Dynamics of Hepatitis B Virus Disease with Infectious Latent and

Vertical Transmission

HELEN O. EDOGBANYA1, ANSELM O. OYEM1,2,*, JOHN O. DOMINIC1,

JESSICA M. GYEGWE1

1Department Mathematics,

Federal University Lokoja,

PMB 1154, Lokoja,

NIGERIA

2Department of Mathematics,

Busitema University,

P.O. Box 236, Tororo,

UGANDA

*Corresponding Author

Abstract: - Hepatitis B has become a major health threat because it is a life-threatening liver disease with an

estimated 0.25 billion people suffering from this infectious disease worldwide. This paper presents a SLITR

(Susceptible-Latent-Infectious-Treatment-Recovery) mathematical model that combines both vaccination and

treatment as a means of controlling the hepatitis B virus (HBV). The nonlinear ordinary differential equations

for the HBV transmission capacities were resolved and the basic reproduction number computed using the

next generation matrix method and simulated numerically using the Runge-Kutta fourth order scheme

implemented using MatLab. The stability points for disease-free equilibrium state (DFE), endemic equilibrium

state (EE), and basic reproduction number 𝑅0 were obtained and the results show that the disease-free

equilibrium was both locally and globally asymptotically stable . Similarly, treatment or vaccine

administered was effective in alleviating the spread of HBV disease, and when both control strategies are

combined, the diseases are quickly controlled and eventually eradicated.

Key-Words: - HBV, Disease, Latent, Vertical transmission, Infectious, Stability.

Received: August 16, 2023. Revised: January 6, 2024. Accepted: February 13, 2024. Published: April 16, 2024.

1 Introduction

A very large variety of organisms exist, including

some which can survive and find the bodies of

people or animals suitable for their development,

and These organisms are an infectious agent (also

called pathogens) which causes infection and illness

through the toxins produced are communicable and

infectious. Waste products from the host, be it

faeces or urine (latency and persistence) develops

within the environment or intermediate host before

making contact with a susceptible person or animal,

[1], [2]. Pathogens are non-infectious during the

latent period but non-latent pathogens are infectious

directly after excretion, [3]. Infected persons

transmit this virus vertically either before or after

birth from an infected mother to the newborn or

through the body fluid of an infected person to an

uninfected person via sharing of non-sterilized

injection syringes, tattoo materials, and through sex,

among others.

Viral hepatitis, an inflammation of the liver is

caused by one or more of five main hepatic viruses:

A, B, C, D, and E. These viruses exhibit similar

symptoms that can potentially cause liver disease to

varying degrees; they however differ significantly in

regards to epidemiology, prevention, diagnosis,

care, and treatment. This virus is categorized as a

major global health problem with over 400 million

patients chronically infected, with death cases of

million deaths per year, [4]. Nigeria is known

to be among the countries with a high affliction of

viral hepatitis with a Hepatitis B Virus (HBV) and

Hepatitis C Virus (HCV) prevalence of 11% and

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2.2%, [5]. Statistics show that there are over 350

million chronic carriers of HBV with 6 million

deaths per year due to HBV related liver disease or

hepatocellular carcinoma and this has become a

global health challenge. Considerable successes

have been recorded in a bid to eliminate HBV

transmission yet; the prevalence of these infectious

diseases still outweighs total elimination. Several

mathematical models by researchers have modeled

the control of the spread of HBV using different

parameters and arriving at different intervention

measures, [6].

The mathematical modelling of infectious

diseases is primarily aimed at studying the spread

and duration of epidemics, understanding the scale

of the disease challenge and the potential impact of

interventions; predicting the spread of the disease,

total number of infected persons, duration of the

epidemic, as well as reproduction numbers and then,

identify the most efficient technique for issuing a

limited number of vaccines in a given population,

[7]. Researchers like [8], proposed that the

infectivity during the incubation period can be a

second way of transmission prompting, [9], [10], to

study the global behavior of the spread of HBV

using an SEIR model with a constant vaccination

rate and the dynamics of Hepatitis B via a

Susceptible Exposed Infectious Recovered (SEIR)

type epidemic model. Their study showed that the

endemic equilibrium was globally asymptotically

stable since the disease persisted in the population

and reproductive number . Researchers have

continued to look into the scientific ways of

mitigating or correcting the continued spread of the

infectious disease, some of which are [11], [12],

[13], [14], [15], [16], [17], [18], [19], [20], [21].

