F1000ResearchF1000Research2046-1402F1000 Research LimitedLondon, UK10.12688/f1000research.72865.1Research ArticleArticlesThe impact of testing and treatment on the dynamics of Hepatitis B virus
Despite the intervention of WHO on vaccination for reducing the spread of Hepatitis B Virus (HBV), there are records of the high prevalence of HBV in some regions. In this paper, a mathematical model was formulated to analyze the acquisition and transmission process of the virus with the view of identifying the possible way of reducing the menace and mitigating the risk of the virus. The models' positivity and boundedness were demonstrated using well-known theorems. Equating the differential equations to zero demonstrates the equilibria of the solutions i.e., the disease-free and endemic equilibrium. The next Generation Matrix method was used to compute the basic reproduction number for the models. Local and global stabilities of the models were shown via linearization and Lyapunov function methods respectively. The importance of testing and treatment on the dynamics of HBV were fully discussed in this paper. It was discovered that testing at the acute stage of the virus and chronic unaware state helps in better management of the virus.
: Positivity and boundedness of solutionsEquilibria of solutionsNext generation matrixLinearizationLyapunov functionslocal and global stabilities.There is no funding for this researchThe author(s) declared that no grants were involved in supporting this work.1. Introduction
Hepatitis is an inflammation/scarring of the liver that contributes to various health complications, including death. It occurs due to an immune system attack by the virus in the liver and damages this vital organ of the body in the process (
Ciupe
et al., 2014). The hepatitis B virus (HBV) can survive outside the body for at least seven days. If the virus enters the body of someone who is not protected by vaccination during this time, it can still cause infection. The average incubation time for HBV is 75 days, but it can range from 30 to 180 days. Within 30 to 60 days of infection, the virus may be detected, persist, and grow into chronic hepatitis B (
CDC, 2019). Hepatitis B is most common in the Western Pacific region with prevalence rate 6.2% and Africa with prevalence rate of 6.1%, with the Americas region (0.7%) having the lowest prevalence (
WHO, 2019).
In highly endemic areas, the most common form of transmission of hepatitis B is from mother to child at birth (vertical transmission) or through horizontal route (contact with infected blood), particularly from infected children to uninfected children during the first five years of life. Chronic infection develops in infants infected by their mothers or before the age of five. It is often transmitted through transdermal or mucosal contact of infected persons to infected blood and different body fluids, such as spittle, catamenial, vaginal and spermatic fluids and, to a lesser degree, perspiration, breast milk, tears, and urine. In particular, hepatitis B can be transmitted through sexual contact in unvaccinated men who have sex with men (MSM) and heterosexual people who have multiple sexual partners or have contact with sex workers. However, adult infection contributes to chronic hepatitis in less than 5% of cases. This transmission may similarly ensue when needles and syringes are reused, whether in healthcare settings or among drug users. Furthermore, an infection can occur during medical, surgical, and dental procedures, such as tattooing or using razors and other similar objects contaminated with infected blood (
Mpeshe and Nyerere, 2019).
2. Mathematical formulation
Some chronic carriers are unaware of their status and as such transmit the virus unknowingly and also at higher risk of cirrhosis and makes treatment less effective (
Niederau, 2014,
Mcpherson
et al., 2013,
Cohen
et al., 2011,
Piorkowsky, 2009,
Lin
et al., 2009,
Meffre
et al., 2004).
In view of this, this model is developed to factor the aforementioned set of people. In the model, the population is divided into the following different groups: the susceptible, the acute, the chronic unaware carriers, the chronic aware carriers, the treated chronic aware and the recovered individuals.
