reducing the number of thosewho are undiagnosed by testing them and enrolling them on ART treatment. This in turn lowerstheir infectivity as well as chances of progressing to the AIDS class.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.21 11
5. Conclusion

In this study, a mathematical model is formulated, based on a system of ordinary differentialequations, incorporating the impact of dual protection and ART adherence in preventing the spreadof HIV among persons at high risk of infection.Stability analysis of the model was done and depicted that when R0 < 1, the disease freeequilibrium is both locally and globally asymptotically stable. The Endemic Equilibrium of themathematical model exists and was shown to be locally asymptotically stable whenever R0 > 1,implying that there is persistence of HIV infection in the population provided that R0 is greaterthan unity. Sensitivity analysis was conducted, depicting that the most sensitive parameter is β1,the mean contact rate with the un-diagnosed infectives. Therefore, in order to control the spreadof HIV among the high risk population, efforts ought to be channeled towards the undiagnosedpopulation by frequently testing and enrolling them on ART treatmentwhich guarantees low viral load within the infected individual, making them less infective. ThusDual protection and ART adherence are essential in the fight against the spread of HIV among thehigh risk population.
References

[1] C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of R0 and its role on global stability, in: C. Castillo-Chavez, S. Blower, P. Van Den Driessche, D. Kirschner, A.-A. Yakubu (Eds.), Mathematical Approaches for Emergingand Reemerging Infectious Diseases: An Introduction, Springer New York, New York, NY, 2002: pp. 229-250.
https://doi.org/10.1007/978-1-4757-3667-0_13.[2] S.C. Weller, K. Davis-Beaty, Condom effectiveness in reducing heterosexual HIV transmission, Cochrane DatabaseSyst. Rev. 2012 (2002). https://doi.org/10.1002/14651858.cd003255.[3] D.K. Smith, J.H. Herbst, X. Zhang, C.E. Rose, Condom effectiveness for hiv prevention by consistency of use amongmen who have sex with men in the United States, JAIDS J. Acq. Immune Defic. Syndr. 68 (2015) 337-344. https:
//doi.org/10.1097/qai.0000000000000461.[4] E.O. Omondi, R.W. Mbogo, L.S. Luboobi, Mathematical modelling of the impact of testing, treatment and control ofHIV transmission in Kenya, Cogent Math. Stat. 5 (2018) 1475590. https://doi.org/10.1080/25742558.2018.
1475590.[5] F.K. Tireito, G.O. Lawi, O.A. Colleta, Mathematical analysis of HIV/AIDS anti-retroviral treatment incorporatingadherence, Asian Res. J. Math. 10 (2018) 1-13. https://doi.org/10.9734/ARJOM/2018/42830.[6] I. Cremin, R. Alsallaq, M. Dybul, P. Piot, G. Garnett, T.B. Hallett, The new role of antiretrovirals in combinationHIV prevention: a mathematical modelling analysis, AIDS. 27 (2013) 447-458. https://doi.org/10.1097/qad.
0b013e32835ca2dd.[7] J.C. Helton, R.L. Iman, J.B. Brown, Sensitivity analysis of the asymptotic behavior of a model for the environmentalmovement of radionuclides, Ecol. Model. 28 (1985) 243-278. https://doi.org/10.1016/0304-3800(85)90077-8.[8] J.M. Baeten, J.E. Haberer, A.Y. Liu, N. Sista. Pre-exposure prophylaxis for HIV prevention: where have we beenand where are we going? J. Acq. Immune Defic. Syndr. 63 (2013) 122-129. https://doi.org/10.1097/QAI.
0b013e3182986f69.[9] J.M. Molina. PrEP failures: diagnosis, resistance, and treatment. Conference on Retroviruses and OpportunisticInfections. 4-7 March 2019. https://www.croiwebcasts.org/p/2019croi/160.

https://doi.org/10.28924/ada/ma.3.21
https://doi.org/10.1007/978-1-4757-3667-0_13
https://doi.org/10.1002/14651858.cd003255
https://doi.org/10.1097/qai.0000000000000461
https://doi.org/10.1097/qai.0000000000000461
https://doi.org/10.1080/25742558.2018.1475590
https://doi.org/10.1080/25742558.2018.1475590
https://doi.org/10.9734/ARJOM/2018/42830
https://doi.org/10.1097/qad.0b013e32835ca2dd
https://doi.org/10.1097/qad.0b013e32835ca2dd
https://doi.org/10.1016/0304-3800(85)90077-8
https://doi.org/10.1097/QAI.0b013e3182986f69
https://doi.org/10.1097/QAI.0b013e3182986f69
https://www.croiwebcasts.org/p/2019croi/160


Eur. J. Math. Anal. 10.28924/ada/ma.3.21 12
[10] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmentalmodels of disease transmission, Math. Biosci. 180 (2002) 29-48. https://doi.org/10.1016/s0025-5564(02)

00108-6.[11] K.R. Amico, M.J. Stirratt, Adherence to preexposure prophylaxis: current, emerging, and anticipated bases of evi-dence, Clin. Infect. Dis. 59 (2014) S55-S60. https://doi.org/10.1093/cid/ciu266.[12] T.T. Yusuf, F. Benyah, Optimal strategy for controlling the spread of HIV/AIDS disease: a case study of SouthAfrica, J. Biol. Dyn. 6 (2012) 475-494. https://doi.org/10.1080/17513758.2011.628700.[13] UNAIDS. Global HIV and AIDS statistics: 2019 Fact Sheet: https://www.unaids.org/en/resources/
fact-sheet[14] U.S Food and Drug Administration (F.D.A).U.S F.D.A Approves First Drug for Reducing the Riskof Sexually Acquired H.I.V InfectionJuly. 2011. https://aidsinfo.nih.gov/news/1254/fda-approves-{\@
@par}rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection[15] X. Wang, A simple proof of Descartes’s rule of signs, Amer. Math. Mon. 111 (2004) 525-526. https://doi.org/
10.2307/4145072.

https://doi.org/10.28924/ada/ma.3.21
https://doi.org/10.1016/s0025-5564(02)00108-6
https://doi.org/10.1016/s0025-5564(02)00108-6
https://doi.org/10.1093/cid/ciu266
https://doi.org/10.1080/17513758.2011.628700
https://www.unaids.org/en/resources/fact-sheet
https://www.unaids.org/en/resources/fact-sheet
https://aidsinfo.nih.gov/news/1254/fda-approves-{\@@par }rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection
https://aidsinfo.nih.gov/news/1254/fda-approves-{\@@par }rst-drug-for-reducing-the-risk-of-sexually-acquired-hiv-infection
https://doi.org/10.2307/4145072
https://doi.org/10.2307/4145072

	1. Introduction
	2. Model Formulation and Description
	3. Model Analysis
	3.1. Existence of the Endemic Steady State
	3.2. Local Stability of the Endemic Equilibrium

	4. Sensitivity Analysis
	5. Conclusion
	References