Since studies have shown that Hepatitis B is

characterized by multiple endemic solutions, a

matter which may be of concern in developing

control strategies, [22], was able to identify the

possible causes of multiple endemic solutions in a

Hepatitis B model and concluded that the

dependence of the probability of carriage

development on the force of infection is

the main reason for multiple endemicity; that is

where a large proportion of infants that are not

vaccinated (ω)). Subsequently, [23], looked into the

SIR epidemic model with a changing population

size leaving the immigration rate constant. The

stability properties of the equilibrium points of the

model were analyzed indicating that the disease-free

equilibrium point was stable and the population

survived and for a stable endemic equilibrium point,

the number of infectives did not change, meaning

the infected rate equals the recovery rate.

Furthermore, working on an SEIRS model with a

time delay on complex networks, [24], was able to

determine and equilibriums of the model from

the mean field theory. Time delay cannot change the

basic reproductive number because it is dependent

on the topology of the underlying networks by

theoretical analyses, and can reduce the endemic

level and weaken the epidemic spreading. [25],

made a comparison of two viruses; Hepatitis B

Virus (HBV) and Human Immunodeficiency Virus

(HIV) infection as a public health problem

worldwide and the study concluded that the

prevalence of HBV infection is higher than that of

HIV among blood donors.

Then, [26], analyzed a class of discrete vertical

and horizontal disease models with constant

vaccination and population size. The obtained

values were input into the eigenvalue determined

from the model equations and discussed the

influence of the coefficient parameter on the

eigenvalues. This prompted [27] and [28], to look

into of an age structured SEIR epidemic model

with latency in its infectivity derived by using

theories of both differential and integral equations.

It was observed that the disease-free equilibrium is

locally and globally asymptotically stable if

and just one endemic equilibrium exists, if .

From literature, researches had focused more on

vertical transmission of HBV infections than

latency, thereby creating a lacunar. Hence, this

research considers the combined effects of latent

and vertical transmission of HBV infections and

their dynamics of treatments, and vaccination. The

objective is to see if the combination of vaccination

and treatment can be an effective intervention

means in mitigating and possibly eradicating the

HBV infection.

2 Problem Formulation

Consider a complex but realistic SLITR model

which captures an additional treatment

compartment. The governing SLITR model

considers an individual’s infection in the latent

category and considers the flow of disease from the

susceptible category to other categories based on the

following assumptions, [29], [30],

i). Members of the population are the same

(homogeneous population);

ii). Recruitment of individuals into the population

is only through birth;

iii). Exiting out of the population is through both

natural death and virus-related death only;

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iv). Individuals who received vaccination may not

necessarily achieve permanent immunity;

v). Infants born by carrier mothers proceed to

either susceptible or latent class immediately;

vi). Treated carriers recover; and

vii). Induced death is only in the infectious class.

The mathematical model compartmentalized the

total human population into; susceptible individuals

, latent individuals , infectious individuals

and recovered patients . Where,

is the rate of recruitment of

population into susceptible group, the

transmission coefficient from susceptible group to

latent class, the transmission rate from latent class

to infectious class, the recruitment rate into

the latent population, and are the natural death

rate which occurs in all the five classes and the

Hepatitis B Virus related death rates respectively in

the model system. Similarly, is the

proportion of birth without vaccination, the

vaccinated proportion, the proportion of birth

vertically infected (children infected during birth),

the proportion of children not vertically

infected, the rate that recovered individuals

become susceptible again, the rate at which

individuals in latent class go for treatment, the

rate at which individuals in infectious class go for

treatment, and the rate at which those that receive

treatment recover.

From the SLITR model in Figure 1 (Appendix),

the nonlinear ordinary differential systems of

equations become:

subject to the axioms:

2.1 Analysis of the Model

The nonlinear differential system, Eq. (1) to Eq. (5),

for all , , , , and will

be positive in since, all the parameters used in

the model are entirely positive. Placing a lower

bound on the equations, gives:

By the method of separation of variables, Eq. (8)

becomes:

and integrating both sides of Eq. (13), gives:

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as and

Applying the same methods and principles of Eqs.

(8) and (13) to Eqs. (9) – (12), gives

From Eqs. (14) – (18), , , , and

are positive in for all .

Lemma: There exist an , , , , such

that , , , , ,

, ,

, for all .

Proof:

The boundedness of the solution for all ,

, , , , and is proved. Since all

the constants used in the system are positive, then

Taking the limit of the supremum of both sides

gives:

Because is bounded, ,

, , , and are all bounded since

.

Then, , , ,

and for all .