The total population at time
t, denoted by
Nt is divided into the six subgroups corresponding to different epidemiological status: susceptible individuals
St, acute
At, unaware chronically infected
Cut, aware chronically infected
Cat, treated
Tct, and removed/recovered class
Rt. The model equation is subject to the initial conditions,
St≥0,At≥0,Cut≥0,Cat≥0,Tct≥0,Rt≥0
Figure 1 represents schematically the epidemiology of HBV infected model. The different disease stages are reproduced by the different circle and the arrows indicate the way individual progress from one stage to the other. It is assumed that at time,
t, susceptible individuals,
S, enter the population at a constant rate,
Π. For all classes, individuals die at a constant natural mortality rate,
μ.HBV chronically infected individuals
(Cut,
Cat)have an additional death rate due to HBV,
ⅆc (Zhang and Zhang (2018)). It is assumed that HBV infected individuals on treatment,
Tct do not transmit HBV infection. Susceptible individuals,
St, may acquire HBV infection when in contact with individuals in A
,Cu, and
Ca,populace at a rate,
λ (force of infection associated with HBV), where
λ=βA+α1Cu+α2CaN
Compartmental flow diagram of HBV model.
Parameter
β represents the probability that a contact will result in an HBV infection while
α1,α2>1 respectively account for modification parameter of chronic HBV-infected individuals.
A proportion of the acute HBV-infected individuals,
σ, spontaneously clear the virus, then return to being susceptible. The HBV acutely infected individuals develop to chronic without been aware if no testing at a rate,
γ. The acutely infected and chronic unaware individual progress to chronic aware stage with a testing
ν1,ν2 respectively and moved to treatment stage after testing at the rate
δ.
ω is the recovery rate of treated infected individual with full immunity.
These assumptions lead to the system of equations in (3)
dSdt=Π−λS+σA−μSdAdt=λS−(σ+γ+ν1)AdCudt=γA−(ν2+μ+dc)CudCadt=ν2Cu+ν1A−(δ+μ+dc)CadTcdt=δCa−(ω+μ)TcdRdt=ωTc−μR
where
λ=βA+α1Cu+α2CaN
2.1 Positivity and boundedness of solutions
For the system of equations
(3) to be epidemiologically meaningful, it is important to prove that all solution with non-negative initial conditions will remain non-negative.
Lemma 1: The initial values of the parameters are
S0≥0A0≥0Cu0≥0Ca0≥0Tc0≥0R0≥0andN0≥0∈Φ
Then the solution of the model
StAtCutCatTctRtNt is positive for all
t≥0.
Proof
Considering the first equation in
(3),
dSdt=Π−λS+σA−μSdSdt≥−λ+μS∫1SdS≥∫−λ+μdtS≥S0e−λ+μt≥0
Hence,
S≥0
with respect to the second equation in (3);
dAdt=λS−σ+γ+ν1AdAdt≥−σ+γ+ν1A∫1AdA≥∫−σ+γ+ν1dtA≥A0e−σ+γ+ν1t≥0
Hence,
A≥0. Same goes for the other compartments
Clearly, the above state variables are positive on bounding plane
ℝ+6.
For the boundedness the following calculation follows:
Nt=St+At+Cut+Cat+Tct+RtN′=S′+A′+Cu′+Ca′+Tc′+R′N′=Π−λS+σA−μS+λS−σ+γ+ν1A+γA−ν2+μ+dcCu+ν2Cu+ν1A−δ+μ+dcCa+δCa−ω+μTc+ωTc-μR
It follows that the solutions of the model system (3) are positive and bounded in the region
T=S+A+Cu+Ca+Tc+R∈ℝ+6:S+A+Cu+Ca+Tc+R≤Πμ
It follows from Lemma 1 that it is sufficient to consider the dynamics of system (3) and the model can be considered to be epidemiologically well-posed.
