DYNAMICAL SYSTEM FOR EBOLA OUTBREAK WITHIN QUARANTINE AND VACCINATION TREATMENTS

ABSTRACT


Ebola Virus Disease (EVD) is a viral infection caused by a virus from the family Filoviridae, the genus
Ebolavirus, that derived from fruit bats, the family Pteropodidae [1]. EVD is a deadly infectious disease that was first discovered in the Democratic Republic of Congo (DRC) near the Ebola River in 1976 [2] [3]. The last case of EVD reappeared in 2014 on March 23, with deaths reaching 11,315 out of 28,637 cases occurring in 6 countries until January 2016 [4]. EVD is spread through direct contact with infected body fluids in the mouth, nose, eyes or through a break in the skin [5] [6]. The virus is not transmitted through the air, by water, or in general, by food [6].
Individuals suffering from EVD do not infect others during the incubation period, which can survive between two days and three weeks. The recovery from EVD relies heavily on the immune response of infected individuals and support for good clinical care. Individuals who have recovered cannot spread the ebola virus. However, Ebola virus can stay in body fluids such as semen and breast milk for some time after recovery [6] [7].
For years, researchers have been trying to develop a model for the spread of EVD since Ebola's disease became a deadly infectious disease. The mathematical model of EVD spread aims to analyze the characteristics of the spread of the disease [8] [9]. Imran et al. [10] developed a model of the spread of ebola disease with SLSHEIHR type by providing treatment in the hospital for infected individuals. The treatment aims to provide recovery to an infected individual or reduce the rate of death caused by EVD.

RESEARCH METHODS
This research modified the SLSHEIHR type of disease model to SLSHVEQIHR type by adding treatment in the form of vaccine against susceptible individuals to reduce the risk of infection and adding quarantine treatment to prevent transmission of the disease from infected individuals. The total population at any time instant t, denoted by ( ), is the sum of individual populations in each compartment that includes low risk susceptible individuals ( ), high risk susceptible individuals ( ), vaccinated individuals ( ), exposed individuals ( ), quarantined individuals ( ), and infected individuals ( ), Hospitalized individuals ( ), and Recovered individuals ( ), such that, ( ) = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ). The flow diagram of Ebola model is shown in Figure 1.
Based on the diagram in Figure 1, we can obtain a system of ordinary differential equations as follows: All parameters are non-negative constants.

RESULTS AND DISCUSSION
In this section, we will determine the equilibrium point, the basic reproduction number, and the stability analysis.

The Equilibrium Point
The disease-free equilibrium of the system of Equation (1) is given by

The Basic Reproduction Number
The basic reproduction number, denoted ( 0 R ) is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual. We calculate the basic reproduction number by using the next generation operator approach by van den Driessche and Watmough [13] [14]. The next generation matrix at the disease-free equilibrium 0 is given by:

The Stability Analysis
The stability of system of Equation (1) is dependent on the basic reproduction number 0 . The stability analysis of both equilibrium 0 T and * T will be provided through the following theorems: Proof. The Jacobian matrix at 0 for system of Equation (1) is given by According to Equation (2), we obtain eight eigenvalues with four of them are negative: According to Routh-Hurwitz criterion [15] [16], Equation (3) on disease-free equilibrium 0 T is stable if the following stability criterion satisfied: Disease-free equilibrium 0 has one zero eigenvalue and seven negative eigenvalues if The zero eigenvalue has right eigenvector ( ,

 
As indicated previously that 5 is arbitrary positive, then, , 0 ) .  , the disease-free equilibrium 0 T stability changes from stable and becomes unstable, while endemic equilibrium * T coordinates changes from negative becomes positive and thus becomes local asymptotically stable. As a consequence, the endemic equilibrium * T is locally asymptotically stable if

CONCLUSION
This study modified mathematical model of Ebola outbreak to consider quarantine and vaccination. The results of the model analysis obtained two equilibrium points, namely, disease-free equilibrium and endemic equilibrium. The basic reproduction number ( ) 0 R was determined. The disease-free equilibrium is locally asymptotically stable on condition 1 0  R , whereas the endemic equilibrium is locally asymptotically stable on condition 1 0  R . The numerical simulation of population dynamics showed similar patterns as expected.