2.1.1 Disease Free Equilibrium (DFE) Point

The point at which there is no infection within the

population in study or the point in which the

population is free of disease is called the Disease-

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Free Equilibrium (DFE) point. Equating the Eqs. (1)

– (4) to zero, gives:

such that,

Using the next generation matrix method to

calculate , the basic reproduction number is given

by where, is the Jacobian of at ,

is the rate of appearance of new infection in the

compartment , is the Jacobian of at , and

represents the rate at which individuals are

transferred into and out of the compartment .

The population infected with the disease is

represented by the following;

Taking ,

where, is equal to the Jacobian of and

Multiplying Eq. (28) and Eq. (29), gives

The basic reproduction number is given by the

spectral radius of the matrix , that is, the

highest absolute value of eigenvalues

2.1.2 Endemic Equilibrium (EE) Point

For this disease to continue in the population, all the

compartments must not be zero that is,

. Thus,

where, exists if and only if .

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2.1.3 Local Stability of the DFE Point

The local stability is calculated using the Jacobian

of the model at . It is achieved using the sign of

the real parts of the eigenvalues of the

corresponding Jacobian matrix.

Theorem: The disease-free equilibrium of the

system of ODEs is locally asymptotically stable if

the reproduction number and unstable if

.

Proof:

Taking the equations of the system (1) – (5), the

Jacobian matrix becomes

And gives;

By inspection, the matrix above shows that

, , and

are the eigenvalues while the

other two remaining eigenvalues are obtained from

the matrix,

Using Routh-Hurwitz criterion, the disease-free

equilibrium point 𝐸0 is locally asymptotic stable if

and . Thus, for ,

, ,

Dividing through Eq. (40) by , gives

since, , implies the disease-free equilibrium

point is asymptotically stable.

2.1.4 Global Stability of the DFE Point

Using Castillo-Chavez approach, Eqs (8) – (11) can

then be expressed as

Where, is the number of non-

infected individuals and is the

infected compartment. Defining the conditions for

global stability of disease-free equilibrium as

i). , is asymptotically stable

ii). ,

for

where, the M–matrix for its off–diagonal elements

are positive in the area, and where the model

equations make epidemiological sense. If the above

two conditions are satisfied by the model system,

then the theorem stated below is true.

Theorem: Provided that and the conditions

and are satisfied, the disease-free

equilibrium point of Eqs. (8) – (11) is

globally asymptotically stable.

Proof:

The DFE is now denoted as where

. Now, the first condition of global

asymptotically stability (GAS) is

Solving the linear differential equation, gives:

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And the solution indicates that,

as regardless of the

values of and . Therefore,

is globally asymptotically stable.

By the second condition,

where

, . Hence the proof is

complete and the disease-free equilibrium is

asymptotically stable, [31].

3 Numerical Solutions

This paper focuses on the possibility of eliminating

infectious HBV by changing the data values

assigned to vaccination and treatment. To obtain the

numerical solution to Eqs. (1) to (5), MatLab is

employed to simulate the systems and investigate

the impact of both vaccination and treatment as a

control strategy against infectious HBV on Latent

and Infectious individuals. Some of the parameters

for analysis were assumed based on the already

established assumptions and also the fact that

vaccination and treatment are combined, [32], [33],

[34], [35], [36], as shown in Table 1 (Appendix).

The parameter values used are sufficiently small

so that the analysis outputs remain relatively

accurate when smaller parameter values are used.

The assumptions made about the population do not

and are not meant, in any way, to influence the

simulation results but rather intended to optimize

the analysis execution. Figure 2 (Appendix) shows

the presence of the Hepatitis B Virus in the carrier

population with vaccination intervention but no

treatment administered to the infectious class. At the

initial point where there are no both vaccinations

and treatment , and . The disease

rises to the peak and decreases slightly and then

remains almost stable all through the populations.

The peak represents the epidemic crisis while the

“almost stable state” indicates how the entire

population is diseased. However, vaccination and

treatment parameters of the Latent class at varying

rates help alter the almost stable state of the

pandemic and reduce it as shown by the green, blue,

and yellow graphs.

Figure 3 (Appendix) indicates that without any

intervention strategy, , and , the

population of the infectious class remains high,

almost constant. However, when relatively high

vaccinations and treatment efforts are made at the

same rate of , , and , the

disease persistence drastically reduces as shown by

the blue graph among the infected. Interestingly,

treating the infectious tends to be more effective

than vaccinating them at the same rate as shown by

the graphs of , , compared to

that of 6, , . This is because of

relapse and resistance of infection to antibodies of

the immune system.

