1. A. A. Momoh, M. O. Ibrahim, A. Tahir / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-august 2012, pp.2248-2250
Modeling the Effects of Detection and Treatment of Latent
Hepatitis B Infection on Transmission Dynamics of Hepatitis B
Disease
*
A. A. Momoh, ** M. O. Ibrahim and *A. Tahir
*
Department of Mathematics, Modibbo Adama University of Technology, Yola. Adamawa
State. Nigeria.
**
Department of Mathematics, Usmanu Danfodiyo University, Sokoto. Sokoto State. Nigeria
Abstract
In this paper, we proposed an SVEIR model Theorem. The model derivation is given the next
to understand the effect of detection and treatment of sections.
Hepatitis B at latent stage on the transmission
dynamics of hepatitis B disease. Mathematical 2. Model Formulation
analysis was carried out that completely determines As with any modeling endeavor, various
the global dynamics of the model. The impact of assumptions about the underlying biology must be
detection and treatment of Hepatitis B at latent stage made. At this stage, we wish to clearly state some
on the transmission dynamics are discussed through assumptions.
the stability analysis of the disease free equilibrium.  We assume that throughout the duration of
the vaccines efficacy, latent Hepatitis B is
Keywords: Hepatitis B Virus (HBV), Mathematical completely undetectable.
Model, Latent Hepatitis, Stability analysis  We also assume that the birth and mortality
rate are equal.
1. Introduction  We assume that the efficacy of the vaccines
Hepatitis B virus is the most common cause wanes out at the rate  .
of cirrhosis and hepatocellular carcinoma in the Individuals enters the population with a
world today. Of approximately 2 billion people who recruitment rate of  . The natural mortality rate is
have been infected with HBV worldwide, more than denoted by  . A proportion, c, of individuals
350 million, or about 5% of the world’s population entering the population are vaccinated as infants and
are chronic carriers, and with an annual incidence of hence enter the vaccinated class, V. the remaining
more than 50 million (Khan et al 2007 & Patel et al
2004 ). Hepatitis B virus accounts for 500,000 to 1.2 proportion, 1  c , enter the susceptible class, S. the
million death per year. efficacy of the vaccines used in the vaccines used in
Compartmental mathematical models have been the vaccination wanes over time at the rate of  and
widely used to gain insight into the spread and result in the movement of individuals from the
control of emerging and re-emerging human disease vaccinated class to the susceptive class. We use the
dating back to the pioneering work of Bernoulli in frequency-dependent description of disease
1760 and likes of Ross, Kermack and McKendrick transmission, with transmission coefficient  .
and others (Anderson et al 1982, 1991). The study of We assume that assume that, upon infection with
infectious diseases has been transformed by the use hepatitis B, an individual can either progress quickly
of mathematical models to gain insight into the to active hepatitis B or develop a latent infection and
dynamics of epidemics, to identify potential public progress slowly to active hepatitis B. we denote the
health interventions, and to assess their impact classes of latently infected and infected individuals
(McCluskey et al 2003). Mathematical models were by E and I respectively. The probability of a
useful in informing policy during the foot and mouth random individual exhibiting fast progression to
disease outbreak in the United Kingdom in 2001, active hepatitis B is denoted by  . latent infections
progress slowly to active disease at the rate  . We
during the Severe Acute Respiratory Syndrome
(SARS) outbreak in 2003 and in recent planning of
assume that individuals in the latent hepatitis class
responses to potential smallpox or pandemic
moves to removed class at the rate  when treated.
influenza outbreak.
In this paper, we propose an SVEIR model to We also assume that active hepatitis B disease clears
understand the effect of detection and treatment of at the rate  .
hepatitis B at latent stage on the transmission Model Diagram
dynamics of the disease. We solved the system of
ordinary differential equation and obtain our disease
free equilibrium. Stability analysis was carried out on
the disease free equilibrium using Routh-Hurtwitz
2248 | P a g e