2.2 Equilibrium points
The disease-free equilibrium of the
equation (3) exists and is given by:
E0=Πμ00000
The endemic steady states are calculated here which is done by setting system of equation in (3.3.3) to zero and setting
S=S∗,A=A∗,Cu=Cu∗,Ca=Ca∗,Tc=Tc∗,R=R∗ so that
S∗=Π(μ3+δ+γ+2dc+υ1+υ2μ2+(dc2+υ1+υ2+δ+γdc+υ2+δγ+υ2+δυ1+δυ2)μ)+δυ1dc+υ2υ1+γυ1+σ+γLA∗=−υ1−β+γ+σdc2+2υ1−2β+2γ+2σμ+υ2+δ−βα1γ+υ2+δ−βα2υ1−β−συ2+δdc+υ1−β+γ+σμ2+υ2+δ−βα1γ+υ2+δ−βα2υ1−β−συ2+δμ+−δα1−α2υ2βΠLCu∗=A∗γCa∗=Cu∗((υ1+υ1μ)+υ2υ1+γ)γTc∗=−Ca∗μ2dc2+μdc+υ2μ2+1υ1+υ2δR∗=ωTc∗
where
L=υ2+μ+dcυ1+γ+σμ3+2υ1+2γ+2σdc+υ1+γ+συ2+β+δγ+β+δυ1+δσμ2+υ1+γ+σdc2+υ1+γ+συ2−γ2+2β+δ−σ−2υ1γ−υ12+2β+δ−συ1+δσdc+β+δγ+β+δυ1+δσυ2+βυ1+γγα1+α2υ1+δμ−υ1+γυ1−β+γ+σdc2+−υ1+γυ1−β+γ+συ2+βα1−δγ2+−δ+α1+α2βυ1+δβ−σγ+δ+α2υ1βυ1dc+βγα1+α2υ1+δυ2+δγα1)υ1+γδ+μ+dcμ+ω
2.3 Basic reproduction number
The basic reproduction number (
Ro) which is the number of secondary infections caused by an infectious individual is determined by the next generation matrix which is given by
ρFV−1
2.4 Local stability analysis of the disease-free equilibrium
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<mml:math display="inline">
<mml:msub>
<mml:mi>E</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:math>
</inline-formula>
Theorem 1:
E0 is locally asymptotically stable if
R0 < 1 and unstable if
R0 > 1.
Proof: The resulting matrix from the linearized model is
dXdt=AXX=x1x2x3x4x5,x6T,x1x2x3x4x5,x6∈R+6,and
The resulting Jacobian matrix at
E0 is
JE0=−μ−λ−β+σ−βα1−βα2000β−σ−γ−ν1−λβα1βα2000γ−dc−μ−ν2−λ0000ν1ν2−dc−μ−δ−λ00000δ−ω−μ−λ00000ω−μ−λ
From (15),
λ1=−μ,λ2=−ω−μ,λ3=−μ
and the resulting quadratic equation is:
β−σ−γ−ν1−λ−dc−μ−ν2−λ−dc−μ−δ−λ−βα1−dc−μ−δ−λγ+βα2−dc−μ−ν2−λν1−γν2fλ=λ3+2μ+ν1+ν2−β+δ+γ+σ+2dcλ2+βα2ν1−βδ−2βμ−2βdc−βν2+δγ+δμ+δσ+δdc+δν1+δν2+2γμ+2γdc+γν2+μ2+2μσ+2μdc+2μν1+μν2+2σdc+σν2+dc2+2dcν1+dcν2+ν1ν2−γβα1λ+γμ2+γdc2+μ2σ+μ2ν1+σdc2+dc2ν1+μσν2+2μdcν1+μν1ν2+σdcν2+dcν1ν2−βδμ−βδdc−βδν2−2βμdc−βμν2−βdcν2−βμ2−βdc2+δγμ+δγdc+δγν2+δμσ+δμν1+δσdc+δσν2+δdcν1+δν1ν2+2γμdc+γμν2+γdcν2+2μσdc−βδγα1−βγμα1−βγα1dc+ν2γβα2+ν1βα2dc+ν1βα2μ+ν1βα2ν2
Now,
λ1,
λ2,λ3 < 0 since the values are assumed positive. If
R0 < 1,
E0 is stable and unstable when
R0 < 1.
2.5 Global stability of the disease-free equilibrium
The global behavior of the equilibrium system (3) is analyzed here in this section.
Theorem 2: For system (3), the disease-free equilibrium
E0 is asymptotically stable globally if
R0<1.