Figure 4 (Appendix) shows that treating and

vaccinating the carriers at the same rate will cause

the disease to decrease but eradication is possible

when the efforts are increased. This is attested to by

the graph of , , and in which

intervention strategies are very high (Table 2,

Appendix). Figure 5 (Appendix) shows infected

population proportions in which only vaccination is

administered and no treatment. The natural immune

system fights the disease up to a relatively stable

state. However increasing vaccination helps

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supplement the immune system in the fight and

reduction of the prevalence of disease among latent

individuals. Figure 6 (Appendix) reveals that by

combining both treatment and vaccination, the

disease will be fully fought out of the population.

The graph also indicates that for us to effectively

bring full control over the infectious HBV disease,

we have to increase the rate of implementing the

two strategies.

Normalized Sensitivity Index

In this paper, the normalized sensitivity index is

employed to measure and determine the best control

measures using the reproduction number of the

HBV model. Thus, quantify the developed model

sensitivity by calculating changes in the

reproduction number input parameters. Normalizing

and comparing the sensitivity of each parameter in

the model, makes it easy to assess the impact of

various parameter input factors. Therefore, in the

quest for best disease control measures, we identify

and determine the most influential factor affecting

the reproduction number output using the

normalized sensitivity index denoted by and

defined by

with

The relative change measure of with respect to

is,

Similarly, substituting the above numerical values,

gives the normalized sensitivity index of the

reproduction numbers as;

Arranging the magnitude of the sensitivity

analysis in descending order, determines the most

sensitive parameter, that is,

,

, , ,

Similarly, increasing the parameter values of

, , ,

and reducing the value of the

parameter values of and , will reduce the

reproduction number and significantly reduce the

spread of HBV. Figure 7 (Appendix) shows the bar

chart MATLAB plot for the normalized sensitivity

index analysis of each parameter in the reproduction

number and which represents rate at which

individuals in latent class go for treatment was

considered and increased by 0.3 in Figure 7(a),

Figure 7(b) and Figure 7(c) in Appendix. Result

shows that all other parameters in the reproduction

number were affected by a slight increase in ,

which implies that as more individuals yield

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themselves for treatment, there is a corresponding

decrease in the transmission rate from the latent

class of HBV disease, which suggests the disease

awareness campaign (DAC).

4 Conclusion

A mathematical study of an infectious Hepatitis B

Virus in which treatment and vaccination were

combined in an attempt to predict the total

eradication of the disease from the susceptible

population was carried out. In this study, the

existence, uniqueness, and boundedness of the

solution were looked into as well as computing for

the disease-free equilibrium point, basic

reproduction number associated with the system,

and endemic equilibrium point. To obtain the

numerical result for the study, the Runge-Kutta

fourth order scheme was used and implemented

using MatLab. The obtained result showed that:

1. Vaccination and treatment can be effective

intervention efforts to mitigate and possibly

eradicate the prevalence of infectious HBV

when administered at a higher rate.

2. A periodic mass vaccination of expecting

mothers and children should always be carried

out as this can set the basis for eliminating the

hepatitis B Virus as shown in Figure 2

(Appendix).

3. Test and increase the level of treatment among

latent individuals (Figure 5, Appendix).

4. In pursuit of total eradication of the Hepatitis B

Virus pandemic, governments should double

vaccination coupled with treatment programs

among the susceptible populations (Figure 6,

Appendix).

5. Continued educating of the nomad communities

which are tightly held to unhealthy cultures like

traditional methods of circumcision. This will

help in eliminating unnecessary transmission of

HBV which is usually contracted as a result of

using non-sterilized objects for tattooing and

circumcision.

For further research, a look into exploring the

application of computational and artificial

intelligence in resolving the complexity of the

problems, would add value and enhance scientific

output.

Acknowledgement:

The authors hereby acknowledge the various

contributions and review comments of the reviewer

and editorial board toward a successful article.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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APPENDIX





























󰇛󰇛󰇜󰇛󰇜

Fig. 1: Schematic diagram of the SLITR model for HBV

Fig. 2: Impacts of vaccination on Latent class without treatment

Fig. 3: Impact of low treatment and vaccination on the Latent class

Fig. 4: Impact of equal rate of vaccination and treatment on Latent class

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Fig. 5: Impact of increased treatment on Infectious individuals

Fig. 6: Impact of both vaccination and treatment on Infectious individuals

(a) (b)

(b)

Fig. 7: Bar chat of Normalized Sensitivity Index Analysis of HBV Reproduction number.

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Table 1. Parameter for Numerical Simulation

Parameter

Value

References

[22], [23]

ASSUMED

[24]

[24], [25]

ASSUMED

[24], [25]

ASSUMED

[25]

ASSUMED

Table 2. Nomenclature of some Parameters

Parameter

Key

(vaccinated children)

(Treatment parameter for infectious class)

(Treatment parameter for Latent class)

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