The Transmission Dynamics and Optimal Control of Hepatitis B in Ethiopia Using SVEIRS Model
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Date
2017-10
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Addis Ababa University
Abstract
An epidemic model is a simplified means of describing the transmission
of communicable disease through individuals. In this paper the mathematics
behind the model and the various tools for judging effectiveness of policies
and control methods on the spread of infectious diseases in populations has
been analyzed mathematically. And it has been applied to specific diseases to
study the propagation of diseases using the mathematical epidemic model.
Epidemic models are many types form that select SVEIRS model and discussed
the dynamics and control of infectious diseases, but quantifying the underlying
epidemic structure can be challenging especially for new and understudied
diseases. SVEIRS model is that generalizes several classical deterministic
epidemic models then apply it for Hepatitis B. Consider compartments of
susceptible, vaccination, exposed, infected, and recovered humans without immunity
and modeled the natural growth, natural death and death due to disease
and the interactions between these populations.
The model has two equilibrium states namely, the disease - free and the endemic
equilibrium points. The stability of each equilibrium point discussed
has been found to be stable or unstable. The basic reproduction number(R0)
estimate the stability, with (R0 > 1) whenever the transmission rate was increased
or the recovery rate reduced but turned to the disease die out with
(R0 < 1) whenever the transmission rate was reduced or the recovery rate increased.
The results of our sensitivity analysis showed that the most sensitive parameter
that controls the spread of Hepatitis B is the initial infection rate of the
susceptible, b and d or death rate. Decreasing the value of b at the same rate
as the other parameter values completely decreases the proportions of both the
infective and the exposed more effectively than any parameter value.
Consider an optimal control problem subject to an SVEIRS Hepatitis B epidemic
model with vaccination controls. Our aim is to find the best optimal
control strategies to make the number of infectious individuals as small as possible
and to keep the vaccination ratio of Hepatitis B as low as possible during
a certain vaccination period that will minimize the cost of control. Pontryagin’s
maximum principle to characterize the optimal levels of the controls. The
resulting optimality system is solved numerically by forward-backward sweep
method. The results show that the optimal vaccination, drug and education using
media differs according to the controlled and uncontrolled individuals and
has a very desirable effect upon the population for reducing the number of
infected individuals. The effect of vaccination on transmission dynamics of
Hepatitis B is studied. The resulting optimality system also showed that, the
use of vaccinating at the highest possible rate to the population as early as
possible is essential for controlling an epidemic of the Hepatitis B disease.
Finally model to simulate the data of Hepatitis B cases in the Ethiopia from
2015 and design a control strategy of the country to eliminate the epidemic for
the future course with optimal control theory.
Keywords: SVEIRS-Model; mathematical models; vital dynamics; vaccinations;
HB, HBV, Herd Immunity; epidemiology number, optimal control.
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Keywords
SVEIRS-Model; mathematical models; vital dynamics; vaccinations; HB, HBV, Herd Immunity; epidemiology number, optimal control.