Proof: Considering the Lyapunov function defined as:
GACuCa=1B0A+βα1B0B1+βα2υ2B0B1B2Cu+βα2B0B2CaG′ACuCa=1B0A′+βα1B0B1+βα2υ2B0B1B2Cu′+βα2B0B2Ca′G′ACuCa=1B0βA+α1Cu+α2CaNS−σ+γ+ν1A+βα1B0B1+βα2υ2B0B1B2γA−ν2+μ+dcCu+βα2B0B2ν2Cu+ν1A−δ+μ+dcCa
At DFE, S=N so that (20) becomes:
G′ACuCa=1B0βA+α1Cu+α2Ca−σ+γ+ν1A+βα1B0B1+βα2υ2B0B1B2γA−ν2+μ+dcCu+βα2B0B2ν2Cu+ν1A−δ+μ+dcCa
Expanding and simplifying (21) gives:
G′=βB0+βα1γB0B1+βα2υ2γB0B1B2+βα2ν1B0B2−1A+βα1B0−βα1B1B0B1−βα2υ2B1B0B1B2+βα2ν2B0B2Cu+βα2B0−βα2B2B0B2CaG′=R0−1A≤0
From
Equation (23), it can be deduced that the DFE is globally stable since
R0 < 1.
2.6 Local stability of endemic equilibrium
Theorem 3: If
R0>1, then the endemic equilibrium is locally asymptotically stable.
Proof:
The endemic equilibria of system (3), denoted by
S∗A∗Cu∗Ca∗Tc∗R∗, can be rewritten as:
LetS=x+S∗,A=y+A∗,Cu=z+Cu∗,Ca=h+Ca∗,Tc=p+Tc∗,R=j+R∗J=B0−μ−λ−B1+σ−B2−B3B4B5−B6B7−σ−γ−ν1−λB8B9−B11−B120γ−dc−μ−ν2−λ0000ν1ν2−dc−μ−δ−λ00000δ−ω−μ−λ00000ω−μ−λ
From (25),
λ1=−μ,λ2=−ω+μ,λ3=−dc+μ+ν2, then;
J=B0−μ−λ−B1+σ−B2−B6B7−σ−γ−ν1−λB80γ−dc−μ−ν2−λ
from (26);
λ3+γ+2μ+σ−B0−B4+dc+ν1+ν2λ2+2γμ−B0γ+γdc+γν2+μ2+2μσ−μB0−2μB4+μdc+2μν1+μν2−σB0+σB3+σdc+σν2+B0B4−B0dc−B0ν1−B0ν2−B3B1−B4dc−B4ν2+dcν1+ν1ν2λ+B5γ+γμν2+μσν2+μν1ν2+B3B2γ+γμ2+μ2σ+μ2ν1+μdcν1+γμdc+μσdc−μ2B4−γμB0−γB0dc−γB0ν2−μσB0+μσB3+μB0B4−μB0ν1−μB3B1−μB4dc−μB4ν2−σB0dc−σB0ν2+σB3dc+σB3ν2+B0B4dc+B0B4ν2−B0dcν1−B0ν1ν2−B3B1dc−B3B1ν2
The result of the determinant of the Jacobian matrix is of the form:
a0λ3+a1λ2+a2λ+a3
where
a0=1a1=γ+2μ+σ−B0−B4+dc+ν1+ν2a2=2γμ−B0γ+γdc+γν2+μ2+2μσ−μB0−2μB4+μdc+2μν1+μν2−σB0+σB3+σdc+σν2+B0B4−B0dc−B0ν1−B0ν2−B3B1−B4dc−B4ν2+dcν1+ν1ν2a3=B5γ+γμν2+μσν2+μν1ν2+B3B2γ+γμ2+μ2σ+μ2ν1+μdcν1+γμdc+μσdc−μ2B4−γμB0−γB0dc−γB0ν2−μσB0+μσB3+μB0B4−μB0ν1−μB3B1−μB4dc−μB4ν2−σB0dc−σB0ν2+σB3dc+σB3ν2+B0B4dc+B0B4ν2−B0dcν1−B0ν1ν2−B3B1dc−B3B1ν2
By the Routh–Hurwitz criterion governing the polynomials of order 3, we have the following:
a2.a3 are positive
a1a2>a3
From
equation (27), 1 and 2 are satisfied.
