Mathematical Model for Ebola Virus Infection in Human with Effectiveness of Drug Usage

In this paper, we formulated a mathematical model of the dynamics of Ebola virus infection incorporating effectiveness of drug usage. The infection free and infection persistence equilibrium points were obtained. The control reproduction number was obtained which was used to analyse the local and global stability of the infectionfree equilibrium. Using the method of linearization, the infection-free equilibrium (IFE) state was found to be locally asymptotically stable if R c 1. By constructing lyapunov function, the infection-free equilibrium was found to be globally asymptotically unstable if R c > 1. Numerical simulation of the model was done. It is observed that, as percentage of effectiveness of drug administration increases, the control reproduction number decreases. This suggests that with the help of drugs usage, the immunes system have the ability to suppress the increase of infected cells, as well as virus load which shown that the virus does not maintain an infection in the system. Keywords : Drug Usage; Ebola virus; Global stability; Immune response; Mathematical model

Ebola virus is a highly infectious and deadly (lethal) disease.The virus is among the most dangerous and devastating threats to human health, causing a large number of fatalities.The virus was first discovered in 1976 in Democratic Republic of Congo (Fritz, 2012).Recently, the virus again resurfaced in some West African Countries like Guinea, Sierra Leone, Liberia and Nigeria.The incubation period of Ebola virus ranges from 2 to 21 days and infectious period ranges from 4 to 10 days (WHO, 2014).Meanwhile, it takes an approximation of 31 days to quarantine a patient under investigation of the Ebola virus.Consequently, the symptoms of the Ebola virus are characterized by headaches, fever, vomiting, bleeding diarrhoea, and rash (Fauci, 2014), in infected person, severe bleeding and shock are usually followed by death.The spread of the virus and eventual death of infected patients was largely contained (reduced) through early detection and effective contact tracing (Beeching et al., 2014).
Ebola virus is known to cause damage to large variety of cell types including monocytes, macrophages, dendritic cells, endothelial cells, fibroblasts, hepatocytes, and several types of epithelial cells.The primary targets of the virus are dendritic, monocytes and macrophage cells (Centre for Disease Control and Prevention (CDC), 2014).Mathematical models have played a key role in the formulation of the dynamics of Ebola virus infectious and control strategies for nonimmune response (Chowell et al., 2015).Mathematical models have played a central role to capture the dynamics of different virus infection (Chowell et al., 2015).The model equations are formulated using ordinary differential equation.The aim of this paper is therefore to formulate a mathematical model of the dynamics of Ebola virus infection in the cell population and immune response by incorporating effectiveness of drug usage, humoral immunity and clearance of the free virus.

Model Formulation:
The human cell population is divided into the following classes: uninfected cells (U), infected cell (I), free virus (V), cytotoxic Tlymphocyte (T) and B-cell (Antibody) (B).Thus, the model indicates the passage of individuals cell is from the virus class, V, infect uninfected class, U, to progress to infected class, I and move back to free virus class for more viruses to be produced in the absence of immune response.Cytotoxic T-lymphocyte class, T, interact with infected cells for the purpose of wipe out the infected cells.The B-cells (Antibody) class, B, interact with the free virus and neutralized the free virus from the system.The corresponding mathematical equations of the above description are given by a system of ordinary equations below:

ANALYSIS OF THE MODEL Equilibrium States of the Model
Therefore, Infection Free equilibrium (IFE) denoted as ( ) : To derive the control reproductive number of Model ( 1)-( 5), we first introduce the method of next generation matrix formulated by (Diekmann and Heesterbeek, 2000;Driessche Van den and Watmough, 2002).Therefore, we have the following The inverse of ( 9) gives (10) The multiplication of ( 8) and ( 10 .Therefore, the control reproductive number is Thus, the equation ( 14) is the control reproductive number of the system (1)-( 5).

RESULTS AND DISCUSSION
Numerical simulations of the model: For the purpose of model validation, in order to ensure that the model is in agreement with reality, numerical simulation is undertaking using the data provided in Table 2.The results are displayed in Figure 1 to Figure 4. Figure 1 and Figure 2 shown that, as effectiveness of drug administration increases its percentage, it decrease the control reproduction number.In Figure 3, it is observed that, as percentage of effectiveness of drug usage increases, the control reproduction number decreases.This shown that an infection does not have the ability to established itself.Figure 4, shown that, as death rate of infected cells increases the control reproduction number decreases.The infection will die out of the system as effort is made to control the infection.Figures 7-8 shown that upon introduction of the virus, the population of uninfected cells decreases relative to the infected cells population, this behavior continues until the virus reaches the peak infected cells point.After reaching this value, the infected cells population begins to decline until it reaches zero.This suggests that with the help of drugs usage, the immunes system have the ability to suppress the increase in infected cells, as well as virus load which shows that the virus does not maintain an infection in the system.Conclusion: In this paper, we developed, analyzed and study effectiveness of drug usage on the dynamics of Ebola virus infection in human cells population.The infection free and infection persistence equilibria were obtained and investigated.The model showed that the infection free equilibrium is locally asymptotically stable G H I and unstable at threshold parameter greater than a unit.While the infection free equilibrium with method of Lyapunov function is globally asymptotically unstable if the threshold parameter is greater than a unit.Finally there is a need for further research work on the age of infection on dynamics of Ebola virus infection.

φ
. The effect of φ is to reduce the production of infected cells, we have the range of effectiveness of drug there is no drug and control intervention against the Ebola virus infection.This model explores the effect of drugs usage on the Ebola virus.

Fig 6 :Fig7:
Fig 6: The relationship between the infected cells, CTLs against time and between the viruses, antibody against time

Fig 8 :
Fig 8: The relationship between the uninfected cells, infected cells against time

Table 1 : Parameters of the Model
Reproduction rate of antibodies Clearance rate of infected cells Clearance rate of the viruses by antibodies Natural death rate of uninfected cell Death rate of infected cell Decay rate of the virus Death rate of Cytotoxic T-cell Death rate of B-cell secreted antibodies Booster of CTL Booster of Antibody Effectiveness of drug usage