Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 842792, 9 pages

http://dx.doi.org/10.1155/2015/842792

## Mathematical Modelling, Simulation, and Optimal Control of the 2014 Ebola Outbreak in West Africa

^{1}Mathématiques pour l’Industrie et la Physique, Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, France^{2}Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received 26 December 2014; Accepted 28 February 2015

Academic Editor: Sanling Yuan

Copyright © 2015 Amira Rachah and Delfim F. M. Torres. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries, it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals. We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.

#### 1. Introduction

Ebola is a lethal virus for humans. It was previously confined to Central Africa but recently was also identified in West Africa [1]. As of October 8, 2014, the World Health Organization (WHO) reported 4656 cases of Ebola virus deaths, with most cases occurring in Liberia [2]. The extremely rapid increase of the disease and the high mortality rate make this virus a major problem for public health. Typically, patients present fever, malaise, abdominal, headache, and asthenia. One week after the onset of symptoms, a rash often appears followed by haemorrhagic complications, leading to death after an average of 10 days in 50%–90% of infections [3–6]. Ebola is transmitted through direct contact with blood, bodily secretions and tissues of infected ill or dead humans and nonhuman primates [7–9].

The main aim of this work is to understand the dynamics of the Liberian population infected by Ebola virus in 2014, by using an appropriate mathematical model. More precisely, we begin by considering a system of ordinary differential equations to describe the 2014 Ebola outbreak, which is nothing else than an epidemic SIR model, that is, a model based on the division of the population into three groups: the susceptible, the infected, and the recovered [10–12]. We simulate the model obtained by parameters estimated on November 4, 2014, by Kaurov [13]. Our objectives are to better understand the outbreak and to predict the effect of vaccination on the infected individuals over time.

In order to deal with the epidemic of Ebola, governments have decided to implement tough measures. For example, some have applied quarantine procedures while others have opted for mass vaccination plans as a precaution to the epidemic [14–17]. In this context, we consider an optimal control problem to study the effect of vaccination on the spread of virus.

The text is organized as follows. In Section 2 we introduce a basic mathematical model to describe the dynamics of the Ebola virus during the 2014 outbreak in Liberia. After the mathematical modelling, we use the obtained model to simulate it in Section 3 with the parameters estimated from recent statistical data based on the WHO report of the 2014 Ebola outbreak [18]. Following [13], in Section 4 we study different cases of vaccination of individuals by adding a vaccination term to the model. Section 5 presents an optimal control problem, which we use to study the impact of a vaccination campaign on the spread of the virus. We end with Section 6 of conclusions and future work, where the results are summarized and some research perspectives presented.

#### 2. The Basic Mathematical Model

The objective of this section is to describe mathematically the dynamics of the population infected by the Ebola virus. The dynamics is described by a system of differential equations. This system is based on the common SIR (susceptible-infectious-recovery) epidemic model, where the population is divided into three groups: the susceptible group, denoted by , the infected group, denoted by , and the recovered group, denoted by . The total population, assumed constant during the short period of time under study, is given by .

To create the equations that describe the population of each group along time, let us start with the fact that the population of the susceptible group will be reduced as the infected come into contact with them with a rate of infection . This means that the change in the population of susceptible is equal to the negative product of with and : Now, let us create the equation that describes the infected group over time, knowing that the population of this group changes in two ways: (i)people leave the susceptible group and join the infected group, thus adding to the total population of infected a term ;(ii)people leave the infected group and join the recovered group, reducing the infected population by . This is written as Finally, the equation that describes the recovery population is based on the individuals recovered from the virus at rate . This means that the recovery group is increased by multiplied by :Figure 1 shows the relationship between the three variables of our SIR model. As shown in the next section, this simple model describes well the 2014 Ebola outbreak in Liberia for suitable chosen parameters and and appropriate initial conditions , , and .