Therefore, endemic equilibrium is locally asymptotically stable.
2.7 Global stability of the endemic equilibrium
Theorem 4: The equations of the model have a positive distinctive endemic equilibrium whenever
R0 > 1, which is said to be globally asymptotically stable.
Proof: Considering the Lyapunov function defined as:
LS∗A∗Cu∗Ca∗Tc∗R∗=S−S∗lnSS∗+A−A∗lnAA∗+Cu−Cu∗lnCuCu∗+Ca−Ca∗lnCaCa∗+Tc−Tc∗lnTcTc∗+R−R∗lnRR∗
where
L takes it derivative along the system directly as:
dLdt=1−S∗SdSdt+1−A∗AdAdt+1−Cu∗CudCudt+1−Ca∗CadCadt+1−Tc∗TcdTcdt+1−R∗RdRdtdLdt=1−S∗SΠ−βA+α1Cu+α2CaNS+σA−μS+1−A∗AβA+α1Cu+α2CaNS−σ+γ+ν1A+1−Cu∗CuγA−ν2+μ+dcCu+1−Ca∗Caν2Cu+ν1A−δ+μ+dcCa+1−Tc∗TcδCa−ω+μTc+1−R∗RωTc−μR
At equilibrium,
Π=(β(A∗+α1Cu∗+α2Ca∗)N∗)S∗−σA∗+μS∗(σ+γ+ν1)=(β(A∗+α1Cu∗+α2Ca∗)AN∗)S∗(ν2+μ+dc)=γA∗Cu∗(δ+μ+dc)=ν2Cu∗Ca∗+ν1A∗Ca∗(ω+μ)=δCa∗Tc∗ω=μR∗Tc∗dLdt=1−S∗SβA∗+α1Cu∗+α2Ca∗N∗S∗−σA∗+μS∗−βA+α1Cu+α2CaNS+σA−μS+1−A∗AβA+α1Cu+α2CaNS−βA∗+α1Cu∗+α2Ca∗AN∗S∗A+1−Cu∗CuγA−γA∗Cu∗Cu+1−Ca∗Caν2Cu+ν1A−ν2Cu∗Ca∗+ν1A∗Ca∗Ca+1−Tc∗TcδCa−δCa∗Tc∗Tc+1−R∗RμR∗Tc∗Tc−μR=1−S∗SβA∗S∗N∗+βα1Cu∗S∗N∗+βα2Ca∗S∗N∗−σA∗+μS∗−βASN−βα1CuSN−βα2CaSN+σA−μS+1−A∗AβASN−βAS∗N∗+βα1CuSN−βα1Cu∗S∗AA∗N∗+βα2CaSN−βα2Ca∗S∗AA∗N∗+1−Cu∗CuγA1−A∗CuACu∗+1−Ca∗Caν2Cu1−Cu∗CaCuCa∗+ν1A1−A∗CaACa∗+δCa1−Tc∗Tc1−Ca∗TcCaTc∗−μR1−R∗R1−R∗TcRTc∗=1−S∗S−βASN1−A∗S∗NASN∗−βα1CuSN1−Cu∗S∗NCuSN∗+βα2CaSN1−Ca∗S∗NCaSN∗−σA1−A∗A−μS−1−S∗S+1−A∗AβASN1−A∗S∗NASN∗−βα1CuSN1−Cu∗S∗ANCuSA∗N∗+βα2CaSN1−Ca∗S∗ANCaSA∗N∗+1−Cu∗CuγA1−A∗CuACu∗+1−Ca∗Caν2Cu1−Cu∗CaCuCa∗+ν1A1−A∗CaACa∗+δCa1−Tc∗Tc1−Ca∗TcCaTc∗−μR1−R∗R1−R∗TcRTc∗=−μS1−S∗S2−βASN1−S∗S1−A∗S∗NASN∗−βα1CuSN1−S∗S1−Cu∗S∗NCuSN∗−βα2CaSN1−S∗S1−Ca∗S∗NCaSN∗−σA1−S∗S1−A∗A+βASN1−A∗A1−S∗NSN∗+βα1CuSN1−A∗A1−Cu∗S∗ANCuSA∗N∗+βα2CaSN1−A∗A1−Ca∗S∗ANCaSA∗N∗+σA1−Cu∗Cu1−A∗CuACu∗+ν2Cu1−Ca∗Ca1−Cu∗CaCuCa∗+ν1A1−Ca∗Ca1−A∗CaACa∗+δCa1−Tc∗Tc1−Ca∗TcCaTc∗−μR1−R∗R1−R∗TcRTc∗=−μS1−S∗S2+P1SACaCuTcR+P2SACaCuTcR
and
P2≤0 whenever
S∗N≥SN∗,Cu∗S∗AN≥CuSA∗N∗,Ca∗S∗AN≥CaSA∗N∗,A∗Cu≥ACu∗,Cu∗Ca≥CuCa∗,A∗Ca≥ACa∗,Ca∗Tc≥CaTc∗
Thus,
dLdt≤0 if the condition in (35) and (36) holds.