2. A. A. Momoh, M. O. Ibrahim, A. Tahir / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-august 2012, pp.2248-2250
dR
  I  R  E
dt (5)
r
 (   ) where r is the population of
1 r
latent infections which are detected and successfully
treated before progression to active hepatitis B or
death.
If a population is free of hepatitis B infection
(i.e. E  I  0), the system reduces to
dS
 (1  c)  V   S
dt
dV
 c  V  V
dt
dR
  R
Model Equations dt
dS  SI dS dV dR
  0
 (1  c)   V   S (1) Recalling that is the
dt N dt dt dt
dV VI necessary for an equilibrium, we see that the derived
 c  V   V (2) as follows
dt N (1  c)  V   S  0
dE  SI  EI c  V  V  0
 (1   )  E     E  E
dt N N  R  0
(3)
From (11), R  0
*
dI  EI  SI VI
 E      I   I Also, from (10), c  (   )V  0
dt N N N
(4) c
V* 
dR
  I  R  E (5)
(   )
dt Substituting (12) into (9) we have,
(1  c)  V   S  0
3. Mathematical Analysis
 c 
In this section, we wish to study the equilibrium state (1  c)      S  0
and analyze for stability of the system.  (   ) 
(1  c)  
Disease Free Equilibrium S* 
To establish analytic thresholds for when  (   )
vaccination is a society’s prudent choice, we consider The disease free equilibrium of the model is given by
the case of a population at the brink of eradicating
 (1  c)    c  
hepatitis B. Mathematically; we must therefore X *  ( S * , V * , E * , I * , R* )   ,  , 0, 0, 0 
consider the disease free equilibrium (DFE).
We recall that the full system is given by
  (   )  (   )  
dS  SI
 (1  c)   V   S (1) 4. Stability Analysis of Disease Free
dt N Equilibrium
dV VI *
We find out that the Jacobian of F about X is
 c  V   V (2)
dt N      0  S * 0 
dE  SI  EI  0 (   )   0  V *
0 
 (1   )  E     E  E DF ( X ) 
* (1   )  I * 0 (     )    E *
0 
dt N N  0 0    E *
(    )   0 
(3)  
dI  EI  SI VI  0 0      
 E      I   I
dt N N N
(4)
2249 | P a g e

3. A. A. Momoh, M. O. Ibrahim, A. Tahir / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-august 2012, pp.2248-2250
At disease free equilibrium, I  E  0, the Conclusion
* *
From our analysis, we found out that the
Jacobian reduce to
disease free equilibrium is asymptotically stable
    0  S * 0  when the population of latent infections which are
 0 (   )   0  V * 0  detected and successfully treated before progression
DF ( X * )   0 0 (     )   0 0  to active hepatitis B or death is less than zero. Thus,
 0 0  (    )   0  if a country is able to detect and treat latent hepatitis
 0
 0      
 B infection at the rate which exceed  , the
discontinuation of HBIB* vaccination will increase
Noticing the form of the Jacobian, we immediately the stability of the disease free equilibrium. If the
have that    is an eigen value and that  is detection and treatment rate of latent hepatitis B is
below  , HBIB* vaccination will result in a more
a double eigen values. The remaining eigen values
stable disease free equilibrium.
are those of the 2  2 sub-matrix.
(     )   0  References (14)

  (   )   
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 g x ( x* , y* ) g y ( x* , y* ) 

(15)
McCluskey C. C., .Van den Driesseche. Global result
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 g ( x, y )  Differentiable mild from moderate-to-advanced
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Benyah, F. (2008). Introduction to epidemiological
Then the critical point ( x* , y* ) modeling. 10th regional college on modeling,
1. is asymptotically stable if trace simulation and optimization. University of cape
coast, Ghana.
( J )  0 and det( J )  0 Ssematimba, A., Mugisha, J. Y. Luboobi, L. S.
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Let 
A  (     )

0
(   ) 
Then from matrix A we have,
det( A)  (     )(   ) and
trace( A)  (     )  (   )
r
Substituting   (   ) we have,
1 r
( (1  r )   (1  r )  r (   ))(    )
1 r
Thus, our model is asymptotically stable when
r  0.
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