Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5104524, 7 pages

https://doi.org/10.1155/2018/5104524

## Mathematical Modeling of Ebola Virus Disease in Bat Population

^{1}Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco^{2}Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco

Correspondence should be addressed to Khalid Hattaf; rf.oohay@fattah.k

Received 4 September 2018; Accepted 14 November 2018; Published 2 December 2018

Academic Editor: J. R. Torregrosa

Copyright © 2018 Zineb EL Rhoubari et al. 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

Many scientific researches have demonstrated that bats can survive Ebola virus and they represent natural reservoirs of this zoonotic virus. In this study, we develop a generalized epizootic model for the transmission dynamics of Ebola virus disease (EVD) in bat population by taking into account the environment contamination. The transmission process is modeled by two general incidence functions that include many incidence rates used in infectious diseases modeling. We first prove that the model is epidemiologically and mathematically well-posed by showing the existence, positivity, and boundedness of solutions. By analyzing the characteristic equations and constructing suitable Lyapunov functionals, the stability analysis of equilibria is investigated.

#### 1. Introduction

Ebola virus (EBOV) is a filovirus that belongs to the Filoviridae family and it causes a severe hemorrhagic disease in human and nonhuman primates. EBOV is transmitted to the human population through close contact with the blood, secretions, organs, or other bodily fluids of infected animals such as fruit bats, chimpanzees, gorillas, monkeys, antelopes, and porcupines found ill or dead in the rainforest [1]. The first outbreak of Ebola virus disease (EVD) appeared in near the Ebola River in the Democratic Republic of Congo formerly Zaire. So far, five species of Ebola have been identified which are Zaire, Sudan, Bundibugyo, Reston, and Taï Forest. The first three species have been associated with large outbreaks in Africa. For example, the virus causing the 2014-2016 West African outbreak belongs to the Zaire ebolavirus species [1]. The EVD dominated international news in and the World Health Organization (WHO) reported more than cases worldwide and over deaths [2].

Bats are recognized as a major reservoir of many viral infections including Ebola virus and other human viral infections, such as measles and mumps [3–5]. These species are long-lived mammals with typically highly synchronous birthing [6, 7]. Other studies have shown that bats can survive Ebola infection without necessarily becoming ill. For instance, Swanpoel et al. [8] experienced that the Ebola virus replicates in bats without being affected. Also, a seropositive bat for Ebola has demonstrated a long-term survival, living healthy over months after sampling [9]. On the other hand, the mechanism of Ebola transmission to nonhuman primates is still unknown, but Leroy et al. [3] think that it occurs with consumption of contaminated fruits by bodily fluids of infected bats.

Over the past years, several mathematical models have been proposed and developed to describe the dynamics of EVD [10–20]. However, these models have traditionally treated the transmission of the disease in human population and ignored the initial source of Ebola in bats. In addition, they do not take into account the contaminated environment by bodily fluids of infected bats. For these mathematical and biological considerations, we propose a generalized epizootic model for Ebola that is given by the following nonlinear system: where , , and represent the numbers of susceptible, infected, and recovered bats at time , respectively. Then the total population of bats is . Further, represents the concentration of EBOV in the environment at time . The susceptible population increases at recruitment rate by births or immigration and decreases at the natural mortality rate . It also decreases and converts into the infected subpopulation by direct contact with infected bats at rate or by contact with contaminated environment at rate . Thus, the term is the total infection rate of susceptible population. Moreover, the infected bats recover from Ebola at rate and die only at the natural mortality rate . Finally, the parameter denotes the deposition rate of EBOV in the environment by infected bats and is the decay rate of EBOV in the environment.

As in [21], we suppose that the general incidence functions and are continuously differentiable in the interior of and satisfy the following hypotheses:Epidemiologically, the above hypotheses are reasonable and consistent with the reality. In fact, the first assumption on the function means that the incidence rate by direct contact with infected bats is equal to zero if there are no susceptible bats. This incidence rate is increasing when the number of infected bats is constant and the number of susceptible bats increases. Also, it is decreasing when the number of susceptible bats is constant and the number of infected bats increases. Similarly, the second assumption on the function means that the incidence rate by contaminated environment is equal to zero if there are no susceptible bats. Besides, this incidence rate is increasing when the concentration of EBOV is constant in the environment and the number of susceptible bats increases. Also, it is decreasing when the number of susceptible is constant and the concentration of EBOV in the environment increases. Therefore, the more susceptible bats are, the more infectious events will occur. However, the higher the number of infected bats or the concentration of EBOV in the environment is, the less infectious events will be. Further, the biological meaning of for other diseases is given in [22, 23]. On the other hand, the schematic representation of model (1) is illustrated in Figure 1.