therefore, by LaSalle asymptotic stability theorem (
LaSalle, 1976), and
Oke
et al. (2020) the positive equilibrium state
dLdt is globally asymptotically stable in the positive region
R+6.
3. Numerical computation
The numerical study is carried out using maple software embedded code for the Runge-Kutta of fourth order. Here, the subsequent default values are assumed for the embedded parameters taken from theoretical studies in literatures
γ=0.9,β=0.008,σ=0.59,dc=0.00693,μ=0.00693,ω=0.1,ν1=0.002,ν2=0.002,α1=0.0016,α2=0.0016,δ=0.0085,Π=0.07. The values remain unchanged throughout* the computations except otherwise indicated.
The effects of varying the testing rate of the acute individuals
υ1, testing rate of chronic individuals (
υ2) and treatment rate of chronic individuals (
δ) on the population dynamics are shown in
Figures 2 to
7. From
Figures 2 and
3, an increase in the parameters values reduces susceptible and acute populations thereby reducing the spread HBV due to low interaction between the host immune system and the virus. Therefore, the appearance of HBV and the pathogenesis reduces, which in so doing, lessens the potential injury on the liver. Hence, the liver is shielded from hepatocellular carcinoma over time. The rate of chronic unaware and chronically aware individuals is examined in
Figures 4 and
5. The parameter variations show a significant decrease in the chronic unaware population which implies that testing at that stage is a great tool for reducing the disease transmission. The transmission process dies down as the time progresses; this discourages liver inflammation as a result of lowering the infected individuals. Meanwhile, the chronic population in
Figure 5, depicts a high significant influence of the acutely infected and chronically unaware infected individuals over time. A chronic infection phase is found at the time range
10<t<20, as such, the individuals are exposed to liver carcinoma or cirrhosis. Hence, the chronic population diminishes as the parameters are increased.
Behavioral dynamics of susceptible population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
Behavioral dynamics of acute population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
Behavioral dynamics of chronic unaware population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
Behavioral dynamics of chronic aware population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
Behavioral dynamics of treated population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
Behavioral dynamics of recovered population when varying testing rate for acute and chronic individuals and treatment for chronic individuals.
In
Figures 6 and
7, the impact of varying the testing rate of the acute individuals
υ1, testing rate of chronic individuals (
υ2) and treatment rate of chronic individuals (
δ) on the treated and the recovered population are presented. The treated population increases with parameters variation along the rising time (
t) as a result of long time effect of parameter values. The recovery rate is enhanced as observed in
Figure 7 due to significant simulation of surface antibodies of Hepatitis B. This is in conformity with the works of
Pang
et al. (2010) and
Ullah
et al. (2019). This result implies that an intensification in testing at all infectious states and rise in treatment of chronic individual will bring about a reduction in the HBV transmission process which is a response to the WHO goal for 2030 that concentrating efforts on awareness program and campaign will sure bring about a decrease or eradication in the transmission process of the virus (
WHO, 2020).
4. Conclusion
A deterministic model of hepatitis B testing was developed and investigated, which included testing in the chronic unaware state as well as testing in all infectious states. The model has disease-free and endemic equilibria. The basic reproduction number was calculated using the next generation matrix method. The equilibria's local and global stability were discussed and shown to be asymptotically stable. The testing and treatment rate effects were thoroughly discussed.
Data availability
No data are associated with this article.
Grant information
None declared.
Competing interests
None declared.
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10.1155/2018/207698310.5256/f1000research.76473.r94650Reviewer response for version 1okeSegun1Refereehttps://orcid.org/0000-0002-4987-6287
Department of Mathematical and Applied Mathematics, University of Pretoria, Pretoria, South Africa
Competing interests: No competing interests were disclosed.
The authors worked on the impact of testing and treatment on the dynamics of Hepatitis B virus. Although the manuscript is well presented and gives new insight into the Hepatitis B virus, especially for unvaccinated men who have sex with men (MSM) and heterosexual people who have multiple sexual partners or have contact with sex workers.
In their paper, a mathematical model was formulated to analyse the acquisition and transmission process of the virus with the view of identifying the possible way of reducing the menace and mitigating the risk of the virus. The models’ positivity and boundedness were demonstrated using well-known theorems. Equating the differential equations to zero demonstrates the equilibria of the solutions i.e., the disease-free and endemic equilibrium. The next Generation Matrix method was used to compute the basic reproduction number for the models. Local and global stabilities of the models were shown via linearization and contrition of Lyapunov function methods respectively. The authors concluded by showing disease-free, endemic equilibria. The basic reproduction number was calculated using the next-generation matrix method. The equilibria's local and global stability were discussed and shown to be asymptotically stable. The testing and treatment rate effects were thoroughly discussed. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
Some minor comments:
The authors need to review more literature to back up their claims.
The introduction is too short and is not well cited. For example:
"In highly endemic areas, the most common form of transmission of hepatitis B is from mother to child at birth (vertical transmission) or through horizontal route (contact with infected blood), particularly from infected children to uninfected children during the first five years of life. vaginal and spermatic fluids and, to a lesser degree, perspiration, breast milk, tears, and urine. In particular, hepatitis B can be transmitted through sexual contact in unvaccinated men who have sex with men (MSM) and heterosexual people who have multiple sexual partners or have contact with sex workers" there is no citation here.
The authors need to add epidemiological interpretation and recommendations to the manuscript.
The conclusion needs to be rewritten.
Is the work clearly and accurately presented and does it cite the current literature?
No
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
No
Are sufficient details of methods and analysis provided to allow replication by others?
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.
10.5256/f1000research.76473.r94651Reviewer response for version 1SalawuSulyman Olakunle1Referee
Department of Mathematics, Landmark University, Omu-aran, Nigeria
Competing interests: No competing interests were disclosed.
Reviewer Reports on the impact of testing and treatment on the dynamics of Hepatitis B virus.
The impacts of different parameters were examined. The manuscript is well written and presented logically. The abstract captures accurately the whole work and the introduction is sound. However, I have the following comments and queries for the authors:
The authors should explain what prompted the choice of method of solution used over other techniques.
The significance and motivation of the study are not clear. The authors should clarify these.
There is a need to reference equation (3) to establish its correctness and source.
Some grammatical mistakes should be checked through.
Why have the authors chosen to engage in this study?
A likely extension of the study should be stated or recommended.
The manuscript should be indexed subject to minor correction.
Is the work clearly and accurately presented and does it cite the current literature?
Yes
If applicable, is the statistical analysis and its interpretation appropriate?
Not applicable
Are all the source data underlying the results available to ensure full reproducibility?
No source data required
Is the study design appropriate and is the work technically sound?
Yes
Are the conclusions drawn adequately supported by the results?
Yes
Are sufficient details of methods and analysis provided to allow replication by others?
Yes
Reviewer Expertise:
Mathematical Modeling and flow dynamic
I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.